## Islands behind the horizon

If you’ve been following the arXiv, or keeping abreast of developments in high-energy theory more broadly, you may have noticed that the longstanding black hole information paradox seems to have entered a new phase, instigated by a pair of papers [1,2] that appeared simultaneously in the summer of 2019. Over 200 subsequent papers have since appeared on the subject of “islands”—subleading saddles in the gravitational path integral that enable one to compute the Page curve, the signature of unitary black hole evaporation. Due to my skepticism towards certain aspects of these constructions (which I’ll come to below), my brain has largely rebelled against boarding this particular hype train. However, I was recently asked to explain them at the HET group seminar here at Nordita, which provided the opportunity (read: forced me) to prepare a general overview of what it’s all about. Given the wide interest and positive response to the talk, I’ve converted it into the present post to make it publicly available.

Well, most of it: during the talk I spent some time introducing black hole thermodynamics and the information paradox. Since I’ve written about these topics at length already, I’ll simply refer you to those posts for more background information. If you’re not already familiar with firewalls, I suggest reading them first before continuing. It’s ok, I’ll wait.

Done? Great; let me summarize the pre-island state of affairs with the following two images, which I took from the post-island review [3] (also worth a read):

On the left is the Penrose diagram for an evaporating black hole in asymptotically Minkowski space. We suppose the black hole formed from some initially pure state of matter (in this case, a star, denoted in yellow), and evaporates completely via the production of Hawking radiation (represented by the pairs of squiggly lines). The key thing to note is that the Hawking modes are pairwise entangled: the interior partner is trapped behind the horizon and eventually hits the singularity, while the exterior partner is free to escape to infinity as Hawking radiation. An observer who remains far outside the black hole and attempts to collect this radiation will then find the final state to be highly mixed (in fact, almost thermal), since along any timeslice (the green slices), she only has access to the modes outside the horizon. But since we took the initial state to be pure (entropy ${S=0}$), that violates unitarity; this is the information paradox in a nutshell.

The evaporation process is depicted in the right figure, which schematically plots various relevant entropies over time. On the one hand, the Bekenstein-Hawking entropy of the black hole starts at some large value given by the surface area, and steadily decreases as the hole evaporates (orange curve). Meanwhile, the entropy of the radiation starts at zero, and then steadily increases over time as more and more modes are collected. This is the result given in Hawking’s original calculation [4], represented by the green curve. The expectation from unitarity is that the entropy of the radiation should instead follow the so-called Page curve, shown in purple. The idea is that at around the halfway point — more accurately, the point at which the entropy of the radiation equals the entropy of the horizon — the emitted modes start to purify the previously-collected radiation, so the entropy starts to decrease. Eventually, when the black hole has completely evaporated and all the “information” has been emitted, the radiation should again be in a pure state with ${S=0}$.

In a nutshell, the island formula is a prescription for computing the entropy of the radiation in such a way as to reproduce the Page curve. The claim is that instead of naïvely tracing-out the black hole interior, one should assign the radiation the generalized entropy

$\displaystyle S_\mathrm{rad}=\mathrm{min}_\mathcal{X}\left\{\mathrm{ext}_\mathcal{X}\left[\frac{A(\mathcal{X})}{4G_N}+S_\mathrm{semi-cl}\left(\Sigma_\mathrm{rad}\cup\Sigma_\mathrm{island}\right)\right]\right\}~, \ \ \ \ \ (1)$

where ${\mathcal{X}}$ is the minimal quantum extremal surface (QES), and ${S_\mathrm{semi-cl}}$ is the semi-classical entropy of matter fields on the union of ${\Sigma_\mathrm{rad}}$ (the exterior portion of the Cauchy slice where the radiation is collected) and ${\Sigma_\mathrm{island}}$ (a new portion of the Cauchy slice that lies inside the black hole). The next section of this post will trace a quasi-historical route through the various ingredients leading up to this formula, in an attempt to provide some context for an expression that otherwise appears rather out of the blue. Those who work on bulk reconstruction in AdS/CFT can probably skip to “Some Flat Space Intuition” below.

Quantum Extremal Surfaces

While the Penrose diagram above (and most of those below) depicts a black hole in asymptotically flat space, the island formula (1) is properly formulated in AdS/CFT, which is where our story begins. In the interest of making this maximally accessible, let’s start at the beginning with my favourite picture of the hyperbolic disc, courtesy of M. C. Escher:

Here we have a gravitational theory living in the bulk anti-de Sitter (AdS) space, which is dual to some conformal field theory (CFT) living on the asymptotic boundary. (Since this picture is necessarily embedded in flat space, the hyperbolic nature of the bulk manifests in the fact that the devils are all the same size). Inherent in the nature of a duality is that every element on one side has an equivalent element on the other, and a major research area in AdS/CFT is completing the so-called holographic dictionary that relates them. One of the major lessons of the past decade is that entanglement entropy appears to play a significant role in this mapping. In particular, the Ryu-Takayanagi formula [5] states that the von Neumann entropy of a boundary subregion — e.g., the blue or red boundary segments I’ve drawn over the image above — is given by the area of the co-dimension 2 minimal surface in the bulk anchored to the boundary of said region that satisfies the homology constraint (see below) — e.g., the blue or red arcs extending towards the center — called the Ryu-Takayanagi or RT surface. More generally, the idea is that all of the physics within the enclosed bulk region can be described by operators living in the corresponding boundary segment (more precisely, its causal domain of dependence).

The most salient aspect of this proposal for our purposes is that it has precisely the same form as the Bekenstein-Hawking entropy, i.e.,

$\displaystyle S_X=\frac{A(X)}{4G_N}~, \ \ \ \ \ (2)$

where ${X}$ is the boundary subregion, and the area is that of the corresponding minimal RT surface (in AdS${_3}$, a spacelike geodesic) in the bulk. As discussed in the post on black hole thermodynamics however, the Bekenstein-Hawking entropy is inadequate if one wishes to capture semi-classical effects, i.e., quantum fields propagating on the black hole (or AdS) background. Rather, one must consider the generalized entropy of the black hole plus radiation:

$\displaystyle S_\mathrm{gen}=\frac{A}{4G_N}+S_\mathrm{semi-cl}~, \ \ \ \ \ (3)$

where for consistency I’ve denoted the entropy of the radiation ${S_\mathrm{semi-cl}}$. As shown in [6], precisely the same formal expression appears in AdS/CFT when one takes into consideration one-loop corrections from quantum fields living between the bulk minimal surface and the corresponding boundary subregion. Intuitively, this plays the same role as the radiation in the black hole picture: just as the external observer must include both the area term and the radiation if she wishes to properly account for the entropy, so too does the entropy of the CFT subregion capture both the area contribution and the entropy of all the enclosed bulk fields.

Formally, the generalized entropy (3) is exactly the quantity that appears in the innermost brackets in the island formula (1), subject to the details of how one selects what semi-classical entropy to compute—more on this below. What about the ${\mathrm{min}_\mathcal{X}}$ condition? This has roots in the Ryu-Takayanagi prescription as well, in the form of the aforementioned homology constraint required for computing holographic entanglement entropy. This is discussed at length in [7], and conceptually summarized by the following images therefrom:

This is again the hyperbolic disc, but with a large AdS black hole whose horizon is demarcated by the inner circle. Now imagine starting with a small boundary subregion at the top of the image, whose corresponding bulk RT surface is drawn in red. As we make this region progressively larger, the RT surface lies deeper and deeper into the bulk—this is represented by the shading from red towards violet. Since the surfaces are defined geometrically, they are sensitive to the presence of the black hole, and get deformed around it as we include more and more of the boundary in the subregion under consideration. Eventually the subregion includes the entire boundary, and the corresponding RT surface is the violet one in the left image. For BTZ black holes, that’s it.

In higher dimensions however, there’s a second RT surface that becomes relevant when the subregion reaches a certain size. Consider the right image, at the point where the RT surface for the subregion is blue. The old solution is the RT surface that loops back around the black hole, on the same side as the boundary subregion. The new solution is the RT surface that connects these same two boundary points on the opposite (that is, lower) side of the black hole, plus a piece that wraps around the horizon itself. The horizon is included because the RT prescription requires the surface to be homologous to the boundary (crudely speaking, we wrap a second piece around the black hole in order to excise the hole from the manifold, so that the solution remains in the same homology class. The fact that one must consider disconnected surfaces may seem rather bizarre, but there’s nothing intrinsically pathological about it; see [7] for more discussion and additional references on the homology constraint in this context). One can see visually that when the boundary subregion reaches a certain size, the combined length of these two pieces will be smaller than the length of the old RT surface that has to go around the black hole the long way. A priori, we now have an ambiguity in which bulk entity we identify with the entropy of the boundary subregion, which is resolved by the requirement that we choose the minimum among all such choices. (While I’m unaware of a satisfactory physical explanation for this, it appears necessary in order for the generalized entropy to satisfy subadditivity [8]). Therefore, as the size of the boundary subregion is continuously increased, a discontinuous (i.e., non-perturbative) switchover occurs to the new minimal surface. As we shall see, a similar switchover plays a crucial role in the island computation of the Page curve.

Now, a quantum extremal surface (QES) is a seemingly mild correction to the above prescription, whereby the semi-classical contribution is taken into account when selecting the surface itself. That is, in the above RT prescription, we selected the minimal surface (which fixes the area term in (3)), and then added the entropy of all the matter fields living between that fixed surface and the boundary (the second term in (3)). As argued in [9] however, the correct procedure is to instead find the surface such that the combination of both terms is extremized, which defines the QES, denoted ${\mathcal{X}}$ (equivalently, ${\mathcal{X}}$ may be defined as a marginally trapped surface in both the past and future null directions, ${\partial_\mathcal{X}S_\mathrm{gen}=0}$). There’s nothing “quantum” about the surface itself (it’s still purely geometrical), the name merely reflects the fact that the quantum corrections to the leading area term (that is, ${S_\mathrm{semi-cl}}$) are taken into account when determining the surface’s location. The classical extremal surface (i.e., RT without quantum corrections, (2)) is then recovered in the limit ${\hbar\rightarrow0}$. Generically however, the QES will be spacelike separated from its classical counterpart, so that it probes more deeply into the bulk. This is illustrated by the picture below, from [9], where ${C_R}$ denotes the causal surface for the boundary subregion ${R}$ (which is equivalent to the classical extremal surface in vacuum spacetimes), and the causal domain of dependence in both the bulk (the wedge ${W_R}$) and the boundary (the domain ${D_R}$) are shown by the shaded green region. The QES ${\mathcal{X}_R}$ shares the same boundary points ${\partial R}$, but extends more deeply into the bulk, since it is defined by extremizing both the area and the quantum corrections together.

In most cases, the difference between the QES and its classical counterpart is relatively small. However, the remarkable feature uncovered in [1,2] is that this can sometimes be quite large, and furthermore, that the QES can sometimes lie deep inside the black hole. This is in sharp contrast to the classical RT surfaces discussed above: as one can see in those rainbow images from [7], they never reach beyond the horizon. (In fact, this appears to be a limitation of all such classical probes, as my colleagues and I analyzed in detail when I was a PhD student [10]: one can get exponentially close to the horizon in some cases, but no better. The corresponding “holographic shadows” seem to represent a puzzle for holographers, since — as per AdS/CFT being a duality — the boundary must have a complete and equivalent description of the physics in the bulk, including the black hole).

These two elements — the minimality requirement, and the ability of QESs to probe behind horizons — are the key to islands.

Some Flat Space Intuition

Let us set holography aside for the moment, and return to the flat space Penrose diagram for an evaporating black hole we introduced above. The review [3] does a great job painting an intuitive picture of how the QES changes the story. In particular, the various ingredients in (1) are shown in the following Penrose diagram:

The basic idea is as follows: suppose one is an external observer moving along the timelike trajectory given by the dashed purple line, who collects Hawking radiation that escapes to infinity. The latter comprises the semi-classical entropy along ${\Sigma_\mathrm{rad}}$, the “radiation” subset of the Cauchy slice ${\Sigma}$. However, the novelty in the island formula is the claim that this does not comprise the full contribution to the semi-classical entropy ${S_\mathrm{semi-cl}}$. Rather, one must also include the interior modes along the “island” subregion of the Cauchy slice, ${\Sigma_\mathrm{island}}$. The latter is defined as the region between the origin and the QES denoted by the blue point (marked ${X}$ in the image).

To understand why this QES appears, consider the following diagram, again from [3]. Suppose we fix a Cauchy slice at some point in the evaporation process (say, time ${t_2}$ in the figure), start with the candidate QES right on the horizon, and imagine moving it along the slice to find the extremum. Initially, the portion of the Cauchy slice denoted ${\Sigma_\mathrm{island}}$ captures all interior modes, some of which have not yet been included in the radiation ${\Sigma_\mathrm{rad}}$, so ${S_\mathrm{semi-cl}}$ has some finite value. As we move the surface ${\mathcal{X}}$ inwards along the Cauchy slice, effectively shrinking ${\Sigma_\mathrm{island}}$, we stop capturing some of the unpaired modes, so ${S_\mathrm{semi-cl}}$ decreases. Eventually however, we start losing modes whose partners are included in the radiation, at which point ${S_\mathrm{semi-cl}}$ will begin to increase. Meanwhile, the area ${A(\mathcal{X})}$ monotonically decreases. The candidate surface becomes a QES at the point where the decrease in the area term precisely cancels the increase in the semi-classical term ${S_\mathrm{semi-cl}}$.

If we repeat this process along an earlier Cauchy slice however, the picture is quite different. Before any radiation has been emitted, shrinking ${\Sigma_\mathrm{island}}$ simply decreases the semi-classical entropy as well as the area, so the QES becomes a vanishing surface at the origin. In this case, as we then move forwards in time, all of the entropy (1) comes from the radiation captured in ${\Sigma_\mathrm{rad}}$, between the observer’s worldline and infinity.

We thus have two possible choices for the QES ${\mathcal{X}}$ that appears in the island formula (1): the vanishing surface at the origin, and the non-vanishing surface that appears at later times. In analogy with the RT prescription above, we are instructed to select whichever yields lower generalized entropy at any given time. Starting at ${t_0}$ in the Penrose diagram above, we have only the vanishing surface, so our plot of the generalized entropy as the black hole evaporates starts looking exactly as it did in the original story summarized at the beginning of this post: since ${\Sigma_\mathrm{island}=\emptyset}$, the only contribution to the generalized entropy is ${S_\mathrm{semi-cl}(\Sigma_\mathrm{rad})}$, which increases monotonically as more and more modes are collected. In particular, as explained above, the issue is that these modes are pairwise entangled with partners trapped behind the horizon. The more unpaired modes we collect, the larger the entropy becomes; thus if we use the vanishing surface ${\mathcal{X}}$, we get the green curve in the schematic plot.

Shortly after the black hole starts to evaporate however, the second, non-vanishing QES appears some distance inside the horizon. The corresponding area contribution is initially very large, but steadily decreases as the hole evaporates. Crucially, the semi-classical contribution also declines, since we count modes in the union ${\Sigma_\mathrm{rad}\cup\Sigma_\mathrm{island}}$, which enables us to capture both partners (see the colored pairs of modes in the Penrose diagram). Using this QES for the generalized entropy then gives us the orange curve in the plot.

Since we’re instructed to always use the minimum QES, a switchover analogous to that we discussed in the context of RT surfaces thus occurs at the Page time: at this point, the area of the non-empty QES (which dominates over the corresponding semi-classical contribution) equals the semi-classical contribution from the empty QES. We therefore follow the black curve in the plot, which is the sought-after Page curve, where the change from an increasing (green) to decreasing (orange) entropy corresponds to a non-perturbative shift in the location of the minimal QES. In this way, the island formula obtains a result for the generalized entropy consistent with our expectations from unitarity.

However, while the exposition above paints a nice intuitive picture, it’s ultimately just a cartoon, and leaves many physical aspects unexplained—for example, how the interior “island” is able to purify the radiation, or the dependence of (the location of) ${\mathcal{X}}$ on the observer’s worldline (since the latter controls the size of ${\Sigma_\mathrm{rad}}$). As mentioned at the beginning of this flat space interlude, the island prescription is properly formulated in AdS/CFT, which is the setting in which we actually know how to perform these computations (indeed, it has not been shown that the crucial QES even exists in asymptotically Minkowski space). Even then, analytical computations to date have only been performed in lower-dimensional toy models. These have the powerful advantage of allowing one finer control over various technical aspects of the construction, and also provide the setting in which the claimed derivations of the island formula (1) via the replica trick are performed [11,12]. However, as alluded at the beginning of this post, these rely on some assumptions that I personally find unphysical, or at least insufficiently justified given the current state of our knowledge. So let’s now return to AdS/CFT, and discuss these in a bit more detail.

Just drink the Kool-Aid

There are two main stumbling blocks that must be overcome for the picture we’ve painted above to work:

1. Large black holes in AdS don’t evaporate.
2. The identification of the island with the radiation is ultimately imposed by hand.

The first of these is due to the fact that large AdS black holes, unlike their flat-space counterparts, are microcanonically stable: they have ${M\sim T}$, rather than ${M\sim T^{-1}}$. Intuitively, AdS acts like a box that reflects the radiation back into the black hole, allowing it to reach thermal equilibrium with the ambient spacetime. Consequently, since the black hole never evaporates, there’s no Page curve to speak of in the first place: it emits radiation until it reaches equilibrium, and then contentedly bathes in it for all eternity.

To circumvent this, [1,2] coupled the system to an auxiliary reservoir in order to force the black hole to evaporate. This is illustrated by the following figure from [2], in which a coupling between the asymptotic boundary and a previously-independent CFT that acts as a bath is turned on at the time indicated by the orange dot. Prior to this, radiation from the AdS${_2}$ black hole (not shown) is reflected back into the bulk as explained above; afterwards however, it is able to propagate out of the bulk and into the bath CFT, which they take to live in Minkowski space. This effectively changes the asymptotic structure so that the black hole evaporates by the collection of radiation in this auxiliary reservoir.

This is approximately the point at which I get off the metaphorical train. In particular, insofar as AdS/CFT is a duality between equivalent descriptions of the same system, not two systems that interact at some physical boundary, the so-called “absorbing boundary conditions” [1,2] claim when performing this operation do not appear to me to be valid. Another way to express this underlying point is that it is incorrect to treat the bulk and boundary as akin to tensor factorizations of a single all-encompassing Hilbert space. Rather, the radiation — which carries some finite energy — has an equivalent description in the boundary CFT. Allowing it to propagate into the boundary simultaneously decreases the energy of the bulk while increasing the energy of the boundary, thereby breaking the duality. Following the initial works above, an interesting attempt was made in [13] to make this construction more plausible by allowing the auxiliary system itself to be holographic. However, while the resulting “double holography” model is more technically refined, it ultimately still features fields propagating between different sides of the duality, so does not actually overcome the fundamental objection above.

Suppose however that one accepts such forced-evaporation models for the sake of argument—after all, the entire point of a toy model is to grant a sufficient deviation from reality that calculations become tractable (the tricky bit, and the focus of my skepticism here, lies in determining how much physics is retained in the result). One then has to contend with the second roadblock, namely identifying the radiation — now contained in the auxiliary reservoir — with the interior island. That is, when we wrote down the island formula (1), the semi-classical term was defined as the entropy over the union ${\Sigma_\mathrm{rad}\cup\Sigma_\mathrm{island}}$, as opposed to the pre-island contribution over only ${\Sigma_\mathrm{rad}}$. The inclusion of ${\Sigma_\mathrm{island}}$ was crucial for recovering the Page curve, since the radiation is never purified without it. Note that the question is not about the QES itself, but rather, why one should include the region between the origin and the QES in the semi-classical contribution to the generalized entropy. As many others in the field have pointed out, this seems to include the interior — the absence of information in which we knew to be the problem from the very start — by hand!

Proponents of the island prescription would disagree with this claim. The review [3] for example addresses it explicitly as follows:

“Now a skeptic would say: Ah, all you have done is to include the interior region. As I have always been saying, if you include the interior region you get a pure state,’ or This is only an accounting trick.’ But we did not include the interior `by hand’. The fine-grained entropy formula is derived from the gravitational path integral through a method conceptually similar to the derivation of the black hole entropy by Gibbons and Hawking…”

The calculation they refer to is the replica trick for computing entanglement entropy, which I’ve written about before. And indeed, at least two papers [11,12] have claimed to derive the island formula via precisely such a replica wormhole calculation. The underlying idea is actually rather beautiful: the switchover to the new QES corresponds to an initially sub-leading saddlepoint in the gravitational path integral which becomes dominant at the Page time. (There’s a pedagogically exemplary talk on YouTube by Douglas Stanford in which he explains how this comes about in the toy model of [11]). The issue here is not that there’s anything wrong with the technical aspects of the above derivations per se. However, as I’ve stressed before, the conclusions one can draw from any derivation are dependent on the assumptions that go into it. In this case, the key assumption is that when sewing the replicas together, we make a cut along what becomes the island. That is, when we perform the replica trick, we make ${n}$ copies of the manifold, and glue them together with cuts along the region whose reduced density matrix we wish to compute, as illustrated in fig. 1 of [14]:

In derivations of the island formula, one makes a disjoint cut along ${\Sigma_\mathrm{rad}\cup\Sigma_\mathrm{island}}$, since this is defined as the subregion of interest; see for example fig. 3 of [12]. In this sense, the island is baked into the derivation in choosing to place a cut along the region ${\Sigma_\mathrm{island}}$ (labeled “I” therein). Essentially, defining the cuts this way simply amounts to defining a reduced density matrix that includes both the exterior ${\Sigma_\mathrm{rad}}$ and the interior ${\Sigma_\mathrm{island}}$, in which case, you will indeed reproduce the Page curve, from the switchover to the new subleading saddle (i.e., the new QES).

A proponent might argue that there are good reasons to join the replicas in this fashion; but producing the correct answer a posteriori should not be one of them. In short, my unfashionable opinion is that the claim in [12] that “[t]his new saddle suggests that we should think of the inside of the black hole as a subsystem of the outgoing Hawking radiation” is, at least at this point in our collective explorations/explanations, backwards: rather, if you think of the inside of the black hole as a subsystem of the outgoing Hawking radiation, you will get a new saddle. The logic works both ways in principle, I’m simply not quite convinced that we’ve justified running it in the popular direction.

Even if the island prescription is (ontologically) correct — and it may well be — it does not suffice to resolve the firewall paradox, for several reasons. Physically, it does not explain how the radiation is purified from an operational perspective, i.e., how unitarity is restored from a pure bulk entropy calculation (e.g., in Minkowski space without any bells and whistles). Additionally, it does not explain where Hawking’s original calculation [4] went wrong. Abstractly, one would say that the error is that he did not include this sub-leading saddle, which arises from the different ways of connecting the replica geometries. But insofar as Hawking’s calculation does not employ a gravitational path integral at all, an interesting open question that would likely shed light on the underlying physics of the island prescription is how to bridge the conceptual gap between the calculation via replica wormholes and the more down-to-earth calculation via Bogolyubov coefficients in the black hole scattering matrix.

To be sure, this is still a very interesting and non-trivial result. It’s remarkable that the bulk gravitational path integral includes these crucial replica wormhole geometries, despite the fact that the ${n}$ copies of the boundary CFT should factorize, and hence would seem to know nothing about them (see for example [15] and related work). This may be telling us something deep about quantum gravity, and I’m excited to see what further research in this direction will uncover.

References

1. G. Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” JHEP 09 (2020) 002, arXiv:1905.08255.
2. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12 (2019) 063, arXiv:1905.08762.
3. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “The entropy of Hawking radiation,” arXiv:2006.06872.
4. S. W. Hawking, “Breakdown of predictability in gravitational collapse,” Phys. Rev. D 14 (Nov, 1976) 2460–2473.
5. S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001.
6. T. Faulkner, A. Lewkowycz, and J. Maldacena, “Quantum corrections to holographic entanglement entropy,” JHEP 11 (2013) 074, arXiv:1307.2892.
7. V. E. Hubeny, H. Maxfield, M. Rangamani, and E. Tonni, “Holographic entanglement plateaux,” JHEP 08 (2013) 092, arXiv:1306.4004.
8. M. Headrick and T. Takayanagi, “A Holographic proof of the strong subadditivity of entanglement entropy,” Phys. Rev. D 76 (2007) 106013, arXiv:0704.3719.
9. N. Engelhardt and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” JHEP 01 (2015) 073, arXiv:1408.3203.
10. B. Freivogel, R. Jefferson, L. Kabir, B. Mosk, and I.-S. Yang, “Casting Shadows on Holographic Reconstruction,” Phys. Rev. D 91 no. 8, (2015) 086013, arXiv:1412.5175.
11. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, “Replica wormholes and the black hole interior,” arXiv:1911.11977.
12. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05 (2020) 013, arXiv:1911.12333.
13. A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP 03 (2020) 149, arXiv:1908.10996.
14. P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42 (2009) 504005, arXiv:0905.4013.
15. D. Marolf and H. Maxfield, “Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information,” JHEP 08 (2020) 044, arXiv:2002.08950.
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## QFT in curved space, part 2: Bogolyubov transformations & the Unruh effect

One of the most important lessons of QFT in curved space is that the notion of a particle is an observer-dependent concept. That’s not to say it isn’t locally useful, but without specifying the details of the mode decomposition and the trajectory/background of the detector/observer, it’s simply not meaningful. In addition to challenging our intuition, this fact lies at the heart of some of the most vexing paradoxes in physics, firewalls chief among them. In this post, I want to elucidate this issue by introducing a very important tool: the Bogolyubov (also transliterated as “Bogoliubov”) transformation.

In a nutshell, a Bogolyubov transformation relates mode decompositions of the field in different states. Consider quantizing the free scalar field ${\phi}$ in ${(1\!+\!1)}$-dimensional Minkowski (i.e., flat) spacetime. From part 1, we have

$\displaystyle \phi(x)=\int\!\frac{\mathrm{d}\mathbf{k}}{\sqrt{4\pi\omega}}\,\left( a_k u_k+a_k^\dagger u_k^*\right)~, \qquad\qquad u_k\equiv \frac{1}{\sqrt{4\pi\omega}}e^{i\mathbf{k} \mathbf{x}-i\omega t}~, \ \ \ \ \ (1)$

where ${x=(t,\mathbf{x})}$, ${k=(\omega,\mathbf{k})}$, and the creation/annihilation operators satisfy the standard equal-time commutation relation, which in our convention is ${\big[a_k,a_{k'}^\dagger\big]=4\pi\omega\delta(\mathbf{k}-\mathbf{k}')}$, with the vacuum state ${\left|\Omega\right>}$ such that ${{a_k\left|\Omega\right>=0}}$. (The derivation of this commutation relation, as well as the reason for splitting the normalization in this particular way when identifying the modes ${u_k}$, will be discussed below). Typically, we choose the positive solution to the Klein-Gordon equation, ${\omega\!>\!0}$, so that the operator ${a_k^\dagger}$ has an interpretation as the creation operator for a positive-frequency mode traveling forwards in time. Note however that this statement depends on the existence of a global timelike Killing vector. That is, the modes ${u_k\sim e^{ikx}}$ are of positive frequency with respect to some time parameter ${t}$, by which we mean they’re eigenfunctions of the operator ${\partial_t}$ with eigenvalue ${\omega\!>\!0}$:

$\displaystyle \partial_t u_k=-i\omega u_k~. \ \ \ \ \ (2)$

In general spacetimes however, there will be no Killing vectors with respect to which we can define positive-frequency modes, so this decomposition depends on the observer’s frame of reference. Another way to say this is that the vacuum state is not invariant under general diffeomorphisms, because these have the potential to mix positive- and negative-frequency modes. Indeed, the spirit of general relativity is that there are no privileged coordinate systems. Hence we could equally-well quantize the field with a different set of modes:

$\displaystyle \phi(x)=\int\!\frac{\mathrm{d} \mathbf{p}}{\sqrt{4\pi\varpi}}\,\left( \bar a_p \bar u_p+\bar a_p^\dagger \bar u_p^*\right)~, \qquad\qquad \bar u_p\equiv\frac{1}{\sqrt{4\pi\varpi}}e^{i\mathbf{p}\mathbf{x}-i\varpi t}~, \ \ \ \ \ (3)$

which represent excitations with respect to some other vacuum state ${\left|\bar\Omega\right>}$, i.e., ${\bar a_p\left|\bar\Omega\right>=0}$. Since both sets of modes are complete, they must be linear combinations of one another, i.e.,

$\displaystyle \bar u_p=\int\!\mathrm{d}\mathbf{k}\,\left(\alpha_{kp}u_k+\beta_{kp}u_k^*\right)~, \qquad\mathrm{and}\qquad u_k=\int\!\mathrm{d}\mathbf{p}\,\left(\alpha_{kp}^*\bar u_p+\beta_{kp}^*\bar u_p^*\right)~. \ \ \ \ \ (4)$

These are the advertised Bogolyubov transformations, and the matrices ${\alpha_{ij}}$, ${\beta_{ij}}$ are called Bogolyubov coefficients. (Note that unlike [1], we’re working directly in the continuum, so the coefficients should be interpreted as continuous functions rather than discrete matrices). These can be determined by appealing to the normalization condition imposed by the Klein-Gordon inner product,

$\displaystyle (f,g)\equiv i\int_t\left[\,\left(\partial_tf\right) g^*-f\partial_tg^*\right]~, \ \ \ \ \ (5)$

where the integral is performed over any Cauchy slice; for the present case of 1d flat space, ${\int_t=\int\!\mathrm{d}\mathbf{x}}$. The idea behind this choice of inner product is that it allows the easy extraction of the Fourier coefficients from (1) as follows:

\displaystyle \begin{aligned} (\phi,u_k)&=i\int\!\mathrm{d}\mathbf{x}\,\left(\partial_t\phi\,u_k^*-\phi\,\partial_tu_k^*\right) =2\omega\int\!\frac{\mathrm{d}\mathbf{p}}{\sqrt{4\pi\omega}}\int\!\mathrm{d}\mathbf{x}\,a_pu_pu_k^*\\ &=\frac{1}{\sqrt{4\pi\omega}}\!\int\!\frac{\mathrm{d}\mathbf{p}}{2\pi}\,a_p\!\int\!\mathrm{d}\mathbf{x}\,e^{i(\mathbf{p}-\mathbf{k})\mathbf{x}} =\frac{1}{\sqrt{4\pi\omega}}\int\!\mathrm{d}\mathbf{p}\,\delta(\mathbf{k}-\mathbf{p})\,a_p =\frac{1}{\sqrt{4\pi\omega}}a_k~, \end{aligned} \ \ \ \ \ (6)

and similarly for ${a_k^\dagger}$; commuting the two yields the standard commutation relation for the creation/annihilation operators above. The modes ${u_k}$ are orthonormal with respect to this inner product, and therefore provide a convenient basis for the solution space:

$\displaystyle (u_k,u_p)=\delta(\mathbf{k}-\mathbf{p})~,\qquad (u_k^*,u_p^*)=-\delta(\mathbf{k}-\mathbf{p})~,\qquad (u_k,u_p^*)=0~, \ \ \ \ \ (7)$

and similarly for ${\bar u_k}$, hence the funny split in the normalization we mentioned below (1) (for our notation/normalization conventions, see part 1). For example,

\displaystyle \begin{aligned} (u_k,u_p)&=i\int\!\mathrm{d}\mathbf{x}\,\left[\,\left(\partial_tu_k\right) u_p^*-u_k\partial_tu_p^*\right] =2\omega\!\int\!\mathrm{d}\mathbf{x}\,u_ku_p^*\\ &=\frac{1}{2\pi}\int\!\mathrm{d}\mathbf{x}\,e^{i(\mathbf{k}-\mathbf{p})\mathbf{x}} =\delta(\mathbf{k}-\mathbf{p})~. \end{aligned} \ \ \ \ \ (8)

Taking the inner product of different sets of modes then allows one to isolate the Bogolyubov coefficients:

$\displaystyle \alpha_{kp}=(\bar u_p,u_k)~,\qquad\mathrm{and}\qquad \beta_{kp}=-(\bar u_p,u_k^*)~. \ \ \ \ \ (9)$

Explicitly:

\displaystyle \begin{aligned} (\bar u_p,u_k)&=i\!\int\!\mathrm{d}\mathbf{x}\,\left[\,\left(\partial_t\bar u_p\right){u_k}^*-\bar u_p\,\partial_t{u_k}^*\right]\\ &=i\!\int\!\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{j}\,\big[\left(\alpha_{jp}\partial_t u_j+\beta_{jp}\partial_tu_j^*\right){u_k}^*-\left(\alpha_{jp}u_j+\beta_{jp}u_j^*\right)\,\partial_t{u_k}^*\big]\\ &=\omega\!\int\!\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{j}\,\big[\left(\alpha_{jp}u_j-\beta_{jp}u_j^*\right){u_k}^*+\left(\alpha_{jp}u_j+\beta_{jp}u_j^*\right)\,u_k^*\big]\\ &=2\omega\!\int\!\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{j}\,\alpha_{jp}u_ju_k^* =\int\!\mathrm{d}\mathbf{j}\,\alpha_{jp}\delta(\mathbf{j}-\mathbf{k}) =\alpha_{kp}~, \end{aligned} \ \ \ \ \ (10)

where the second equality in the last line follows from (8); similarly for ${\beta_{kp}}$. Equating the two mode expansions (1) and (3) for the same field ${\phi}$ then leads to the following relationship between the creation/annihilation operators:

$\displaystyle a_k=\int\!\mathrm{d}\mathbf{p}\,\left(\alpha_{kp}\bar a_p+\beta_{kp}^*\bar a_p^\dagger\right)~, \qquad\mathrm{and}\qquad \bar a_p=\int\!\mathrm{d}\mathbf{k}\,\left(\alpha_{kp}^* a_k+\beta_{kp}a_k^\dagger\right)~, \ \ \ \ \ (11)$

For example, the first of these is obtained by expressing everything in terms of ${u_k}$:

\displaystyle \begin{aligned} \int\!\mathrm{d}\mathbf{k}\,\left( a_k u_k+a_k^\dagger u_k^*\right)=&\int\!\mathrm{d}\mathbf{p}\,\left( \bar a_p \bar u_p+\bar a_p^\dagger \bar u_p^*\right)\\ =&\int\!\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{j}\,\big[\bar a_p \left(\alpha_{jp}u_j+\beta_{jp}u_j^*\right)+\bar a_p^\dagger\left(\alpha_{jp}^*u_j^*+\beta_{jp}^*u_j\right)\big]~. \end{aligned} \ \ \ \ \ (12)

Taking the Klein-Gordon product with ${u_q}$ (recall this is how we isolate the coefficients, cf. (6)), and using the fact that the modes are orthonormal (7), we then have

$\displaystyle \int\!\mathrm{d}\mathbf{k}\,a_k \delta(\mathbf{k}-\mathbf{q})=\int\!\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{j}\,\left(\alpha_{jp}\bar a_p+\beta_{jp}^*\bar a_p^\dagger\right)\delta(\mathbf{j}-\mathbf{q}) \;\implies\; a_k=\int\!\mathrm{d}\mathbf{p}\,\left(\alpha_{kp}\bar a_p+\beta_{kp}^*\bar a_p^\dagger\right)~. \ \ \ \ \ (13)$

Note that had we taken the Klein-Gordon product with ${u_q^*}$, we’d simply have obtained ${a_k^\dagger}$. To solve for the second relation, ${\bar a_p}$, we repeat this process, instead rewriting the left-hand side of (12) in terms of ${\bar u_p}$. Importantly, one can see immediately from (11) that these two quantizations, (1) and (3), lead to different Fock spaces (i.e., different vacuum states) unless ${\beta_{kp}=0}$. Said differently, ${\beta_{kp}\neq0}$ implies that the alternative quantization ${\bar u_p}$ is a linear combination of both positive- and negative-frequency modes with respect to the Killing vector ${\partial_t}$. In this case, ${a_k\left|\Omega\right>=0}$ does not imply ${\bar a_p\left|\Omega\right>=0}$, so the latter quantization will detect particles when the former does not! In some sense, this is just the same spirit of relativity (i.e., no privileged reference frames) carried over to QFT, but it’s a radical conclusion nonetheless.

To make things more concrete, let’s apply this machinery to Rindler space, which describes a uniformly accelerating observer in the Minkowski background. The metric is obtained via a hyperbolic transformation of the flat space coordinates ${(t,\mathbf{x})}$ to ${(\eta,\boldsymbol{\xi})}$:

$\displaystyle t=\frac{1}{a}e^{a\boldsymbol{\xi}}\sinh a\eta~,\qquad\qquad \mathbf{x}=\frac{1}{a}e^{a\boldsymbol{\xi}}\cosh a\eta~,\qquad\qquad \ \ \ \ \ (14)$

which yields

$\displaystyle \mathrm{d} s^2=e^{2a\boldsymbol{\xi}}\left(-\mathrm{d}\eta^2+\mathrm{d}\xi^2\right)~, \ \ \ \ \ (15)$

where ${a\!>\!0}$ is some constant that parametrizes the acceleration. Note however that these coordinates cover only a single quadrant of Minkowski space, namely the wedge ${\mathbf{x}\!>\!|t|}$. The line ${\mathbf{x}=t}$ is the Rindler horizon, the physics of which well-approximates the near-horizon geometry of a black hole (when the curvature and radial directions can be ignored). This last fact underlies the central importance of Rindler space in the study of black holes, and QFT in curved space in general, as it captures many of the key features of horizons (e.g., their thermodynamic properties). Rindler time-slices — that is, lines of constant ${\eta}$ — are straight, ${\mathbf{x}\propto t}$, while lines of constant ${\eta}$ — which represent a fixed position from the horizon in these coordinates — are hyperbolae, ${\mathbf{x}^2-t^2=a^{-2}e^{2a\boldsymbol{\xi}}}$. The proper acceleration is given by ${ae^{-a\boldsymbol{\xi}}}$, so that the closer one gets to the horizon, the harder one has to accelerate (note again the analogy with black holes here). These hyperbolae asymptote to the null rays ${\mathbf{x}\pm t}$, so that the accelerated observer approaches the speed of light in the limit ${\eta\pm\infty}$; see the image below (I found this online, so don’t let the different labels confuse you):

Now, suppose we want to determine the particle spectrum that a Rindler observer would detect in the Minkowski vacuum. That is, if you were a constantly accelerating observer in empty flat space, would you still see vacuum? To answer this question, let’s compute the expectation value of the number operator for the Rindler modes in the Minkowski vacuum state, ${\left|0_\textrm{\tiny{M}}\right>}$. This requires expressing the Rindler creation/annihilation operators in terms of their Minkowski counterparts, in order to figure out how they act on this state.

There’s a trick in Birrell and Davies [1], credited to Unruh, for deducing the linear combination of Minkowski modes that corresponds to a given Rindler mode that bypasses the need to work out the Bogolyubov coefficients explicitly (though the normalization must be inserted by hand, and the switch to a second, alternative set of Minkowski modes appears rather out of the blue). The basic idea is to write down the combination of Minkowski modes with the same positive-frequency analyticity properties as the Rindler modes in the desired wedged. While this is sufficient if one is only interested in knowing how the Rindler modes act on the Minkowski vacuum, working out the Bogolyubov coefficients explicitly is required in many other scenarios, and is a character-building exercise besides. Since I had to do this in the course of a previous project anyway (we wanted to compute the overlap between two states, which required commuting Rindler and Minkowski modes), I may as well share the calculation here. For simplicity, in what follows I’m going to restrict to the case of a massless scalar field. Performing the computation in ${(t,\mathbf{x})}$ and ${(\eta,\boldsymbol{\xi})}$ coordinates is rather tedious, and requires a case-by-case analysis depending on the signs of various momenta. A more elegant method, presented in [2] (beware, wrong sign convention!), is to first rotate to lightcone coordinates

$\displaystyle x_\pm = (\mathbf{x}\pm t)/\sqrt{2}~,\qquad\qquad \bar x_\pm = (\boldsymbol{\xi}\pm \eta)/\sqrt{2}~, \ \ \ \ \ (16)$

where we’ve chosen the signs so that ${x_\pm>0}$ in the wedge under consideration. Since we’re working with massless fields, these coordinates have the significant advantage of not mixing positive and negative frequencies. That is, observe that in these coordinates, the Minkowski and Rindler metrics become, respectively,

$\displaystyle \mathrm{d} s^2=\mathrm{d} x_+\mathrm{d} x_- \qquad\mathrm{and}\qquad \mathrm{d} s^2=e^{\sqrt{2}a(\bar x_++\bar x_-)}\mathrm{d}\bar x_+\mathrm{d}\bar x_-~, \ \ \ \ \ (17)$

which makes the transformation between them much simpler:

$\displaystyle x_\pm = \frac{1}{\sqrt{2}a}e^{\sqrt{2}a\bar x_\pm}~. \ \ \ \ \ (18)$

The key point is the following: since the wave equation is conformally invariant, and Rindler is conformal to Minkowski, we can write the mode expansion in terms of plane waves just as we did in the flat space case above, cf. (1). The additional simplification afforded by the use of lightcone coordinates is that the contribution from positive-frequency modes in this expansion can be directly matched to the corresponding positive-frequency contribution in flat space (similarly for the negative frequencies). The lack of mode mixing in these coordinates greatly shortens the computation of the Bogolyubov coefficients, because — in addition to avoiding the case-by-case treatment — it enables us to isolate the desired modes via a simple Fourier transform, rather than needing to perform the full Klein-Gordon inner product as above.

So, let’s split the Minkowski mode decomposition into positive- and negative-frequency parts; from (1), we have

\displaystyle \begin{aligned} \phi&=\int_{-\infty}^\infty\!\frac{\mathrm{d}\mathbf{k}}{4\pi\omega}\left( a_ke^{i\mathbf{k}\mathbf{x}-i\omega t}+a_k^\dagger e^{-i\mathbf{k}\mathbf{x}+i\omega t}\right)\\ &=\int_0^\infty\!\frac{\mathrm{d}\mathbf{k}}{4\pi\mathbf{k}}\left( a_ke^{i\mathbf{k}(\mathbf{x}-t)}+a_k^\dagger e^{-i\mathbf{k}(\mathbf{x}-t)}\right) +\int_{-\infty}^0\!\frac{\mathrm{d}\mathbf{k}}{4\pi|\mathbf{k}|}\left( a_ke^{-i|\mathbf{k}|(\mathbf{x}+t)}+a_k^\dagger e^{i|\mathbf{k}|(\mathbf{x}+t)}\right)\\ &=\int_0^\infty\!\frac{\mathrm{d}\mathbf{k}}{4\pi\mathbf{k}}\left( a_ke^{i\mathbf{k}(\mathbf{x}-t)}+a_k^\dagger e^{-i\mathbf{k}(\mathbf{x}-t)}+a_{-k}e^{-i\mathbf{k}(\mathbf{x}+t)}+a_{-k}^\dagger e^{i\mathbf{k}(\mathbf{x}+t)}\right)~, \end{aligned} \ \ \ \ \ (19)

where in the second line, we used the on-shell condition ${\omega=|\mathbf{k}|>0}$, and then in the last line changed integration variables ${\mathbf{k}\rightarrow-\mathbf{k}}$ in the second term in order to combine the integrals. In lightcone coordinates (16), this becomes

$\displaystyle \phi=\int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{4\pi\mathbf{p}}\left( a_pe^{i\mathbf{p} x_-}+a_p^\dagger e^{-i\mathbf{p} x_-}+a_{-p}e^{-i\mathbf{p} x_+}+a_{-p}^\dagger e^{i\mathbf{p} x_+}\right)~, \ \ \ \ \ (20)$

where we’ve absorbed the pesky factor of ${\sqrt{2}}$ via the rescaling ${\mathbf{p}\equiv\sqrt{2}\mathbf{k}}$. Observe how the final expression naturally decomposes into creation/annihilation operators for left- and right-movers. And since the Rindler metric is conformally flat, it admits a formally identical expression within the wedge ${|x|\!>\!t}$:

$\displaystyle \phi=\int_0^\infty\!\frac{\mathrm{d}\mathbf{q}}{4\pi\mathbf{q}}\left( b_qe^{i\mathbf{q} \bar x_-}+b_q^\dagger e^{-i\mathbf{q} \bar x_-}+b_{-q}e^{-i\mathbf{q} \bar x_+}+b_{-q}^\dagger e^{i\mathbf{q} \bar x_+}\right)~, \ \ \ \ \ (21)$

where ${\mathbf{q}}$ is the rescaled Rindler momentum, and ${b_q\left|0_\textrm{\tiny{R}}\right>=b_{-q}\left|0_\textrm{\tiny{R}}\right>=0}$. Now comes the chief advantage of this approach mentioned above: since the notion of positive/negative momenta is preserved under the conformal transformation from Minkowski to Rindler space, we can directly identify

\displaystyle \begin{aligned} \int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{4\pi\mathbf{p}}\left( a_pe^{i\mathbf{p} x_-}+a_p^\dagger e^{-i\mathbf{p} x_-}\right)&=\int_0^\infty\!\frac{\mathrm{d}\mathbf{q}}{4\pi\mathbf{q}}\left( b_qe^{i\mathbf{q} \bar x_-}+b_q^\dagger e^{-i\mathbf{q} \bar x_-}\right)~,\\ \int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{4\pi\mathbf{p}}\left( a_{-p}e^{-i\mathbf{p} x_+}+a_{-p}^\dagger e^{i\mathbf{p} x_+}\right)&=\int_0^\infty\!\frac{\mathrm{d}\mathbf{q}}{4\pi\mathbf{q}}\left( b_{-q}e^{-i\mathbf{q} \bar x_+}+b_{-q}^\dagger e^{i\mathbf{q} \bar x_+}\right)~. \end{aligned} \ \ \ \ \ (22)

Now, what we’re after is an expression for the Rindler operators ${b_{\pm q},b_{\pm q}^\dagger}$ in terms of the Minkowski operators ${a_{\pm p},a_{\pm p}^\dagger}$, so that we can compute the expectation value of the number operator, ${\langle0_\textrm{\tiny{M}}\big|\hat N_b\big|0_\textrm{\tiny{M}}\rangle}$. To that end, we first isolate the right-moving annihilation mode ${b_q}$ by an appropriate Fourier transform of the first of these equations:

\displaystyle \begin{aligned} \int_{-\infty}^{\infty}\!\mathrm{d}\bar x_-\!&\int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{4\pi\mathbf{p}}\left( a_pe^{i\mathbf{p} x_--i\mathbf{q}'\bar x_-}+a_p^\dagger e^{-i\mathbf{p} x_--i\mathbf{q}'\bar x_-}\right)\\ &=\int_{-\infty}^\infty\!\mathrm{d}\bar x_-\!\int_0^\infty\!\frac{\mathrm{d}\mathbf{q}}{4\pi\mathbf{q}}\left( b_qe^{i(\mathbf{q}-\mathbf{q}')\bar x_-}+b_q^\dagger e^{-i(\mathbf{q}+\mathbf{q}') \bar x_-}\right)~,\\ &=\int_0^\infty\!\frac{\mathrm{d}\mathbf{q}}{2\mathbf{q}}\left[ b_q\,\delta(\mathbf{q}-\mathbf{q}')+b_q^\dagger\,\delta(\mathbf{q}+\mathbf{q}')\right] =\frac{1}{2\mathbf{q}'}\times\begin{cases} b_{q'}\quad&\mathbf{q}'>0~,\\ b_{-q'}^\dagger\quad&\mathbf{q}'<0~, \end{cases}~, \end{aligned} \ \ \ \ \ (23)

where, a priori, we allowed the momentum ${\mathbf{q}'}$ in the Fourier transform to take any sign. Recalling however that the expressions (22) were derived for positive momenta, we must have

$\displaystyle b_{q}=\int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{2\pi}\frac{\mathbf{q}}{\mathbf{p}}\left[ a_p\,F(\mathbf{p},\mathbf{q})+a_p^\dagger\,F(-\mathbf{p},\mathbf{q})\right]~, \ \ \ \ \ (24)$

where we have defined

$\displaystyle F(\mathbf{p},\mathbf{q})\equiv\int_{-\infty}^{\infty}\!\mathrm{d} \bar x_-\,e^{i\mathbf{p} x_--i\mathbf{q}\bar x_-} =\int_{-\infty}^{\infty}\!\mathrm{d}\bar x_-\, \mathrm{exp}\left(\frac{i\mathbf{p}}{\sqrt{2}a}e^{\sqrt{2}a\bar x_-}-i\mathbf{q} \bar x_-\right)~, \ \ \ \ \ (25)$

where in the second equality we have substituted the definition (18). We can then obtain ${b_q^\dagger}$ by simply taking the hermitian conjugate of this expression, noting that by definition,

$\displaystyle F^\dagger(\mathbf{p},\mathbf{q})=F(-\mathbf{p},-\mathbf{q})~. \ \ \ \ \ (26)$

Similarly, by Fourier transforming the second equation in (22), we obtain

$\displaystyle b_{-q}^\dagger=\int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{2\pi}\frac{\mathbf{q}}{\mathbf{p}}\left[ a_{-p}\,F(-\mathbf{p},\mathbf{q})+a_{-p}^\dagger\,F(\mathbf{p},\mathbf{q})\right]~, \ \ \ \ \ (27)$

the hermitian conjugate of which gives ${b_{-q}}$.

With the expressions (24), (27) for the Rindler creation/annihilation operators in terms of their Minkowski counterparts in hand, we can compute the expectation value of the Rindler number operator ${\hat N_b=\int\!\frac{\mathrm{d}\mathbf{q}}{4\pi\omega}b_q^\dagger b_q}$ in the Minkowski vacuum, to determine what a uniformly accelerating observer would observe. For compactness, we shall denote ${\langle0_\textrm{\tiny{M}}|\ldots|0_\textrm{\tiny{M}}\rangle\equiv\langle\ldots\rangle}$ in the following computation:

\displaystyle \begin{aligned} \langle\hat N_b\rangle=\int\!\frac{\mathrm{d}\mathbf{q}}{4\pi\omega}\langle b_q^\dagger b_q\rangle &=\int\!\frac{\mathrm{d}\mathbf{q}}{4\pi\omega}\int_0^\infty\!\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{k}\,\beta_{pq}^*\beta_{kq}\langle a_pa_k^\dagger\rangle\\ &=\int\!\frac{\mathrm{d}\mathbf{q}}{4\pi\omega}\int_0^\infty\!\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{k}\,\beta_{pq}^*\beta_{kq}\,4\pi\mathbf{p}\,\delta(\mathbf{p}-\mathbf{k})\\ &=\int\!\mathrm{d}\mathbf{q}\int_0^\infty\!\mathrm{d}\mathbf{p}\,\frac{\mathbf{p}}{|\mathbf{q}|}|\beta_{pq}|^2\\ &=\int\!\mathrm{d}\mathbf{q}\int_0^\infty\!\frac{\mathrm{d}\mathbf{p}}{(2\pi)^2}\,\frac{|\mathbf{q}|}{\mathbf{p}}\big|F(-\mathbf{p},\mathbf{q})\big|^2~. \end{aligned} \ \ \ \ \ (28)

Now, the seemingly straightforward way to evaluate the integral over ${\mathbf{p}}$ is to explicitly work out ${F}$, but the latter requires a contour integral, and one is left with an integral over a product of complex Gamma functions for the former (I’ve included this at the end of this post, for the masochistic among you). A more clever alternative, taking a tip from [2], is to use the normalization condition on the Bogolyubov coefficients one obtains from the canonical commutation relations:

\displaystyle \begin{aligned} {}[b_q,b_{q'}^\dagger]&=\int\!\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{p}'\left(\alpha_{pq}^*\alpha_{p'q'}-\beta_{p'q'}^*\beta_{pq}\right) [a_p,a_{p'}^\dagger]\\ &=\int\!\mathrm{d}\mathbf{p}\,4\pi|\mathbf{p}|\left(\alpha_{pq}^*\alpha_{pq'}-\beta_{pq'}^*\beta_{pq}\right)\\ &=\int\!\frac{\mathrm{d}\mathbf{p}}{\pi}\frac{\mathbf{q}\mathbf{q}'}{|\mathbf{p}|}\left[F(\mathbf{p},\mathbf{q})F^*(\mathbf{p},\mathbf{q}')-F(-\mathbf{p},\mathbf{q})F^*(-\mathbf{p},\mathbf{q}')\right]\\ &=\int\!\frac{\mathrm{d}\mathbf{p}}{\pi}\frac{\mathbf{q}\mathbf{q}'}{|\mathbf{p}|}\left[\left( e^{\pi \mathbf{q}/\sqrt{2}a}+e^{\pi \mathbf{q}'/\sqrt{2}a}\right)-1\right]F(-\mathbf{p},\mathbf{q})F^*(-\mathbf{p},\mathbf{q}') \\ \end{aligned} \ \ \ \ \ (29)

where in the last step we used the fact that ${F(\mathbf{p},\mathbf{q})=e^{\pi \mathbf{q}/\sqrt{2}a}F(-\mathbf{p},\mathbf{q})}$ (exercise for the reader, or see the appendix below). Since the left-hand side is equal to ${4\pi|\mathbf{q}|\delta(\mathbf{q}-\mathbf{q}')}$, we must have

$\displaystyle \int\!\frac{\mathrm{d}\mathbf{p}}{(2\pi)^2}\frac{\mathbf{q}'}{|\mathbf{p}|}\left[\left( e^{\pi \mathbf{q}/\sqrt{2}a}+e^{\pi \mathbf{q}'/\sqrt{2}a}\right)-1\right]F(-\mathbf{p},\mathbf{q})F^*(-\mathbf{p},\mathbf{q}')=\delta(\mathbf{q}-\mathbf{q}')~, \ \ \ \ \ (30)$

and therefore, setting ${\mathbf{q}=\mathbf{q}'}$, we obtain

$\displaystyle \int\!\frac{\mathrm{d}\mathbf{p}}{(2\pi)^2}\frac{\mathbf{q}}{|\mathbf{p}|}\big|F(-\mathbf{p},\mathbf{q})\big|^2=\left( e^{\pi\sqrt{2} \mathbf{q}/a}-1\right)^{-1}\delta(0)~, \ \ \ \ \ (31)$

which is exactly the integral we needed in (28)! Therefore,

$\displaystyle \langle\hat N_b\rangle =\delta(0)\int\!\mathrm{d}\mathbf{q}\left( e^{\pi\sqrt{2}\mathbf{q}/a}-1\right)^{-1} =\delta(0)\int\!\mathrm{d}\mathbf{k}\left( e^{\frac{2\pi}{a}\mathbf{k}}-1\right)^{-1}~, \ \ \ \ \ (32)$

where in the last step, we have undone our rescaling below (20): ${\mathbf{q}=\sqrt{2}\mathbf{k}}$ (that is, here ${\mathbf{k}}$ plays the role of the original Rindler momentum in (3). Note that the integration measure doesn’t pick up a factor of ${\sqrt{2}}$, because we really should have integrated with respect to ${\mathbf{k}}$ from the very start of (28)).

This is the final answer, except for two divergences: an IR divergence ${\delta(0)}$, and a UV divergence due to the fact that we integrated over arbitrarily high momenta. The first is rather silly, and arises from the infinite spatial volume ${V}$:

$\displaystyle 2\pi\delta(0)=\lim_{L\rightarrow\infty}\int_{-L/2}^{L/2}\!\mathrm{d}\mathbf{x}\,e^{-i\mathbf{p}\mathbf{x}}\Big|_{\mathbf{p}=0}=\lim_{L\rightarrow\infty}\int_{-L/2}^{L/2}\!\mathrm{d}\mathbf{x}=V~. \ \ \ \ \ (33)$

Meanwhile, the UV divergence is to be expected, since there are an infinite number of modes. We could impose a momentum cutoff on the integral to regulate this, but what we’re really after is the number density ${\langle \hat n_b\rangle}$, where ${\hat N_b=\frac{1}{V}\int\!\frac{\mathrm{d}\mathbf{k}}{2\pi}\,\hat n_b}$. Hence, stripping off the divergences in this manner, we arrive at

$\displaystyle \langle \hat n_b\rangle = \left( e^{2\pi\mathbf{k}/a}-1\right)^{-1}~, \ \ \ \ \ (34)$

which is precisely the Plank spectrum for a black body at temperature ${T=a/2\pi}$.

This result — that a uniformly accelerating observer in Minkowski vacuum observes a thermal spectrum — is called the Unruh effect, and is nothing short of remarkable. From an operationalist standpoint, one can say that the acceleration of the Rindler observer is what provides the energy for these excitations, but this sidesteps the question of their ontological status (indeed, the closely related question of exactly how pairwise entangled Hawking modes evolve to particles in the inertial frame at infinity is a non-trivial problem to which I hope to return). In the context of the maximum entropy procedure we’ve discussed before, the thermal spectrum may be regarded as a consequence of the fact that the Rindler observer has traced-out everything on the other side of the horizon, so the only information she has left is the temperature set by her own acceleration. If we restore units, this temperature reads

$\displaystyle T=\frac{\hbar a}{2\pi c k_B}~. \ \ \ \ \ (35)$

The Unruh effect thus represents a deep interplay between statistical thermodynamics née information theory (${k_B}$), relativity (${c}$), and quantum field theory (${\hbar}$). It is another manifestation of the thermal nature of horizons we mentioned in part 1, and arises as an inevitable consistency condition between inertial and Rindler frames. Furthermore, since the Rindler frame well-approximates the near-horizon geometry of a static black hole, the Unruh effect is intimately related to Hawking radiation, and my personal suspicion is that when we understand things more deeply, we’ll find the same underlying explanation for both. Indeed, in the Bekenstein-Hawking formula for black hole entropy ${S=\frac{k_BA}{4\ell_P^2}}$, the only additional ingredient in this exceptional confluence is the presence of gravity (${G}$)—the nature of which we’re far from having grasped.

References

1. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 1984. http://www.cambridge. org/mw/academic/subjects/physics/theoretical-physics-and-mathematical-physics/ quantum-fields-curved-space?format=PB.
2. V. F. Mukhanov and S. Winitzki, Introduction to Quantum Fields in Classical Backgrounds. 2004. https://uwaterloo.ca/physics-of-information-lab/sites/ca. physics-of-information-lab/files/uploads/files/text.pdf. (freely-available draft of the published textbook).

As an exercise in contour integration, one can obtain an explicit expression for the integral (25). The trick is to massage it into a gamma function:

$\displaystyle \Gamma(\zeta,a,b)=\int_a^b\!\mathrm{d} \chi\,e^{-\chi}\chi^{\zeta-1} =\begin{cases} \Gamma(\zeta)~\qquad &a\!=\!0~,\;\;b\rightarrow\infty~,\\ \Gamma(\zeta,s)~\qquad &a\!=\!s~,\;\;b\rightarrow\infty~,\\ \gamma(\zeta,s)~\qquad &a\!=\!0~,\;\;b\!=\!s~, \end{cases} \ \ \ \ \ (36)$

where ${\Gamma(\zeta,s)}$ and ${\gamma(\zeta,s)}$ are the upper and lower incomplete gamma functions, respectively, and ${\zeta\in\mathbb{C}}$ with ${\mathrm{Re}(\zeta)\!>\!0}$. We then begin by defining a new variable ${z=e^{\sqrt{2}a\bar x_-}}$, so that

$\displaystyle F=\frac{1}{\sqrt{2}a}\int_0^\infty\!\mathrm{d} z\,e^{-ip z}z^{-iq-1}~, \ \ \ \ \ (37)$

where we have defined the rescaled momentum vectors ${\sqrt{2}a\,p=\mathbf{p}}$, ${\sqrt{2}a\,q=\mathbf{q}}$ (and broken the boldface convention, because I didn’t want to introduce even more letters). This is pretty close to the desired form already, but the complex exponent ${e^{-ipz}}$ necessitates that we promote ${z}$ to a complex variable, and treat this as a contour integral in the complex plane.

Recall that we can write any ${z\in\mathbb{C}}$ in the form ${z^\alpha=r^\alpha e^{i\alpha\theta}}$ with ${r\in\mathbb{R}}$, which has branch points at both ${z=0}$ and ${z=\infty}$, since there’s a non-trivial monodromy ${z^\alpha\rightarrow z^\alpha e^{2\pi i\alpha}}$ along a closed path that encircles either point in the complex plane. Since the integral converges as ${z\rightarrow i\infty}$ due to the exponential damping, let’s choose the branch cut to run from ${z=0}$ to ${z=-\infty}$ along the negative real axis, so that the following contour encloses no poles:

$\displaystyle \oint=\int_0^\infty+\int_\mathrm{arc}+\int_{i\infty}^0=0~, \ \ \ \ \ (38)$

where the arc term is a quarter-circle that connects the positive real and complex axes at ${|z|\rightarrow\infty}$, which vanishes since ${q,p>0}$. Thus, taking ${z=re^{i\pi/2}}$ along the positive complex axis, the integral (37) becomes

$\displaystyle F=\frac{1}{\sqrt{2}a}\int_0^{i\infty}\!\mathrm{d} z\,e^{-ipz}z^{-iq-1} =\frac{1}{\sqrt{2}a}e^{\pi q/2}\int_0^{\infty}\!\mathrm{d} r\,e^{pr}r^{-iq-1}~, \ \ \ \ \ (39)$

where ${\left( e^{i\pi/2}\right)^{-iq-1}=e^{\pi q/2}e^{-i\pi/2}=-i\,e^{\pi q/2}}$. Comparing with the form of the gamma function (36), we identify ${\chi=pr}$ and ${\zeta=-iq}$, whence we find

$\displaystyle F(\mathbf{p},\mathbf{q}) =\frac{1}{\sqrt{2}a}e^{\pi q/2}p^{iq}\,\Gamma(-iq)~. \ \ \ \ \ (40)$

As an aside, while the form of this expression makes it non-obvious, the property (26) is preserved, as one can easily show by noting that

$\displaystyle e^{-\pi q/2}(-p)^{-iq} =e^{-\pi q/2}\left( e^{i\pi}p\right)^{-iq} =\left( e^{-i \pi/2}e^{i\pi}p\right)^{-iq} =\left( e^{i \pi/2}p\right)^{-iq} =e^{\pi q/2}p^{-iq}~. \ \ \ \ \ (41)$

Substituting this result back into (24), we at last obtain

$\displaystyle b_{q}=\int_0^\infty\!\mathrm{d}\mathbf{p}\,\left(\alpha_{pq}^*a_p+\beta_{pq}a_p^\dagger\,\right) \quad\implies\quad b_q^\dagger=\int_0^\infty\!\mathrm{d}\mathbf{p}\,\left(\beta_{pq}^*a_p+\alpha_{pq}a_p^\dagger\,\right)~, \ \ \ \ \ (42)$

where the Bogolyubov coefficients are

$\displaystyle \alpha_{pq}^*=\frac{\mathbf{q}}{2\pi\mathbf{p}}F(\mathbf{p},\mathbf{q})~, \qquad \qquad \beta_{pq}=\frac{\mathbf{q}}{2\pi\mathbf{p}}F(-\mathbf{p},\mathbf{q})~. \ \ \ \ \ (43)$

The left-movers ${b_{-q},b_{-q}^\dagger}$ are similarly obtained from (27).

Posted in Physics | 3 Comments

## QFT in curved space, part 1: Green functions

I was recently** asked to give a lecture on black hole thermodynamics and the associated quantum puzzles, which provided a perfect excuse to spend some time reviewing one of my favourite subjects: quantum field theory (QFT) in curved spacetime. I’ll mostly follow the canonical reference by Birrell and Davies [1], and will use this series of posts to highlight a number of important and/or interesting aspects along the way. I spent many a happy hour with this book as a graduate student, and warmly recommend it to anyone desiring a more complete treatment.

**(That is, “recently” when I started this back in February. I had intended to finish the series before posting the individual parts, to avoid retroactive edits as my understanding and plans for future segments evolves, but alas the constant pressure to publish (or, perhaps more charitably, the fact that I don’t get paid to teach a course on this stuff) means that studies of this sort are frequently pushed down my priority list (and that was before an international move in the midst of a global pandemic, for those wondering why I haven’t posted in so long). Since such time constraints are likely to continue — on top of which, I have no fixed end-point in mind for this vast subject — I’ve decided to release the first few parts I’ve been sitting on, lest they never see the light of day. I hope to add more installments as time permits.)

I’m going to start by discussing Green functions (commonly but improperly called “Green’s functions”), which manifest one of the deepest relationships between gravity, quantum field theory, and thermodynamics, namely the thermodynamic character of the vacuum state. Specifically, the fact that Green functions are periodic in imaginary time — also known as the KMS or Kubo-Martin-Schwinger condition — hints at an intimate relationship between Euclidean field theory and statistical mechanics, and underlies the thermal nature of horizons (including, but not limited to, those of black holes).

For simplicity, I’ll stick to Green functions of free scalar fields ${\phi(x)}$, where the ${D}$-vector ${x=(t,\mathbf{x})}$. As a notational aside, I will depart from [1] in favour of the modern convention in which the spacetime dimension is denoted ${D=d\!+\!1}$, with Greek indices running over the full ${D}$-dimensional spacetime, and Latin indices restricted to the ${d}$-dimensional spatial component. While I’m at it, I should also warn you that [1] uses the exceedingly unpalatable mostly-minus convention ${\eta_{\mu\nu}=(+,-,-,-)}$, whereas I’m going to use mostly-plus ${\eta_{\mu\nu}=(-,+,+,+)}$. The former seems to be preferred by particle physicists, because they share with small children a preference for timelike 4-vectors to have positive magnitude. But the latter is generally preferred by relativists and most workers in high-energy theory and quantum gravity, because 3-vectors have no minus signs (i.e., it’s consistent with the non-relativistic case, whereas mostly-plus yields a negative-definite metric), raising and lowering indices involves flipping only a single sign (arguably the most important for our purposes, since we’ll be Wick rotating between Lorentzian and Euclidean signature; mostly-plus would again lead to a negative-definite Euclidean metric), and the extension to general dimensions contains only a single ${-1}$ in the determinant (as opposed to a factor of ${(-1)^D}$ in mostly-minus).

Notational disclaimers dispensed with, the Lagrangian density is

$\displaystyle \mathcal{L}(x)=-\frac{1}{2}\left(\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+m^2\phi^2\right)~, \ \ \ \ \ (1)$

where ${\eta^{\mu\nu}}$ is the Minkowski metric (no curved space just yet). By applying the variational principle ${\delta S=0}$ to the action

$\displaystyle S=\int\!\mathrm{d}^Dx\mathcal{L}~, \ \ \ \ \ (2)$

we obtain the familiar Klein-Gordon equation

$\displaystyle \left(\square-m^2\right)\phi=0~, \ \ \ \ \ (3)$

where ${\square\equiv\eta^{\mu\nu}\partial_\mu\partial_\nu}$. The general solution, upon imposing that ${\phi}$ be real-valued, is

$\displaystyle \phi(x)=\int\!\frac{\mathrm{d}^dk}{2\omega(2\pi)^d}\left[a_k e^{i\mathbf{k}\mathbf{x}-i\omega t}+a_k^\dagger e^{-i\mathbf{k}\mathbf{x}+i\omega t}\right]~, \ \ \ \ \ (4)$

where ${k=(\omega,\mathbf{k})}$ (note that [1] restricts to the case of a discrete spectrum, as though the system were in a box; useful for imposing an IR regulator, but unnecessary for our purposes, and potentially problematic if we want to consider Lorentz boosts or Euclidean continuations). Here ${a_k}$ is the annihilation operator that kills the vacuum state, i.e., ${a_k|0\rangle=0}$ (so ${\phi}$, by extension, is a real-valued field operator).

One last (lengthy, but important) notational aside: different authors make different choices for the integration measure ${\int\!\frac{\mathrm{d}^dk}{f(k)}}$, which affects a number of later formulas, and can cause confusion when comparing different sources. The convention I’m using is physically well-motivated in that it makes the measure Lorentz invariant while encoding the on-shell condition ${k^2\!=\!-m^2}$. That is, the Lorentz invariant measure in the full ${D}$-dimensional spacetime is ${\int\!\mathrm{d}^Dk}$. If we then impose the on-shell condition along with ${\omega>0}$ (in the form of the Heaviside function ${\Theta(\omega)}$), we have

$\displaystyle \int\!\mathrm{d}^Dk\,\delta(k^2+m^2)\Theta(\omega) =\int\!\mathrm{d}\omega\int\!\mathrm{d}^dk\,\delta(\mathbf{k}^2-\omega^2+m^2)\Theta(\omega)~. \ \ \ \ \ (5)$

We now use the following trick: if a smooth function ${g(x)}$ has a root at ${x_0}$, then we may write

$\displaystyle \int\!\mathrm{d} x\,\delta(g(x))=\int\!\mathrm{d} x\,\frac{\delta(x-x_0)}{|g'(x)|} =\frac{1}{|g'(x_0)|} \ \ \ \ \ (6)$

where the prime denotes the derivative with respect to ${x}$. In the present case, ${{g(\omega)=\mathbf{k}^2-\omega^2+m^2}}$, and ${x_0^2=\mathbf{k}^2+m^2}$ (note that the Heaviside function will select the positive root). Thus

$\displaystyle \int\!\mathrm{d}\omega\!\int\!\mathrm{d}^dk\,\delta(\mathbf{k}^2-\omega^2+m^2)\Theta(\omega) =\int\!\frac{\mathrm{d}^dk}{2\sqrt{\mathbf{k}^2+m^2}}~. \ \ \ \ \ (7)$

Finally, since ${k}$ and ${x}$ are related by a Fourier transform, we must adopt a convention for the associated factor of ${(2\pi)^d}$. Mathematicians seem to prefer splitting this so that both ${\mathrm{d}^dk}$ and ${\mathrm{d}^d x}$ get a factor of ${(2\pi)^{d/2}}$, but physicists favour simply attaching it all to the momentum, so that

$\displaystyle \hat\phi(k)=\int\!\mathrm{d}^dx\,e^{-ikx}\phi(x) \qquad\mathrm{and}\qquad \phi(x)=\int\!\frac{\mathrm{d}^dk}{(2\pi)^d}\,e^{ikx}\hat\phi(k)~. \ \ \ \ \ (8)$

which further implies the convention

$\displaystyle (2\pi)^d\delta^d(k-p)=\int\!\mathrm{d}^dx\,e^{-i(k-p)x} \qquad \mathrm{and} \qquad \delta^d(x-y)=\int\!\frac{\mathrm{d}^dk}{(2\pi)^d}\,e^{ik(x-y)}~, \ \ \ \ \ (9)$

as one can readily verify by substituting ${\phi(x)}$ into ${\hat\phi(k)}$ (or vice versa):

\displaystyle \begin{aligned} \hat\phi(k)&=\int\!\mathrm{d}^dx\,e^{-ikx}\!\int\!\frac{\mathrm{d}^dp}{(2\pi)^d}\,e^{ipx}\hat\phi(p) =\int\!\frac{\mathrm{d}^dp}{(2\pi)^d}\int\!\mathrm{d}^dx\,e^{-i(k-p)x}\hat\phi(p)\\ &=\int\!\mathrm{d}^dp\,\delta^d(k-p)\hat\phi(p) =\hat\phi(k)~. \end{aligned} \ \ \ \ \ (10)

Thus our choice for the measure in (4):

$\displaystyle \int\!\frac{\mathrm{d}^dk}{f(k)}\equiv\int\!\frac{\mathrm{d}^dk}{2\sqrt{\mathbf{k}^2+m^2}(2\pi)^d} =\int\!\frac{\mathrm{d}^dk}{2\omega(2\pi)^d}~. \ \ \ \ \ (11)$

(I realize that was a bit tedious, but setting one’s conventions straight will pay dividends later. Trust me: I’ve lost hours trying to sort out factors of ${2\pi}$ and the like for failure to invest this time at the start).

We can now consider vacuum expectation values of products of field operators ${\phi}$. For free scalar fields, these can always be decomposed into two-point functions, which therefore play a defining role. In particular, we can construct various Green functions of the wave operator ${(\square-m^2)}$ from the two-point correlator ${\langle\phi(x)\phi(x')\rangle}$, including the familiar Feynman propagator. Following [1], we’ll denote the expectation values of the commutator and anticommutator as follows:

\displaystyle \begin{aligned} iG(x,x')&=\langle\left[\phi(x),\phi(x')\right]\rangle=G^+\!(x,x')-G^-\!(x,x')~,\\ G^{(1)}(x,x')&=\langle\left\{\phi(x),\phi(x')\right\}\rangle=G^+\!(x,x')+G^-\!(x,x')~, \end{aligned} \ \ \ \ \ (12)

where ${G^{\pm}}$ on the far right-hand sides are the so-called positive/negative frequency Wightman functions,

\displaystyle \begin{aligned} G^+\!(x,x')&=\langle\phi(x)\phi(x')\rangle~,\\ G^-\!(x,x')&=\langle\phi(x')\phi(x)\rangle~. \end{aligned} \ \ \ \ \ (13)

Note that while physicists call all of these Green functions, they’re technically kernels, i.e.,

$\displaystyle \left(\square_x-m^2\right)\mathcal{G}(x,x')=0~,\qquad\qquad\mathcal{G}\in\{G,\,G^{(1)}\!,G^{\pm}\}~. \ \ \ \ \ (14)$

One can immediately verify this by observing that since ${\square_x}$ acts only on ${\phi(x)}$ (that is, ${\square_x\phi(x')=0}$), it reduces to the Klein-Gordon equation above for the Wightman functions, from which the others follow.

Using these building blocks, we can consider the true Green functions

$\displaystyle iG_F(x,x')=\langle\mathcal{T}\phi(x)\phi(x')\rangle=\Theta(t-t')G^+\!(x,x')+\Theta(t'-t)G^-\!(x,x')~, \ \ \ \ \ (15)$

which is the familiar (time-ordered, ${\mathcal{T}}$) Feynman propagator, and

\displaystyle \begin{aligned} G_R(x,x')&=-\Theta(t-t')G(x,x')~,\\ G_A(x,x')&=\Theta(t'-t)G(x,x')~, \end{aligned} \ \ \ \ \ (16)

which are the retarded (R) and advanced (A) propagators. All three of these are Green functions of the wave operator, i.e.,

$\displaystyle \left(\square_x-m^2\right)\mathcal{G}(x,x')=\delta^D(x-x')~,\qquad\qquad\mathcal{G}\in\{G_F,G_R,G_A\}~. \ \ \ \ \ (17)$

Let’s verify this for the Feynman propagator; the others are similar. Using the fact that ${\eta^{\mu\nu}\partial_\nu\Theta(t-t')=\eta^{\mu0}\partial_0\Theta(t-t')=\eta^{\mu0}\delta(t-t')}$, we have

\displaystyle \begin{aligned} \square_x G_F=&-i\eta^{\mu0}\partial_\mu\left[\delta(t-t')G^+\!(x,x')-\delta(t'-t)G^-\!(x,x')\right]\\ &-i\eta^{\mu\nu}\partial_\mu\left[\Theta(t-t')\partial_\nu G^+\!(x,x')+\Theta(t'-t)\partial_\nu G^-\!(x,x')\right]~. \end{aligned} \ \ \ \ \ (18)

Now observe that by virtue of the delta function, the equal-time commutator ${{[\phi(t,\mathbf{x}),\phi(t,\mathbf{x}')]=0}}$ means that in the first line, ${G^+=G^-}$. And since the delta function itself is even, this implies that the first two terms cancel, so we continue with just the second line:

\displaystyle \begin{aligned} \square_x G_F=&-i\eta^{00}\left[\delta(t-t')\partial_0 G^+\!(x,x')-\delta(t'-t)\partial_0 G^-\!(x,x')\right]\\ &-i\eta^{\mu\nu}\left[\Theta(t-t')\partial_\mu\partial_\nu G^+\!(x,x')+\Theta(t'-t)\partial_\mu\partial_\nu G^-\!(x,x')\right]\\ =&\,\,i\delta(t-t')\left[\pi(x)\phi(x')-\phi(x')\pi(x)\right]\\ &-i\left[\Theta(t-t')\square_x G^+\!(x,x')+\Theta(t'-t)\square_x G^-\!(x,x')\right]~, \end{aligned} \ \ \ \ \ (19)

where in the second step, we have used the fact that the delta function is even, and identified the conjugate momentum ${\pi(x)=\partial_0\phi(x)}$. Then by (14), the second line will vanish for all values of ${t\!-\!t'}$ when we add in the ${-m^2}$ term of the wave operator, and the first line is just (minus) the equal-time commutator ${[\phi(t,\mathbf{x}),\pi(t,\mathbf{x}')]=i\delta^d(\mathbf{x}-\mathbf{x}')}$. Hence

$\displaystyle \left(\square_x-m^2\right) G_F=\delta(t-t')\delta^d(\mathbf{x}-\mathbf{x}')=\delta^D\!(x-x')~. \ \ \ \ \ (20)$

Thus the Feynman propagator is indeed a Green function of the wave operator ${(\square_x\!-\!m^2)}$; similarly for ${G_R}$ and ${G_A}$.

The reason I’ve been calling the Green functions ${G_F,G_R,G_A}$ “propagators” is that, unlike the kernels ${G,G^{(1)},G^{\pm}}$, they represent the transition amplitude for a particle (virtual or otherwise) propagating from ${x}$ to ${x'}$, subject to appropriate boundary conditions. To see this, consider the integral representation

$\displaystyle \mathcal{G}(x,x')=\int\!\frac{\mathrm{d}^Dk}{(2\pi)^D}\frac{e^{ik(x-x')}}{-k_0^2+\mathbf{k}^2+m^2}~, \ \ \ \ \ (21)$

where ${k^2=k^\mu k_\mu=-k_0^2+\mathbf{k}^2}$. Due to the poles at ${k_0=\omega=\pm\sqrt{\mathbf{k}^2+m^2}}$, we need to choose a suitable contour for the integral to be well-defined (analytically continuing to ${{k\in\mathbb{C}}}$). The particular choice of contour determines which of the kernels ${G,G^{(1)},G^{\pm}}$ or Green functions ${G_F,G_R,G_A}$ we obtain. (As for how we obtained (21) in the first place, one can directly substitute in the mode expansion (4) to the definitions, and convert the Heaviside functions into an appropriate integral. An easier way, at least for the Green functions, is to simply Fourier transform the wave equation (17):

\displaystyle \begin{aligned} \left(\square_x-m^2\right)\int\!\frac{\mathrm{d}^Dk}{(2\pi)^D}\,\tilde{\mathcal{G}}(k)\,e^{ik(x-x')}&=\delta^D(x-x')\\ \implies \int\!\frac{\mathrm{d}^Dk}{(2\pi)^D}\left(-k^2-m^2\right)\,e^{ik(x-x')}\tilde{\mathcal{G}}(k)&=\int\!\frac{\mathrm{d}^Dp}{(2\pi)^D}\,e^{ip(x-x')}~. \end{aligned} \ \ \ \ \ (22)

Since this expression (i.e., the delta function) is even in ${x\!-\!x'}$, we may absorb the sign into the integration variable, and identify

$\displaystyle \tilde{\mathcal{G}}(k)=\frac{1}{k^2+m^2}~, \ \ \ \ \ (23)$

whereupon Fourier transforming back to position space yields (21). As alluded above however, these expressions don’t make sense without specifying a pole prescription, so this argument isn’t very rigorous; it’s just a quick-and-dirty way of convincing yourself that (21) is plausible.)

To make sense of this expression, we split the integral based on the two poles of ${k_0}$:

\displaystyle \begin{aligned} \mathcal{G}(x,x')&=\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\oint\mathrm{d} k_0\frac{e^{-ik_0(x_0-x'_0)}}{-k_0^2+\mathbf{k}^2+m^2}\\ &=\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\oint\mathrm{d} k_0\frac{e^{-ik_0(x_0-x'_0)}}{-2k_0}\left(\frac{1}{k_0-\sqrt{\mathbf{k}^2+m^2}}+\frac{1}{k_0+\sqrt{\mathbf{k}^2+m^2}}\right)~. \end{aligned} \ \ \ \ \ (24)

Now, the boundary conditions of the propagator at hand determines the ${i\epsilon}$ prescription, i.e., which of the poles we want to enclose with the choice of contour. Consider first the retarded propagator ${G_R}$: the boundary condition implicit in (16) is that the function should vanish when ${x_0\!<\!x_0'}$ (where ${x_0\!=\!t}$). Conversely, when ${x_0\!>\!x_0'}$, we must close the contour in the negative half-plane so that ${e^{-ik_0(x_0-x'_0)}\rightarrow e^{-i(-i\infty)(x_0-x'_0)}=e^{-\infty(x_0-x'_0)}=e^{-\infty}}$, and the integral converges. Thus we should introduce factors of ${i\epsilon}$ such that both poles are slightly displaced into the lower half-plane. We can then apply Cauchy’s integral formula to correctly capture the poles at ${k_0=\pm\omega-i\epsilon}$, and then take ${\epsilon\rightarrow0}$:

\displaystyle \begin{aligned} G_R(x,x')&=\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\oint\mathrm{d} k_0\frac{e^{-ik_0(x_0-x'_0)}}{-2k_0}\left(\frac{1}{k_0-\sqrt{\mathbf{k}^2+m^2}+i\epsilon}+\frac{1}{k_0+\sqrt{\mathbf{k}^2+m^2}+i\epsilon}\right)\\ &=\Theta(x_0-x'_0)\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}(2\pi i)\left(\frac{e^{-i\omega(x_0-x'_0)}}{-2\omega}+\frac{e^{i\omega(x_0-x'_0)}}{2\omega}\right)\\ &=-i\Theta(x_0-x'_0)\int\frac{\mathrm{d}^dk}{2\omega(2\pi)^d}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\left( e^{-i\omega(x_0-x'_0)}-e^{i\omega(x_0-x'_0)}\right)\\ &=-i\Theta(x_0-x'_0)\int\frac{\mathrm{d}^dk}{2\omega(2\pi)^d}\left[e^{ik(x-x')}-e^{-ik(x-x')}\right]\\ &=i\Theta(x_0-x'_0)\langle\left[\phi(x),\phi(x')\right]\rangle =-\Theta(t-t')G(x,x')~, \end{aligned} \ \ \ \ \ (24)

where in the penultimate line, we have taken ${\mathbf{k}\rightarrow-\mathbf{k}}$ in the second term, using the fact that the integration over all (momentum) space is even; in the last line, we have used the mode expansion (4) and the commutation relation ${[a_k,a_k'^\dagger]=2\omega(2\pi)^d\delta^d(\mathbf{k}-\mathbf{k}')}$. Note that to yield the correct signs, we’ve chosen the contour to run counter-clockwise (note the factor of ${+2\pi i}$), which means that it runs from ${+\infty}$ to ${-\infty}$ along the real axis. The prescription for the advanced propagator is precisely similar, except we deform both poles in the positive complex direction (so that the integral vanishes when we close the contour below, as required for ${x_0-y_0>0}$), and the non-vanishing contribution comes from closing the contour in the positive half-plane, encircling both poles clockwise rather than counter-clockwise (so that the integral again runs from ${+\infty}$ to ${-\infty}$ along the real axis).

Note that ${G_R,G_A}$ are superpositions of both positive (${\omega\!>\!0}$) and negative (${\omega\!<\!0}$) energy modes, which is necessary in order for them to vanish outside their prescribed lightcones (past and future, respectively). In contrast, the Heaviside functions in the Feynman propagator are tantamount to imposing boundary conditions such that it picks up only positive or negative frequencies, depending on the sign of ${t\!-\!t'}$. For ${t\!>\!t'}$, we close the contour in the lower-half plane for convergence (${e^{-ik^0(t-t')}=e^{-i(-i\infty)(t-t')}=e^{-\infty(t-t')}}$), and enclose ${k_0=\omega}$ counter-clockwise (in the present conventions, we’re again going from ${+\infty}$ to ${-\infty}$ along the real axis); conversely, we close the contour clockwise in the upper-half plane to converge with ${k_0=-\omega}$ when ${t\!<\!t'}$. Hence the corresponding ${i\epsilon}$ prescription is

\displaystyle \begin{aligned} iG_F(x,x')&=i\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\oint\mathrm{d} k_0\frac{e^{-ik_0(x_0-x'_0)}}{-2k_0}\left(\frac{1}{k_0-\sqrt{\mathbf{k}^2+m^2}-i\epsilon}+\frac{1}{k_0+\sqrt{\mathbf{k}^2+m^2}+i\epsilon}\right)\\ &=i\int\frac{\mathrm{d}^dk}{(2\pi)^D}e^{i\mathbf{k}(\mathbf{x}-\mathbf{x}')}\left[(2\pi i)\Theta(x_0-x_0')\frac{e^{-i\omega(x_0-x'_0)}}{-2\omega}+(-2\pi i)\Theta(x_0'-x_0)\frac{e^{i\omega(x_0-x'_0)}}{2\omega}\right]\\ &=\int\frac{\mathrm{d}^dk}{2\omega(2\pi)^d}\left[\Theta(x_0-x_0')e^{ik(x-x')}+\Theta(x_0'-x_0)e^{-ik(x-x')}\right]\\ &=\Theta(t-t')G^+(x,x')+\Theta(t'-t)G^-(x,x')~, \end{aligned} \ \ \ \ \ (25)

as desired. It is in this sense that the time-ordering is automatically encoded by the Feynman propagator: for ${t\!>\!t'}$, it corresponds to a positive-energy particle propagating forwards in time, while for ${t\!, we have a negative-energy particle (i.e., an antiparticle) propagating backwards.

(I won’t go into the pole prescriptions for the kernels here, but the contours are illustrated in fig. 3 of [1]. The essential difference is that unlike the Green functions, the contours for the kernels are all closed loops, so these don’t correspond to propagating amplitudes.)

So far everything I’ve reviewed is for zero-temperature field theory; as alluded in the introduction of this post however, finite-temperature is where things get really interesting. Recall from quantum mechanics that a mixed state can be thought of as a statistical ensemble of pure states, so rather than computing expectation values with respect to the vacuum state, we compute them with respect to the mixed state given by the thermal density matrix

$\displaystyle \rho=\sum_ip_i\left|\psi_i\rangle\langle\psi_i\right|=\frac{1}{Z}e^{-\beta H}~, \ \ \ \ \ (26)$

where the system, governed by the Hamiltonian ${H}$, is in any of the states ${|\psi_i\rangle}$ with (classical) probability

$\displaystyle p_i=\frac{1}{Z}e^{-\beta E_i}~. \ \ \ \ \ (27)$

Of course, not all mixed states are thermal, but the latter is the correct state to use in the absence of any additional constraints. (One way to think of this is that the mixedness of a quantum state is a measure of our ignorance, which is why pure states are states of minimum entropy). Expectation values of operators ${\mathcal{O}}$ with respect to (26) are then ensemble averages at fixed temperature ${T=\beta^{-1}}$:

$\displaystyle \langle\mathcal{O}\rangle_\beta=\mathrm{tr}\left(\rho\,\mathcal{O}\right) =\sum_ip_i\langle\psi_i\left|\mathcal{O}\right|\psi_i\rangle~. \ \ \ \ \ (28)$

Note that we’re in the canonical ensemble (fixed temperature), rather than the microcanonical ensemble (fixed energy), because the energy — that is, the expectation value of the hamiltonian operator ${H=\sum_k \omega\,a_k^\dagger a_k}$ — will fluctuate as quanta are created or destroyed. Strictly speaking I should also include the chemical potential, since the number operator ${N=\sum_ka_k^\dagger a_k}$ also fluctuates, but it doesn’t play any important role in what follows. (The distinction is worth keeping in mind when discussing black hole thermodynamics, where one should use the microcanonical ensemble instead, because the negative specific heat makes the canonical ensemble unstable).

The thermal Green functions (and kernels), which we denote with the subscript ${\beta}$, are then obtained by replacing the vacuum expectation value with the expectation value in the thermal state, (28); for example, the Wightman functions become

\displaystyle \begin{aligned} G_\beta^+\!(x,x')&=\langle\phi(x)\phi(x')\rangle_\beta~,\\ G_\beta^-\!(x,x')&=\langle\phi(x')\phi(x)\rangle_\beta~. \end{aligned} \ \ \ \ \ (29)

The aforementioned KMS condition can then be obtained from the Heisenberg equation of motion,

$\displaystyle \phi(t_1,\mathbf{x})=e^{iH(t_1-t_0)}\phi(t_0,\mathbf{x})e^{-iH(t_1-t_0)}~, \ \ \ \ \ (30)$

by evolving in Euclidean time by ${t_1-t_0=i\beta}$:

\displaystyle \begin{aligned} G_\beta^+&=\frac{1}{Z}\mathrm{tr}\left[e^{-\beta H}\phi(t,\mathbf{x})\phi(t,\mathbf{x}')\right] =\frac{1}{Z}\mathrm{tr}\left[e^{-\beta H}\phi(t,\mathbf{x})e^{\beta H}e^{-\beta H}\phi(t,\mathbf{x}')\right]\\ &=\frac{1}{Z}\mathrm{tr}\left[\phi(t+i\beta,\mathbf{x})e^{-\beta H}\phi(t,\mathbf{x}')\right] =\frac{1}{Z}\mathrm{tr}\left[e^{-\beta H}\phi(t',\mathbf{x}')\phi(t+i\beta,\mathbf{x})\right]~, \end{aligned} \ \ \ \ \ (31)

where the last step relied on the cyclic property of the trace; similarly for ${G_\beta^-}$. Thus we arrive at the KMS condition

$\displaystyle G_\beta^\pm(t,\mathbf{x};t',\mathbf{x}')=G_\beta^\mp(t+i\beta,\mathbf{x};t',\mathbf{x}')~. \ \ \ \ \ (32)$

Note that this is a statement about expectation values of operators in the particular state (26) (indeed, this can easily be formulated for a general observable ${\mathcal{O}}$, we’re just sticking with scalar fields for concreteness; for a slightly more rigorous treatment, with suitable comments about boundedness and whatnot, see for example [2]). More generally however, any state which satisfies (32) is called a KMS state, and describes a system in thermal equilibrium. Similar relations hold for the other Green functions / kernels as well; e.g.,

$\displaystyle G_\beta^{(1)}(t,\mathbf{x};t',\mathbf{x}')=G_\beta^{(1)}(t+i\beta,\mathbf{x};t',\mathbf{x}')~. \ \ \ \ \ (33)$

As an exception to this however, note that since the commutator of free scalar fields is a c-number, ${G}$ in (12) remains unchanged, i.e., ${G_\beta=G}$.

In arriving at (32), we evolved in imaginary time ${\tau=it}$ by an amount given by the (inverse) temperature ${\beta}$. This is none other than the usual Wick rotation from Minkowski to Euclidean space, except that the periodicity of the Green functions implies that the Euclidean or thermal time direction is compact, with period ${\beta}$. That is, if the original field theory lived on ${\mathbb{R}^{d+1}}$, the finite-temperature field theory lives on ${\mathbb{R}^d\times S_\beta^1}$, where ${\beta}$ denotes the (inverse) circumference of the ${S^1}$ (observe that as ${\beta\rightarrow\infty}$, we recover the zero temperature Euclidean theory on ${\mathbb{R}^{d+1}}$). Thus in general, a Wick rotation in which Euclidean time is periodic makes an intimate connection between QFT and statistical thermodynamics, where the compact direction controls the temperature.

So what does this have to do with black holes, or horizons more generally? As I hope to cover in a future part of this sequence, the spacetime outside a horizon is also described by a thermal state. From the statistical thermodynamics or information theory perspective, one can think of this as due to the fact that we traced over the states on the other side, so the mixed density matrix that now describes the part of the vacuum to which we have access is a reflection of our ignorance. As alluded in the previous paragraph however, the thermodynamic character of the vacuum in the black hole state is already encoded in the periodicity of the Euclidean time direction, and emerges quite neatly in the case of the Schwarzschild black hole,

$\displaystyle \mathrm{d} s^2=-f(r)\mathrm{d} t^2+\frac{1}{f(r)}\mathrm{d} r^2+r^2\mathrm{d}\Omega_{d-1}^2~, \quad\quad f(r)=1-\frac{r_s}{r}~, \ \ \ \ \ (34)$

where ${r_s}$ is the Schwarzschild radius, and ${\mathrm{d}\Omega_{d-1}^2}$ is the metric on the ${(d\!-\!1)-}$sphere, which we’ll ignore since it just comes along for the ride. Recall from my very first blog post that after Wick rotating to Euclidean time, one can make a coordinate change so that the near-horizon metric becomes

$\displaystyle \mathrm{d} s^2=\mathrm{d}\rho^2+\frac{\rho^2}{4r_s^2}\mathrm{d}\tau^2~, \ \ \ \ \ (35)$

where ${\rho}$ is the radial direction, and — since these are polar coordinates — ${\tau}$ takes on the role of the angular coordinate, which must be periodic to avoid a conical singularity; that is, for any integer ${n}$,

$\displaystyle \frac{\tau}{2r_s}\sim\frac{\tau}{2r_s}+2\pi n \quad\implies\quad \tau\sim\tau+4\pi r_s n~, \ \ \ \ \ (36)$

and thus we identify the period ${4\pi r_s=\beta}$.

As a closing comment, the density matrix for KMS states has deeper relations to the idea of time translation symmetry via Tomita-Takesaki theory, through the modular hamiltonian that generates this 1-parameter family of automorphisms of the algebra of operators in the corresponding region. See for example [3]; this strikes me as a surprisingly under-researched direction, and I hope to revisit it in glorious detail soon.

References

1. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 1984. http://www.cambridge.org/mw/academic/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-fields-curved-space?format=PB.
2. S. Fulling and S. Ruijsenaars, “Temperature, periodicity and horizons,” Physics Reports 152 no. 3, (1987) 135 – 176.
3. A. Connes and C. Rovelli, “Von neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories,” Classical and Quantum Gravity 11 no. 12, (Dec, 1994) 2899–2917, https://arxiv.org/abs/gr-qc/9406019
Posted in Physics | 2 Comments

## Criticality in deep neural nets

In the previous post, we introduced mean field theory (MFT) as a means of approximating the partition function for interacting systems. In particular, we used this to determine the critical point at which the system undergoes a phase transition, and discussed the characteristic divergence of the correlation function. In this post, I’d like to understand how recent work in machine learning has leveraged these ideas from statistical field theory to yield impressive training advantages in deep neural networks. As mentioned in part 1, the key feature is the divergence of the correlation length at criticality, which controls the depth of information propagation in such networks.

I’ll primarily follow a pair of very interesting papers [1,2] on random neural nets, where the input (i.e., pre-activation) ${z_i^l}$ of neuron ${i}$ in layer ${l}$ is given by

$\displaystyle z_i^l=\sum_jW_{ij}^ly_j^{l-1}+b_i^l~, \ \ \ \ \ (1)$

where ${y_j^{l-1}=\phi(z_j^{l-1})}$ is some non-linear activation function of the neurons in the previous layer, and ${W_{ij}^l}$ is an ${N_l\times N_{l-1}}$ matrix of weights. The qualifier “random” refers to the fact that the weights and biases are i.i.d. according to the normal distributions

\displaystyle \begin{aligned} W_{ij}^l\sim\mathcal{N}\left(0,\sigma_w^2/N_{l-1}\right) &\implies p\left( w_{ij}^l\right)=\sqrt{\frac{N_{l-1}}{2\pi\sigma_w^2}}\,e^{-\frac{1}{2}\left(\!\frac{w_{ij}^l}{\sigma_w}\right)^2}~,\\ b_i^l\sim\mathcal{N}\left(0,\sigma_b^2\right) &\implies p\left( b_i^l\right)=\frac{1}{\sqrt{2\pi}\sigma_b}\,e^{-\frac{1}{2}\left(\!\frac{b_i^l}{\sigma_b}\right)^2}~, \end{aligned} \ \ \ \ \ (2)

where the factor of ${N_{l_1}}$ has been introduced to ensure that the contribution from the previous layer remains ${O(1)}$ regardless of layer width.

Given this, what is the probability distribution for the inputs ${z_i^l}$? Treating ${y_i^{l-1}}$ as fixed, this is a linear combination of independent, normally-distributed variables, so we expect it to be Gaussian as well. Indeed, we can derive this rather easily by considering the characteristic function, which is a somewhat more fundamental way of characterizing probability distributions than the moment and cumulant generating functions we’ve discussed before. For a random variable ${x\in\mathbb{R}}$, the characteristic function is defined as

$\displaystyle \phi_x(t)=\langle e^{itx}\rangle=\int_\mathbb{R}\!\mathrm{d} x\,e^{itx}p(x) \ \ \ \ \ (3)$

where in the second equality we assume the probability density (i.e., distribution) function ${p(x)}$ exists. Hence for a Gaussian random variable ${x\sim\mathcal{N}(\mu,\sigma^2)}$, we have

$\displaystyle \phi_x(t)=\frac{1}{\sqrt{2\pi}\sigma}\int\!\mathrm{d} x\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2+itx} =e^{-\frac{1}{2}\sigma^2t^2+it\mu}~. \ \ \ \ \ (4)$

A pertinent feature is that the characteristic function of a linear combination of independent random variables can be expressed as a product of the characteristic functions thereof:

$\displaystyle \phi_{a_1x_1+\ldots+a_nx_n}(t)=\phi_{x_1}(a_1t)\ldots\phi_{x_n}(a_nt)~, \ \ \ \ \ (5)$

where ${\{x_i\}}$ are independent random variables and ${\{a_i\}}$ are some constants. In the present case, we may view each ${y_j^{l-1}}$ as a constant modifying the random variable ${W_{ij}^l}$, so that

\displaystyle \begin{aligned} \phi_{z_i^l}(t)&=\phi_{b_i^l}(t)\prod_j\phi_{W_{ij}^l}(y_j^{l-1}t) =e^{-\frac{1}{2}\sigma_b^2t^2}\prod_je^{-\frac{1}{2N_{l-1}}\sigma_w^2(y_j^{l-1})^2}\\ &=\exp\left[-\frac{1}{2}\left(\sigma_w^2\frac{1}{N_{l-1}}\sum_j(y_j^{l-1})^2+\sigma_b^2\right) t^2\right]~. \end{aligned} \ \ \ \ \ (6)

Comparison with (4) then implies that ${z_i^l}$ indeed follow a Gaussian,

$\displaystyle z_i^l\sim\mathcal{N}(0,\sigma_z^2)~,\qquad\mathrm{with}\qquad \sigma_z^2=\sigma_w^2\frac{1}{N_{l-1}}\sum_j(y_j^{l-1})^2+\sigma_b^2~. \ \ \ \ \ (7)$

This is the probability distribution that describes the pre-activations of the neurons in layer ${l}$. Note that since the neurons are indistinguishable (i.e., all identical), in the large-${N}$ limit ${\sigma_z^2}$ becomes the variance of the distribution of inputs across the entire layer. The MFT approximation is then to simply replace the actual neurons with values drawn from this distribution, which becomes exact in the large-${N}$ limit (EDIT: there’s a subtlety here, see “MFT vs. CLT” after the references below). For consistency with [1] (and because tracking the ${l}$-dependence by writing ${\sigma_{z^l}}$ is cumbersome), we’ll introduce the notation ${q^l:=\sigma_z^2}$ to denote the variance of the pre-activations in layer ${l}$.

Now, observe that in the large-${N}$ limit, the average over the activations of the previous layer becomes an integral:

$\displaystyle \frac{1}{N_{l-1}}\sum_j(y_j^{l-1})^2 =\frac{1}{N_{l-1}}\sum_j\phi(z_j^{l-1})^2 \;\overset{N\rightarrow\infty}{\longrightarrow}\; \int\!\mathcal{D} z_j\,\phi(z_j^{l-1})^2 =\int\!\frac{\mathrm{d} z}{\sqrt{2\pi}}\,e^{-\frac{z^2}{2}}\,\phi\left(\!\sqrt{q^{l-1}}z\right)^2~, \ \ \ \ \ (12)$

where we’ve used ${\mathcal{D} z}$ to denote the standard normal Gaussian measure, and we’ve defined ${z:= z_j^{l-1}/\sqrt{q^{l-1}}}$ so that ${z\sim\mathcal{N}(0,1)}$. That is, we can’t just write

$\displaystyle \lim_{N\rightarrow\infty}\frac{1}{N}\sum_if(z_i)=\int\!\mathrm{d} z_i\,f(z_i)~, \ \ \ \ \ (13)$

because this makes an assumption on the measure, ${\mathrm{d} z_i}$. Specifically, it assumes that we’re integrating over the real line, whereas we know that ${z_i}$ in fact lie along a Gaussian. Thus we need a Lebesgue integral (rather than the more familiar Riemann integral) with an appropriate measure—in this case, ${{\mathcal{D} z=\tfrac{1}{\sqrt{2\pi}}e^{-z^2/2}\mathrm{d} z}}$. (We could alternatively have written ${{\mathcal{D} z_j=\tfrac{1}{\sqrt{2\pi q^{l-1}}}e^{-\tfrac{1}{2}\left( z_j^{l-1}/q^{l-1}\right)^2}}}$, but it’s cleaner to simply rescale things). Substituting this back into (7), we thus obtain a recursion relation for the variance:

$\displaystyle q^l=\sigma_w^2\int\!\mathcal{D} z\,\phi\left(\!\sqrt{q^{l-1}}z\right)^2+\sigma_b^2~, \ \ \ \ \ (14)$

which is eqn. (3) of [1]. Very loosely speaking, this describes the spread of information at a single neuron, with smaller values implying a more tightly peaked distribution. We’ll return to the importance of this expression momentarily.

As you might have guessed from all the song-and-dance about correlation functions in part 1 and above, a better probe of information propagation in the network is the correlator of two inputs (in this case, the correlation between different inputs in two identically prepared copies of the network). Let us thus add an additional Latin index ${a,b,\ldots}$ to track particular inputs through the network, so that ${z_{i,a}^l}$ is the value of the ${i^\mathrm{th}}$ neuron in layer ${l}$ in response to the data ${\mathbf{x}_a=\{x_{i,a}\}}$, ${z_{i,b}^l}$ is its value in response to data ${\mathbf{x}_b}$, etc., where the data is fed into the network by setting ${\mathbf{y}_a^0=\mathbf{x}_a}$ (here boldface letters denote the entire layer, treated as a vector in ${\mathbb{R}^l}$). We then consider the two-point correlator between these inputs at a single neuron:

\displaystyle \begin{aligned} \langle z_{i,a}^lz_{i,b}^l\rangle&= \left<\left(\sum_jW_{ij}^ly_{j,a}^{l-1}+b_i\right)\left(\sum_kW_{ik}^ly_{k,b}^{l-1}+b_i\right)\right>\\ &=\sigma_w^2\frac{1}{N_{l-1}}\sum_jy_{j,a}^{l-1}y_{j,b}^{l-1}+\sigma_b^2~, \end{aligned} \ \ \ \ \ (15)

where we have used the fact that ${\langle W_{ij}^lW_{ik}^l\rangle=\delta_{ij}\sigma_w^2/N_{l-1}}$, and that the weights and biases are independent with ${\langle W_{ij}^l\rangle=0=\langle b_i^l\rangle}$. Note that since ${{\langle z_{i,a}^l\rangle\!=\!0}}$, this also happens to be the covariance ${{\mathrm{cov}(z_{i,a}^l,z_{i,b}^l)\!=\!\langle z_{i,a}^lz_{i,b}^l\rangle-\langle z_{i,a}^l\rangle\langle z_{i,b}^l\rangle}}$, which we shall denote ${q_{ab}^l}$ in accordance with the notation from [1] introduced above.

As in the single-input case above, we again take the continuum (i.e., large-${N}$) limit, whereupon the sum becomes

$\displaystyle \frac{1}{N_{l-1}}\sum_jy_{j,a}^{l-1}y_{j,b}^{l-1} \;\overset{N\rightarrow\infty}{\longrightarrow}\; \int\!\mathcal{D} z_a\mathcal{D} z_b\,\phi(z_a^{l-1})\phi(z_b^{l-1})~. \ \ \ \ \ (16)$

Unlike the previous case however, we can’t just use independent standard Gaussian measures for ${\mathcal{D} z_a}$ and ${\mathcal{D} z_b}$, since that would preclude any correlation in the original inputs ${\mathbf{x}_a,~\mathbf{x}_b}$. Rather, any non-zero correlation in the data will propagate through the network, so the appropriate measure is the bivariate normal distribution,

$\displaystyle \mathcal{D} z_a\mathcal{D} z_b =\frac{\mathrm{d} z_a\mathrm{d} z_b}{2\pi\sqrt{q_a^{l-1}q_b^{l-1}(1-\rho^2)}}\, \exp\!\left[-\frac{1}{2(1-\rho^2)}\left(\frac{z_a^2}{q_a^{l-1}}+\frac{z_b^2}{q_b^{l-1}}-\frac{2\rho\,z_az_b}{\sqrt{q_a^{l-1}q_b^{l-1}}}\right)\right]~, \ \ \ \ \ (17)$

where the (Pearson) correlation coefficient is defined as

$\displaystyle \rho^l:=\frac{\mathrm{cov}(z_a^l,z_b^l)}{\sigma_{z_a}\sigma_{z_b}} =\frac{q_{ab}^l}{\sqrt{q_a^lq_b^l}}~, \ \ \ \ \ (18)$

and we have denoted the squared variance in layer ${l}$ corresponding to the input ${\mathbf{x}_a}$ by ${{q_a^l:=\sigma_{z_a^l}^2}}$. If the inputs are uncorrelated (i.e., ${{\langle \mathbf{x}_a\mathbf{x}_b\rangle=\langle\mathbf{x}_a\rangle\langle\mathbf{x}_b\rangle}}$), then ${{\mathrm{cov}(\mathbf{x}_a,\mathbf{x}_b)\!=\!0\implies\rho\!=\!0}}$, and the measure reduces to a product of independent Gaussian variables, as expected. (I say “as expected” because the individual inputs ${z_i^l}$ are drawn from a Gaussian. In general however, uncorrelatedness does not necessarily imply independence! The basic reason is that the correlation coefficient ${\rho(x,y)}$ is only sensitive to linear relationships between random variables ${x}$ and ${y}$; if these are related non-linearly, then it’s possible for them to be dependent (${{p(x,y)\neq p(x)p(y)}}$) despite being uncorrelated (${{\rho(x,y)\!=\!0}}$). See Wikipedia for a simple example of such a case.)

Following [1], the clever trick is then to define new integration variables ${z_1,z_2}$ such that

$\displaystyle z_a=\sqrt{q_a^{l-1}}\,z_1~,\qquad\qquad z_b=\sqrt{q_b^{l-1}}\left(\rho^{l-1} z_1+\sqrt{1-(\rho^{l-1})^2}\,z_2\right)~, \ \ \ \ \ (19)$

(Do not confuse the standard normal integration variables ${z_1,z_2}$ with the inputs ${z_{i,a}^l}$; the former are distinguished by the lack of any layer label ${l}$). With this substitution, the integral (16) becomes

$\displaystyle \int\!\mathcal{D} z_a\mathcal{D} z_b\,\phi(z_a^{l-1})\phi(z_b^{l-1}) =\int\!\frac{\mathrm{d} z_1\mathrm{d} z_2}{2\pi}e^{-\frac{1}{2}\left( z_1^2+z_2^2\right)}\phi(z_a)\phi(z_b)~, \ \ \ \ \ (20)$

and thus we obtain the recursion relation for the covariance, cf. eqn. (5) of [1]:

$\displaystyle q_{ab}^l=\sigma_w^2\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi(z_a)\phi(z_b)+\sigma_b^2~, \ \ \ \ \ (21)$

with ${z_a,z_b}$ given by (19). Note that since ${z_1,z_2}$ are Gaussian, their independence (i.e., the fact that ${p(z_1,z_2)=p(z_1)p(z_2)}$) follows from their uncorrelatedness, ${\rho(z_1,z_2)\!=\!0}$; the latter is easily shown by direct computation, and recalling that ${\langle z_a^2\rangle\!=\!\sigma_a^2}$.

Now, the interesting thing about the recursion relations (14) and (21) is that they exhibit fixed points for particular values of ${\sigma_w}$, ${\sigma_b}$, which one can determine graphically (i.e., numerically) for a given choice of non-linear activation ${\phi(z)}$. For example, the case ${\phi(z)=\tanh(z)}$ is plotted in fig. 1 of [1] (shown below), demonstrating that the fixed point condition

$\displaystyle q^*=\sigma_w^2\int\!\frac{\mathrm{d} z}{\sqrt{2\pi}}\,e^{-\frac{z^2}{2}}\tanh(\sqrt{q^*}z)^2+\sigma_b^2 \ \ \ \ \ (22)$

is satisfied for a range of ${(\sigma_w,\sigma_b)}$. Therefore, when the parameters of the network (i.e., the distributions (2)) are tuned appropriately, the variance remains constant as signals propagate through the layers. This is useful, because it means that non-zero biases at each layer prevent signals from decaying to zero. Furthermore, provided the non-linearity ${\phi}$ is concave, the convergence to the fixed point occurs relatively rapidly as a function of depth (i.e., layer ${l}$). One can verify this numerically for the case ${\phi(z)=\tanh(z)}$ considered above: since ${\tfrac{\mathrm{d}^2 q^{l}}{\mathrm{d}(q^{l-1})^2}\!<\!0}$, the dynamics ensures convergence to ${q^*}$ for all initial values. To visualize this, consider the following figure, which I took from [1] (${\mathcal{V}}$ is their notation for the recursion relation (14)). In the left plot, I’ve added the red arrows illustrating the convergence to the fixed point for ${\sigma_w=4.0}$. For the particular starting point I chose, it takes about 3 iterations to converge, which is consistent with the plot on the right.

Similarly, the swift convergence to ${q^*}$ for the variance of individual neurons implies a fixed point for (21). Observe that when ${q_a^{l-1}\!=\!q_b^{l-1}\!=\!q^*}$, ${\rho^{l-1}\!=\!q_{ab}^{l-1}/q^*}$. Hence by dividing (21) by ${q^*}$, we have

$\displaystyle \rho^l=\frac{\sigma_w^2}{q^*}\int\!\mathcal{D} z_1\mathcal{D} z_2\, \phi\left(\sqrt{q^*}z_1\right) \phi\left(\sqrt{q^*}\left[\rho^{l-1}z_1+\sqrt{1-(\rho^{l-1})^2}z_2\right]\right) +\frac{\sigma_b^2}{q^*}~. \ \ \ \ \ (23)$

This equation always has a fixed point at ${\rho^*\!=\!1}$:

$\displaystyle \rho^l=\frac{\sigma_w^2}{q^*}\int\!\mathcal{D} z_1\mathcal{D} z_2\, \phi\left(\sqrt{q^*}z_1\right)^2 +\frac{\sigma_b^2}{q^*} =\frac{1}{q^*}q^{l}=1~,\ \ \ \ \ \ (24)$

where ${\int\!\mathcal{D} z_2\!=\!1}$, and we have identified the single-neuron variance ${q^l}$, cf. (14), which is ${q_a^l\!=\!q_b^l\!=\!q^*}$.

Let us now ask about the stability of this fixed point. As observed in [1], this is probed by the first derivative evaluated at ${\rho^*}$:

$\displaystyle \chi_1:= \frac{\partial \rho^l}{\partial\rho^{l-1}}\bigg|_{\rho^{l-1}=1} =\;\sigma_w^2\int\!\mathcal{D} z\,\phi'\left(\sqrt{q^*}z\right)^2~, \ \ \ \ \ (25)$

where ${\phi'(u)=\tfrac{\partial\phi(u)}{\partial u}}$ (the second equality is not obvious; see the derivation at the end of this post if you’re curious). The reason is that for monotonic functions ${\phi(z)}$, ${\chi<1}$ implies that the curve (23) must approach the unity line from above (i.e., it’s concave, or has a constant slope with a vertical offset at ${\rho^{l-1}\!=\!0}$ set by the biases), and hence in this case ${\rho^l}$ is driven towards the fixed point ${\rho^*}$. Conversely, if ${\chi>1}$, the curve must be approaching this point from below (i.e., it’s convex), and hence in this case ${\rho^l}$ is driven away from unity. Depending on the strengths of the biases relative to the weights, the system will then converge to some new fixed point that approaches zero as ${\sigma_b\rightarrow 0}$. (It may help to sketch a few examples, like I superimposed on the left figure above, to convince yourself that this slope/convergence argument works).

The above implies the existence of an order-to-chaos phase transition in the ${(\sigma_w,\sigma_b)}$ plane: ${\chi>1}$ corresponds to the chaotic phase, in which the large random weights cause any correlation between inputs to shrink, while ${\chi<1}$ corresponds to the ordered phase, in which the inputs become perfectly aligned and hence ${\rho\rightarrow1}$. At exactly ${\chi\!=\!1}$, the system lies at a critical point, and should therefore be characterized some divergence(s) (recall that, in the limit ${N\rightarrow\infty}$, a phase transition is defined by the discontinuous change of some quantity; one often hears of an “${n^\mathrm{th}}$ order phase transition” if the discontinuity lies in the ${n^\mathrm{th}}$ derivative of the free energy).

We saw in part 1 that at a critical point, the correlation length that governs the fall-off of the two-point correlator diverges. This is the key physics underlying the further development of this mean field framework in [2], who used it to show that random networks are trainable only in the vicinity of this phase transition. Roughly speaking, the intuition is that for such a network to be trainable, information must be able to propagate all the way through the network to establish a relationship between the input (data) and the output (i.e., the evaluation of the cost function) (actually, it needs to go both ways, so that one can backpropagate the gradients through as well). In general, correlations in the input data will exhibit some fall-off behaviour as a function of depth, which limits how far information can propagate through these networks. However, borrowing our physics terminology, we may say that a network at criticality exhibits fluctuations on all scales, so there’s no fall-off behaviour limiting the propagation of information with depth.

So, what is the correlation length in the present problem? In the previous post on mean field theory, we determined this by examining the fall-off of the connected 2-point correlator (i.e., the Green function), which in this case is just ${\langle z_a^lz_b^l\rangle=q_{ab}^l}$. To determine the fall-off behaviour, we expand near the critical point, and examine the difference ${|\rho^l-\rho^*|}$ as ${l\!\rightarrow\!\infty}$ (where recall ${\rho^l=q_{ab}^l/q^*}$. The limit simply ensures that ${q^*,\rho^*}$ exist; as shown above, we don’t have to wait that long in practice). This is done explicitly in [2], so I won’t work through all the details here; the result is that upon expanding ${\rho^l=\rho^*+\epsilon^l}$, one finds the recurrence relation

$\displaystyle \epsilon^{l+1}=\epsilon^l\,\sigma_w^2\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi'(z_a)\phi'(z_b)~, \ \ \ \ \ (26)$

up to terms of order ${(\epsilon^l)^2}$. This implies that, at least asymptotically, ${\epsilon^l\sim e^{-l/\xi}}$, which one can verify by substituting this into (26) and identifying

$\displaystyle \xi^{-1}=-\ln\left[\sigma_w^2\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi'(z_a)\phi'(z_b)\right]~. \ \ \ \ \ (27)$

Thus ${\xi}$ is the correlation length that governs the propagation of information — i.e., the decay of the two-point correlator — through the network. And again, we know from our discussion in part 1 that the correlation length diverges at the critical point, which is indeed what happens here: denoting the value of the correlation length at the critical point ${\rho^*\!=\!1}$ by ${\xi_1}$, observe that the argument of (27) reduces to (25), i.e.,

$\displaystyle \xi_1^{-1}=-\ln\chi_1~, \ \ \ \ \ (28)$

which goes to infinity precisely at the order-to-chaos phase transition, since the latter occurs at exactly ${\chi\!=\!1}$.

In practice of course, both ${l}$ and the number of neurons ${N}$ are finite, which precludes a genuine discontinuity at the phase transition (since the partition function is then a finite sum of analytic functions). Nonetheless, provided ${\xi\gg1}$, we expect that the basic conclusion about trainability above still holds even if the correlation length remains finite. And indeed, the main result of [2] is that at the critical point, not only can information about the inputs propagate all the way through the network, but information about the gradients can also propagate backwards (that is, exploding/vanishing gradients happen only in the ordered/chaotic phases, respectively, but are stable at the phase transition), and consequently these random networks are trainable provided ${l\lesssim\xi}$.

Incidentally, you might worry about the validity of these MFT results, given that — as stressed in part 1 — it’s precisely at a critical point that MFT may break down, due to the relevance of higher-order terms that the approximation ignores. But in this case, we didn’t neglect anything: we saw in (7) that the distribution of inputs already follows a Gaussian, so the only “approximation” lies in ignoring any finite-size effects associated with backing away from the ${N\rightarrow\infty}$ limit (EDIT: actually, this isn’t true! See “MFT vs. CLT” below).

The application of MFT to neural networks is actually a rather old idea, with the presence of an order-to-chaos transition known in some cases since at least the 1980’s [3]. More generally, the idea that the critical point may offer unique benefits to computation has a long and eclectic history, dating back to an interesting 1990 paper [4] on cellular automata that coined the phrase “computation at the edge of chaos”. However, only in the past few years has this particular interdisciplinary ship really taken off, with a number of interesting papers demonstrating the power of applying physics to deep learning. The framework developed in [1,2] that I’ve discussed here has already been generalized to convolutional [5] and recurrent [6,7] neural networks, where one again finds that initializing the network near the phase transition results in significant improvements in training performance. It will be exciting to see what additional insights we can discover in the next few years.

References

1. B. Poole, S. Lahiri, M. Raghu, J. Sohl-Dickstein, and S. Ganguli, “Exponential expressivity in deep neural networks through transient chaos,” arXiv:1606.05340 [stat.ML].
2. S. S. Schoenholz, J. Gilmer, S. Ganguli, and J. Sohl-Dickstein, “Deep Information Propagation,” arXiv:1611.01232 [stat.ML].
3. H. Sompolinsky, A. Crisanti, and H. J. Sommers, “Chaos in random neural networks,” Phys. Rev. Lett. 61 (Jul, 1988) 259–262.
4. C. G. Langton, “Computation at the edge of chaos: Phase transitions and emergent computation,” Physica D: Nonlinear Phenomena 42 no. 1, (1990) 12 – 37.
5. L. Xiao, Y. Bahri, J. Sohl-Dickstein, S. S. Schoenholz, and J. Pennington, “Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks,” arXiv:1806.05393 [stat.ML].
6. M. Chen, J. Pennington, and S. S. Schoenholz, “Dynamical Isometry and a Mean Field Theory of RNNs: Gating Enables Signal Propagation in Recurrent Neural Networks,” arXiv:1806.05394 [stat.ML].
7. D. Gilboa, B. Chang, M. Chen, G. Yang, S. S. Schoenholz, E. H. Chi, and J. Pennington, “Dynamical Isometry and a Mean Field Theory of LSTMs and GRUs,” arXiv:1901.08987 [cs.LG].

The second equality in (25), namely

$\displaystyle \frac{\partial \rho^l}{\partial\rho^{l-1}}\bigg|_{\rho^{l-1}=1} =\;\sigma_w^2\int\!\mathcal{D} z\,\phi'\left(\sqrt{q^*}z\right)^2~, \ \ \ \ \ (29)$

is not obvious, but requires an exercise in basic calculus. Since all the functional dependence on ${\rho^{l-1}}$ — denoted simply ${\rho}$ here for brevity — is contained in ${z_b}$ (cf.(19)), we have from (23)

\displaystyle \begin{aligned} \frac{\partial \rho^l}{\partial\rho} &=\frac{\sigma_w^2}{q^*}\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi(z_a)\frac{\partial}{\partial\rho}\phi(z_b) =\frac{\sigma_w^2}{q^*}\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi(z_a)\phi'(z_b)\frac{\partial z_b}{\partial\rho}\\ &=\frac{\sigma_w^2}{\sqrt{q^*}}\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi(z_a)\phi'(z_b)\left( z_1-\frac{\rho}{\sqrt{1-\rho^2}}z_2\right)~, \end{aligned} \ \ \ \ \ (30)

where the prime denotes the derivative of ${\phi}$ with respect to its argument. We now use the hint given in the appendix of [1], namely that for the standard Gaussian measure, integration by parts allows us to write ${{\int\!\mathcal{D} z\,zf(z)=\int\!\mathcal{D} z\,f'(z)}}$. We therefore have the identities

$\displaystyle \int\!\mathcal{D} z_1\,z_1\phi(z_a) =\int\!\mathcal{D} z_1\frac{\partial\phi(z_a)}{\partial z_1} =\int\!\mathcal{D} z_1\,\phi'(z_a)\frac{\partial z_a}{\partial z_1} =\sqrt{q^*}\int\!\mathcal{D} z_1\,\phi'(z_a) \ \ \ \ \ (31)$

and

\displaystyle \begin{aligned} \int\!\mathcal{D} z_2\,z_2\phi'(z_b) &=\int\!\mathcal{D} z_2\,z_2\frac{\partial\phi(z_b)}{\partial z_2}\left(\frac{\partial z_b}{\partial z_2}\right)^{-1} =\sqrt{q^*(1-\rho^2)}\int\!\mathcal{D} z_2\,z_2\frac{\partial\phi(z_b)}{\partial z_2}\\ &=\sqrt{q^*(1-\rho^2)}\int\!\mathcal{D} z_2\,(z_2^2-1)\phi(z_b)~, \end{aligned} \ \ \ \ \ (32)

where in the last step we have again applied integration by parts over the Gaussian measure. Substituting these results into (30), we find

$\displaystyle \frac{\partial \rho^l}{\partial\rho} =\sigma_w^2\int\!\mathcal{D} z_1\mathcal{D} z_2\,\phi'(z_a)\phi'(z_b) +\sigma_w^2\rho\int\!\mathcal{D} z_1\mathcal{D} z_2\,(z_2^2-1)\phi(z_a)\phi(z_b)~. \ \ \ \ \ (33)$

If we evaluate this at ${\rho^*\!=\!1}$, we can integrate freely over ${z_2}$, whereupon the second term vanishes,

$\displaystyle \sigma_w^2\rho\int\!\mathcal{D} z_1\mathcal{D} z_2\,(z_2^2-1)\phi\left(\sqrt{q^*}z_1\right)^2=0 \ \ \ \ \ (34)$

and the first returns the claimed result:

$\displaystyle \frac{\partial \rho^l}{\partial\rho}\bigg|_{\rho\!=\!1} =\sigma_w^2\int\!\mathcal{D} z_1\,\phi'\left(\sqrt{q^*}z_1\right)^2=\chi_1~. \ \ \ \ \ (35)$

MFT vs. CLT (edit 2021-12-09)

Throughout this post, I referred to the approximation employed in [1,2] as MFT, since they and related works in the machine learning literature advertise it under this name. It turns out however that this isn’t quite right: the results above actually follow directly from the central limit theorem (CLT), which — perhaps surprisingly — is not necessarily the same as MFT. I discovered this together with my excellent collaborator Kevin Grosvenor while working on the nascent NN-QFT correspondence. I might write a dedicated blog article about this at some point, but the basic idea is to develop the connections between physics and machine learning that I’ve alluded to in various places here in more detail, and use techniques from QFT in particular to understand deep neural networks.

In our latest paper, we made this correspondence concrete by explicitly constructing the QFT corresponding to a general class of networks including both recurrent and feedforward architectures. The MFT approximation corresponds to the leading saddle point of the theory, as I’ve explained before. But we can also compute perturbative corrections in ${T/N}$, the ratio of depth to width (this works for most deep networks of practical interest, which typically have ${T/N\ll1}$). Finite-width corrections appear at ${\mathcal{O}(T/N)}$, which physically correspond to effective interactions that arise upon marginalizing over hidden degrees of freedom; see my post on deep learning and the renormalization group for details.

However, we also discovered an ${\mathcal{O}(1)}$ correction to the MFT (i.e., tree level) result, which is independent of network size, and therefore survives in the ${N\rightarrow\infty}$ limit. Physically, this corresponds to fluctuations in the ensemble of random networks under study; some intuition for this is given in section 5 of the aforementioned paper. Since MFT ignores all fluctuations (not just those arising at finite width), it fails to capture this contribution. Results from the CLT might implicitly include this ${\mathcal{O}(1)}$ effect, but this has yet to be worked-out in detail.

Posted in Minds & Machines | 2 Comments

## Mean field theory: from physics to deep neural nets

In a previous post, I alluded to the question of whether criticality played any role in deep neural networks. The question I originally had in mind was whether the fact that the correlation length diverges at a critical point implies an advantage in terms of information propagation in such systems. In particular, would a neural net operating at or near the critical point (e.g., by savvy initialization of weights and biases) exhibit advantages in training or performance? As it turns out, a few papers have actually addressed this issue using a prescription from physics known as mean field theory (MFT). In this two-part sequence, I’d like to first explain briefly what MFT is, and then in part 2 understand how machine learning researchers have applied it to obtain impressive real-world improvements in training performance.

In a nutshell, the idea behind MFT is that most partition functions (containing interactions) are too hard to evaluate explicitly, but can be made tractable by replacing each degree of freedom, together with its interactions, with an effective degree of freedom — the mean field — in which these interactions have been averaged over. Note that MFT is an approximation, for two reasons: it ignores higher-order fluctuations, and the averaging prescription necessarily washes-out some fine-grained information. We’ll cover both these points in more detail below, when discussing the situations under which MFT fails. Perhaps ironically given the previous paragraph, MFT breaks down precisely at the critical point, so it’s important to understand the conditions under which the associated predictions are valid.

To make our discussion concrete, consider the ${d}$-dimensional Ising hamiltonian with ${N}$ spins total (i.e., ${N/d}$ spins per direction):

$\displaystyle H=-\frac{1}{2}\hat J^{ij}\sigma_i\sigma_j-\hat h^i\sigma_i~, \ \ \ \ \ (1)$

where ${\sigma_i=\pm1}$, and for compactness I’m employing Einstein’s summation convention with ${i=\{1,\ldots,N\}}$ and ${\hat h^i=\hat h\mathbf{1}}$. Since all directions are spacelike, there’s no difference between raised and lowered indices (e.g., ${\sigma_i=\sigma^i}$), so I’ll denote the inverse matrix ${\hat J^{-1}}$ explicitly to avoid any possible confusion, i.e., ${\hat J\hat J^{-1}=\mathbf{1}}$ and ${{\hat J^{ij}\hat J_{ij}^{-1}=\delta^i_{~i}=N}}$. In a 1d lattice, one would typically avoid boundary effects by joining the ends into an ${S^1}$ by setting ${\sigma_{N+1}=\sigma_1}$, but this issue won’t be relevant for our purposes, as we’ll be interested in the thermodynamic limit ${N\rightarrow\infty}$ anyway.

One approach to constructing a MFT for this model is to observe that for a given spin ${\sigma_i}$, the effect of all the other spins acts like an external magnetic field. That is, observe that we may write (1) as

$\displaystyle H=\sum_i\sigma_i\left(-\frac{1}{2}\sum_j\hat J_{ij}\sigma_j-\hat h\right)~. \ \ \ \ \ (2)$

We then replace ${\sigma_j}$ by the average value, ${\langle\sigma_j\rangle\equiv s}$. We’ll give a more thorough Bayesian treatment of MFT below, but the idea here is that since no spin is special, the most likely value of ${\sigma_j}$ is the mean. This allows us to define an effective magnetic field at site ${i}$:

$\displaystyle \hat h^\textrm{\tiny{eff}}_i\equiv\frac{s}{2}\sum_j\hat J_{ij}+\hat h~, \ \ \ \ \ (3)$

so that the hamiltonian becomes

$\displaystyle H\approx-\sum_i\sigma_i\hat h_i^\textrm{\tiny{eff}}~, \ \ \ \ \ (4)$

where the interactions have been absorbed into the effective magnetic field. Thus we’ve reduced an interacting many-body problem to a non-interacting one-body problem, which is much easier to solve!

As mentioned above however, this result is an approximation, which in the present case amounts to neglecting the correlation between spins at different sites. That is, when we replaced ${\sigma_j}$ by the mean, we secretly discarded terms of order ${\delta s^2}$. This is illustrated explicitly in John McGreevy’s entertaining notes on the Renormalization Group, and runs as follows. As mentioned above, one can think of MFT as the replacement of the degrees of freedom by the average value plus fluctuations:

\displaystyle \begin{aligned} \sigma_i\sigma_i&=\left[s+(\sigma_i-s)\right]\left[s+(\sigma_j-s)\right] =\left( s+\delta s_i\right)\left( s+\delta s_j\right)\\ &=s^2+s\left( \delta s_i+\delta s_j\right)+O(\delta s^2) =-s^2+s\left(\sigma_i+\sigma_j\right)+O(\delta s^2)~, \end{aligned} \ \ \ \ \ (5)

where ${\delta s_i\equiv\sigma_i-s}$. We then substitute this into the hamiltonian (1); taking ${J_{ij}=J_{ji}}$ for simplicity, we obtain

$\displaystyle H=\frac{s^2}{2}\sum_{ji}\hat J_{ij} +\sum_i\sigma_i\left(-\sum_j\hat J_{ij}s-\hat h\right)+O(\delta s^2)~. \ \ \ \ \ (6)$

Note that the first term is some constant factor of the lattice size, and hence doesn’t affect the dynamics (we simply absorb it into the normalization of the partition function). If we then define an effective action as in (3) (with a suitable factor of 2) and work to linear order in the fluctuations, we recover the effective one-body hamiltonian (4). Thus, in the course of our mean field approximation, we averaged over the fluctuations, but lost some information about the interactions between spins.

Despite its approximate nature, the hamiltonian (6) (equivalently, (1)) is quite useful insofar as it can be used to obtain upper bound on the free energy. To understand this, let us introduce the Bayesian perspective promised above. In MFT, we’re ignoring some information about the system at hand, but we want to make inferences that are as accurate as possible subject to the available constraints. Recall from our discussion of entropy that if we do not know the underlying distribution with respect to which a particular expectation value is computed, the most rational choice is obtained by maximizing the von Neumann entropy. In particular, if we know the average energy, ${\langle H\rangle=E}$, this procedure yields the Boltzmann distribution

$\displaystyle p_i\equiv p(x_i)=\frac{1}{Z[\beta]}e^{-\beta E_i}~, \qquad\qquad\sum_ip_i=1~, \ \ \ \ \ (7)$

where ${E_i\equiv\langle H(x_i)\rangle}$, and we identify the inverse temperature ${\beta}$ as the Lagrange multiplier arising from the constraint on the energy.

Now suppose that instead of knowing the average energy, we know only the temperature (i.e., we consider the canonical rather than microcanonical ensemble). As explained in a previous post, this amounts to a constraint in the (canonically) dual space, so the appropriate extremization procedure is instead to minimize the free energy,

$\displaystyle F=E-\beta^{-1}S~. \ \ \ \ \ (8)$

(Note that for a given energy configuration, the free energy is minimized when the entropy is maximized). One finds again (7), with ${Z[\beta]=e^{\beta\lambda}}$ where ${\lambda}$ is the Lagrange multiplier for the normalization constraint, cf. the max-ent procedure here.

The upshot is that the max-ent distribution (7) has minimum free energy: had we used any other distribution, it would amount to the imposition of additional constraints on the system, thereby reducing the entropy and increasing ${F}$ in (8). This is essentially what happens in MFT, since we select a more tractable distribution with respect to which we can compute expectation values (i.e., a simpler hamiltonian, with a constraint on the fluctuations). In the present case, this implies that the normalized free energy obtained via the mean-field hamiltonian (6), denoted $f_\textrm{\tiny{MF}}$, provides an upper bound on the true (equilibrium) free energy ${f}$:

$\displaystyle f_\textrm{\tiny{MF}}\geq f~, \ \ \ \ \ (9)$

where ${f=F/N}$ (henceforth I’ll refer to this simply as the free energy, without the “normalized” qualifier). This statement, sometimes referred to as the Bogolyubov inequality, can be easily shown to follow from Gibb’s inequality. (This apparently standard derivation is however not a sufficient explanation per se, since it simply assumes that expectation values are taken with respect to the mean-field distribution. Had we chosen to take them with respect to the equilibrium (max-ent) distribution, the bound would be reversed!)

Working to linear order in the fluctuations, the mean-field partition function for (6) is

\displaystyle \begin{aligned} Z_\textrm{\tiny{MF}}&=\sum_{\{\sigma\}}e^{-\beta H_\textrm{\tiny{MF}}} =\prod_{k=1}^N\sum_{\sigma_k=\pm1}\exp\left[-NdJs^2+\left(2dJs+h\right)\sum_i\sigma_i\right]\\ &=e^{-NdJs^2}\prod_{k=1}^N\exp\left[-\left(2dJs+h\right)+\left(2dJs+h\right)\right]\\ &=2^Ne^{-NdJs^2}\cosh^N\left(2dJs+h\right)~. \end{aligned} \ \ \ \ \ (10)

where for simplicity we have restricted to homogeneous nearest-neighbor interactions (in ${d}$ spatial dimensions, each of the ${N}$ spins has ${2d}$ neighbors with coupling strength ${J\equiv\beta\hat J}$ and ${h\equiv\beta \hat h}$). The corresponding free energy is then

$\displaystyle f_\textrm{\tiny{MF}}(s)=-\frac{1}{N\beta}{\ln Z} =\beta^{-1}dJs^2-\beta^{-1}\ln\cosh\left(2dJs+h\right)~, \ \ \ \ \ (11)$

where we have dropped the ${\beta^{-1}\ln 2}$ term, since this doesn’t contribute to any of the observables for which ${F}$ serves as a generating function (that is, it’s just a normalization factor).

Now, as per our discussion above, (11) provides an upper bound on the true free energy. Thus we can obtain the tightest possible bound (given our ansatz (6)) by minimizing over ${s}$:

$\displaystyle \frac{\partial f_{\textrm{\tiny{MF}}}}{\partial s}=0 \quad\implies\quad s=\tanh\left(2dJs+h\right)~. \ \ \ \ \ (12)$

This last is referred to as the self-consistency condition. The reason is that it’s precisely what we would have obtained had we computed the average spin via the single-site hamiltonian (6), or equivalently (4): since the linear term has been absorbed into the effective magnetic field ${\hat h^\textrm{\tiny{eff}}}$, it looks as though ${J=0}$, and therefore

$\displaystyle s=\langle\sigma\rangle=\sum_{\sigma=\pm1}\!\sigma\,p(\sigma) =\frac{1}{Z}\sum_{\sigma=\pm1}\sigma e^{ h^{\textrm{\tiny{eff}}}\sigma} =\frac{\sum_{\sigma=\pm1}\sigma e^{h^{\textrm{\tiny{eff}}}\sigma}}{\sum_{\sigma=\pm1} e^{h^{\textrm{\tiny{eff}}}\sigma}} =\tanh h^\textrm{\tiny{eff}}~, \ \ \ \ \ (13)$

where ${h^\textrm{\tiny{eff}}\equiv\beta\hat h^\textrm{\tiny{eff}}}$. Substituting in (3) for the homogeneous nearest-neighbor case at hand (that is, without the factor of ${1/2}$, cf. (6)) then gives precisely (12).

The self-consistency equation can be solved graphically, i.e., the critical points are given by the intersections of the left- and right-hand sides; see [1] for a pedagogical treatment. In brief, for ${h\!\neq\!0}$, the global minimum of ${f_\textrm{\tiny{MF}}}$ is given by the intersection at positive ${s}$, regardless of temperature. For ${h\!=\!0}$ in contrast, there’s a single minimum at ${s\!=\!0}$ for high temperatures, and two degenerate minima at ${\pm s_0}$ at low temperatures (depending on whether ${\tanh(2d\beta\hat Js)}$ crosses ${s}$ for ${s\neq0}$; to see this, recall that for small ${x}$, ${\tanh x\approx x-x^3/3+\ldots}$, so a sufficiently small value of ${\beta}$ makes this an approximately straight line whose slope is less than ${s}$). The critical temperature that divides these two regimes is found by imposing that

$\displaystyle s=\tanh(2d\beta_c\hat Js)\overset{!}{=}\tanh(s) \quad\implies\quad T_c=2d\hat J~. \ \ \ \ \ (14)$

Note that this is completely wrong for low dimensions! For ${d\!=\!1}$, ${T_c\!=0\!}$, while for ${d\!=\!2}$, ${T_c\approx2.269\hat J}$; we’ll have more to say about this failure below.

Let’s concentrate on the ${h\!=\!0}$ case henceforth: note that the critical point ${s_0}$ will always be small (${|s_0|\!<\!1}$) independent of ${T}$ (since ${\lim\nolimits_{x\rightarrow\pm\infty}\tanh x=\pm1}$), so we can expand

$\displaystyle \ln\cosh(2dJs_0)=\frac{1}{2}(2dJ)^2s_0^2-\frac{1}{12}(2dJ)^4s_0^4+O(s_0^6)~, \ \ \ \ \ (15)$

whence the free energy (11) near the critical point is approximately

$\displaystyle f_\textrm{\tiny{MF}}(s_0)\approx \frac{r}{2}s_0^2+\frac{g}{4!}s_0^4~, \ \ \ \ \ (16)$

where we have defined

\displaystyle \begin{aligned} r\equiv 2d&J\beta^{-1}\left(1-2dJ\right) =\frac{T_c}{T}\left( T-T_c\right)~,\\ g&\equiv 32\beta^{-1}d^4J^4 =\frac{2T_c^4}{T^3}~, \end{aligned} \ \ \ \ \ (17)

and dropped all higher-order terms. Observe that the sign of ${r}$ changes at the critical temperature ${T\!=\!T_c}$, which determines whether the global minimum of ${f_\textrm{\tiny{MF}}}$ lies at ${s_0\!=\!0}$ (${T\!>\!T_c}$) or ${\pm s_0\!>\!0}$ (${T\!<\!T_c}$). The physical interpretation is that below the critical temperature, it is energetically favourable for the spins to align, resulting in a non-zero magnetization (which is what the average spin ${\langle\sigma\rangle=s}$ is). Above the critical temperature however, thermal fluctuations disrupt this ordering, so the net magnetization is zero. For this reason, the magnetization ${s}$ is an example of an order parameter, since it parametrizes which “order” — that is, which phase — we’re in on either side of the critical point.

As alluded above however, there’s a problem with the MFT results for the critical point, namely that it’s precisely at the critical point where MFT breaks down! The reason is that at the critical point, fluctuations at all scales are important, whereas MFT includes only fluctuations to linear order (cf. (5)). The contribution from all scales is related to the statement we made in the introductory paragraph, namely that the correlation length diverges at the critical point. To properly understand this, we need to go beyond the MFT approach above. In particular, while the discrete lattice is a helpful starting point, we can gain further insight by considering a continuum field theory. We’ll see that MFT corresponds to the leading-order saddle point approximation, and that the first corrections to this expansion can qualitatively change these results.

To proceed, we’ll map our square-lattice Ising model to an equivalent theory of scalar fields. (If you like, you can just jump to the Ginzburg-Landau action (32) and take it as an ansatz, but I find the mapping both neat and instructive). Starting again from (1), the partition function is

$\displaystyle Z=\sum_{\{\sigma\}}e^{-\beta H} =\prod_{k=1}^N\sum_{\sigma_k=\pm1}\exp\left(\frac{1}{2} J^{ij}\sigma_i\sigma_j+h^i\sigma_i\right)~, \ \ \ \ \ (18)$

where as before we have absorbed the pesky factor of ${\beta}$ by defining ${J=\beta\hat J}$, ${h=\beta\hat h}$. The first step is to apply the Hubbard-Stratanovich transformation,

$\displaystyle e^{\frac{1}{2}K^{ij}s_is_j}=\left[\frac{\mathrm{det}K}{(2\pi)^N}\right]^{1/2}\!\int\!\mathrm{d}^N\phi\exp\left(-\frac{1}{2}K^{ij}\phi_i\phi_j+K^{ij}s_i\phi_j\right)~, \qquad\forall\,s_i\in\mathbb{R}~. \ \ \ \ \ (19)$

(We used this before in our post on restricted Boltzmann machines; the difference here is that we want to allow ${h\neq0}$). Applying this transformation to the first term in the partition function, we have

$\displaystyle Z=\left[\frac{\mathrm{det}J}{(2\pi)^N}\right]^{1/2}\!\int\!\mathrm{d}^N\phi\sum_{\{\sigma\}}\exp\left[-\frac{1}{2}J^{ij}\phi_i\phi_j+\left( J^{ij}\phi_j+h^i\right)\sigma_i\right]~. \ \ \ \ \ (20)$

At a computational level, the immediate advantage of the Hubbard-Stratanovich transformation in the present case is that we can sum over the binary spins ${\sigma}$, leaving us with an expression entirely in terms of the new field variables ${\phi}$. Observe that for each spin,

$\displaystyle \sum_{\sigma_i=\pm1}\exp\left[\left( J^{ij}\phi_j+h^i\right)\sigma_i\right]=2\cosh\left( J^{ij}\phi_j+h^i\right)~, \ \ \ \ \ (21)$

and therefore by re-exponentiating this expression, the partition function becomes

$\displaystyle Z=\left[\left(\frac{2}{\pi}\right)^N\!\mathrm{det}J\right]^{1/2}\!\int\!\mathrm{d}^N\phi\exp\!\left[-\frac{1}{2}J^{ij}\phi_i\phi_j+\sum_i\ln\cosh\left( J^{ij}\phi_j+h^i\right)\right]~. \ \ \ \ \ (22)$

We now observe that ${J^{ij}\phi_j+h^j\equiv\mu^i}$ can be thought of as the mean field ${\langle\phi_i\rangle}$ at site ${i}$, incorporating the interaction with all other sites as well as the external magnetic field. We can then express the partition function in terms of the mean field ${\mu^i}$ by inverting this identification:

\displaystyle \begin{aligned} \phi^i=J^{-1}_{ij}\left(\mu^j-h^j\right) \quad\implies\quad J^{ij}\phi_i\phi_j&=J^{ij}J^{-1}_{in}J^{-1}_{jm}\left(\mu^n-h^n\right)\left(\mu^m-h^m\right)\\ &=J^{-1}_{ij}\left(\mu^i\mu^j-h^i\mu^j-h^j\mu^i+h^ih^j\right)~. \end{aligned} \ \ \ \ \ (23)

As for the change in the measure, it follows from the anti-symmetry of the wedge product, together with the fact that ${J_{ii}=0}$, that

$\displaystyle \mathrm{d}\phi_i=J^{-1}_{ij}\mathrm{d}\mu^j \quad\implies\quad \mathrm{d}^N\!\phi=\mathrm{det}J^{-1}\mathrm{d}^N\!\mu~. \ \ \ \ \ (24)$

Hence the partition function may be equivalently expressed as

$\displaystyle Z=\left[\left(\frac{2}{\pi}\right)^N\!\mathrm{det}J^{-1}\right]^{1/2}\!e^{-\frac{1}{2}J^{-1}_{ij}h^ih^j}\!\int\!\mathrm{d}^N\!\mu\exp\!\left[-\frac{1}{2}J^{-1}_{ij}\mu^i\mu^j+J^{-1}_{ij}h^i\mu^j+\sum_i\ln\cosh\mu_i\right]~, \ \ \ \ \ (25)$

where I’ve assumed ${J_{ij}=J_{ji}}$. While this only becomes a proper (mean) field theory in the thermodynamic limit, it’s worth emphasizing that up to this point, the transformation from the original lattice model (18) to is exact!

Now comes the approximation: to obtain a more tractable expression, let’s consider the case where the external magnetic field is very small as we did above. In this case, since the spin interactions don’t induce any preferred direction, we expect the mean field to be centered near zero, i.e., ${|\mu_i|\ll1}$. We can then expand

$\displaystyle \ln\cosh\mu_i=\frac{1}{2}\mu_i^2-\frac{1}{12}\mu_i^4+O(\mu_i^6)~, \ \ \ \ \ (26)$

whereupon the partition function becomes

$\displaystyle Z\approx\left[\left(\frac{2}{\pi}\right)^N\!\mathrm{det}J^{-1}\right]^{1/2}\!e^{-\frac{1}{2}J^{-1}_{ij}h^ih^j}\!\int\!\mathrm{d}^N\mu\exp\!\left[-\frac{1}{2}J^{-1}_{ij}\mu^i\mu^j +J^{-1}_{ij}h^i\mu^j +\sum_i\left(\frac{1}{2}\mu_i^2-\frac{1}{12}\mu_i^4\right)\right]~. \ \ \ \ \ (27)$

Finally, we take the continuum limit, in which we label the field at each site by the {${d}$-dimensional} vector ${\mathbf{x}}$ (i.e., ${\mu_i\rightarrow\mu(\mathbf{x})}$ and ${\sum\nolimits_i\rightarrow\int\!\mathrm{d}^dx=\int\!\mathrm{d}\mathbf{x}}$), and obtain the path-integral measure

$\displaystyle \left[\left(\frac{2}{\pi}\right)^N\!\mathrm{det}J^{-1}\right]^{1/2}\!e^{-\frac{1}{2}J^{-1}_{ij}h^ih^j}\!\int\!\mathrm{d}^N\mu \;\longrightarrow\;\mathcal{N}\!\int\!\mathcal{D}\mu~. \ \ \ \ \ (28)$

Thus the continuum field theory for the Ising model is

\displaystyle \begin{aligned} Z\approx\mathcal{N}\!\int\!\mathcal{D}\mu\exp\bigg\{\!&-\frac{1}{2}\int\!\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y}\,\mu(\mathbf{x})J^{-1}(\mathbf{x}-\mathbf{y})\big[\mu(\mathbf{y})-h\big]\\ &+\frac{1}{2}\int\!\mathrm{d}\mathbf{x}\left[\mu(\mathbf{x})^2-\frac{1}{6}\mu(\mathbf{x})^4\right]\bigg\}~, \end{aligned} \ \ \ \ \ (29)

where ${h(\mathbf{x})=h}$, since the external magnetic field is the same for all lattice sites, cf. the definition of ${\hat h^i}$ below (1).

In obtaining (29), I haven’t made any assumptions about the form of the coupling matrix ${J}$, except that it be a symmetric invertible matrix, with no self-interactions (${J_{ii}=0}$). Typically however, we’re interested in the case in which the hamiltonian (1) includes only nearest-neighbor interactions — as we eventually specified in our lattice model, cf. (10) — and we’d like to preserve this notion of locality in the field theory. To do this, we take ${|\mathbf{y}-\mathbf{x}|\ll 1}$ and Taylor expand the field ${\phi(\mathbf{y})}$ around ${\mathbf{x}}$:

$\displaystyle \mu(\mathbf{y})=\mu(\mathbf{x})+(y^i-x^i)\partial_i\mu(\mathbf{x})+\frac{1}{2}(y^i-x^i)(y^j-x^j)\partial_i\partial_j\mu(\mathbf{x})+O\left((\mathbf{y}-\mathbf{x})^3\right)~. \ \ \ \ \ (30)$

That is, we view ${J^{-1}(\mathbf{y}-\mathbf{x})}$ as mediating interactions between fields at infinitesimally separated points in space, with increasingly non-local (i.e., higher-derivative) terms suppressed by powers of the separation. Upon substituting this expansion into (29), and working to second-order in this local expansion, one obtains a partition function of the form

$\displaystyle Z\approx\mathcal{N}\!\int\!\mathcal{D}\mu\,e^{-S[\mu]}~, \ \ \ \ \ (31)$

with

$\displaystyle S[\mu]=\int\!\mathrm{d}^d\mathbf{x}\left[\frac{1}{2}\kappa(\nabla\mu)^2-h\mu+\frac{1}{2}\tilde r\mu^2+\frac{\tilde g}{4!}\mu^4\right]~, \ \ \ \ \ (32)$

where the coefficients ${\kappa}$, ${\tilde r}$, and ${\tilde g}$ are some (analytic) functions of the physical parameters, and can be expressed in terms of the zero modes of the inverse coupling matrix. I’m not going to go through the details of that computation here, since a great exposition is already available in the answer to this post on Stack Exchange (note however that they do not keep the linear ${h\mu}$ term).

The main lesson of this field-theoretic exercise is that MFT is nothing more than the leading saddle point of (31). Denoting the minimum ${\mu_0}$, and expanding the action to second order in the fluctuations ${(\mu-\mu_0)}$, we have

$\displaystyle Z\approx\mathcal{N} e^{-S[\mu_0]}\int\!\mathcal{D}\mu\,e^{-\frac{1}{2}(\mu-\mu_0)^2S''[\mu_0]}~, \ \ \ \ \ (33)$

where the prime denotes variation with respect to ${\mu}$, and the linear term has vanished by definition, i.e., ${\mu_0}$ is given by

$\displaystyle \frac{\delta S}{\delta\mu}=0 \quad\implies\quad \kappa\nabla^2\mu_0=-h+\tilde r\mu_0+\frac{\tilde g}{6}\mu_0^3~, \ \ \ \ \ (34)$

where we have applied integration by parts and assumed that the field vanishes at infinity. If we then keep only the leading-order saddle point, the partition function is given entirely by the prefactor

$\displaystyle Z\approx\mathcal{N} e^{-S[\mu_0]} \quad\mathrm{with}\quad S[\mu_0]=\int\!\mathrm{d}^d\mathbf{x}\left(\tilde r\mu_0^2+\frac{\tilde g}{8}\mu_0^4-\frac{3}{2}h\mu_0\right)~, \ \ \ \ \ (35)$

so that the free energy is

$\displaystyle f_\textrm{\tiny{sp}}=\frac{1}{\beta}\left(\tilde r\mu_0^2+\frac{\tilde g}{2}\mu_0^4-\frac{3}{4}h\mu_0\right) =\frac{\hat r}{2}\hat\mu_0^2+\frac{\hat g}{4!}\hat\mu_0^4-\hat h\mu_0~, \ \ \ \ \ (36)$

where the subscript “sp” stands for “saddle point”, and we have dropped the non-dynamical ${\ln\mathcal{N}}$ term. In the second equality, we have simply extracted the factor of ${\beta}$ from ${\mu}$ by defining ${\mu=\beta\hat\mu}$ (cf. the absorption of ${\beta}$ into the coefficients ${J,h}$ in (18), and the definition of ${\mu}$ below (22)), and defined ${2\hat r\equiv\beta\tilde r}$, ${\hat g\equiv12\tilde g\beta^3}$, and ${4\hat h\equiv3h}$. For ${h\!=\!0}$, this is formally identical to ${f_\textrm{\tiny{MF}}}$ obtained above, cf. (16)! (By this point the two theories are technically different, though Kopietz et al. [1] do give an argument as to how one might match the coefficients; otherwise one can compute them explicitly via Fourier transform as mentioned above).

Now suppose we kept the leading correction to the MFT result, given by the quadratic term in the path integral (33). For ${S''[\mu_0]}$, we have the operator

$\displaystyle K\equiv\frac{\delta^2S}{\delta\mu_x\delta\mu_y}=\left(-\kappa\nabla^2+\tilde r+\frac{\tilde g}{2}\mu_0^2\right)\delta^d(\mathbf{x}-\mathbf{y})~, \ \ \ \ \ (37)$

where ${\mu_x\equiv\mu(\mathbf{x})}$. Substituting this into (33) and doing the Gaussian integral, one finds that the contribution from this term is given by the sum of the eigenvalues of the operator ${K}$. I’m not going to go through this in detail, since this post is getting long-winded, and McGreevy’s notes already do a great job. The result is an additional contribution to the free energy that shifts the location of the critical point. Whether or not these higher-order corrections are important depends not only on the size of the fluctuations, but also on the spatial dimension of the system. It turns out that for systems in the Ising universality class (that is, systems whose critical points are characterized by the same set of critical exponents), the MFT result is good enough in ${d\!\geq\!4}$, but the fluctuations diverge in ${d\!<\!4}$ and hence render its conclusions invalid. We’ll give a better explanation for this dimensional-dependent validity below.

There’s another way to see the breakdown of MFT at the critical point in a manner that makes more transparent the precise role of the higher-order terms in the expansion, via the renormalization group. Suppose we’d included all higher-order terms in (32)—that is, all terms consistent with the symmetries of the problem (rotation & translation invariance, and ${\mathbb{Z}_2}$ symmetry if ${h\!=\!0}$). The result is called the Ginzburg-Landau action, after the eponymous authors who first used it to study systems near critical points. Now observe that the field ${\mu}$ has mass dimension ${\tfrac{d-2}{2}}$, so the squared mass ${\tilde r}$ has dimension 2, the quartic coupling ${\tilde g}$ has dimension ${4\!-\!d}$, a sextic coupling would have dimension ${6\!-\!2d}$, and so on. Recall that a coupling is relevant if its mass dimension ${\Delta\!>\!0}$ (since the dimensionless coupling carries a factor of ${E^{-\Delta}}$, e.g., ${m^2=\tilde r/E^2}$), irrelevant if ${\Delta\!<\!0}$ (since it runs like ${E^\Delta}$), and marginal if ${\Delta\!=\!0}$. Thus we see that the quadratic term is always relevant, and that higher-order corrections are increasingly suppressed under RG in a dimension-dependent manner.

So, a more sophisticated alternative to our particular MFT attempt above — where we kept the quartic ${\mu_0^4}$ term in the saddle point — is to compute the Gaussian path integral consisting of only the quadratic contribution, and treat the quartic and higher terms perturbatively. (As these originally arose from higher-order terms in the Taylor expansion, this is morally in line with simply taking ${\mu_0\ll1}$ in the MFT result ${f_\textrm{\tiny{sp}}}$). Treating the action as a Gaussian integral also allows us to obtain a simple expression for the two-point correlator that captures the limiting behaviour in which we’re primarily interested. That is, tying all this back to the information theory / neural network connections alluded in the introduction, we’re ultimately interested in understanding the propagation of information near the critical point, so understanding how correlation functions behave in the leading-order / MFT / saddle point approximation — and how perturbative corrections from fluctuations might affect this — is of prime importance.

We thus consider the partition function (31), with the quadratic action

$\displaystyle S[\mu]=\int\!\mathrm{d}^d\mathbf{x}\left[\frac{1}{2}(\nabla\mu(\mathbf{x}))^2+\frac{m^2}{2}\mu(\mathbf{x})^2-h\mu(\mathbf{x})\right]~, \ \ \ \ \ (38)$

where we’ve set ${\kappa\!=\!1}$ and relabelled the quadratic coefficient ${m^2}$. Evaluating partition functions of this type is a typical exercise in one’s first QFT course, since the action now resembles that of a free massive scalar field, where ${-h}$ plays the role of the source (normally denoted ${J}$). The basic prescription is to Fourier transform to momentum space, where the modes decouple, and then absorb the remaining source-independent term into the overall normalization. The only difference here is that we’re in Euclidean rather than Lorentzian signature, so there’s no issues of convergence; see for example Tong’s statistical field theory notes for a pedagogical exposition. The result is

$\displaystyle Z\simeq\exp\frac{1}{2}\!\int\!\frac{\mathrm{d}^dk}{(2\pi)^d}\frac{\tilde h_{\mathbf{k}}\tilde h_{-\mathbf{k}}}{k^2+m^2} =\exp\frac{1}{2}\!\int\!\mathrm{d}^dx\mathrm{d}^dy\,h(\mathbf{x})G(\mathbf{x}-\mathbf{y})h(\mathbf{y})~, \ \ \ \ \ (39)$

where the position-dependence of the external field ${h(\mathbf{x})}$ merely serves as a mnemonic, and ${\tilde h_\mathbf{k}}$ is the same field in momentum space. In the second equality, we’ve simply Fourier transformed back to real space by identifying the propagator

$\displaystyle G(\mathbf{x}-\mathbf{y})=\int\!\frac{\mathrm{d}^dk}{(2\pi)^2}\frac{e^{-i\mathbf{k}\mathbf{x}}}{k^2+m^2}~, \ \ \ \ \ (40)$

which describes the correlation between the field ${\mathbf{x}}$ and ${\mathbf{y}}$. To see that this is indeed a correlation function, recall that the variance is given by the second cumulant:

$\displaystyle \kappa_2=\frac{\partial^2\ln Z}{\partial h^2} \overset{\tiny{(38)}}{=} \langle\mu_x\mu_y\rangle-\langle\mu_x\rangle\langle\mu_y\rangle \overset{\tiny{(40)}}{=} G(\mathbf{x}-\mathbf{y})~, \ \ \ \ \ (41)$

and thus ${G}$ is indeed the connected 2-point correlator. (I should mention that in the present case, there’s a special name for this which seems to be preferred by condensed matter theorists: it’s the magnetic susceptibility, defined as the sensitivity of ${s}$ with respect to ${h}$,

$\displaystyle \chi=\partial_hs~, \ \ \ \ \ (42)$

where the connection arises by observing that the magnetization is none other than the mean (i.e., the first cumulant),

$\displaystyle s=-\frac{\partial\ln Z}{\partial h}=\langle\mu\rangle~. \ \ \ \ \ (43)$

But I’ll continue to refer to it as the correlation function, or the connected Green function, since calling it the “susceptibility” obscures its deeper physical and information-theoretic significance. Actually seeing that this is a Green function does however require slightly more work.)

The evaluation of (40) is treated very nicely in the aforementioned notes by Tong. In brief, we proceed by defining a length scale ${\xi^2=m^{-2}}$, and use the identity

$\displaystyle \int_0^\infty\!\mathrm{d} t\,e^{-t(k^2+\xi^{-2})}=\frac{1}{k^2+1/\xi^2} \ \ \ \ \ (44)$

to massage the integral into the following form:

$\displaystyle G(r)=\frac{1}{(4\pi)^{d/2}}\int_0^\infty\!\mathrm{d} t\,t^{-d/2}e^{-r^2/4t-t/\xi^2} \ \ \ \ \ (45)$

which is obtained by completing the square in the exponential and performing the integral over ${\mathrm{d}^dk}$; we’ve also used rotation invariance with ${r=|\mathbf{x}|}$ (not to be confused with the old name for the quadratic coefficient).

As will shortly become apparent, ${\xi}$ is the correlation length that determines the size of fluctuations, and hence the spatial structure of correlations. Since we’re primarily interested in the limiting cases where ${r\gg\xi}$ and ${r\ll\xi}$, it is more illuminating to evaluate the integral via saddle point, rather than to preserve the exact form (which, as it turns out, can be expressed as a Bessel function). We thus exponentiate the ${t^{-d/2}}$ factor to write

$\displaystyle G(r)=\frac{1}{(4\pi)^{d/2}}\int_0^\infty\!\mathrm{d} t\,e^{-P(t)}~, \qquad P(t)\equiv\frac{r^2}{4t}+\frac{t}{\xi^2}+\frac{d}{2}\ln t~, \ \ \ \ \ (46)$

so that we have, to second order,

$\displaystyle G(r)\sim \sqrt{\frac{\pi}{2S''(t_*)}}\,e^{-P(t_*)}~, \ \ \ \ \ (47)$

where the saddle point ${t_*}$ is given by

$\displaystyle S'(t_*)=0\implies t_*=\frac{\xi^2}{2}\left(-\frac{d}{2}+\sqrt{\frac{d^2}{4}+\frac{r^2}{\xi^2}}\right) \approx\begin{cases} \frac{r^2}{2d}\; & r\ll\xi~,\\ \frac{r\xi}{2}\; & r\gg\xi~. \end{cases} \ \ \ \ \ (48)$

Substituting these case values into (47), we find the following limiting behaviour for the correlation function:

$\displaystyle G(r)\sim\begin{cases} \frac{1}{r^{d-2}}\; & r\ll\xi~,\\ \frac{e^{-r/\xi}}{r^{(d-1)/2}}\; & r\gg\xi~. \end{cases} \ \ \ \ \ (49)$

Recalling that ${m^2\sim|T-T_c|}$ near the critical point, we see that the correlation length diverges as

$\displaystyle \xi\sim\frac{1}{|T-T_c|^{1/2}} \ \ \ \ \ (50)$

as the system approaches criticality. This means that at the critical point, we are always in the regime ${r\ll\xi}$, and hence the correlator exhibits a power law divergence. Another way to say this is that there is no longer any length scale in the problem (since that role was played by ${\xi}$, which has gone to infinity). This is why the divergence of the correlator at criticality must be a power law: any other function would require a length scale on dimensional grounds.

In the previous MFT treatment, we mentioned that fluctuations can change the results. From the RG perspective, this is because the quadratic coupling (which determines the location of the critical point) may be adjusted in the renormalization process as we integrate out UV modes. (In fact, we saw an explicit example of this in our post on deep learning and the renormalization group). The lower the dimension, the more relevant operators we need to take into account; in particular, all operators are relevant in ${d\!=\!2}$, so the saddle point approximation is exceedingly poor. In contrast, as the dimension increases, more and more operators are suppressed under RG flow. In the lattice picture, we can understand the fact that MFT gets more accurate in higher dimensions by noting that more dimensions means more neighbors, and hence approximating the degrees of freedom by the mean field is more likely to be accurate.

Finally, let us return to the comment we made at the beginning of this post, namely that the correlation length diverges at a critical point. This is another way of understanding the breakdown of MFT, since a divergent correlation length implies that fluctuations on all scales are important (and hence we neglect them at our peril). Explicitly, MFT (broadly interpreted) is valid when the fluctuations are much smaller than the mean or background field around which they’re fluctuating, i.e., ${\langle\mu^2\rangle\ll\langle\mu\rangle^2}$. Tong offers a clean way to see the dimensional dependence explicitly: simply integrate these expectation values over a ball of radius ${\xi}$ and compare the ratio

$\displaystyle R\equiv\frac{\int_0^\xi\!\mathrm{d}^dx\langle\mu(\mathbf{x})\mu(0)\rangle}{\int_0^\xi\!\mathrm{d}^dx\langle\mu^2\rangle} \simeq \frac{1}{\mu_0^2\xi^d}\int_0^\xi\!\mathrm{d} r\frac{r^{d-1}}{r^{d-2}} =\frac{\xi^{2-d}}{\mu_0^2} \sim|T-T_c|^{(d-4)/2}~, \ \ \ \ \ (51)$

where ${\mu_0=\langle\mu\rangle}$ is the mean field from above, and in the last step we have used the scaling behaviours of ${\xi\sim|T-T_c|^{-1/2}}$ and ${\mu_0\sim|T-T_c|^{1/2}}$ (the latter can be obtained by minimizing the quartic result for ${f_\textrm{\tiny{MF}}}$ (16)). Upon demanding that this ratio be much less than unity, we see that the MFT results (for the Ising universality class) are only trustworthy in ${d\geq4}$ dimensions. (The case ${d\!=\!4}$ actually requires a more careful RG treatment due to the logarithmic divergence; see Tong’s notes for more details).

To summarize: MFT is a useful approximation method that averages over interactions and enables one to obtain closed-form expressions of otherwise intractable partition functions. It is tantamount to the saddle point approximation, and — in the context of RG — may be qualitatively altered by any relevant higher-order terms. While these corrections can potentially shift the location of the critical point however, the basic fact that the correlation function diverges at criticality remains unchanged. As we’ll see in part 2, it is this feature that makes phase transitions interesting from a computational perspective, since it means that the propagation of information at this point is especially stable.

References

1. P. Kopietz, L. Bartosch, and F. Schütz, “Mean-field Theory and the Gaussian Approximation,” Lect. Notes Phys. 798 (2010).

## Black hole thermodynamics, quantum puzzles, and the holographic principle

I was asked to give a lecture on “quantum puzzles and black holes” at the 20th Jürgen Ehlers Spring School, which was to be hosted at AEI this week. Unfortunately the school was cancelled due to the SARS-CoV-2 pandemic, but since I enjoyed researching the topic so much, I thought I’d make a post of it instead. Part of what made preparing for this lecture so interesting is that the students — primarily undergrads bordering on Masters students — hadn’t had quantum field theory (QFT), which meant that if I wanted to elucidate, e.g., the firewall paradox or the thermal nature of horizons in general, I’d have to do so without recourse to the standard toolkit. And while there’s a limit to how far one can get without QFT in curved spacetime, it was nice to go back and revisit some of the things with which long familiarity has made me take for granted.

Accordingly, I’ve endeavored to make this post maximally pedagogical, assuming only basic general relativity (GR) and a semblance of familiarity with undergraduate quantum mechanics and statistical thermodynamics. I’ll start by introducing black hole thermodynamics, which leads to the conclusion that black holes have an entropy given by a quarter the area of their event horizons in Planck units. Then in the second section, I’ll discuss some quantum puzzles that arise in light of Hawking’s discovery that black holes radiate, which seems to imply that information is lost as they evaporate, in violation of quantum mechanics. In the third and final section, I’ll explain how the considerations herein gave rise to the holographic principle, one of the deepest revelations in physics to date, which states that the three-dimensional world we observe is described by a two-dimensional hologram.

1. Black hole thermodynamics

Classically, black hole thermodynamics is a formal analogy between black holes and statistical thermodynamics. It was originally put forth by Jacob Bekenstein in his landmark 1973 paper [1], in which he proposed treating black holes thermodynamically, and argued that the entropy should be proportional to the area of the event horizon. Let’s start be examining the idea of black holes as thermodynamic objects, and build up to the (in)famous entropy-area relation as we go.

As I’ve mentioned before, black holes must be endowed with an entropy in order to avoid violating the second law of thermodynamics; otherwise, one could decrease the entropy of the universe simply by dropping anything into the black hole. Taking entropy as a measure of our ignorance — equivalently, as a measure of the inaccessibility of the internal configuration — this is intuitive, since the degrees of freedom comprising whatever object one dropped in are now hidden behind the horizon and should thus be counted among the internal microstates of the black hole. Furthermore, one knows from Hawking’s area theorem [2] that the surface area of a classical black hole is non-decreasing, and thus the dynamics of black holes appears to select a preferred direction in time, analogous to the thermodynamic arrow of time consequent of the fact that entropy (of any closed thermodynamic system) always increases. This led Bekenstein [1] to propose that one could “develop a thermodynamics of black holes”, in which entropy is precisely related to the area of the horizon, ${S\sim A}$ (here “${\sim}$” means “proportional to”; we’ll fix the coefficient later).

Thermodynamically, entropy is an extensive property, so associating the entropy to some function of the size of the black hole makes sense. But why ${S\sim A}$, specifically? In statistical mechanics, the entropy generally scales with the volume of the system, so one might naïvely have expected ${S\sim V}$. Indeed, one of the most remarkable aspects of black holes is that the entropy scales with the area instead of the volume. Insofar as black holes represent the densest possible configuration of energy — and hence of information — this implies a drastic reduction in the (maximum) number of degrees of freedom in the universe, as I’ll discuss in more detail below. However, area laws for entanglement entropy are actually quite common; see for example [3] for a review. And while the ultimate source of black hole entropy (that is, the microscopic degrees of freedom it’s counting) is an ongoing topic of current research, the entanglement between the interior and exterior certainly plays an important role. But that’s a QFT calculation, whereas everything I’ve said so far is purely classical. Is there any way to see that the entropy must scale with ${A}$ instead of ${V}$, without resorting to QFT in curved space or the full gravitational path integral?

In fact, there’s a very simple reason the entropy must scale with the area: the interior volume of a black hole is ill-defined. Consider the Schwarzschild metric

$\displaystyle \mathrm{d} s^2=-f(r)\mathrm{d} t^2+\frac{\mathrm{d} r^2}{f(r)}+r^2\mathrm{d}\Omega^2~, \quad\quad f(r)=1-\frac{r_s}{r}~, \ \ \ \ \ (1)$

where ${r_s=2M}$ is the Schwarzschild radius and ${\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\!\theta\,\mathrm{d}\phi^2}$. For ${r>r_s}$, the metric is static: the spatial components look the same for any value of ${t}$. But inside the black hole, ${r, and hence ${f(r)<0}$. This makes the ${\mathrm{d} t^2}$ component positive and the ${\mathrm{d} r^2}$ component negative, so that space and time switch roles in the black hole interior. Consequently, the “spatial” components are no longer static inside the black hole, since they will continue to evolve with ${t}$. Thus the “volume” of the black hole interior depends on time, and in fact on one’s choice of coordinates in general. (This isn’t too strange, if you think about it: the lesson of general relativity is that spacetime is curved, so your quantification of “space” will generally depend on your choice of “time”).

The issue is clarified nicely in a paper by Christodoulou and Rovelli [4] (be warned however that while the GR calculations in this paper are totally solid, the discussion of entropy in section VIII is severely flawed). The crux of the matter is that our usual definition of “volume” doesn’t generalize to curved spacetime. In flat (Minkowski) spacetime, we define volume by picking a Cauchy slice, and consider the spacelike 3d hypersurface ${\Sigma}$ bounded by some 2d sphere ${B}$ on that slice. But when space is curved, there are many different constant-${t}$ slices we can choose, none of which has any special status (in GR, the coordinates don’t matter). Suppose for example we tried to calculate the interior volume in Schwarzschild coordinates (1). Our flat-space intuition says to pick a constant-${t}$ slice bounded by some surface ${B}$ (in this case, the horizon itself), and integrate over the enclosed hypersurface ${\Sigma}$:

$\displaystyle V=\int_\Sigma\!\mathrm{d}^3x\sqrt{g}~, \ \ \ \ \ (2)$

where ${g}$ is the determinant of the (induced) metric on ${\Sigma}$. Along a timeslice, ${\mathrm{d} t=0}$, so we have

$\displaystyle V=4\pi\int\!\mathrm{d} r\,r^2\left(1-\tfrac{r_s}{r}\right)^{-1/2}~. \ \ \ \ \ (3)$

But the Schwarzschild coordinates break-down at the horizon, so the upper and lower limits of the remaining integral are the same, ${r=r_s}$, and the integral vanishes. Thus the Schwarzschild metric would lead one to conclude that the “volume” of the black hole is zero! (Technically the integral is ill-defined at ${r=r_s}$, but one obtains the same result by changing the outer limit to ${r=r_s+\epsilon}$ and taking the limit ${\epsilon\rightarrow0}$ [5]).

Let’s try a different coordinate system, better suited to examining the interior. Define the new time variable

$\displaystyle T=t+r_s\left(2\sqrt{\frac{r}{r_s}}+\ln\left|\frac{\sqrt{\tfrac{r}{r_s}}-1}{\sqrt{\tfrac{r}{r_s}}+1}\right|\right)~, \ \ \ \ \ (4)$

in terms of which the metric (1) becomes

$\displaystyle \mathrm{d} s^2=-f(r)\mathrm{d} T^2+2\sqrt{\frac{r_s}{r}}\,\mathrm{d} T\mathrm{d} r+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2~. \ \ \ \ \ (5)$

These are Gullstrand-Painlevé (GP) coordinates. They’re relatively unfamiliar, but have some useful properties; see for example [6], in which my colleagues and I utilized them in the context of the firewall paradox during my PhD days. Unlike the Schwarzschild coordinates, they cover both the exterior region and the black hole interior. They look like this:

where constant ${T}$ slices are in yellow, and constant ${r}$ slices are in green. One neat thing about these coordinates is that ${T}$ is the proper time of a free-falling observer who starts from rest at infinity. (Somewhat poetically, they’re the natural coordinates that would be associated to a falling drop of rain, and are sometimes called “rain-frame coordinates” for this reason). Another neat thing about them is that the constant-${T}$ slices are flat! Thus if we attempt to calculate the interior volume along one such Cauchy slice, we simply recover the flat-space result,

$\displaystyle V=4\pi\int_0^{r_s}\mathrm{d} r^2\,r^2=\frac{4}{3}\pi r_s^3~, \ \ \ \ \ (6)$

and thus the volume is constant, no matter what ${T}$-slice we choose; in other words, the observer can fall forever and never see less volume! See [5] for a pedagogical treatment of the volume calculation in some other coordinate systems, which again yield different results.

The above examples illustrate the fact that in general, there are many (in fact, infinitely many!) different choices of ${\Sigma}$ within the boundary sphere ${B}$, and we need a slightly more robust notion of volume to make sense in curved spacetime. As Christodoulou and Rovelli point out, a better definition for ${\Sigma}$ is the largest spherically symmetric surface bounded by ${B}$. This reduces to the familiar definition above in Minkowski space, but extends naturally and unambiguously to curved spacetime as well. Thus the basic idea is to first fix the boundary sphere ${B}$, and then extremize over all possible interior 3d hypersurfaces ${\Sigma}$. For a Schwarzschild black hole, in the limit where the null coordinate ${v=r+t}$ is much greater than ${M}$ (i.e., at late times), one finds [4]

$\displaystyle V\rightarrow3\sqrt{3}\pi m^2 v~. \ \ \ \ \ (7)$

Thus, the interior volume of a black hole continues to grow linearly for long times, and can even exceed the volume of the visible universe!

Whether one thinks of entropy as a measure of one’s ignorance of the interior given the known exterior state, or a quantification of all possible microstates given the constraints of mass (as well as charge and angular momentum for a Kerr-Newman black hole), it should not depend on the choice of coordinates, or continue growing indefinitely while the surface area (i.e., the boundary between the known and unknown regions) remains fixed. Thus if we want a sensible, covariant quantification of the size of the black hole, it must be the area. (Note that the area is more fundamental than the radius: the latter is defined in terms of the former (equivalently, in terms of the mass), rather than by measuring the distance from ${r\!=\!0}$, for the same reasons we encountered when attempting to define volume above). Since the event horizon is a null-surface, the area is coordinate-invariant; fixing ${t}$ and ${r}$ in the Schwarzschild metric then simply yields the area element of the 2-sphere,

$\displaystyle \mathrm{d} s^2=r^2\mathrm{d}\Omega^2 \quad\longrightarrow\quad A=\int\!\mathrm{d} s^2=\int\!\mathrm{d}\Omega\sqrt{g} =4\pi r_s^2~. \ \ \ \ \ (8)$

Thus areas, rather than volumes, provide the only covariant, well-defined measures of the spatial “size” of black holes.

Technically, this doesn’t prove that ${S\sim A}$, of course; it might logically have been some other function or power of the area, but this would be less natural on physical grounds (though ${S\sim A^{1/2}}$ can be easily ruled out by considering a black hole merger [1]). And, while it’s a nice consistency check on the universe, it doesn’t really give any insight into why the degrees of freedom are ultimately bounded by the surface area, beyond the necessity-of-curved-space argument above.

There is however one problem with this identification: the entropy, in natural units, is dimensionless, while the area has units of length squared, so the mismatch must be remedied by the hitherto undetermined proportionality factor. As Bekenstein pointed out, there is no universal constant in GR alone that has the correct units; the only fundamental constant that fits the bill is the Planck length,

$\displaystyle \ell_P=\frac{G\hbar}{c^3}~. \ \ \ \ \ (9)$

As Hawking was quick to show [7], the correct result is

$\displaystyle S_{BH}=\frac{A}{4\ell_P}~, \ \ \ \ \ (10)$

which is the celebrated Bekenstein-Hawking entropy of black holes. This is one of the most remarkable expressions in all of physics, insofar as it’s perhaps the only known example in which gravity (${G}$), quantum mechanics (${\hbar}$), and special relativity (${c}$) all come together (thermodynamics too, if you consider that we set ${k_B=1}$).

Hawking’s calculation, and the myriad alternative derivations put forward since, require a full QFT treatment, so I’m not going to go into them here. If you’re interested, I’ve covered one such derivation based on the gravitational path integral before, and the case of a collapsing black hole that Hawking considered is reviewed in the classic textbook [8]. In the original paper [1] however, Bekenstein provides a very cute derivation which barely even requires quantum mechanics, and yet gets surprisingly close to the right answer. The basic idea is to calculate the minimum possible increase in the size of the black hole which, classically, would occur when we gently drop in a particle whose size is of order its own Compton wavelength (this is where the ${\hbar}$ comes in). This can be related to the entropy on the basis that the loss of information is the entropy of a single bit, ${\ln 2}$, i.e., the answer to the yes-or-no question, “does the black hole contain the particle?” This line of reasoning yields ${\tfrac{1}{2\pi}\ln2\,S_{BH}}$; not bad, given that we ignored QFT entirely!

By now I hope I’ve convinced you of two facts: (1) black holes have an entropy, and (2) the entropy is given by the area of the horizon. This is the foundation on which black hole thermodynamics is built.

We are now in position to write down the analogue of the fundamental thermodynamic relation for black holes. Recall that for a closed system in equilibrium,

$\displaystyle \mathrm{d} U=T\mathrm{d} S-P\mathrm{d} V~, \ \ \ \ \ (11)$

where ${U}$ is the internal energy, ${P}$ is the pressure, and ${T}$, ${S}$, and ${V}$ are the temperature, entropy, and volume as above. The second term on the right-hand side represents the work done on the system by the environment (in this context, “closed” refers only to the transfer of mass or particles; the transfer of energy is still allowed). Supposing that this term is zero, the first term can be regarded as the definition of entropy for reversible processes, i.e., ${\mathrm{d} S=\mathrm{d} Q/T}$ where ${Q}$ is the heat.

Now consider, for generality, a charged, rotating black hole, described by the Kerr-Newman metric; in Boyer-Lindquist coordinates, this reads:

\displaystyle \begin{aligned} \mathrm{d} s^2=&~\frac{\Delta-a^2\sin^2\theta}{\rho^2}\,\mathrm{d} t^2 -\frac{\rho^2}{\Delta}\mathrm{d} r^2 -\rho^2\,\mathrm{d}\theta^2\\ &-\sin^2\!\theta\,\frac{\left( a^2+r^2\right)^2-a^2\Delta\sin^2\theta}{\rho^2}\,\mathrm{d}\phi^2 +\frac{2a\left( a^2+r^2-\Delta\right)\sin^2\theta}{\rho^2}\,\mathrm{d} t\mathrm{d}\phi~, \end{aligned} \ \ \ \ \ (12)

which reduces to the Schwarzschild black hole above when the charge ${Q}$ and angular momentum ${J}$ go to zero (after rescaling by the radius). For compactness, we have defined

$\displaystyle a\equiv\frac{J}{M}~,\quad \Delta\equiv r^2-2 M r+a^2+Q^2~,\quad \rho^2\equiv r^2+a^2\cos^2\theta~. \ \ \ \ \ (13)$

The ${g_{rr}}$ component diverges when ${\Delta=0}$, which yields an inner (${r_-}$) and outer (${r_+}$) horizon:

$\displaystyle r_\pm=M\pm\sqrt{M^2-a^2-Q^2}~. \ \ \ \ \ (14)$

The inner horizon is generally thought to be unstable, while the outer is the event horizon whose area we’re interested in calculating. Setting ${r}$ and ${t}$ to constant values as above, the induced metric on the resulting 2d surface ${B}$ is

$\displaystyle \mathrm{d} s^2= -\rho^2\,\mathrm{d}\theta^2 -\sin^2\!\theta\,\frac{\left( a^2+r^2\right)^2-a^2\Delta\sin^2\theta}{\rho^2}\,\mathrm{d}\phi^2 \ \ \ \ \ (15)$

We can then consider the case where the radius ${r=r_+}$, whereupon ${\Delta=0}$, and the area is simply

$\displaystyle A=\int\!\mathrm{d}\Omega\sqrt{g} =4\pi\left( r_+^2+a^2\right)~, \ \ \ \ \ (16)$

which is fairly intuitive: we get the Schwarzschild result, plus an additional contribution from the angular momentum.

Now, the area depends only on the mass ${M}$, charge ${Q}$, and angular momentum ${J}$ (cf. the no-hair theorem), so a generic perturbation takes the form

$\displaystyle \mathrm{d} A=\frac{\partial A}{\partial M}\,\mathrm{d} M+\frac{\partial A}{\partial Q}\,\mathrm{d} Q+\frac{\partial A}{\partial J}\,\mathrm{d} J~. \ \ \ \ \ (17)$

Performing the derivatives and solving for ${\mathrm{d} M}$, one obtains the analogue of the fundamental thermodynamic relation (11) for black holes:

$\displaystyle \mathrm{d} M=\frac{\kappa}{8\pi}\,\mathrm{d} A+\Omega\,\mathrm{d} J+\Phi\,\mathrm{d} Q~, \ \ \ \ \ (18)$

where ${\Omega}$ and ${\Phi}$ are some functions of ${M}$, ${Q}$, and ${J}$ that I’m not going to bother writing down, and ${\kappa}$ is the surface gravity,

$\displaystyle \kappa=\frac{r_+-r_-}{2\left( r_+^2+a^2\right)}=\frac{\sqrt{M^2-a^2-Q^2}}{2M^2-Q^2+2M\sqrt{M^2-a^2-Q^2}}~, \ \ \ \ \ (19)$

which is the (normalized) gravitational acceleration experienced at the equator. (“Normalized”, because the Newtonian acceleration diverges at the horizon, so a meaningful value is obtained by dividing the proper acceleration by the gravitational time dilation factor).

Each term in this expression for ${\mathrm{d} M}$ has a counterpart in (11). We already identified the area with the entropy, cf. (10), and since the mass is the only relevant parameter in the problem, it plays the role of the internal energy ${U}$. The surface gravity ${\kappa}$ corresponds to the temperature. So if we restricted to a Schwarzschild black hole, we’d have

$\displaystyle \mathrm{d} U=T\mathrm{d} S\qquad \longleftrightarrow\qquad \mathrm{d} M=\frac{\kappa}{4\pi}\mathrm{d} A~, \ \ \ \ \ (20)$

which just canonizes the relationship between entropy and area we uncovered above, with ${\kappa\sim T}$. What about the other terms? As mentioned above, the ${-P\mathrm{d} V}$ term in (11) corresponds to the work done to the system. And as it turns out, there’s a way of extracting energy from a (charged, rotating) black hole, known as the Penrose process. I don’t have the spacetime to go into this here, but the upshot is that the parameters ${\Omega}$ and ${\Phi}$ in (18) correspond to the rotational angular momentum and electric potential, respectively, so that ${\Omega\,\mathrm{d} J+\Phi\,\mathrm{d} Q}$ is indeed the analogue of the work that the black hole could perform on some external system; i.e.,

$\displaystyle -P\mathrm{d} V\qquad \longleftrightarrow\qquad \Omega\,\mathrm{d} J+\Phi\,\mathrm{d} Q~. \ \ \ \ \ (21)$

And of course, energy that can’t be extracted as work is another way of describing entropy, so if even if you could extract all the angular momentum and charge from the black hole, you’d still be left with what Bekenstein calls the “degradation energy” [1], which is the area term (20) (determined by the irreducible mass).

That’s all I wanted to say about black hole thermodynamics here, though the analogy we’ve established above can be fleshed out more thoroughly, complete with four “laws of black hole thermodynamics” in parallel to the classic set. See for example my earlier post on firewalls, or the review by Jacobson [9], for more details. However, I’ve been glossing over a critical fact, namely that at the classical level, black holes are, well, black: they don’t radiate, and hence a classical black hole has zero temperature. This is the reason I’ve been careful to refer to black hole thermodynamics as an analogy. Strictly speaking, one cannot regard the temperature ${T}$ as the physical temperature of a single black hole, but rather as referring to the equivalence class of all possible black holes subject to the same (observable) constraints of mass, charge, and angular momentum. In other words, the “temperature” of a Schwarzschild black hole is just a quantification of how the entropy — which measures the number of possible internal microstates — changes with respect to the mass, ${T^{-1}=\mathrm{d} S/\mathrm{d} M}$.

2. Quantum black holes

Of course, Hawking’s greatest claim to fame was the discovery [7] that when quantum field theory is properly taken into account, black holes aren’t black after all, but radiate with a temperature

$\displaystyle T=\frac{1}{8\pi M}~. \ \ \ \ \ (22)$

(This result is for a Schwarzschild black hole in thermal equilibrium, and is precisely what we obtain when taking ${Q=J=0}$ in the expression for the surface gravity ${\kappa}$ (19)). Hawking’s calculation, and many other derivations since, require the machinery of QFT, so I won’t go into the details here. There is however a cute hack for obtaining the identification (22), whereby one Wick rotates to Euclidean signature so that the ${(3\!+\!1)}$-dimensional Schwarzschild geometry becomes ${\mathbb{R}^3\times S^1}$, whereupon the temperature appears as a consequence of the periodicity in Euclidean time; see my first post for a sketch of the resulting “cigar geometry”, or my upcoming post on QFT in curved space for a more detailed discussion about the relationship between periodicity and horizons.

Hawking radiation is sometimes explained as the spontaneous fluctuation of a particle-antiparticle pair from the vacuum across the horizon; the particle escapes to infinity as Hawking radiation, while the antiparticle is captured by the black hole. This is a cute cartoon, except that it’s wrong, and an over-reliance on the resulting intuition can get one into trouble. I’ve already devoted an entire post to this issue, so I’ll refer you there if you’re interested; if you’ve got a QFT background, you can also find some discussion of the physical aspects of black hole emission in chapter eight of [8]. In a nutshell, the basic point is that radiation comes out in momentum-space modes with wavelength ${\lambda\sim r_s}$, which can’t be Fourier transformed back to position space to yield anything localized near the horizon. In other words, near the horizon of a black hole, the meaning of “particles” employed by an external observer breaks down. The fact that black holes can radiate away energy means that if you stop throwing in matter, the black hole will slowly shrink, which seems to contradict Hawking’s area theorem above. The catch is that this theorem relies on the weak energy condition, which states that the matter density along every timelike vector field is non-negative; this is no longer necessarily true once quantum fluctuations are taking into account, so there’s no mathematical contradiction. It does however mean that our formulation of the “second law” of black hole thermodynamics was too naïve: the area (and hence entropy) of a black hole can decrease, but only by emitting Hawking radiation which increases the entropy of the environment by at least as much. This motivates us to introduce the generalized entropy

$\displaystyle S_\mathrm{gen}=\frac{A}{4\ell_P}+S_\mathrm{ext}~, \ \ \ \ \ (23)$

where the first term is the black hole entropy ${S_\mathrm{BH}}$ (10), and the second is the entropy of the thermal radiation. In full generality, the Second Law of (Black Hole) Thermodynamics is then statement is that the entropy (10) of all black holes, plus the entropy of the rest of the universe, never decreases:

$\displaystyle \mathrm{d} S_\mathrm{gen}\geq 0~. \ \ \ \ \ (24)$

Evaporating black holes have some peculiar properties. For example, since the temperature of a Schwarzschild black hole is inversely proportional to the mass, the specific heat capacity ${C}$ is negative:

$\displaystyle \frac{\mathrm{d} T}{\mathrm{d} M}=-\frac{1}{8\pi M^2} \qquad\implies\qquad C=\frac{1}{M}\frac{\mathrm{d} Q}{\mathrm{d} T}=-8\pi M~. \ \ \ \ \ (25)$

(We’re working in natural units, so ${c\!=\!1}$ and hence the heat ${Q=M}$). Consequently, throwing matter into a black hole to increase its size actually makes it cooler! Conversely, as the black hole emits Hawking radiation, its temperature increases, causing it to emit more radiation, and so on in a feedback loop that causes the black hole to get hotter and hotter as it shrinks away to nothing. (Precisely what happens in the final moments of a black hole’s lifespan is an open question, likely requiring a more developed theory of quantum gravity to answer. Here I’m going to take the majority view that it indeed evaporates away completely). Note that this means that whenever one speaks about black holes thermodynamically, one should use the microcanonical ensemble rather than the canonical ensemble, because the latter is unstable to any quantum fluctuation that changes the mass of the black hole.

The fact that black holes radiate when quantum field theory is taken into account transforms black hole thermodynamics from a formal analogy to an ontologically meaningful description, where now the temperature ${T=\mathrm{d} M/\mathrm{d} S}$ is indeed the physical (thermodynamic) temperature of a single black hole. In this sense, quantum effects were required to resolve the tension between the fact that the information-theoretic interpretation of entropy as the measure of possible internal microstates was applicable to a single black hole — and hence had physical significance — while the temperature had no meaningful physical interpretation in the non-radiating (classical) case. The combination of seemingly disparate regimes in the expression for the entropy (10) is not a coincidence, but represents a truly remarkable unification. It’s perhaps the first thing a successful theory of quantum gravity should be expected to explain.

The fact that black holes evaporate also brings into focus the need for such a unification of general relativity and quantum field theory: a black hole is one of the only known regimes (the other being the Big Bang singularity) that falls within the purview of both theories, but attempts to combine them yield nonsensical infinities that have thus far resisted all attempts to tame them. This leads me to the main quantum puzzle I wanted to discuss: the information paradox. (The firewall paradox is essentially just a more modern sharpening of the underlying conflict, but is more difficult to sketch without QFT).

The information paradox, in a nutshell, is a conflict between the apparent ability of black holes to destroy information, and the quantum mechanical postulate of unitarity. Recall that unitarity is the statement that the time-evolution of a quantum state via the Schrödinger equation is described by a unitary operator, which have the property that they preserve the inner product. Physically, this ensures that probabilities continue to sum to one, i.e., that no information is lost. While evolution in open systems can be non-unitary due to decoherence with the environment, the evolution of any closed quantum mechanical system must be unitary, i.e., pure states evolve to pure states only, never to mixed states. This means that if we create a black hole by collapsing some matter in an initially pure state, let it evaporate, and then collect all the Hawking radiation, the final state must still be pure. The problem is that the Hawking radiation is, to a very good approximation, thermal, meaning it has the Planckian spectrum characteristic of black-body radiation, and thermal radiation contains no information.

The situation is often depicted by the Page curve [10,11], which is a plot of entropy with respect to time as the black hole evaporates. Suppose we collect all the Hawking radiation from a black hole that starts in a pure state; call the entropy of this radiation ${S_R}$. Initially, ${S_R=0}$, because our subsystem is empty. As the black hole evaporates, ${S_R}$ steadily increases as we collect more and more radiation. Eventually the black hole evaporates completely, and we’re left with a thermal bath of radiation in a maximally mixed state, so ${S_R=1}$ (after normalizing): a maximal loss of information has occurred! This is the information paradox in a single graph. In sharp contrast, what quantum mechanics expects to happen is that after the halfway point in the black hole’s lifespan, the late-time radiation starts to purify the early-time radiation we’ve already collected, so the entropy curve should turn around and head back to 0 when the black hole disappears. This is illustrated in the figure below, from Page’s paper [11]. (The lack of symmetry in the upwards and downwards parts is due to the fact that the emission of different particles (in this calculation, just photons and gravitons) affect the change in the black hole entropy and the change in the radiation entropy slightly differently. The turnover isn’t at exactly half the lifetime either, but rather around ${53.81\%}$.)

The fundamental issue is that quantum mechanics demands that the information escape the black hole, but there doesn’t seem to be any way of enabling this. (For more discussion, see my earlier post on firewalls. I should also mention that there are alternative proposals for what happens at the end of a black hole’s lifetime, but these are generally disfavoured for a variety of reasons, most notably AdS/CFT). That said, just within the past year, it was discovered that in certain AdS/CFT set-ups, one can obtain a Page curve for the entropy by including the contributions from wormhole geometries connecting different replicas that arise as subleading saddles in the gravitational path integral; see for example [12], or the talks by Douglas Stanford and Juan Maldacena as part of my research group’s QGI seminar series. While this doesn’t quite solve the paradox insofar as it doesn’t explain how the information actually escapes, it’s encouraging that — at least in AdS/CFT — there does seem to be a mechanism for correctly tracking the entropy as the black hole evaporates.

3. The holographic principle

To close this lecture/post, I’d be remiss if I didn’t mention the most remarkable and far-reaching consequence of the black hole investigations above: the holographic principle. Put forth by ‘t Hooft [13], and given a formulation in terms of string theory by Susskind [14], this is essentially the statement that the ultimate theory of quantum gravity must exhibit a dimensional reduction (from 3 to 2 spatial dimensions in our ${(3\!+\!1)}$-dimensional universe) in the number of fundamental degrees of freedom. This developed from the arguments of Bekenstein, that the black hole entropy (10) represents a bound on the amount of information that can be localized within any given region. The basic idea is that any attempt to cram more information into a region of fixed size will cause the system to collapse into a black hole, and therefore the dimension of the Hilbert space associated to any region must scale with the area of the boundary.

The review by Bousso [15] contains an excellent modern introduction to this principle; I’ll only give a quick summary of the main idea here. Recall that in quantum mechanics, the number of degrees of freedom ${N}$ is given by the log of the dimension of the Hilbert space ${\mathcal{H}}$. For example, in a system with 100 spins, there are ${2^{100}}$ possible states, so ${\mathrm{dim}\,\mathcal{H}=2^{100}}$ and ${N=100\ln 2}$, i.e., the system contains 100 bits of information. One can crudely think of quantum field theory as a continuum theory with a harmonic oscillator at every spacetime point; a single harmonic oscillator already has ${N=\infty}$, so one would expect an infinite number of degrees of freedom for any region. However, one can’t localize more than a Planck energy ${M_P=1.3\times 10^{19}\,\mathrm{GeV}}$ into a Planck cube ${\ell_P^3}$ without forming a black hole, which provides an ultra-violet (UV) cutoff on the spectrum. And since any finite volume imposes an infra-red (IR) cutoff, we can take the degrees of freedom in field theory to scale like the volume of the region, with one oscillator per Planck cell. In other words, we think of space as a grid with lattice spacing ${\ell_P=1.6\times 10^{-33}\,\mathrm{cm}}$; the total number of oscillators thus scales like the volume ${V}$, and each one has a finite number of states ${n}$ due to the UV cutoff mentioned above. Hence ${\mathrm{dim}\,\mathcal{H}=n^V}$ and ${N=V\ln n}$. Thus, since ${e^S=\mathrm{dim}\,\mathcal{H}}$, ${S\!=\!V\ln n\!\sim\! V}$, and we expect entropy to scale with volume just as in classical mechanics.

The lesson from black hole thermodynamics is that gravity fundamentally alters this picture. Consider a Schwarzschild black hole: the mass scales like ${M\sim r_s}$, not ${M\sim r_s^3}$, so the energy can’t scale with the volume: the vast majority of the states which QFT would naïvely allow can’t be reached in a gravitational theory, because we form a black hole when we’ve excited only a small fraction of them. The maximum number of states we can excite is ${A/4}$ (in Planck units).

You might object that the argument I gave above as to why the black hole entropy must scale with area rather than volume was based on the fact that the interior volume of the black hole is ill-defined, and that a volume law might still apply in other situations. Bousso [15] gives a nice argument as to why this can’t be true: it would violate unitarity. That is, suppose a region of space had an entropy that scaled with the volume, i.e., ${\mathrm{dim}\,\mathcal{H}=e^V}$. If we then collapse that region into a black hole, the Hilbert space dimension would have to suddenly drop to ${e^{A/4}}$. It would then be impossible to recover the initial state from the final state (e.g., after allowing the black hole to evaporate). Thus in order to preserve unitarity, the dimension of the Hilbert space must have been ${e^{A/4}}$ from the start.

I’ve glossed over an important technical issue in introducing this “holographic entropy bound” however, namely that the spatial bound doesn’t actually work: it’s violated in all sorts of scenarios. For example, consider a region of our universe, which is well-approximated as a flat (${\mathbb{R}^3}$), homogeneous, isotropic space with some average entropy density ${\sigma}$. Then the entropy scales like

$\displaystyle S=\sigma V=\frac{\sigma}{6\sqrt{\pi}}A^{3/2}~, \ \ \ \ \ (26)$

which exceeds the bound ${A/4}$ when ${r\geq 3\sigma/4}$. The proper way to generalize black hole entropy to the sort of bound we want is to recall that the area of the event horizon is a null hypersurface, and it is the formulation in terms of such light-sheets which is consistent with all known examples. This is known as the covariant entropy bound, and states that the entropy on (or rather, contained within) non-expanding light-sheets of some spacetime codimension-2 surface ${B}$ does not exceed the area of ${B}$. A thorough discussion would be another lecture in itself, so do check out Bousso’s review [15] if you’re interested in more details. Here I merely wanted to bring attention to the fact that the holographic principle is properly formulated on null, rather than spacelike, hypersurfaces.

The holographic principle represents a radical departure from our intuition, and implies that reality is fundamentally nonlocal. One further expects that this feature should be manifest in the ultimate theory of quantum gravity. AdS/CFT provides a concrete realization of this principle, and its success is such that the unqualified “holography” is taken to refer to it in the literature, but it’s important to remember that the holographic principle itself is more general, and applies to our universe as well.

References

1. J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (Apr, 1973) 2333–2346.
2. S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26 (May, 1971) 1344–1346.
3. J. Eisert, M. Cramer, and M. B. Plenio, “Area laws for the entanglement entropy – a review,” Rev. Mod. Phys. 82 (2010) 277–306, arXiv:0808.3773 [quant-ph].
4. M. Christodoulou and C. Rovelli, “How big is a black hole?,” Phys. Rev. D91 no. 6, (2015) 064046, arXiv:1411.2854 [gr-qc].
5. B. S. DiNunno and R. A. Matzner, “The Volume Inside a Black Hole,” Gen. Rel. Grav. 42 (2010) 63–76, arXiv:0801.1734 [gr-qc].
6. B. Freivogel, R. Jefferson, L. Kabir, and I.-S. Yang, “Geometry of the Infalling Causal Patch,” Phys. Rev. D91 no. 4, (2015) 044036, arXiv:1406.6043 [hep-th].
7. S. W. Hawking, “Particle creation by black holes,” Comm. Math. Phys. 43 no. 3, (1975) 199–220.
8. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 1984.
9. T. Jacobson, “Introductory Lectures on Black Hole Thermodynamics,”. http://www.physics.umd.edu/grt/taj/776b/lectures.pdf.
10. D. N. Page, “Information in black hole radiation,” Phys. Rev. Lett. 71 (1993) 3743–3746, arXiv:hep-th/9306083 [hep-th].
11. D. N. Page, “Time Dependence of Hawking Radiation Entropy,” JCAP 1309 (2013) 028, arXiv:1301.4995 [hep-th].
12. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” arXiv:1911.12333 [hep-th].
13. G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C930308 (1993) 284–296, arXiv:gr-qc/9310026 [gr-qc].
14. L. Susskind, “The World as a hologram,” J. Math. Phys. 36 (1995) 6377–6396, arXiv:hep-th/9409089 [hep-th].
15. R. Bousso, “The Holographic principle,” Rev. Mod. Phys. 74 (2002) 825–874, arXiv:hep-th/0203101 [hep-th].

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In a previous post, I mentioned that the firewall paradox could be phrased as a question about the existence of interior operators that satisfy the correct thermal correlation functions, namely

$\displaystyle \langle\Psi|\mathcal{O}(t,\mathbf{x})\tilde{\mathcal{O}}(t',\mathbf{x}')|\Psi\rangle =Z^{-1}\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t,\mathbf{x})\mathcal{O}(t'+i\beta/2,\mathbf{x}')\right]~, \ \ \ \ \ (1)$

where ${\tilde{\mathcal{O}}}$ and ${\mathcal{O}}$ operators inside and outside the black hole, respectively; cf. eqn. (2) here. In this short post, I’d like to review the basic argument leading up to this statement, following the original works [1,2].

Consider the eternal black hole in AdS as depicted in the following diagram, which I stole from [1]:

The blue line connecting the two asymptotic boundaries is the Cauchy slice on which we’ll construct our states, denoted ${\Sigma_I}$ in exterior region ${I}$ and ${\Sigma_{III}}$ in exterior region ${III}$. Note that, modulo possible UV divergences at the origin, either half serves as a complete Cauchy slice if we restrict our inquiries to the associated exterior region. But if we wish to reconstruct states in the interior (henceforth just ${II}$, since we don’t care about ${IV}$), then we need the entire slice. Pictorially, one can see this from the fact that only the left-moving modes from region ${I}$, and the right-moving modes from region ${III}$, cross the horizon into region ${II}$, but we need both left- and right-movers to have a complete mode decomposition.

To expand on this, imagine we proceed with the quantization of a free scalar field in region ${I}$. We need to solve the Klein-Gordon equation,

$\displaystyle \left(\square-m^2\right)\phi=\frac{1}{\sqrt{-g}}\,\partial_\mu\left( g^{\mu\nu}\sqrt{-g}\,\partial_\nu\phi\right)-m^2\phi=0 \ \ \ \ \ (2)$

on the AdS black brane background,

$\displaystyle \mathrm{d} s^2=\frac{1}{z^2}\left[-h(z)\mathrm{d} t^2+\mathrm{d} z^2+\mathrm{d}\mathbf{x}^2\right]~, \quad\quad h(z)\equiv1-\left(\frac{z}{z_0}\right)^d~. \ \ \ \ \ (3)$

where, in Poincaré coordinates, the asymptotic boundary is at ${z\!=\!0}$, and the horizon is at ${z\!=\!z_0}$. We work in ${(d+1)-}$spacetime dimensions, so ${\mathbf{x}}$ is a ${(d\!-\!1)}$-dimensional vector representing the transverse coordinates. Note that we’ve set the AdS radius to 1. Substituting the usual plane-wave ansatz

$\displaystyle f_{\omega,\mathbf{k}}(t,\mathbf{x},z)=e^{-i\omega t+i\mathbf{k}\mathbf{x}}\psi_{\omega,\mathbf{k}(z)} \ \ \ \ \ (4)$

into the Klein-Gordon equation results in a second order ordinary differential equation for the radial function ${\psi_{\omega,\mathbf{k}}(z)}$, and hence two linearly independent solutions. As usual, we then impose normalizable boundary conditions at infinity, which leaves us with a single linear combination for each ${(\omega,\mathbf{k})}$. Note that we do not impose boundary conditions at the horizon. Naïvely, one might have thought to impose ingoing boundary conditions there; however, as remarked in [1], this precludes the existence of real ${\omega}$. More intuitively, I think of this as simply the statement that the black hole is evaporating, so we should allow the possibility for outgoing modes as well. (That is, assuming a large black hole in AdS, the black hole is in thermal equilibrium with the surrounding environment, so the outgoing and ingoing fluxes are precisely matched, and it maintains constant size). The expression for ${\psi_{\omega,\mathbf{k}}(z)}$ is not relevant here; see [1] for more details.

We thus arrive at the standard expression of the (bulk) field ${\phi}$ in terms of creation and annihilation operators,

$\displaystyle \phi_I(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{\sqrt{2\omega}(2\pi)^d}\,\bigg[ a_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c.}\bigg]~, \ \ \ \ \ (5)$

where the creation/annihilation operators for the modes may be normalized with respect to the Klein-Gordon norm, so that

$\displaystyle [a_{\omega,\mathbf{k}},a^\dagger_{\omega',\mathbf{k}'}]=\delta(\omega-\omega')\delta^{d-1}(\mathbf{k}-\mathbf{k}')~. \ \ \ \ \ (6)$

Of course, a similar expansion holds for region ${III}$:

$\displaystyle \phi_{III}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{\sqrt{2\omega}(2\pi)^d}\,\bigg[\tilde a_{\omega,\mathbf{k}}\,g_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c.}\bigg]~, \ \ \ \ \ (7)$

where the mode operators ${\tilde a_{\omega,\mathbf{k}}}$ commute with all ${a_{\omega,\mathbf{k}}}$ by construction.

Now, what of the future interior, region ${II}$? Unlike the exterior regions, we no longer have any boundary condition to impose, since every Cauchy slice which crosses this region is bounded on both sides by a future horizon. Consequently, we retain both the linear combinations obtained from the Klein-Gordon equation, and hence have twice as many modes as in either ${I}$ or ${III}$—which makes sense, since the interior receives contributions from both exterior regions. Nonetheless, it may be a bit confusing from the bulk perspective, since any local observer would simply arrive at the usual mode expansion involving only a single set of creation/annihilation operators, and I don’t have an intuition as to how ${a_{\omega,\mathbf{k}}}$ and ${\tilde a_{\omega,\mathbf{k}}}$ relate vis-à-vis their commutation relations in this shared domain. However, the entire framework in which the interior is fed by two exterior regions is only properly formulated in AdS/CFT, in which — it is generally thought — the interior region emerges from the entanglement structure between the two boundaries, so I prefer to uplift this discussion to the CFT before discussing the interior region in detail. This avoids the commutation confusion above — since the operators live in different CFTs — and it was the next step in our analysis anyway. (Incidentally, appendix B of [1] performs the mode decomposition in all three regions explicitly for the case of Rindler space, which provides a nice concrete example in which one can get one’s hands dirty).

So, we want to discuss local bulk fields from the perspective of the boundary CFT. From the extrapolate dictionary, we know that local bulk operators become increasingly smeared over the boundary (in both space and time) the farther we move into the bulk. Thus in region ${I}$, we can construct the operator

$\displaystyle \phi^I_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\mathcal{O}_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathcal{O}^\dagger_{\omega,\mathbf{k}}f^*_{\omega,\mathbf{k}}(t,\mathbf{x},z)\bigg]~, \ \ \ \ \ (8)$

which, while a non-local operator in the CFT (constructed from local CFT operators ${\mathcal{O}_{\omega,\mathbf{k}}}$ which act as creation operators of light primary fields), behaves like a local operator in the bulk. Note that from the perspective of the CFT, ${z}$ is an auxiliary coordinate that simply parametrizes how smeared-out this operator is on the boundary.

As an aside, the critical difference between (8) and the more familiar HKLL prescription [3-5] is that the former is formulated directly in momentum space, while the latter is defined in position space as

$\displaystyle \phi_{\mathrm{CFT}}(t,\mathbf{x},z)=\int\!\mathrm{d} t'\mathrm{d}^{d-1}\mathbf{x}'\,K(t,\mathbf{x},z;t',\mathbf{x}')\mathcal{O}(t',\mathbf{x}')~, \ \ \ \ \ (9)$

where the integration kernel ${K}$ is known as the “smearing function”, and depends on the details of the spacetime. To solve for ${K}$, one performs a mode expansion of the local bulk field ${\phi}$ and identifies the normalizable mode with the local bulk operator ${\mathcal{O}}$ in the boundary limit. One then has to invert this relation to find the bulk mode operator, and then insert this into the original expansion of ${\phi}$. The problem now is that to identify ${K}$, one needs to swap the order of integration between position and momentum space, and the presence of the horizon results in a fatal divergence that obstructs this maneuver. As discussed in more detail in [1] however, working directly in momentum space avoids this technical issue. But the basic relation “smeared boundary operators ${\longleftrightarrow}$ local bulk fields” is the same.

Continuing, we have a similar bulk-boundary relation in region ${III}$, in terms of operators ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ living in the left CFT:

$\displaystyle \phi^{III}_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\tilde{\mathcal{O}}_{\omega,\mathbf{k}}\,f_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\tilde{\mathcal{O}}^\dagger_{\omega,\mathbf{k}}f^*_{\omega,\mathbf{k}}(t,\mathbf{x},z)\bigg]~. \ \ \ \ \ (10)$

Note that even though I’ve used the same coordinate labels, ${t}$ runs backwards in the left wedge, so that ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ plays the role of the creation operator here. From the discussion above, the form of the field in the black hole interior is then

$\displaystyle \phi^{II}_{\mathrm{CFT}}(t,\mathbf{x},z)=\int_{\omega>0}\frac{\mathrm{d}\omega\mathrm{d}^{d-1}\mathbf{k}}{(2\pi)^d}\,\bigg[\mathcal{O}_{\omega,\mathbf{k}}\,g^{(1)}_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\tilde{\mathcal{O}}_{\omega,\mathbf{k}}g^{(2)}_{\omega,\mathbf{k}}(t,\mathbf{x},z)+\mathrm{h.c}\bigg]~, \ \ \ \ \ (11)$

where ${\mathcal{O}_{\omega,\mathbf{k}}}$ and ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ are the (creation/annihilation operators for the) boundary modes in the right and left CFTs, respectively. The point is that in order to construct a local field operator behind the horizon, both sets of modes — the left-movers ${\mathcal{O}_{\omega,\mathbf{k}}}$ from ${I}$ and the right-movers ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$ from ${III}$ — are required. In the eternal black hole considered above, the latter originate in the second copy of the CFT. But in the one-sided case, we would seem to have only the left-movers ${\mathcal{O}_{\omega,\mathbf{k}}}$, hence we arrive at the crucial question: for a one-sided black hole — such as that formed from collapse in our universe — what are the interior modes ${\tilde{\mathcal{O}}_{\omega,\mathbf{k}}}$? Equivalently: how can we represent the black hole interior given access to only one copy of the CFT?

To answer this question, recall that the thermofield double state,

$\displaystyle |\mathrm{TFD}\rangle=\frac{1}{\sqrt{Z_\beta}}\sum_ie^{-\beta E_i/2}|E_i\rangle\otimes|E_i\rangle~, \ \ \ \ \ (12)$

is constructed so that either CFT appears exactly thermal when tracing out the other side, and that this well-approximates the late-time thermodynamics of a large black hole formed from collapse. That is, the exterior region will be in the Hartle-Hawking vacuum (which is to Schwarzschild as Rindler is to Minkowski), with the temperature ${\beta^{-1}}$ of the CFT set by the mass of the black hole. This implies that correlation functions of operators ${\mathcal{O}}$ in the pure state ${|\mathrm{TFD}\rangle}$ may be computed as thermal expectation values in their (mixed) half of the total Hilbert space, i.e.,

$\displaystyle \langle\mathrm{TFD}|\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}(t_n,\mathbf{x}_n)|\mathrm{TFD}\rangle =Z^{-1}_\beta\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}(t_n,\mathbf{x}_n)\right]~. \ \ \ \ \ (13)$

The same fundamental relation remains true in the case of the one-sided black hole as well: given the Hartle-Hawking state representing the exterior region, we can always obtain a purification such that expectation values in the original, thermal state are equivalent to standard correlators in the “fictitious” pure state, by the same doubling formalism that yielded the TFD. (Of course, the purification of a given mixed state is not unique, but as pointed out in [2] “the correct way to pick it, assuming that expectation values [of the operators] are all the information we have, is to pick the density matrix which maximizes the entropy.” That is, we pick the purification such that the original mixed state is thermal, i.e., ${\rho\simeq Z^{-1}_\beta e^{-\beta H}}$ up to ${1/N^2}$ corrections. The reason this is the “correct” prescription is that it’s the only one which does not impose additional constraints.) Thus (13) can be generally thought of as the statement that operators in an arbitrary pure state have the correct thermal expectation values when restricted to some suitably mixed subsystem (e.g., the black hole exterior dual to a single CFT).

Now, what if we wish to compute a correlation function involving operators across the horizon, e.g., ${\langle\mathcal{O}\tilde{\mathcal{O}}\rangle}$? In the two-sided case, we can simply compute this correlator in the pure state ${|\mathrm{TFD}\rangle}$. But in the one-sided case, we only have access to the thermal state representing the exterior. Thus we’d like to know how to compute the correlator using only the available data in the CFT corresponding to region ${I}$. In order to do this, we re-express all operators ${\tilde{\mathcal{O}}}$ appearing in the correlator with analytically continued operators ${\mathcal{O}}$ via the KMS condition, i.e., we make the replacement

$\displaystyle \tilde{\mathcal{O}}(t,\mathbf{x}) \longrightarrow \mathcal{O}(t+i\beta/2,\mathbf{x})~. \ \ \ \ \ (14)$

This is essentially the usual statement that thermal Green functions are periodic in imaginary time; see [1] for details. This relationship allows us to express the desired correlator as

$\displaystyle \langle\mathrm{TFD}|\mathcal{O}(t_1,\mathbf{x}_1)\ldots\tilde{\mathcal{O}}(t_n,\mathbf{x}_n)|\mathrm{TFD}\rangle =Z^{-1}_\beta\mathrm{tr}\left[e^{-\beta H}\mathcal{O}(t_1,\mathbf{x}_1)\ldots\mathcal{O}_{\omega,\mathbf{k}}(t_n+i\beta/2,\mathbf{x}_n)\right]~, \ \ \ \ \ (15)$

which is precisely eqn. (2) in our earlier post, cf. the two-point function (1) above. Note the lack of tilde’s on the right-hand side: this thermal expectation value can be computed entirely in the right CFT.

If the CFT did not admit operators which satisfy the correlation relation (15), it would imply a breakdown of effective field theory across the horizon. Alternatively, observing deviations from the correct thermal correlators would allow us to locally detect the horizon, in contradiction to the equivalence principle. In this sense, this expression may be summarized as the statement that the horizon is smooth. Thus, for the CFT to represent a black hole with no firewall, it must contain a representation of interior operators ${\tilde{\mathcal{O}}}$ with the correct behaviour inside low-point correlators. This last qualifier hints at the state-dependent nature of these so-called “mirror operators”, which I’ve discussed at length elsewhere [6].

References

[1]  K. Papadodimas and S. Raju, “An Infalling Observer in AdS/CFT,” JHEP 10 (2013) 212,arXiv:1211.6767 [hep-th].

[2]  K. Papadodimas and S. Raju, “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity,” Phys. Rev. D89 no. 8, (2014) 086010, arXiv:1310.6335 [hep-th].

[3]  A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev. D74 (2006) 066009, arXiv:hep-th/0606141 [hep-th].

[4]  A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Local bulk operators in AdS/CFT: A Boundary view of horizons and locality,” Phys. Rev. D73 (2006) 086003,arXiv:hep-th/0506118 [hep-th].

[5]  A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, “Local bulk operators in AdS/CFT: A Holographic description of the black hole interior,” Phys. Rev. D75 (2007) 106001,arXiv:hep-th/0612053 [hep-th]. [Erratum: Phys. Rev.D75,129902(2007)].

[6]  R. Jefferson, “Comments on black hole interiors and modular inclusions,” SciPost Phys. 6 no. 4, (2019) 042, arXiv:1811.08900 [hep-th].

## Free energy, variational inference, and the brain

In several recent posts, I explored various ideas that lie at the interface of physics, information theory, and machine learning:

• We’ve seen, à la Jaynes, how the concepts of entropy in statistical thermodynamics and information theory are unified, perhaps the quintessential manifestation of the intimate relationship between the two.
• We applied information geometry to Boltzmann machines, which led us to the formalization of “learning” as a geodesic in the abstract space of machines.
• In the course of introducing VAEs, we saw that the Bayesian inference procedure can be understood as a process which seeks to minimizes the variational free energy, which encodes the divergence between the approximate and true probability distributions.
• We examined how the (dimensionless) free energy serves as a generating function for the cumulants from probability theory, which manifest as the connected Green functions from quantum field theory.
• We also showed how the cumulants from hidden priors control the higher-order interactions between visible units in an RBM, which underlies their representational power.
• Lastly, we turned a critical eye towards the analogy between deep learning and the renormalization group, through a unifying Bayesian language in which UV degrees of freedom correspond to hidden variables over which a low-energy observer must marginalize.

Collectively, this led me to suspect that ideas along these lines — in particular, the link between variational Bayesian inference and free energy minimization in hierarchical models — might provide useful mathematical headway in our attempts to understand learning and intelligence in both minds and machines. It turns out this has been explored before by many scientists; at least in the context of biological brains, the most well-known is probably the approach by the neuroscientist Karl Friston, which provides a convenient starting point for exploring these ideas.

The aptly-named free energy principle (for the brain) is elaborated upon in a series of about ten papers spanning as many years. I found [1-5] most helpful, but insofar as a great deal of text is copied verbatim (yes, really; never trust the h-index) it doesn’t really matter which one you read. I’m going to mostly draw from [3], because it seems the earliest in which the basic idea is fleshed-out completely. Be warned however that the notation varies slightly from paper to paper, and I find his distinction between states and parameters rather confusingly fuzzy; but we’ll make this precise below.

The basic idea is actually quite simple, and proceeds from the view of the brain as a Bayesian inference machine. In a nutshell, the job of the brain is to infer, as accurately as possible, the probability distribution representing the world (i.e., to build a model that best accords with sensory inputs). In a sense, the brain itself is a probabilistic model in this framework, so the goal is to bring this internal model of the world in line with the true, external one. But this is exactly the same inference procedure we’ve seen before in the context of VAEs! Thus the free energy principle is just the statement that the brain minimizes the variational free energy between itself (that is, its internal, approximate model) and its sensory inputs—or rather, the true distribution that generates them.

To elucidate the notation involved in formulating the principle, we can make an analogy with VAEs. In this sense, the goal of the brain is to construct a map between our observations (i.e., sensory inputs ${x}$) and the underlying causes (i.e., the environment state ${z}$). By Bayes’ theorem, the joint distribution describing the model can be decomposed as

$\displaystyle p_\theta(x,z)=p_\theta(x|z)p(z)~. \ \ \ \ \ (1)$

The first factor on the right-hand side is the likelihood of a particular sensory input ${x}$ given the current state of the environment ${z}$, and plays the role of the decoder in this analogy, while the second factor is the prior distribution representing whatever foreknowledge the system has about the environment. The subscript ${\theta}$ denotes the variational or “action parameters” of the model, so named because they parametrize the action of the brain on its substrate and surroundings. That is, the only way in which the system can change the distribution is by acting to change its sensory inputs. Friston denotes this dependency by ${x(\theta)}$ (with different variables), but as alluded above, I will keep to the present notation to avoid conflating state/parameter spaces.

Continuing this analogy, the encoder ${p_\theta(z|x)}$ is then a map from the space of sensory inputs ${X}$ to the space of environment states ${Z}$ (as modelled by the brain). As in the case of VAEs however, this is incomputable in practice, since we (i.e., the brain) can’t evaluate the partition function ${p(x)=\sum_zp_\theta(x|z)p(z)}$. Instead, we construct a new distribution ${q_\phi(z|x)}$ for the conditional probability of environment states ${z}$ given a particular set of sensory inputs ${x}$. The variational parameters ${\phi}$ for this ensemble control the precise hamiltonian that defines the distribution, i.e., the physical parameters of the brain itself. Depending on the level of resolution, these could represent, e.g., the firing status of all neurons, or the concentrations of neurotransmitters (or the set of all weights and biases in the case of artificial neural nets).

Obviously, the more closely ${q_\phi(z|x)}$ approximates ${p_\theta(z|x)}$, the better our representation — and hence, the brain’s predictions — will be. As we saw before, we quantify this discrepancy by the Kullback-Leibler divergence

$\displaystyle D_z(q_\phi(z|x)||p_\theta(z|x))=\sum_zq_\phi(z|x)\ln\frac{q_\phi(z|x)}{p_\theta(z|x)}~, \ \ \ \ \ (2)$

which we can re-express in terms of the variational free energy

\displaystyle \begin{aligned} F_{q|}&=-\langle\ln p_\theta(x|z)\rangle_{q|}+D_z(q_\phi(z|x)||p(z))\\ &=-\sum_zq_\phi(z|x)\ln\frac{p_\theta(x,z)}{q_\phi(z|x)} =\langle E_{p|}\rangle_{q|}-S_{q|}~, \end{aligned} \ \ \ \ \ (3)

where the subscripts ${p|,q|}$ denote the conditional distributions ${p_\theta(z|x)}$, ${q_\phi(z|x)}$. On the far right-hand side, ${E_{p|}=-\ln p_\theta(x,z)}$ is the energy or hamiltonian for the ensemble ${p_\theta(z|x)}$ (with partition function ${Z=p(x)}$), and ${S_{q|}=-\int\!\mathrm{d} z\,q_\phi(z|x)\ln q_\phi(z|x)}$ is the entropy of ${q_\phi(z|x)}$ (see the aforementioned post for details).

However, at this point we must diverge from our analogy with VAEs, since what we’re truly after is a model of the state of the world which is independent of our current sensory inputs. Consider that from a selectionist standpoint, a brain that changes its environmental model when a predator temporarily moves out of sight is less likely to pass on the genes for its construction! Said differently, a predictive model of reality will be more successful when it continues to include the moon, even when nobody looks at it. Thus instead of ${q_\phi(z|x)}$, we will compare ${p_\theta(x|z)}$ with the ensemble density ${q_\lambda(z)}$, where — unlike in the case of ${p(x)}$ or ${p(z)}$ — we have denoted the variational parameters ${\lambda}$ explicitly, since they will feature crucially below. Note that ${\lambda}$ is not the same as ${\theta}$ (and similarly, whatever parameters characterize the marginals ${p(x)}$, ${p(z)}$ cannot be identified with ${\theta}$). One way to see this is by comparison with our example of renormalization in deep networks, where the couplings in the joint distribution (here, ${\phi}$ in ${q_\phi(x,z)}$) get renormalized after marginalizing over some degrees of freedom (here, ${\lambda}$ in ${q_\lambda(z)}$, after marginalizing over all possible sensory inputs ${x}$). Friston therefore defines the variational free energy as

\displaystyle \begin{aligned} \mathcal{F}_q&=-\langle\ln p_\theta(x|z)\rangle_q+D_z(q_\lambda(z)||p(z))\\ &=-\sum_zq_\lambda(z)\ln\frac{p_\theta(x,z)}{q_\lambda(z)} =\langle E_{p|}\rangle_{q}-S_{q}~, \end{aligned} \ \ \ \ \ (4)

where we have used a curly ${\mathcal{F}}$ to distinguish this from ${F}$ above, and note that the subscript ${q}$ (no vertical bar) denotes that expectation values are computed with respect to the distribution ${q_\lambda(z)}$. The first equality expresses ${\mathcal{F}_q}$ as the log-likelihood of sensory inputs given the state of the environment, minus an error term that quantifies how far the brain’s internal model of the world ${q_\lambda(z)}$ is from the model consistent with our observations, ${p(z)}$, cf. (1). Equivalent, comparing with (2) (with ${q_\lambda(z)}$ in place of ${q_\phi(z|x)}$), we’re interested in the Kullback-Leibler divergence between the brain’s model of the external world, ${q_\lambda(z)}$, and the conditional likelihood of a state therein given our sensory inputs, ${p_\theta(z|x)}$. Thus we arrive at the nutshell description we gave above, namely that the principle is to minimize the difference between what is and what we think there is. As alluded above, there is a selectionist argument for this principle, namely that organisms whose beliefs accord poorly with reality tend not to pass on their genes.

As an aside, it is perhaps worth emphasizing that both of these variational free energies are perfectly valid: unlike the Helmholtz free energy, which is uniquely defined, one can define different variational free energies depending on which ensembles one wishes to compare, provided it admits an expression of the form ${\langle E\rangle-S}$ for some energy ${E}$ and entropy ${S}$ (in case it wasn’t clear by now, we’re working with the dimensionless or reduced free energy, equivalent to setting ${\beta=1}$; the reason for this general form involves a digression on Legendre transforms). Comparing (4) and (3), one sees that the difference in this case is simply a difference in entropies and expectation values with respect to prior ${q_\lambda(z)}$ vs. conditional distributions ${q_\phi(z|x)}$ (which makes sense, since all we did was replace the latter by the former in our first definition).

Now, viewing the brain as an inference machine means that it seeks to optimize its predictions about the world, which in this context, amounts to minimizing the free energy by varying the parameters ${\theta,\,\lambda}$. As explained above, ${\theta}$ corresponds to the actions the system can take to alter its sensory inputs. From the first equality in (4), we see that the dependence on the action parameters is entirely contained in the log-likelihood of sensory inputs: the second, Kullback-Leibler term contains only priors (cf. our discussion of gradient descent in VAEs). This, optimizing the free energy with respect to ${\theta}$ means that the system will act in such a way as to fulfill its expectations with regards to sensory inputs. Friston neatly summarizes this philosophy as the view that “we may not interact with the world to maximize our reward but simply to ensure it behaves as we think it should” [3]. While this might sound bizarre at first glance, the key fact to bear in mind is that the system is limited in the actions it can perform, i.e., in its ability to adapt. In other words, a system with low free energy is per definition adapting well to changes in its environment or its own internal needs, and therefore is positively selected for relative to systems whose ability to model and adapt to their environment is worse (higher free energy).

What about optimization with respect to the other set of variational parameters, ${\lambda}$? As mentioned above, these correspond to the physical parameters of the system itself, so this corresponds to adjusting the brain’s internal parameters — connection strengths, neurotransmitter levels, etc. — to ensure that our perceptions are as accurate as possible. By applying Bayes rule to the joint distribution ${p_\theta(x,z)}$, we can re-arrange the expression for the free energy to isolate this dependence in a single Kullback-Leibler term:

$\displaystyle \mathcal{F}_q=-\ln p_\theta(x)+D_z\left( q_\lambda(z)||p_\theta(z|x)\right)~. \ \ \ \ \ (5)$

where we have used the fact that ${\langle \ln p_\theta(x)\rangle_q=\ln p_\theta(x)}$. This form of the expression shows clearly that minimization with respect to ${\lambda}$ directly corresponds to minimizing the Kullback-Leibler divergence between the brain’s internal model of the world, ${q_\lambda(z)}$, and the posterior probability of the state giving rise to its sensory inputs, ${p_\theta(z|x)}$. That is, in the limit where the second, Kullback-Leibler term vanishes, we are correctly modelling the causes of our sensory inputs. The selectionist interpretation is that systems which are less capable of accurately modelling their environment by correctly adjusting internal, “perception parameters” ${\lambda}$ will have higher free energy, and hence will be less adept in bringing their perceptions in line with reality.

Thus far everything is quite abstract and rather general. But things become really interesting when we apply this basic framework to hierarchical models with both forward and backwards connections — such as the cerebral cortex — which leads to “recurrent dynamics that self-organize to suppress free energy or prediction error, i.e., recognition dynamics” [3]. In fact, Friston makes the even stronger argument that it is precisely the inability to invert the recognition problem that necessitates backwards (as opposed to purely feed-forwards) connections. In other words, the selectionist pressure to accurately model the (highly non-linear) world requires that brains evolve top-down connections from higher to lower cortical layers. Let’s flesh this out in a bit more detail.

Recall that ${Z}$ is the space of environmental states as modelled by the brain. Thus we can formally associate the encoder, ${p_\theta(z|x)}$, with forwards connections, which propagate sensory data up the cortical hierarchy; Friston refers to this portion as the recognition model. That is, the recognition model should take a given data point ${x}$, and return the likelihood of a particular cause (i.e., world-state) ${z}$. In general however, the map from causes to sensory inputs — captured by the so-called generative model ${p_\theta(x|z)}$ — is highly non-linear, and the brain must essentially invert this map to find contextually invariant causes (e.g., the continued threat of a predator even when it’s no longer part of our immediate sensory input). This is the intractable problem of computing the partition function above, the workaround for which is to instead postulate an approximate recognition model ${q_\lambda(z)}$, whose parameters ${\lambda}$ are encoded in the forwards connections. The role of the generative model ${p_\theta(x|z)}$ is then to modulate sensory inputs (or their propagation and processing) based on the prevailing belief about the environment’s state, the idea being that these effects are represented in backwards (and lateral) connections. Therefore, the role of these backwards or top-down connections is to modulate forwards or bottom-up connections, thereby suppressing prediction error, which is how the brain operationally minimizes its free energy.

The punchline is that backwards connections are necessary for general perception and recognition in hierarchical models. As mentioned above, this is quite interesting insofar as it offers, on the one hand, a mathematical explanation for the cortical structure found in biological brains, and on the other, a potential guide to more powerful, neuroscience-inspired artificial intelligence.

There are however a couple technical exceptions to this claim of necessity worth mentioning, which is why I snuck in the qualifier “general” in the punchline above. If the abstract generative model can be inverted exactly, then there’s no need for (expensive and time-consuming) backwards connections, because one can obtain a perfectly suitable recognition model that reliably predicts the state of the world given sensory inputs, using a purely feed-forward network. Mathematically, this corresponds to simply taking ${q_\lambda(z)=p_\theta(z|x)}$ in (4) (i.e., zero Kullback-Leibler divergence (2)), whereupon the free energy reduces to the negative log-likelihood of sensory inputs,

$\displaystyle \mathcal{F}_{p}=-\ln p(x)~, \ \ \ \ \ (6)$

where we have used the fact that ${\langle\ln p(x)\rangle_{p|}=\ln p(x)}$. Since real-world models are generally non-linear in their inputs however, invertibility is not something one expects to encounter in realistic inference machines (i.e., brains). Indeed, our brains evolved under strict energetic and space constraints; there simply isn’t enough processing power to brute-force the problem by using dedicated feed-forward networks for all our recognition needs. The other important exception is when the recognition process is purely deterministic. In this case one replaces ${q_\lambda(z)}$ by a Kronecker delta function ${\delta(z(x)-x)}$, so that upon performing the summation, the inferred state ${z}$ becomes a deterministic function of the inputs ${x}$. Then the second expression for ${\mathcal{F}}$ in (4) becomes the negative log-likelihood of the joint distribution

$\displaystyle \mathcal{F}_\delta=-\ln p_\theta(x,z(x)) =-\ln p_\theta(x|z(x))-\ln p(z(x))~, \ \ \ \ \ (7)$

where we have used the fact that ${\ln\delta(0)=0}$. Note that the invertible case, (6), corresponds to maximum likelihood estimation (MLE), while the deterministic case (7) corresponds to so-called maximum a posteriori estimation (MAP), the only difference being that the latter includes a weighting based on the prior distribution ${p(z(x))}$. Neither requires the conditional distribution ${p_\theta(z|x)}$, and so skirts the incomputability issue with the path integral above. The reduction to these familiar machine learning metrics for such simple models is reasonable, since only in relatively contrived settings does one ever expect deterministic/invertible recognition.

In addition to motivating backwards connections, the hierarchical aspect is important because it allows the brain to learn its own priors through a form of empirical Bayes. In this sense, the free energy principle is essentially an elegant (re)formulation of predictive coding. Recall that when we introduced the generative model in the form of the decoder ${p_\theta(x|z)}$ in (1), we also necessarily introduced the prior distribution ${p(z)}$: the liklihood of a particular sensory input ${x}$ given (our internal model of) the state of the environment (i.e., the cause) ${z}$ only makes sense in the context of the prior distribution of causes. Where does this prior distribution come from? In artificial models, we can simply postulate some (e.g., Gaussian or informative) prior distribution and proceed to train the model from there. But a hierarchical model like the brain enables a more natural option. To illustrate the basic idea, consider labelling the levels in such a cortical hierarchy by ${i\in{0,\ldots,n}}$, where 0 is the bottom-most layer and ${n}$ is the top-most layer. Then ${x_i}$ denotes sensory data at the corresponding layer; i.e., ${x_0}$ corresponds to raw sensory inputs, while ${x_n}$ corresponds to the propagated input signals after all previous levels of processing. Similarly, let ${z_i}$ denote the internal model of the state of the world assembled (or accessible at) the ${i^\mathrm{th}}$ layer. Then

$\displaystyle p(z_i)=\sum_{z_{i-1}}p(z_i|z_{i-1})p(z_{i-1})~, \ \ \ \ \ (8)$

i.e., the prior distribution ${p(z_i)}$ implicitly depends on the knowledge of the state at all previous levels, analogous to how the IR degrees of freedom implicitly depend on the marginalized UV variables. The above expression can be iterated recursively until we reach ${p(z_0)}$. For present purposes, this can be identified with ${p(x_0)}$, since at the bottom-most level of the hierarchy, there’s no difference between the raw sensory data and the inferred state of the world (ignoring whatever intralayer processing might take place). In this (empirical Bayesian) way, the brain self-consistently builds up higher priors from states at lower levels.

The various works by Friston and collaborators go into much more detail, of course; I’ve made only the crudest sketch of the basic idea here. In particular, one can make things more concrete by examining the neural dynamics in such models, which some of these works seek to explore via something akin to a mean field theory (MFT) approach. I’d originally hoped to have time to dive into this in detail, but a proper treatment will have to await another post. Suffice to say however that an approach along the lines of the free energy principle may provide an elegant formulation which, as in the other topics mentioned at the beginning of this post, allows us to apply ideas from theoretical physics to understand the structure and dynamics of neural networks, and may even prove a fruitful mathematical framework for both theoretical neuroscience and (neuro-inspired) artificial intelligence.

References:

[1] K. Friston, “Learning and inference in the brain,” Neural Networks (2003) .

[2] K. Frison, “A theory of cortical responses,” Phil. Trans. R. Soc. B (2005) .

[3] K. Friston, J. Kilner, and L. Harrison, “A free energy principle for the brain,” J. Physiology (Paris) (2006) .

[4] K. J. Friston and K. E. Stephan, “Free-energy and the brain,” Synthese (2007) .

[5] K. Friston, “The free-energy principle: a unified brain theory?,” Nature Rev. Neuro. (2010) .

Posted in Minds & Machines | 3 Comments

## Deep learning and the renormalization group

In recent years, a number of works have pointed to similarities between deep learning (DL) and the renormalization group (RG) [1-7]. This connection was originally made in the context of certain lattice models, where decimation RG bears a superficial resemblance to the structure of deep networks in which one marginalizes over hidden degrees of freedom. However, the relation between DL and RG is more subtle than has been previously presented. The “exact mapping” put forth by [2], for example, is really just a formal analogy that holds for essentially any hierarchical model! That’s not to say there aren’t deeper connections between the two: in my earlier post on RBMs for example, I touched on how the cumulants encoding UV interactions appear in the renormalized couplings after marginalizing out hidden degrees of freedom, and we’ll go into this in much more detail below. But it’s obvious that DL and RG are functionally distinct: in the latter, the couplings (i.e., the connection or weight matrix) are fixed by the relationship between the hamiltonian at different scales, while in the former, these connections are dynamically altered in the training process. There is, in other words, an important distinction between structure and dynamics which seems to have been overlooked. Understanding both these aspects is required to truly understand why deep learning “works”, but “learning” itself properly refers to the latter.

That said, structure is the first step to dynamics, so I wanted to see how far one could push the analogy. To that end, I started playing with simple Gaussian/Bernoulli RBMs, to see whether understanding the network structure — in particular, the appearance of hidden cumulants, hence the previous post in this two-part sequence — would shed light on, e.g., the hierarchical feature detection observed in certain image recognition tasks, the propagation of structured information more generally, or the relevance of criticality to both deep nets and biological brains. To really make the RG analogy precise, one would ideally like a beta function for the network, which requires a recursion relation for the couplings. So my initial hope was to derive an expression for this in terms of the cumulants of the marginalized neurons, and thereby gain some insight into how correlations behave in these sorts of hierarchical networks.

To start off, I wanted a simple model that would be analytically solvable while making the analogy with decimation RG completely transparent. So I began by considering a deep Boltzmann machine (DBM) with three layers: a visible layer of Bernoulli units ${x_i}$, and two hidden layers of Gaussian units ${y_i,z_i}$. The total energy function is

\displaystyle \begin{aligned} H(x,y,z)&=-\sum_{i=1}^na_ix_i+\frac{1}{2}\sum_{j=1}^my_j^2+\frac{1}{2}\sum_{k=1}^pz_k^2-\sum_{ij}A_{ij}x_iy_j-\sum_{jk}B_{jk}y_jz_k\\ &=-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\left( \mathbf{y}^\mathrm{T}\mathbf{y}+\mathbf{z}^\mathrm{T}\mathbf{z}\right)-\mathbf{x}^\mathrm{T} A\,\mathbf{y}-\mathbf{y}^\mathrm{T} B\,\mathbf{z}~, \end{aligned} \ \ \ \ \ (1)

where on the second line I’ve switched to the more convenient vector notation; the dot product between vectors is implicit, i.e., ${\mathbf{a}\,\mathbf{x}=\mathbf{a}\cdot\mathbf{x}}$. Note that there are no intra-layer couplings, and that I’ve stacked the layers so that the visible layer ${x}$ is connected only to the intermediate hidden layer ${y}$, which in turn is connected only to the final hidden layer ${z}$. The connection to RG will be made by performing sequential marginalizations over first ${z}$, and then ${y}$, so that the flow from UV to IR is ${z\rightarrow y\rightarrow x}$. There’s an obvious Bayesian parallel here: we low-energy beings don’t have access to complete information about the UV, so the visible units are naturally identified with IR degrees of freedom, and indeed I’ll use these terms interchangeably throughout.

The joint distribution function describing the state of the machine is

$\displaystyle p(x,y,z)=Z^{-1}e^{-\beta H(x,y,z)}~, \quad\quad Z[\beta]=\prod_{i=1}^n\sum_{x_i=\pm1}\int\!\mathrm{d}^my\mathrm{d}^pz\,e^{-\beta H(x,y,z)}~, \ \ \ \ \ (2)$

where ${\int\!\mathrm{d}^my=\int\!\prod_{i=1}^m\mathrm{d} y_i}$, and similarly for ${z}$. Let us now consider sequential marginalizations to obtain ${p(x,y)}$ and ${p(x)}$. In Bayesian terms, these distributions characterize our knowledge about the theory at intermediate- and low-energy scales, respectively. The first of these is

$\displaystyle p(x,y)=\int\!\mathrm{d}^pz\,p(x,y,z)=Z^{-1}e^{-\beta\,\left(-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\mathbf{x}^\mathrm{T} A\,\mathbf{y}\right)}\int\!\mathrm{d}^pz\,e^{-\beta\mathbf{z}^\mathrm{T}\mathbf{z}+\beta\mathbf{y}^\mathrm{T} B\,\mathbf{z}}~. \ \ \ \ \ (3)$

In order to establish a relationship between couplings at each energy scale, we then define the hamiltonian on the remaining, lower-energy degrees of freedom ${H(x,y)}$ such that

$\displaystyle p(x,y)=Z^{-1}e^{-\beta H(x,y)}~, \ \ \ \ \ (4)$

which implies

$\displaystyle H(x,y)=-\frac{1}{\beta}\ln\int\!\mathrm{d}^pz\,e^{-\beta H(x,y,z)} =-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\mathbf{x}^\mathrm{T} A\,\mathbf{y} -\frac{1}{\beta}\ln\int\!\mathrm{d}^pz\,e^{-\beta\mathbf{z}^\mathrm{T}\mathbf{z}+\beta\mathbf{y}^\mathrm{T} B\,\mathbf{z}}~. \ \ \ \ \ (5)$

This is a simple multidimensional Gaussian integral:

$\displaystyle \int\!\mathrm{d}^pz\,\mathrm{exp}\left(-\frac{1}{2}\mathbf{z}^\mathrm{T} M\,\mathbf{z}+J^\mathrm{T}\mathbf{z}\right) =\sqrt{\frac{(2\pi)^p}{|M|}}\,\mathrm{exp}\left(\frac{1}{2}J^\mathrm{T} M^{-1} J\right)~, \ \ \ \ \ (6)$

where in the present case ${M=\beta\mathbf{1}}$ and ${J=\beta B^\mathrm{T}\mathbf{y}}$. We therefore obtain

\displaystyle \begin{aligned} H(x,y)&=-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\mathbf{x}^\mathrm{T} A\,\mathbf{y} -\frac{1}{\beta}\ln\sqrt{\frac{(2\pi)^p}{\beta}}\exp\left(\frac{\beta}{2}\mathbf{y}^\mathrm{T} BB^\mathrm{T}\mathbf{y}\right)\\ &=f(\beta)-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T} \left(\mathbf{1}-B'\right)\mathbf{y}-\mathbf{x}^\mathrm{T} A\,\mathbf{y}~, \end{aligned} \ \ \ \ \ (7)

where we have defined

$\displaystyle f(\beta)\equiv\frac{1}{2\beta}\ln\frac{\beta}{(2\pi)^p} \qquad\mathrm{and}\qquad B_{ij}'\equiv\sum_{\mu=1}^pB_{i\mu}B_{\mu j}~. \ \ \ \ \ (8)$

The key point to note is that the interactions between the intermediate degrees of freedom ${y}$ have been renormalized by an amount proportional to the coupling with the UV variables ${z}$. And indeed, in the context of deep neural nets, the advantage of hidden units is that they encode higher-order interactions through the cumulants of the associated prior. To make this connection explicit, consider the prior distribution of UV variables

$\displaystyle q(z)=\sqrt{\frac{\beta}{(2\pi)^p}}\,\mathrm{exp}\left(-\frac{1}{2}\beta \,\mathbf{z}^\mathrm{T} \mathbf{z}\right)~. \ \ \ \ \ (9)$

The cumulant generating function for ${\mathbf{z}}$ with respect to this distribution is then

$\displaystyle K_{z}(t)=\ln\langle e^{\mathbf{t}\mathbf{z}}\rangle =\ln\sqrt{\frac{\beta}{(2\pi)^p}}\int\!\mathrm{d}^pz\,\mathrm{exp}\left(-\frac{1}{2}\beta \mathbf{z}^\mathrm{T} \mathbf{z}+\mathbf{t}\mathbf{z}\right) =\frac{1}{2\beta}\mathbf{t}\mathbf{t}^\mathrm{T}~, \ \ \ \ \ (10)$

cf. eqn. (4) in the previous post. So by choosing ${\mathbf{t}=\beta\mathbf{y}^\mathrm{T} B}$, we have

$\displaystyle K_{z}\left(\beta\mathbf{y}^\mathrm{T} B\right)=\frac{\beta}{2}\mathbf{y}^\mathrm{T} BB^\mathrm{T}\mathbf{y} \ \ \ \ \ (11)$

and may therefore express (7) as

$\displaystyle H(x,y)=f(\beta)-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\frac{1}{\beta}K_{z}\left(\beta\mathbf{y}^\mathrm{T} B\right)~. \ \ \ \ \ (12)$

From the cumulant expansion in the aforementioned eqn. (4), in which the ${n^\mathrm{th}}$ cumulant is ${\kappa_n=K_X^{(n)}(t)\big|_{t=0}}$, we then see that the effect of the marginalizing out UV (i.e., hidden) degrees of freedom is to induce higher-order couplings between the IR (i.e., visible) units, with the coefficients of the interaction weighted by the associated cumulant:

\displaystyle \begin{aligned} H(x,y)&=f(\beta)-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\frac{1}{\beta}\left(\kappa_1\mathbf{t}+\frac{\kappa_2}{2}\mathbf{t}^2+\ldots\right)\\ &=f(\beta)-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\mathbf{y}-\kappa_1\mathbf{y}^\mathrm{T} B -\frac{\kappa_2}{2}\beta\mathbf{y}^\mathrm{T} BB^\mathrm{T} \mathbf{y}-\ldots~. \end{aligned} \ \ \ \ \ (13)

For the Gaussian prior (9), one immediately sees from (10) that all cumulants except for ${\kappa_2=1/\beta}$ (the variance) vanish, whereupon (13) reduces to (7) above.

Now, let’s repeat this process to obtain the marginalized distribution of purely visible units ${p(x)}$. In analogy with Wilsonian RG, this corresponds to further lowering the cutoff scale in order to obtain a description of the theory in terms of low-energy degrees of freedom that we can actually observe. Hence, tracing out ${y}$, we have

$\displaystyle p(x)=\int\!\mathrm{d}\mathbf{y}\,p(x,y)=Z^{-1}e^{-\beta\,\left( f-\mathbf{a}\mathbf{x}\right)}\int\!\mathrm{d}\mathbf{y} \,\mathrm{exp}\left(-\frac{1}{2}\beta\mathbf{y}^\mathrm{T}\left(\mathbf{1}-B'\right)\mathbf{y}+\beta\mathbf{x}^\mathrm{T} A\,\mathbf{y}\right)~. \ \ \ \ \ (14)$

Of course, this is just another edition of (6), but now with ${M=\beta\left(\mathbf{1}-B'\right)}$ and ${J=\beta A^\mathrm{T}\mathbf{x}}$. We therefore obtain

\displaystyle \begin{aligned} p(x)&=Z^{-1}\sqrt{\frac{(2\pi)^m}{\beta\left(1-|B'|\right)}}\mathrm{exp}\left[-\beta\left( f(\beta)-\mathbf{a}\mathbf{x}+\frac{1}{2}\mathbf{x}^\mathrm{T} A\left(\mathbf{1}-B'\right)^{-1}A^\mathrm{T}\mathbf{x}\right)\right]\\ &=Z^{-1}\sqrt{\frac{(2\pi)^m}{\beta\left(1-|B'|\right)}}\mathrm{exp}\left[-\beta\left( f(\beta)-\mathbf{a}\mathbf{x}+\frac{1}{2}\mathbf{x}^\mathrm{T} A'\mathbf{x}\right)\right] \end{aligned} \ \ \ \ \ (15)

where we have defined ${A\equiv A\,\left(\mathbf{1}-B'\right)^{-1}\!A^\mathrm{T}}$. As before, we then define ${H(x)}$ such that

$\displaystyle p(x)=Z^{-1}e^{-\beta H(x)}~, \ \ \ \ \ (16)$

which implies

$\displaystyle H(x)=g(\beta)-\mathbf{a}\mathbf{x}+\frac{1}{2}\mathbf{x}^\mathrm{T} A'\mathbf{x}~, \ \ \ \ \ (17)$

where

$\displaystyle g(\beta)\equiv f(\beta)+\frac{1}{2\beta}\ln\frac{\beta\left(1-|B'|\right)}{(2\pi)^m} =\frac{1}{2\beta}\ln\frac{\beta^2\left(1-|B'|\right)}{(2\pi)^{p+m}}~, \ \ \ \ \ (18)$

where ${f(\beta)}$ and ${B'}$ are given in (8).

Again we see that marginalizing out UV information induces new couplings between IR degrees of freedom; in particular, the hamiltonian ${H(x)}$ contains a quadratic interaction terms between the visible units. And we can again write this directly in terms of a cumulant generating function for hidden degrees of freedom by defining a prior of the form (9), but with ${z\rightarrow w\in\{y,z\}}$ and ${p\rightarrow m\!+\!p}$. This will be of the form (10), where in this case we need to choose ${\mathbf{t}=\beta\mathbf{x}^\mathrm{T} A\left(\mathbf{1}-B'\right)^{-1/2}}$, so that

$\displaystyle K_z(t)=\frac{\beta}{2} \mathbf{x}^\mathrm{T} A\left(\mathbf{1}-B'\right)^{-1} A^\mathrm{T}\mathbf{x} \ \ \ \ \ (19)$

(where, since ${B'^\mathrm{T}=B'}$, the inverse matrix ${(\mathbf{1}-B')^{-1}}$ is also invariant under the transpose; at this stage of exploration, I’m being quite cavalier about questions of existence). Thus the hamiltonian of visible units may be written

$\displaystyle H(x)=g(\beta)-\mathbf{a}\mathbf{x}+\frac{1}{\beta}K_z(t)~, \ \ \ \ \ (20)$

with ${t}$ and ${g(\beta)}$ as above. Since the prior with which these cumulants are computed is again Gaussian, only the second cumulant survives in the expansion, and we indeed recover (17).

To summarize, the sequential flow from UV (hidden) to IR (visible) distributions is, from top to bottom,

\displaystyle \begin{aligned} p(x,y,z)&=Z^{-1}e^{-\beta H(x,y,z)}~,\qquad &H(x,y,z)&=-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\left( \mathbf{y}^\mathrm{T}\mathbf{y}+\mathbf{z}^\mathrm{T}\mathbf{z}\right)-\mathbf{x}^\mathrm{T} A\,\mathbf{y}-\mathbf{y}^\mathrm{T} B\,\mathbf{z}~,\\ p(x,y)&=Z^{-1}e^{-\beta H(x,y)}~,\qquad &H(x,y)&=f(\beta)-\mathbf{a}\,\mathbf{x}+\frac{1}{2}\mathbf{y}^\mathrm{T}\left(\mathbf{1}-BB^\mathrm{T}\right)\mathbf{y}-\mathbf{x}^\mathrm{T} A\,\mathbf{y}~,\\ p(x)&=Z^{-1}e^{-\beta H(x)}~,\qquad &H(x)&=g(\beta)-\mathbf{a}\mathbf{x}+\frac{1}{2}\mathbf{x}^\mathrm{T} A\left(\mathbf{1}-BB^\mathrm{T}\right)^{-1}\!A^\mathrm{T}\mathbf{x}~, \end{aligned} \ \ \ \ \ (21)

where upon each marginalization, the new hamiltonian gains additional interaction terms/couplings governed by the cumulants of the UV prior (where “UV” is defined relative to the current cutoff scale, i.e., ${q(z)}$ for ${H(x,y)}$ and ${q(w)}$ for ${H(x)}$), and the renormalization of the partition function is accounted for by

$\displaystyle f(\beta)=\frac{1}{2\beta}\ln\frac{\beta}{(2\pi)^p}~, \quad\quad g(\beta)=-\frac{1}{2\beta}\ln\frac{\beta^2\left(1-|BB^\mathrm{T}|\right)}{(2\pi)^{p+m}}~. \ \ \ \ \ (22)$

As an aside, note that at each level, fixing the form (4), (16) is equivalent to imposing that the partition function remain unchanged. Ultimately, this is required in order to preserve low-energy correlation functions. The two-point correlator ${\langle x_1x_2\rangle}$ between visible (low-energy) degrees of freedom, for example, does not depend on which distribution we use to compute the expectation value, so long as the energy scale thereof is at or above the scale set by the inverse lattice spacing of ${\mathbf{x}}$:

\displaystyle \begin{aligned} \langle x_ix_j\rangle_{p(x,y,z)}&=\prod_{k=1}^n\sum_{x_k=\pm1}\int\!\mathrm{d}^my\mathrm{d}^pz\,x_ix_j\,p(x,y,z)\\ &=\prod_{k=1}^n\sum_{x_k=\pm1}x_ix_j\int\!\mathrm{d}^my\,p(x,y) =\langle x_ix_j\rangle_{p(x,y)}\\ &=\prod_{k=1}^n\sum_{x_k=\pm1}x_ix_j\,p(x) =\langle x_ix_j\rangle_{p(x)}~. \end{aligned} \ \ \ \ \ (23)

Failure to satisfy this requirement amounts to altering the theory at each energy scale, in which case there would be no consistent renormalization group relating them. In information-theoretic terms, this would represent an incorrect Bayesian inference procedure.

Despite (or perhaps, because of) its simplicity, this toy model makes manifest the fact that the RG prescription is reflected in the structure of the network, not the dynamics of learning per se. Indeed, Gaussian units aside, the above is essentially nothing more than real-space decimation RG on a 1d lattice, with a particular choice of couplings between “spins” ${\sigma\in\{x,y,z\}}$. In this analogy, tracing out ${y}$ and then ${x}$ maps to a sequential marginalization over even spins in the 1d Ising model. “Dynamics” in this sense are determined by the hamiltonian ${H(\sigma)}$, which is again one-dimensional. When one speaks of deep “learning” however, one views the network as two-dimensional, and “dynamics” refers to the changing values of the couplings as the network attempts to minimize the cost function. In short, RG lies in the fact that the couplings at each level in (21) encode the cumulants from hidden units in such a way as to ensure the preservation of visible correlations, whereas deep learning then determines their precise values in such a way as to reproduce a particular distribution. To say that deep learning itself is an RG is to conflate structure with function.

Nonetheless, there’s clearly an intimate parallel between RG and hierarchical Bayesian modeling at play here. As mentioned above, I’d originally hoped to derive something like a beta function for the cumulants, to see what insights theoretical physics and machine learning might yield to one another at this information-theoretic interface. Unfortunately, while one can see how the higher UV cumulants from ${q(z)}$ are encoded in those from ${q(w)}$, the appearance of the inverse matrix makes a recursion relation for the couplings in terms of the cumulants rather awkward, and the result would only hold for the simple Gaussian hidden units I’ve chosen for analytical tractability here.

Fortunately, after banging my head against this for a month, I learned of a recent paper [8] that derives exactly the sort of cumulant relation I was aiming for, at least in the case of generic lattice models. The key is to not assume a priori which degrees of freedom will be considered UV/hidden vs. IR/visible. That is, when I wrote down the joint distribution (2), I’d already distinguished which units would survive each marginalization. While this made the parallel with the familiar decimation RG immediate — and the form of (1) made the calculations simple to perform analytically — it’s actually a bit unnatural from both a field-theoretic and a Bayesian perspective: the degrees of freedom that characterize the theory in the UV may be very different from those that we observe in the IR (e.g., strings vs. quarks vs. hadrons), so we shouldn’t make the distinction ${x,y,z}$ at this level. Accordingly, [8] instead replace (2) with

$\displaystyle p(\chi)=\frac{1}{Z}\,e^{\mathcal{K}(\chi)} \ \ \ \ \ (24)$

where ${\mathcal{K}\equiv-\beta H}$ is the so-called reduced (i.e., dimensionless) hamiltonian, cf. the reduced/dimensionless free energy ${\tilde F\equiv\beta F}$ in the previous post. Note that ${\chi}$ runs over all the degrees of freedom in the theory, which are all UV/hidden variables at the present level.

Two words of notational warning ere we proceed: first, there is a sign error in eqn. (1) of [8] (in version 1; the negative in the exponent has been absorbed into ${\mathcal{K}}$ already). More confusingly however, their use of the terminology “visible” and “hidden” is backwards with respect to the RG analogy here. In particular, they coarse-grain a block of “visible” units into a single “hidden” unit. For reasons which should by now be obvious, I will instead stick to the natural Bayesian identifications above, in order to preserve the analogy with standard coarse-graining in RG.

Let us now repeat the above analysis in this more general framework. The real-space RG prescription consists of coarse-graining ${\chi\rightarrow\chi'}$, and then writing the new distribution ${p(\chi')}$ in the canonical form (24). In Bayesian terms, we need to marginalize over the information about ${\chi}$ contained in the distribution of ${\chi'}$, except that unlike in my simple example above, we don’t want to make any assumptions about the form of ${p(\chi,\chi')}$. So we instead express the integral — or rather, the discrete sum over lattice sites — in terms of the conditional distribution ${p(\chi'|\chi)}$:

$\displaystyle p(\chi')=\sum_\chi p(\chi,\chi') =\sum_\chi p(\chi'|\chi)\,p(\chi)~. \ \ \ \ \ (25)$

where ${\sum\nolimits_\chi=\prod_{i=1}^m\sum_{\chi_i}}$, ${\chi=\{\chi_1,\ldots,\chi_m\}}$. Denoting the new dimensionless hamiltonian ${\mathcal{K}'(\chi')}$, with ${\chi'=\{\chi_1',\ldots,\chi_n'\}}$, ${n, we therefore have

$\displaystyle e^{\mathcal{K}'(\chi')}=\sum_\chi p(\chi'|\chi)\,e^{\mathcal{K}(\chi)}~. \ \ \ \ \ (26)$

So far, so familiar, but now comes the trick: [8] split the hamiltonian ${\mathcal{K}(\chi)}$ into a piece containing only intra-block terms, ${\mathcal{K}_0(\chi)}$ (that is, interactions solely among the set of hidden units which is to be coarse-grained into a single visible unit), and a piece containing the remaining, inter-block terms, ${\mathcal{K}_1(\chi)}$ (that is, interactions between different aforementioned sets of hidden units).

Let us denote a block of hidden units by ${\mathcal{H}_j\ni\chi_i}$, such that ${\chi=\bigcup_{j=1}^n\mathcal{H}_j}$ (note that since ${\mathrm{dim}(\chi)=m}$, this implies ${\mathrm{dim}(\mathcal{H}_j)=m/n}$ degrees of freedom ${\chi_i}$ per block). To each ${\mathcal{H}_j}$, we associate a visible unit ${\mathcal{V}_j=\chi'_j}$, into which the constituent UV variables ${\chi_i}$ have been coarse-grained. (Note that, for the reasons explained above, we have swapped ${\mathcal{H}\leftrightarrow\mathcal{V}}$ relative to [8]). Then translation invariance implies

$\displaystyle p(\chi'|\chi)=\prod_{j=1}^np(\mathcal{V}_j|\mathcal{H}_j) \qquad\mathrm{and}\qquad \mathcal{K}_0(\chi)=\sum_{j=1}^n\mathcal{K}_b(\mathcal{H}_j)~, \ \ \ \ \ (27)$

where ${\mathcal{K}_b(\chi_i)}$ denotes a single intra-block term of the hamiltonian. With this notation in hand, (26) becomes

$\displaystyle e^{\mathcal{K}'(\chi')}=\sum_\chi e^{\mathcal{K}_1(\chi)}\prod_{j=1}^np(\mathcal{V}_j|\mathcal{H}_j)\,e^{\mathcal{K}_b(\mathcal{H}_j)}~. \ \ \ \ \ (28)$

Now, getting from this to the first line of eqn. (13) in [8] is a bit of a notational hazard. We must suppose that for each block ${\mathcal{H}_j}$, we can define the block-distribution ${p_j=Z_b^{-1}e^{\mathcal{K}_b(\mathcal{H}_j)}}$, where ${Z_b=\sum_{\chi_i\in\mathcal{H}_j}e^{\mathcal{K}_b(\mathcal{H}_j)}=\prod_{i=1}^{m/n}\sum_{\chi_i}e^{\mathcal{K}_b(\chi_i)}}$. Given the underlying factorization of the total Hilbert space, we furthermore suppose that the distribution of all intra-block contributions can be written ${p_0=Z_0^{-1}e^{\mathcal{K}_0(\chi)}}$, so that ${Z_0=\prod_{i=1}^m\sum_{\chi_i\in\chi}e^{\mathcal{K}_0(\chi)}}$. This implies that

$\displaystyle Z_0=\sum_{\chi_1}\ldots\sum_{\chi_m}e^{\mathcal{K}_b(\mathcal{H}_1)}\ldots\,e^{\mathcal{K}_b(\mathcal{H}_n)} =\prod_{j=1}^n\sum_{\chi_i\in\mathcal{H}_j}e^{\mathcal{K}_b(\mathcal{H}_j)} =\prod_{j=1}^nZ_b~. \ \ \ \ \ (29)$

Thus we see that we can insert a factor of ${1=Z_0\cdot\left(\prod\nolimits_jZ_b\right)^{-1}}$ into (28), from which the remaining manipulations are straightforward: we identify

$\displaystyle p(\mathcal{V}_j|\mathcal{H}_j)\frac{1}{Z_b}\,e^{\mathcal{K}_b(\mathcal{H}_j)} =p(\mathcal{V}_j|\mathcal{H}_j)\,p(\mathcal{H}_j) =p(\mathcal{H}_j|\mathcal{V}_j)\,p(\mathcal{V}_j) \ \ \ \ \ (30)$

and ${p(\chi')=\prod_{j=1}^np(\mathcal{V}_j)}$ (again by translation invariance), whereupon we have

$\displaystyle e^{\mathcal{K}'(\chi')}=Z_0\,p(\chi')\sum_\chi p(\chi|\chi')\,e^{\mathcal{K}_1(\chi)}=Z_0\,p(\chi')\langle e^{\mathcal{K}_1(\chi)}\rangle~, \ \ \ \ \ (31)$

where the expectation value is defined with respect to the conditional distribution ${p(\chi|\chi')}$. Finally, by taking the log, we obtain

$\displaystyle \mathcal{K}'(\chi')=\ln Z_0+\ln p(\chi')+\ln\langle e^{\mathcal{K}_1(\chi)}\rangle~, \ \ \ \ \ (32)$

which one clearly recognizes as a generalization of (12): ${\ln Z_0}$ accounts for the normalization factor ${f(\beta)}$; ${\ln p(\chi')}$ gives the contribution from the un-marginalized variables ${x,y}$; and the log of the expectation values is the contribution from the UV cumulants, cf. eqn. (1) in the previous post. Note that this is not the cumulant generating function itself, but corresponds to setting ${t\!=\!1}$ therein:

$\displaystyle K_X(1)=\ln\langle e^{X}\rangle=\sum_{n=1}\frac{\kappa_n}{n!}~. \ \ \ \ \ (33)$

Within expectation values, ${\mathcal{K}_1}$ becomes the dimensionless energy ${-\beta\langle E_1\rangle}$, so the ${n^\mathrm{th}}$ moment/cumulant picks up a factor of ${(-\beta)^n}$ relative to the usual energetic moments in eqn. (11) of the previous post. Thus we may express (32) in terms of the cumulants of the dimensionless hamiltonian ${\mathcal{K}_1}$ as

$\displaystyle \mathcal{K}'(\chi')=\ln Z_0+\ln p(\chi')+\sum_{n=1}\frac{(-\beta)^n}{n!}\kappa_n~, \ \ \ \ \ (34)$

where ${\kappa_n=K_{E_1}^{(n)}(t)\big|_{t=0}}$, and the expectation values in the generating functions are computed with respect to ${p(\chi|\chi')}$.

This is great, but we’re not quite finished, since we’d still like to determine the renormalized couplings in terms of the cumulants, as I did in the simple Gaussian DBM above. This requires expressing the new hamiltonian in the same form as the old, which allows one to identify exactly which contributions from the UV degrees of freedom go where. (See for example chapter 13 of [9] for a pedagogical exposition of this decimation RG procedure for the 1d Ising model). For the class of lattice models considered in [8] — by which I mean, real-space decimation with the imposition of a buffer zone — one can write down a formal expression for the canonical form of the hamiltonian, but expressions for the renormalized couplings themselves remain model-specific.

There’s more cool stuff in the paper [8] that I won’t go into here, concerning the question of “optimality” and the behaviour of mutual information in these sorts of networks. Suffice to say that, as alluded in the previous post, the intersection of physics, information theory, and machine learning is potentially rich yet relatively unexplored territory. While the act of learning itself is not an RG in a literal sense, the two share a hierarchical Bayesian language that may yield insights in both directions, and I hope to investigate this more deeply (pun intended) soon.

References

[1] C. Beny, “Deep learning and the renormalization group,” arXiv:1301.3124.

[2]  P. Mehta and D. J. Schwab, “An exact mapping between the Variational Renormalization Group and Deep Learning,” arXiv:1410.3831.

[3]  H. W. Lin, M. Tegmark, and D. Rolnick, “Why Does Deep and Cheap Learning Work So Well?,” arXiv:1608.08225.

[4]  S. Iso, S. Shiba, and S. Yokoo, “Scale-invariant Feature Extraction of Neural Network and Renormalization Group Flow,” arXiv:1801.07172.

[5]  M. Koch-Janusz and Z. Ringel, “Mutual information, neural networks and the renormalization group,” arXiv:1704.06279.

[6]  S. S. Funai and D. Giataganas, “Thermodynamics and Feature Extraction by Machine Learning,” arXiv:1810.08179.

[7]  E. Mello de Koch, R. Mello de Koch, and L. Cheng, “Is Deep Learning an RG Flow?,” arXiv:1906.05212.

[8]  P. M. Lenggenhager, Z. Ringel, S. D. Huber, and M. Koch-Janusz, “Optimal Renormalization Group Transformation from Information Theory,” arXiv:1809.09632.

[9]  R. K. Pathria, Statistical Mechanics. 1996. Butterworth-Heinemann, Second edition.

Posted in Minds & Machines, Physics | 1 Comment

## Cumulants, correlators, and connectivity

Lately, I’ve been spending a lot of time exploring the surprisingly rich mathematics at the intersection of physics, information theory, and machine learning. Among other things, this has led me to a new appreciation of cumulants. At face value, these are just an alternative to the moments that characterize a given probability distribution function, and aren’t particularly exciting. Except they show up all over statistical thermodynamics, quantum field theory, and the structure of deep neural networks, so of course I couldn’t resist trying to better understand the information-theoretic connections to which this seems to allude. In the first part of this two-post sequence, I’ll introduce them in the context of theoretical physics, and then turn to their appearance in deep learning in the next post, where I’ll dive into the parallel with the renormalization group.

The relation between these probabilistic notions and statistical physics is reasonably well-known, though the literature on this particular point unfortunately tends to be slightly sloppy. Loosely speaking, the partition function corresponds to the moment generating function, and the (Helmholtz) free energy corresponds to the cumulant generating function. By way of introduction, let’s make this identification precise.

The moment generating function for a random variable ${X}$ is

$\displaystyle M_X(t)\equiv \langle e^{tX}\rangle~,\quad\quad\forall t\in\mathbb{R}~, \ \ \ \ \ (1)$

where ${\langle\ldots\rangle}$ denotes the expectation value for the corresponding distribution. (As a technical caveat: in some cases, the moments — and correspondingly, ${M_X}$ — may not exist, in which case one can resort to the characteristic function instead). By series expanding the exponential, we have

$\displaystyle M_X(t)=1+t\langle X\rangle+\frac{t^2}{2}\langle X^2\rangle+\ldots\,=1+\sum_{n=1}m_n\frac{t^n}{n!}~, \ \ \ \ \ (2)$

were ${m_n}$ is the ${n^\mathrm{th}}$ moment, which we can obtain by taking ${n}$ derivatives and setting ${t\!=\!0}$, i.e.,

$\displaystyle m_n=M_X^{(n)}(t)\Big|_{t=0}=\langle X^n\rangle~. \ \ \ \ \ (3)$

However, it is often more convenient to work with cumulants instead of moments (e.g., for independent random variables, the cumulant of the sum is the sum of the cumulants, thanks to the log). These are uniquely specified by the moments, and vice versa—unsurprisingly, since the cumulant generating function is just the log of the moment generating function:

$\displaystyle K_X(t)\equiv\ln M_X(t)=\ln\langle e^{tX}\rangle \equiv\sum_{n=1}\kappa_n\frac{t^n}{n!}~, \ \ \ \ \ (4)$

where ${\kappa_n}$ is the ${n^\mathrm{th}}$ cumulant, which we again obtain by differentiating ${n}$ times and setting ${t=0}$:

$\displaystyle \kappa_n=K_X^{(n)}(t)\big|_{t=0}~. \ \ \ \ \ (5)$

Note however that ${\kappa_n}$ is not simply the log of ${m_n}$!

Now, to make contact with thermodynamics, consider the case in which ${X}$ is the energy of the canonical ensemble. The probability of a given energy eigenstate ${E_i}$ is

$\displaystyle p_i\equiv p(E_i)=\frac{1}{Z[\beta]}e^{-\beta E_i}~, \quad\quad \sum\nolimits_ip_i=1~. \ \ \ \ \ (6)$

The moment generating function for energy is then

$\displaystyle M_E(t)=\langle e^{tE}\rangle=\sum_i p(E_i)e^{tE_i} =\frac{1}{Z[\beta]}\sum_ie^{-(\beta\!-\!t)E_i} =\frac{Z[\beta-t]}{Z[\beta]}~. \ \ \ \ \ (7)$

Thus we see that the partition function ${Z[\beta]}$ is not the moment generating function, but there’s clearly a close relationship between the two. Rather, the precise statement is that the moment generating function ${M_E(t)}$ is the ratio of two partition functions at inverse temperatures ${\beta-t}$ and ${\beta}$, respectively. We can gain further insight by considering the moments themselves, which are — by definition (3) — simply expectation values of powers of the energy:

$\displaystyle \langle E^n\rangle=M^{(n)}(t)\Big|_{t=0} =\frac{1}{Z[\beta]}\frac{\partial^n}{\partial t^n}Z[\beta\!-\!t]\bigg|_{t=0} =(-1)^n\frac{Z^{(n)}[\beta\!-\!t]}{Z[\beta]}\bigg|_{t=0} =(-1)^n\frac{Z^{(n)}[\beta]}{Z[\beta]}~. \ \ \ \ \ (8)$

Note that derivatives of the partition function with respect to ${t}$ have, at ${t=0}$, become derivatives with respect to inverse temperature ${\beta}$ (obviously, this little slight of hand doesn’t work for all functions; simple counter example: ${f(\beta-t)=(\beta-t)^2}$). Of course, this is simply a more formal expression for the usual thermodynamic expectation values. The first moment of energy, for example, is

$\displaystyle \langle E\rangle= -\frac{1}{Z[\beta]}\frac{\partial Z[\beta]}{\partial\beta} =\frac{1}{Z[\beta]}\sum_i E_ie^{-\beta E_i} =\sum_i E_i\,p_i~, \ \ \ \ \ (9)$

which is the ensemble average. At a more abstract level however, (8) expresses the fact that the average energy — appropriately normalized — is canonically conjugate to ${\beta}$. That is, recall that derivatives of the action are conjugate variables to those with respect to which we differentiate. In classical mechanics for example, energy is conjugate to time. Upon Wick rotating to Euclidean signature, the trajectories become thermal circles with period ${\beta}$. Accordingly, the energetic moments can be thought of as characterizing the dynamics of the ensemble in imaginary time.

Now, it follows from (7) that the cumulant generating function (4) is

$\displaystyle K_E(t)=\ln\langle e^{tE}\rangle=\ln Z[\beta\!-\!t]-\ln Z[\beta]~. \ \ \ \ \ (10)$

While the ${n^\mathrm{th}}$ cumulant does not admit a nice post-derivative expression as in (8) (though I suppose one could write it in terms of Bell polynomials if we drop the adjective), it is simple enough to compute the first few and see that, as expected, the first cumulant is the mean, the second is the variance, and the third is the third central moment:

\displaystyle \begin{aligned} K^{(1)}(t)\big|_{t=0}&=-\frac{Z'[\beta]}{Z[\beta]}=\langle E\rangle~,\\ K^{(2)}(t)\big|_{t=0}&=\frac{Z''[\beta]}{Z[\beta]}-\left(\frac{Z'[\beta]}{Z[\beta]}\right)^2=\langle E^2\rangle-\langle E\rangle^2\\ K^{(3)}(t)\big|_{t=0}&=-2\left(\frac{Z'[\beta]}{Z[\beta]}\right)^3+3\frac{Z'[\beta]Z''[\beta]}{Z[\beta]^2}-\frac{Z^{(3)}[\beta]}{Z[\beta]}\\ &=-2\langle E\rangle^3+3\langle E\rangle\langle E^2\rangle-\langle E^3\rangle =-\left\langle\left( E-\langle E\rangle\right)^3\right\rangle~. \end{aligned} \ \ \ \ \ (11)

where the prime denotes the derivative with respect to ${\beta}$. Note that since the second term in the generating function (10) is independent of ${t}$, the normalization drops out when computing the cumulants, so we would have obtained the same results had we worked directly with the partition function ${Z[\beta]}$ and taken derivatives with respect to ${\beta}$. That is, we could define

$\displaystyle K_E(\beta)\equiv-\ln Z[\beta] \qquad\implies\qquad \kappa_n=(-1)^{n-1}K_E^{(n)}(\beta)~, \ \ \ \ \ (12)$

where, in contrast to (5), we don’t need to set anything to zero after differentiating. This expression for the cumulant generating function will feature more prominently when we discuss correlation functions below.

So, what does the cumulant generating function have to do with the (Helmholtz) free energy, ${F[\beta]=-\beta^{-1}\ln Z[\beta]}$? Given the form (12), one sees that they’re essentially one and the same, up to a factor of ${\beta}$. And indeed the free energy is a sort of “generating function” in the sense that it allows one to compute any desired thermodynamic quantity of the system. The entropy, for example, is

$\displaystyle S=-\frac{\partial F}{\partial T}=\beta^2\frac{\partial F}{\partial\beta} =\beta\langle E\rangle+\ln Z=-\langle\ln p\rangle~, \ \ \ \ \ (13)$

where ${p}$ is the Boltzmann distribution (6). However, the factor of ${\beta^{-1}}$ in the definition of free energy technically prevents a direct identification with the cumulant generating function above. Thus it is really the log of the partition function itself — i.e., the dimensionless free energy ${\beta F}$ — that serves as the cumulant generating function for the distribution. We’ll return to this idea momentarily, cf. (21) below.

So much for definitions; what does it all mean? It turns out that in addition to encoding correlations, cumulants are intimately related to connectedness (in the sense of connected graphs), which underlies their appearance in QFT. Consider, for concreteness, a real scalar field ${\phi(x)}$ in 4 spacetime dimensions. As every student knows, the partition function

$\displaystyle Z[J]=\mathcal{N}\int\mathcal{D}\phi\,\exp\left\{i\!\int\!\mathrm{d}^dx\left[\mathcal{L}(\phi,\partial\phi)+J(x)\phi(x)\right]\right\} \ \ \ \ \ (14)$

is the generating function for the ${n}$-point correlator or Green function ${G^{(n)}(x_1,\ldots,x_n)}$:

$\displaystyle G^{(n)}(x_1,\ldots,x_n)=\frac{1}{i^n}\frac{\delta^nZ[J]}{\delta J(x_1)\ldots\delta J(x_n)}\bigg|_{J=0}~, \ \ \ \ \ (15)$

where the normalization ${\mathcal{N}}$ is fixed by demanding that in the absence of sources, we should recover the vacuum expectation value, i.e., ${Z[0]=\langle0|0\rangle=1}$. In the language of Feynman diagrams, the Green function contains all possible graphs — both connected and disconnected — that contribute to the corresponding transition amplitude. For example, the 4-point correlator of ${\phi^4}$ theory contains, at first order in the coupling, a disconnected graph consisting of two Feynman propagators, another disconnected graph consisting of a Feynman propagator and a 1-loop diagram, and an irreducible graph consisting of a single 4-point vertex. But only the last of these contributes to the scattering process, so it’s often more useful to work with the generating function for connected diagrams only,

$\displaystyle W[J]=-i\ln Z[J]~, \ \ \ \ \ (16)$

from which we obtain the connected Green function ${G_c^{(n)}}$:

$\displaystyle G_c^{(n)}(x_1,\ldots,x_n)=\frac{1}{i^{n-1}}\frac{\delta^nW[J]}{\delta J(x_1)\ldots\delta J(x_n)}\bigg|_{J=0}~. \ \ \ \ \ (17)$

The fact that the generating functions for connected vs. disconnected diagrams are related by an exponential, that is, ${Z[J]=\exp{i W[J]}}$, is not obvious at first glance, but it is a basic exercise in one’s first QFT course to show that the coefficients of various diagrams indeed work out correctly by simply Taylor expanding the exponential ${e^X=\sum_n\tfrac{X^n}{n!}}$. In the example of ${\phi^4}$ theory above, the only first-order diagram that contributes to the connected correlator is the 4-point vertex. More generally, one can decompose ${G^{(n)}}$ into ${G_c^{(n)}}$ plus products of ${G_c^{(m)}}$ with ${m. The factor of ${-i}$ in (16) goes away in Euclidean signature, whereupon we see that ${Z[J]}$ is analogous to ${Z[\beta]}$ — and hence plays the role of the moment generating function — while ${W[J]}$ is analogous to ${\beta F[\beta]}$ — and hence plays the role of the cumulant generating function in the form (12).

Thus, the ${n^\mathrm{th}}$ cumulant of the field ${\phi}$ corresponds to the connected Green function ${G_c^{(n)}}$, i.e., the contribution from correlators of all ${n}$ fields only, excluding contributions from lower-order correlators among them. For example, we know from Wick’s theorem that Gaussian correlators factorize, so the corresponding ${4}$-point correlator ${G^{(4)}}$ becomes

$\displaystyle \langle\phi_1\phi_2\phi_3\phi_4\rangle= \langle\phi_1\phi_2\rangle\langle\phi_3\phi_4\rangle +\langle\phi_1\phi_3\rangle\langle\phi_2\phi_4\rangle +\langle\phi_1\phi_4\rangle\langle\phi_2\phi_3\rangle~. \ \ \ \ \ (18)$

What this means is that there are no interactions among all four fields that aren’t already explained by interactions among pairs thereof. The probabilistic version of this statement is that for the normal distribution, all cumulants other than ${n=2}$ are zero. (For a probabilist’s exposition on the relationship between cumulants and connectivity, see the first of three lectures by Novak and LaCroix [1], which takes a more graph-theoretic approach).

There’s one more important function that deserves mention here: the final member of the triumvirate of generating functions in QFT, namely the effective action ${\Gamma[\phi]}$, defined as the Legendre transform of ${W[J]}$:

$\displaystyle \Gamma[\phi]=W[J]-\int\!\mathrm{d} x\,J(x)\phi(x)~. \ \ \ \ \ (19)$

The Legendre transform is typically first encountered in classical mechanics, where it relates the hamiltonian and lagrangian formulations. Geometrically, it translates between a function and its envelope of tangents. More abstractly, it provides a map between the configuration space (here, the sources ${J}$) and the dual vector space (here, the fields ${\phi}$). In other words, ${\phi}$ and ${J}$ are conjugate pairs in the sense that

$\displaystyle \frac{\delta\Gamma}{\delta\phi}=-J \qquad\mathrm{and}\qquad \frac{\delta W}{\delta J}=\phi~. \ \ \ \ \ (20)$

As an example that connects back to the thermodynamic quantities above: we already saw that ${E}$ and ${\beta}$ are conjugate variables by considering the partition function, but the Legendre transform reveals that the free energy and entropy are conjugate pairs as well. This is nicely explained in the lovely pedagogical treatment of the Legendre transform by Zia, Redish, and McKay [2], and also cleans up the disruptive factor of ${\beta}$ that prevented the identification with the cumulant generating function above. The basic idea is that since we’re working in natural units (i.e., ${k_B=1}$), the thermodynamic relation in the form ${\beta F+S=\beta E}$ (13) obscures the duality between the properly dimensionless quantities ${\tilde F\equiv\beta F}$ and ${\tilde S=S/k_B}$. From this perspective, it is more natural to work with ${\tilde F}$ instead, in which case we have both an elegant expression for the duality in terms of the Legendre transform, and a precise identification of the dimensionless free energy with the cumulant generating function (12):

$\displaystyle \tilde F(\beta)+\tilde S(E)=\beta E~, \qquad\qquad K_E(\beta)=\tilde F=\beta F~. \ \ \ \ \ (21)$

Now, back to QFT, in which ${\Gamma[\phi]}$ generates one-particle irreducible (1PI) diagrams. A proper treatment of this would take us too far afield, but can be found in any introductory QFT book, e.g., [3]. The basic idea is that in order to be able to cut a reducible diagram, we need to work at the level of vertices rather than sources (e.g., stripping off external legs, and identifying the bare propagator between irreducible parts). The Legendre transform (19) thus removes the dependence on the sources ${J}$, and serves as the generator for the vertex functions of ${\phi}$, i.e., the fundamental interaction terms. The reason this is called the effective action is that in perturbation theory, ${\Gamma[\phi]}$ contains the classical action as the leading saddle-point, as well as quantum corrections from the higher-order interactions in the coupling expansion.

In information-theoretic terms, the Legendre transform of the cumulant generating function is known as the rate function. This is a core concept in large deviations theory, and I won’t go into details here. Loosely speaking, it quantifies the exponential decay that characterizes rare events. Concretely, let ${X_i}$ represent the outcome of some measurement or operation (e.g., a coin toss); then the mean after ${N}$ independent trials is

$\displaystyle M_N=\frac{1}{N}\sum_{i=1}^N X_i~. \ \ \ \ \ (22)$

The probability that a given measurement deviates from this mean by some specified amount ${x}$ is

$\displaystyle P(M_N>x)\approx e^{-N I(x)} \ \ \ \ \ (23)$

where ${I(x)}$ is the aforementioned rate function. The formal similarity with the partition function in terms of the effective action, ${Z=e^{-\Gamma}}$, is obvious, though the precise dictionary between the two languages is not. I suspect that a precise translation between the two languages — physics and information theory — can be made here as well, in which the increasing rarity of events as one moves along the tail of the distribution correspond to increasingly high-order corrections to the quantum effective action, but I haven’t worked this out in detail.

Of course, the above is far from the only place in physics where cumulants are lurking behind the scenes, much less the end of the parallel with information theory more generally. In the next post, I’ll discuss the analogy between deep learning and the renormalization group, and see how Bayesian terminology can provide an underlying language for both.

References

[1] J. Novak and M. LaCroix, “Three lectures on free probability,” arXiv:1205.2097.

[2] R. K. P. Zia, E. F. Redish, and S. R. McKay, “Making sense of the Legendre transform, arXiv:0806.1147.

[3] L. H. Ryder, Quantum Field Theory. Cambridge University Press, 2 ed., 1996.

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