Ro's blog

In the context of firewalls, the crux of the paradox boils down to whether black holes have smooth horizons (as required by the equivalence principle). It turns out that this is intimately related to the question of how the interior of the black hole can be reconstructed by an external observer. AdS/CFT is particularly useful in this regard, because it enables one to make such questions especially sharp. Specifically, one studies the eternal black hole dual to the thermofield double (TFD) state, which cleanly captures the relevant physics of real black holes formed from collapse.

To construct the TFD, we take two copies of a CFT and entangle them such that tracing out either results in a thermal state. Denoting the energy eigenstates of the left and right CFTs by and , respectively, the state is given by

where is the partition function at inverse temperature . The AdS dual of this state is the eternal black hole, the two sides of which join the left and right exterior bulk regions through the wormhole. Incidentally, one of the fascinating questions inherent to this construction is how the bulk spacetime emerges in a manner consistent with the tensor product of boundary CFTs. For our immediate purposes, the important fact is that operators in the left source states behind the horizon from the perspective of the right (and vice-versa). The requirement from general relativity that the horizon be smooth then imposes conditions on the relationship between these operators.

A noteworthy approach in this vein is the so-called “state-dependence” proposal developed by Kyriakos Papadodimas and Suvrat Raju over the course of several years [1,2,3,4,5] (referred to as PR henceforth). Their collective florilegium spans several hundred pages, jam-packed with physics, and any summary I could give here would be a gross injustice. As alluded above however, the salient aspect is that they phrased the smoothness requirement precisely in terms of a condition on correlation functions of CFT operators across the horizon. Focusing on the two-point function for simplicity, this condition reads:

Here, is an exterior operator in the right CFT, while is an interior operator in the left—that is, it represents an excitation localized behind the horizon from the perspective of an observer in the right wedge (see the diagram above). The analytical continuation arises from the KMS condition (i.e., the periodicity of thermal Green functions in imaginary time). Physically, this is essentially the statement that one should reproduce the correct thermal expectation values when restricted to a single copy of the CFT.

The question then becomes whether one can find such operators in the CFT that satisfy this constraint. That is, we want to effectively construct interior operators by acting only in the exterior CFT. PR achieve this through their so-called “mirror operators” , defined by

While appealingly compact, it’s more physically insightful to unpack this into the following two equations:

The key point is that these operators are defined via their action on the state , i.e., they are state-dependent operators. For example, the second equation does not say that the operators commute; indeed, as operators, . But the commutator does vanish in this particular state, . This may seem strange at first sight, but it’s really just a matter of carefully distinguishing between equations that hold as operator statements and those that hold only at the level of states. Indeed, this is precisely the same crucial distinction between localized states vs. localized operators that I’ve discussed before.

PR’s work created some backreaction, most of which centered around the nature of this “unusual” state dependence, which generated considerable confusion. Aspects of PR’s proposal were critiqued in a number of papers, particularly [6,7], which led many to claim that state dependence violates quantum mechanics. Coincidentally, I had the good fortune of being a visiting grad student at the KITP around this time, where these issues where hotly debated during a long-term workshop on quantum gravity. This was a very stimulating time, when the firewall paradox was still center-stage, and the collective confusion was almost palpable. Granted, I was a terribly confused student, but the fact that the experts couldn’t even agree on language — let alone physics — certainly didn’t do me any favours. Needless to say, the debate was never resolved, and the field’s collective attention span eventually drifted to other things. Yet somehow, the claim that state dependence violates quantum mechanics (or otherwise constitutes an unusual or potentially problematic modification thereof) has since risen to the level of dogma, and one finds it regurgitated again and again in papers published since.

Motivated in part by the desire to understand the precise nature of state dependence in this context (though really, it was the interior spacetime I was after), I wrote a paper [8] last year in an effort to elucidate and connect a number of interesting ideas in the emergent spacetime or “It from Qubit” paradigm. At a technical level, the only really novel bit was the application of modular inclusions, which provide a relatively precise framework for investigating the question of how one represents information in the black hole interior, and perhaps how the bulk spacetime emerges more generally. The relation between Tomita-Takesaki theory itself (a subset of algebraic quantum field theory) and state dependence was already pointed out by PR [3], and is highlighted most succinctly in Kyriakos’ later paper in 2017 [9], which was the main stimulus behind my previous post on the subject. However, whereas PR arrived at this connection from more physical arguments (over the course of hundreds of pages!), I took essentially the opposite approach: my aim was to distill the fundamental physics as cleanly as possible, to which end modular theory proves rather useful for demystifying issues which might otherwise remain obfuscated by details. The focus of my paper was consequently decidedly more conceptual, and represents a personal attempt to gain deeper physical insight into a number of tantalizing connections that have appeared in the literature in recent years (e.g., the relationship between geometry and entanglement represented by Ryu-Takayanagi, or the ontological basis for quantum error correction in holography).

I’ve little to add here that isn’t said better in [8] — and indeed, I’ve already written about various aspects on other occasions — so I invite you to simply read the paper if you’re interested. Personally, I think it’s rather well-written, though card-carrying members of the “shut up and calculate” camp may find it unpalatable. The paper touches on a relatively wide range of interrelated ideas in holography, rather than state dependence alone; but the upshot for the latter is that, far from being pathological, state dependence (precisely defined) is

1. a natural part of standard quantum field theory, built-in to the algebraic framework at a fundamental level, and
2. an inevitable feature of any attempt to represent information behind horizons.

I hasten to add that “information” is another one of those words that physicists love to abuse; here, I mean a state sourced by an operator whose domain of support is spacelike separated from the observer (e.g., excitations localized on the opposite side of a Rindler/black hole horizon). The second statement above is actually quite general, and results whenever one attempts to reconstruct an excitation outside its causal domain.

So why I am devoting an entire post to this, if I’ve already addressed it at length elsewhere? There were essentially two motivations for this. One is that I recently had the opportunity to give a talk about this at the YITP in Kyoto (the slides for which are available from the program website here), and I fell back down the rabbit hole in the course of reviewing. In particular, I wanted to better understand various statements in the literature to the effect that state dependence violates quantum mechanics. I won’t go into these in detail here — one can find a thorough treatment in PRs later works — but suffice to say the primary issue seems to lie more with language than physics: in the vast majority of cases, the authors simply weren’t precise about what they meant by “state dependence” (though in all fairness, PR weren’t totally clear on this either), and the rare exceptions to this had little to nothing to do with the unqualified use of the phrase here. I should add the disclaimer that I’m not necessarily vouching for every aspect of PR’s approach—they did a hell of a lot more than just write down (3), after all. My claim is simply that state dependence, in the fundamental sense I describe, is a feature, not a bug. Said differently, even if one rejects PR’s proposal as a whole, the state dependence that ultimately underlies it will continue to underlie any representation of the black hole interior. Indeed, I had hoped that my paper would help clarify things in this regard.

And this brings me to the second reason, namely: after my work appeared, a couple other papers [10,11] were written that continued the offense of conflating the unqualified phrase “state dependence” with different and not-entirely-clear things. Of course, there’s no monopoly on terminology: you can redefine terms however you like, as long as you’re clear. But conflating language leads to conflated concepts, and this is where we get into trouble. Case in point: both papers contain a number of statements which I would have liked to see phrased more carefully in light of my earlier work. Indeed, [11] goes so far as to write that “interior operators cannot be encoded in the CFT in a state-dependent way.” On the contrary, as I had explained the previous year, it’s actually the state independent operators that lead to pathologies (specifically, violations of unitarity)! Clearly, whatever the author means by this, it is not the same state dependence at work here. So consider this a follow-up attempt to stop further terminological misuse confusion.

As I’ll discuss below, both these works — and indeed most other proposals from quantum information — ultimately rely on the Hayden-Preskill protocol [12] (and variations thereof), so the real question is how the latter relates to state dependence in the unqualified use of the term (i.e., as defined via Tomita-Takesaki theory; I refer to this usage as “unqualified” because if you’re talking about firewalls and don’t specify otherwise, then this is the relevant definition, as it underlies PR’s introduction of the phrase). I’ll discuss this in the context of Beni’s work [10] first, since it’s the clearer of the two, and comment more briefly on Geof’s [11] below.

In a nutshell, the classic Hayden-Preskill result [12] is a statement about the ability to decode information given only partial access to the complete quantum state. In particular, one imagines that the proverbial Alice throws a message comprised of bits of information into a black hole of size . The black hole will scramble this information very quickly — the details are not relevant here — such that the information is encoded in some complicated manner among the (new) total bits of the black hole. For example, if we model the internal dynamics as a simple permutation of Alice’s -bit message, it will be transformed into one of possible -bit strings—a huge number of possibilities!

Now suppose Bob wishes to reconstruct the message by collecting qubits from the subsequent Hawking radiation. Naïvely, one would expect him to need essentially all bits (i.e., to wait until the black hole evaporates) in order to accurately determine among the possibilities. The surprising result of Hayden-Preskill is that in fact he needs only slightly more than bits. The time-scale for this depends somewhat on the encoding performed by the black hole, but in principle, this means that Bob can recover the message just after the scrambling time. However, a crucial aspect of this protocol is that Bob knows the initial microstate of the black hole (i.e., the original -bit string). This is the source of the confusing use of the phrase “state dependence”, as we’ll see below.

Of course, as Hayden and Preskill acknowledge, this is a highly unrealistic model, and they didn’t make any claims about being able to reconstruct the black hole interior in this manner. Indeed, the basic physics involved has nothing to do with black holes per se, but is a generic feature of quantum error correcting codes, reminiscent of the question of how to share (or decode) a quantum “secret” [13]. The novel aspect of Beni’s recent work [10] is to try to apply this to resolving the firewall paradox, by explicitly reconstructing the interior of the black hole.

Beni translates the problem of black hole evaporation into the sort of circuit language that characterizes much of the quantum information literature. One the one hand, this is nice in that it enables him to make very precise statements in the context of a simple qubit model; and indeed, at the mathematical level, everything’s fine. The confusion arises when trying to lift this toy model back to the physical problem at hand. In particular, when Beni claims to reconstruct state-independent interior operators, he is — from the perspective espoused above — misusing the terms “state-independent”, “interior”, and “operator”.

Let’s first summarize the basic picture, and then try to elucidate this unfortunate linguistic hat-trick. The Hayden-Preskill protocol for recovering information from black holes is illustrated in the figure from Beni’s paper below. In this diagram, is the black hole, which is maximally entangled (in the form of some number of EPR pairs) with the early radiation . Alice’s message corresponds to the state , which we imagine tossing into the black hole as . One then evolves the black hole (which now includes Alice’s message ) by some unitary operator , which scrambles the information as above. Subsequently, represents some later Hawking modes, with the remaining black hole denoted . Bob’s task is to reconstruct the state by acting on and (since he only has access to the exterior) with some operator .

Now, Beni’s “state dependence” refers to the fact that the technical aspects of this construction relied on putting the initial state of the black hole radiation in the form of a collection of EPR pairs . This can be done by finding some unitary operator , such that

(Here, one imagines that is further split into a Hawking mode and its partner just behind the horizon, so that acts on the interior mode while affects only the new Hawking mode and the early radiation; see [10] for details). This is useful because it enables the algorithm to work for arbitrary black holes: for some other initial state , one can find some other which results in the same state . The catch is that Bob’s reconstruction depends on , and therefore, on the initial state . But this is to be expected: it’s none other than the Hayden-Preskill requirement above that Bob needs to know the exact microstate of the system in order for the decoding protocol to work. It is in this sense that the Hayden-Preskill protocol is “state-dependent”, which clearly references something different than what we mean here. The reason I go so far as to call this a misuse of terminology is that Beni explicitly conflates the two, and regurgitates the claim that these “state-dependent interior operators” lead to inconsistencies with quantum mechanics, referencing work above. Furthermore, as alluded above, there’s an additional discontinuity of concepts here, namely that the “state-dependent” operator is obviously not the “interior operator” to which we’re referring: it’s support isn’t even restricted to the interior, nor does it source any particular state localized therein!

Needless to say, I was in a superposition of confused and unhappy with the terminology in this paper, until I managed to corner Beni at YITP for a couple hours at the aforementioned workshop, where he was gracious enough to clarify various aspects of his construction. It turns out that he actually has in mind something different when he refers to the interior operator. Ultimately, the identification still fails on these same counts, but it’s worth following the idea a bit further in order to see how he avoids the “state dependence” in the vanilla Hayden-Preskill set-up above. (By now I shouldn’t have to emphasize that this form of “state dependence” isn’t problematic in any fundamental sense, and I will continue to distinguish it from the latter, unqualified use of the phrase with quotation marks).

One can see from the above diagram that the state of the black hole — before Alice & Bob start fiddling with it — can be represented by the following diagram, also from [10]:

where , , and are again the early radiation, later Hawking mode, and remaining black hole, respectively. The problem Beni solves is finding the “partner” — by which he means, the purification — of in . Explicitly, he wants to find the operator such that

Note that there’s yet another language ergo conceptual discontinuity here, namely that Beni uses “qubit”, “mode”, and “operator” interchangeably (indeed, when I pressed him on this very point, he confirmed that he regards these as synonymous). These are very different beasts in the physical problem at hand; however, for the purposes of Beni’s model, the important fact is that one can push the operator (which one should think of as some operator that acts on ) through the unitary to some other operator that acts on both and :

He then goes on to show that one can reconstruct this operator independently of the initial state of the black hole (i.e., the operator ) by coupling to an auxiliary system. Of course, I’m glossing over a great number of details here; in particular, Beni transmutes the outgoing mode into a representation of the interior mode in his model, and calls whatever purifies it the “partner” . Still, I personally find this a bit underwhelming; but then, from my perspective, the Hayden-Preskill “state dependence” wasn’t the issue to begin with; quantum information people may differ, and in any case Beni’s construction is still a neat toy model in its own domain.

However, the various conflations above are problematic when one attempts to map back to the fundamental physics we’re after: is not the “partner” of the mode in the relevant sense (namely, the pairwise entangled modes required for smoothness across the horizon), nor does it correspond to PR’s mirror operator (since its support actually straddles both sides of the horizon). Hence, while Beni’s construction does represent a non-trivial refinement of the original Hayden-Preskill protocol, I don’t think it solves the problem.

So if this model misses the point, what does Hayden-Preskill actually achieve in this context? Indeed, even in the original paper [12], they clearly showed that one can recover a message from inside the black hole. Doesn’t this mean we can reconstruct the interior in a state-independent manner, in the proper use of the term?

Well, not really. Essentially, Hayden-Preskill (in which I’m including Beni’s model as the current state-of-the-art) & PR (and I) are asking different questions: the former are asking whether it’s possible to decode messages to which one would not normally have access (answer: yes, if you know enough about the initial state and any auxiliary systems), while the latter are asking whether physics in the interior of the black hole can be represented in the exterior (answer: yes, if you use state-dependent operators). Reconstructing information about entangled qubits is not quite the same things as reconstructing the state in the interior. Consider a single Bell pair for simplicity, consisting of an exterior qubit (say, in Beni’s model) and the interior “partner” that purifies it. Obviously, this state isn’t localized to either side, and so does not correspond to an interior operator.

The distinction is perhaps a bit subtle, so let me try to clarify. Let us define the operator with support behind the horizon, whose action on the vacuum creates the state in which Alice’s message has been thrown into the black hole; i.e., let

denote the state of the black hole containing Alice’s message, where the identity factor acts on the early radiation. Now, the fundamental result of PR is that if Bob wishes to reconstruct the interior of the black hole (concretely, the excitation behind the horizon corresponding to Alice’s message), he can only do so using state-dependent operators. In other words, there is no operator with support localized to the exterior which precisely equals ; but Bob can find a state that approximates arbitrarily well. This is more than just an operational restriction, but rather stems from an interesting trade-off between locality and unitarity which seems built-in to the theory at a fundamental level; see [8] for details.

Alternatively, Bob might not care about directly reconstructing the black hole interior (since he’s not planning on following Alice in, he’s not concerned about verifying smoothness as we are). Instead he’s content to wait for the “information” in this state to be emitted in the Hawking radiation. In this scenario, Bob isn’t trying to reconstruct the black hole interior corresponding to (7)—indeed, by now this state has long-since been scrambled. Rather, he’s only concerned with recovering the information content of Alice’s message—a subtly related but crucially distinct procedure from trying to reconstruct the corresponding state in the interior. And the fundamental result of Hayden-Preskill is that, given some admittedly idealistic assumptions (i.e., to the extent that the evaporating black hole can be viewed as a simple qubit model) this can also be done.

In the case of Geof’s paper [11], there’s a similar but more subtle language difference at play. Here the author means “state dependence” to mean something different from both Beni and PR/myself; specifically, he means “state dependence” in the context of quantum error correction (QEC). This is more clearly explained his earlier paper with Hayden [14], and refers to the fact that in general, a given boundary operator may only reconstruct a given bulk operator for a single black hole microstate. Conversely, a “state-independent” boundary operator, in their language, is one which approximately reconstructs a given bulk operator in a larger class of states—specifically, all states in the code subspace. Note that the qualifier “approximate” is crucial here. Otherwise, schematically, if represents some small perturbation of the vacuum (where “small” means that the backreaction is insufficient to move us beyond the code subspace), then an exact reconstruction of the operator that sources the state would instead produce some other state . So at the end of the day, I simply find the phrasing in [11] misleading; the lack of qualifiers makes many of his statements about “state-(in)dependence” technically erroneous, even though they’re perfectly correct in the context of approximate QEC.

At the end of the day however, these [10,11,14] are ultimately quantum information-theoretic models, in which the causal structure of the original problem plays no role. This is obvious in Beni’s case [10], in which Hayden-Preskill boils down to the statement that if one knows the exact quantum state of the system (or approximately so, given auxiliary qubits), then one can recover information encoded non-locally (e.g., Alice’s bit string) from substantially fewer qubits than one would naïvely expect. It’s more subtle in [11,14], since the authors work explicitly in the context of entanglement wedge reconstruction in AdS/CFT, which superficially would seem to include aspects of the spacetime structure. However, they take the black hole to be included in the entanglement wedge (i.e., code subspace) in question, and ask only whether an operator in the corresponding boundary region “works” for every state in this (enlarged) subspace, regardless of whether the bulk operator we’re trying to reconstruct is behind the horizon (i.e., ignoring the localization of states in this subspace). And this is where super-loading the terminology “state-(in)dependence” creates the most confusion. For example, when Geof writes that “boundary reconstructions are state independent if, and only if, the bulk operator is contained in the entanglement wedge” (emphasis added), he is making a general statement that holds only at the level of QEC codes. If the bulk operator lies behind the horizon however, then simply placing the black hole within the entanglement wedge does not alter the fact that a state-independent reconstruction, in the unqualified use of the phrase, does not exist.

Of course, as the authors of [14] point out in this work, there is a close relationship between state-dependent in QEC and in PR’s use of the term. Indeed, one of the closing thoughts of my paper [8] was the idea that modular theory may provide an ontological basis for the epistemic utility of QEC in AdS/CFT. Hence I share the authors’ view that it would be very interesting to make the relation between QEC and (various forms of) state-dependence more precise.

I should add that in Geof’s work [11], he seems to skirt some of the interior/exterior objections above by identifying (part of) the black hole interior with the entanglement wedge of some auxiliary Hilbert space that acts as a reservoir for the Hawking radiation. Here I can only confess some skepticism as to various aspects of his construction (or rather, the legitimacy of his interpretation). In particular, the reservoir is artificially taken to lie outside the CFT, which would normally contain a complete representation of exterior states, including the radiation. Consequently, the question of whether it has a sensible bulk dual at all is not entirely clear, much less a geometric interpretation as the “entanglement wedge” behind the horizon, whose boundary is the origin rather than asymptotic infinity.

A related paper [15] by Almeiri, Engelhardt, Marolf, and Maxfield appeared on the arXiv simultaneously with Geof’s work. While these authors are not concerned with state-dependence per se, they do provide a more concrete account of the effects on the entanglement wedge in the context of a precise model for an evaporating black hole in AdS/CFT. The analogous confusion I have in this case is precisely how the Hawking radiation gets transferred to the left CFT, though this may eventually come down to language as well. In any case, this paper is more clearly written, and worth a read (happily, Henry Maxfield will speak about it during one of our group’s virtual seminars in August, so perhaps I’ll obtain greater enlightenment about both works then).

Having said all that, I believe all these works are helpful in strengthening our understanding, and exemplify the productive confluence of quantum information theory, holography, and black holes. A greater exchange of ideas from various perspectives can only lead to further progress, and I would like to see more work in all these directions.

I would like to thank Beni Yoshida, Geof Penington, and Henry Maxfield for patiently fielding my persistent questions about their work, and beg their pardon for the gross simplifications herein. I also thank the YITP in Kyoto for their hospitality during the Quantum Information and String Theory 2019 / It from Qubit workshop, where most of this post was written amidst a great deal of stimulating discussion.

As part of my ongoing love affair with black holes, I’ve been digging more deeply into what it means for them to have entropy, which of course necessitates investigating how this is assigned in the first place. This is a notoriously confusing issue — indeed, one which lies at the very heart of the firewall paradox — which is further complicated by the fact that there are a priori three distinct physical entropies at play: thermodynamic, entanglement, and gravitational. (Incidentally, lest my previous post on entropy cause confusion, let me stress that said post dealt only with the relation between thermodynamic and information-theoretic a.k.a. Shannon entropy, at a purely classical level: neither entanglement nor gravity played any role there. I also didn’t include the Shannon entropy in the list above, because — as explained in the aforementioned post — this isn’t an objective/physical entropy in the sense of the other three; more on this below.)

My research led me to a review on black hole entanglement entropy by Solodukhin [1], which is primarily concerned with the use of the conical singularity method (read: replica trick) to isolate the divergences that arise whenever one attempts to compute entanglement entropy in quantum field theory. The structure of these divergences turns out to provide physical insight into the nature of this entropy, and sheds some light on the relation to thermodynamic/gravitational entropy as well, so these sorts of calculations are well-worth understanding in detail.

While I’ve written about the replica trick at a rather abstract level before, for present purposes we must be substantially more concrete. To that end, the main technical objective of this post is to elucidate one of the central tools employed by these computations, known as the heat kernel method. This is a rather powerful method with applications scattered throughout theoretical physics, notably the calculation of 1-loop divergences and the study of anomalies. Our exposition will mostly follow the excellent pedagogical review by Vassilevich [2]. Before diving into the details however, let’s first review the replica trick à la [1], in order to see how the heat kernel arises in the present context.

Consider some quantum field in -dimensional Euclidean spacetime, where with . The Euclidean time is related to Minkowski time via the usual Wick rotation, and we have singled-out one of the transverse coordinates for reasons which will shortly become apparent. For simplicity, let us consider the wavefunction for the vacuum state, which we prepare by performing the path integral over the lower () half of Euclidean spacetime with the boundary condition :

where we have used to denote the Euclidean action for the matter field, so as to reserve for the entropy.

Now, since we’re interested in computing the entanglement entropy of a subregion, let us divide the surface into two halves, and , by defining a codimension-2 surface by the condition , . Correspondingly, let us denote the boundary data

The reduced density matrix that describes the subregion of the vacuum state is then obtained by tracing over the complementary set of boundary fields . In the Euclidean path integral, this corresponds to integrating out over the entire spacetime, but with a cut from negative infinity to along the surface (i.e., along ). We must therefore impose boundary conditions for the remaining field as this cut is approached from above () and below (). Hence:

Formally, this object simply says that the transition elements are computed by performing the path integral with the specified boundary conditions along the cut.

Unfortunately, explicitly computing the von Neumann entropy is an impossible task for all but the very simplest systems. Enter the replica trick. The basic idea is to consider the -fold cover of the above geometry, introduce a conical deficit at the boundary of the cut , and then differentiate with respect to the deficit angle, whereupon the von Neumann entropy is recovered in the appropriate limit. To see this in detail, it is convenient to represent the subspace in polar coordinates , where and , such that the cut corresponds to for . In constructing the -sheeted cover, we glue sheets along the cut such that the fields are smoothly continued from to . The resulting space is technically a cone, denoted , with angular deficit at , on which the partition function for the fields is given by

where is the power of the density matrix (3). At this point in our previous treatment of the replica trick, we introduced the Rényi entropy

which one can think of as the entropy carried by the copies of the chosen subregion, and showed that the von Neumann entropy is recovered in the limit . Equivalently, we may express the von Neumann entropy directly as

Explicitly:

where in the last step we took the path integral to be appropriately normalized such that , whereupon the second term vanishes. Et voilà! As I understand it, the aforementioned “conical singularity method” is essentially an abstraction of the replica trick to spacetimes with conical singularities. Hence, re-purposing our notation, consider the effective action for fields on a Euclidean spacetime with a conical singularity at . The cone is defined, in polar coordinates, by making periodic with period , and taking the limit in which the deficit . The entanglement entropy for fields on this background is then

As a technical aside: in both these cases, there is of course the subtlety of analytically continuing the parameter resp. to non-integer values. We’ve discussed this issue before in the context of holography, where one surmounts this by instead performing the continuation in the bulk. We shall not digress upon this here, except to note that the construction relies on an abelian (rotational) symmetry , where is an arbitrary constant. This is actually an important constraint to bear in mind when attempting to infer general physical lessons from our results, but we’ll address this caveat later. Suffice to say that given this assumption, the analytical continuation can be uniquely performed without obstruction; see in particular section 2.7 of [1] for details.

We have thus obtained, in (8), an expression for the entanglement entropy in terms of the effective action in the presence of a conical singularity. And while this is all well and pretty, in order for this expression to be of any practical use, we require a means of explicitly computing . It is at this point that the heat kernel enters the game. The idea is to represent the Green function — or rather, the (connected) two-point correlator — as an integral over an auxiliary “proper time” of a kernel satisfying the heat equation. This enables one to express the effective action as

where is the (trace of the) heat kernel for the Laplacian operator . We’ll now spend some time unpacking this statement, which first requires that we review some basic facts about Green functions, propagators, and all that.

Consider a linear differential operator acting on distributions with support on . If admits a right inverse, then the latter defines the Green function as the solution to the inhomogeneous differential equation

where represent vectors in . We may also define the kernel of as the solution to the homogeneous differential equation

The Green function is especially useful for solving linear differential equations of the form

To see this, simply multiply both sides of (10) by and integrate w.r.t ; by virtue of the delta function (and the fact that is independent of ), one identifies

The particular boundary conditions we impose on then determine the precise form of (e.g., retarded vs. advanced Green functions in QFT).

As nicely explained in this Stack Exchange answer, the precise relation to propagators is ambiguous, because physicists use this term to mean either the Green function or the kernel, depending on context. For example, the Feynman propagator for a scalar field is a Green function for the Klein-Gordon operator, since it satisfies the equation

In contrast, the corresponding Wightman functions

are kernels for this operator, since they satisfy

The reader is warmly referred to section 2.7 of Birrell & Davies classic textbook [3] for a more thorough explanation of Green functions in this context (see also part 1 of my QFT in curved space sequence).

In the present case, we shall be concerned with the heat kernel

where is some Laplacian operator, by which we mean that it admits a local expression of the form

for some matrix-valued functions . In the second equality, we’ve written the operator in the so-called canonical form for a Laplacian operator on a vector bundle, where is an endomorphism on the bundle over the manifold, and the covariant derivative includes both the familiar Riemann part as well as the contribution from the gauge (bundle) part; the field strength for the latter will be denoted . Fibre bundles won’t be terribly important for our purposes, but we’ll need some of this notation later; see section 2.1 of [2] for details.

The parameter in (17) is some auxiliary Euclidean time variable (note that if we were to take , the right-hand side of (17) would correspond to the transition amplitude for some unitary operator ). is so-named because it satisfied the heat equation,

with the initial condition

where the subscript on is meant to emphasize the fact that the operator acts only on the transverse variables, not on the auxiliary time . Our earlier claim that the Green function can be expressed in terms of an integral over the latter is then based on the observation that one can invert (17) to obtain the propagator

Note that here, “propagator” indeed refers to the Green function of the field in the path integral representation, insofar as the latter serves as the generating functional for the former. Bear with me a bit longer as we elucidate this last claim, as this will finally bring us full-circle to (9)

Denote the Euclidean path integral for the fields on some fixed background by

where is the source for the matter field . The heat kernel method applies to one-loop calculations, in which case it suffices to expand the action to quadratic order in fluctuations around the classical saddle-point , whence the Gaussian integral may be written

(We’re glossing over some mathematical caveats/assumptions here, notably that be self-adjoint w.r.t. to the scalar product of the fields; see [2] for details). Thus we see that taking two functional derivatives w.r.t. the source brings down the operator , thereby identifying it with the two-point correlator for ,

which is trivially (by virtue of the far r.h.s.) a Green function of in the sense of (10).

We now understand how to express the Green function for the operator in terms of the heat kernel. But we’re after the connected two-point correlator (sometimes called the connected Green function), which encapsulates the one-loop contributions. Recall that the connected Feynman diagrams are generated by the effective action introduced above. After properly normalizing, we have only the piece which depends purely on :

Vassilevich [2] then provides a nice heuristic argument that relates this to the heat kernel (as well as a more rigorous treatment from spectral theory, for the less cavalier among you), which relies on the identity

for (strictly speaking, this identity is only correct up to a constant, but we may normalize away this inconvenience anyway; did I mention I’d be somewhat cavalier?). We then apply this identity to every (positive) eigenvalue of , whence

where

Let us pause to take stock: so far, we’ve merely elucidated eq. (9). And while this expression itself is valid in general (not just for manifolds with conical singularities) the physical motivation for this post was the investigation of divergences in the entanglement entropy (8). And indeed, the expression for the effective action (27) is divergent at both limits! In the course of regulating this behaviour, we shall see that the UV divergences in the entropy (8) are captured by the so-called heat kernel coefficients.

To proceed, we shall need the fact that on manifolds without boundaries (or else with suitable local boundary conditions on the fields), — really, the self-adjoint operator — admits an asymptotic expansion of the form

cf. eq. (67) of [1]. A couple technical remarks are in order. First, recall that in contrast to a convergent series — which gives finite results for arbitrary, fixed in the limit — an asymptotic series gives finite results for fixed in the limit . Second, we are ignoring various subtleties regarding the rigorous definition of the trace, wherein both (29) and (28) are properly defined via the use of an auxiliary function; cf. eq. (2.21) of [2] (n.b., Vassilevich’s coefficients do not include the normalization from the volume integrals; see below).

The most important property of the heat kernel coefficients is that they can be expressed as integrals of local invariants—tensor quantities which remain invariant under local diffeomorphisms; e.g., the Riemann curvature tensor and covariant derivatives thereof. Thus the first step in the procedure for calculating the heat kernel coefficients is to write down the integral over all such local invariants; for example, the first three coefficients are

where and were introduced in (18). A word of warning, for those of you cross-referencing with Solodukhin [1] and Vassilevich [2]: these coefficients correspond to eqs. (4.13) – (4.15) in [2], except that we have already included the normalization in our expansion coefficients (29), consistent with [1]. Additionally, note that Vassilevich’s coefficients are labeled with even integers, while ours/Solodukhin’s include both even and odd—cf. (2.21) in [2]. The reason for this discrepancy is that all odd coefficients in Vassilevich’s original expansion vanish, as a consequence of the fact that there are no odd-dimensional invariants on manifolds without boundary; Solodukhin has simply relabeled the summation index for cleanliness, and we have followed his convention in (30).

It now remains to calculate the constants . This is a rather involved technical procedure, but is explained in detail in section 4.1 of [2]. One finds

Substituting these into (30), and doing a bit of rearranging, we have

where we have suppressed the integration measure for compactness, i.e., ; we have also set the gauge field strength for simplicity, since we will consider only free scalar fields below. These correspond to what Solodukhin refers to as regular coefficients, cf. his eq. (69). If one is working on a background with conical singularities, then there are additional contributions from the singular surface [1]:

where in the last expression and , where are orthonormal vectors orthogonal to . In this case, if the manifold in (32) is the -fold cover constructed above, then the Riemannian curvature invariants are actually computed on the regular points , and are related to their flat counterparts by eq. (55) of [1]. Of course, here and in (33), refers to the conical deficit , not to be confused with the constants in (31).

Finally, we are in position to consider some of the physical applications of this method discussed in the introduction to this post. As a warm-up to the entanglement entropy of black holes, let’s first take the simpler case of flat space. Despite the lack of conical singularities in this completely regular, seemingly boring spacetime, the heat kernel method above can still be used to calculate the leading UV divergences in the entanglement entropy. While in this very simple case, there are integral identities that make the expansion into heat kernel coefficients unnecessary (specifically, the Sommerfeld formula employed in section 2.9 of [1]), the conical deficit method is more universal, and will greatly facilitate our treatment of the black hole below.

Consider a free scalar field with , where is some scalar function (e.g., for a massive non-interacting scalar, , on some background spacetime in dimensions, with a conical deficit at the codimension-2 surface . The leading UV divergence (we don’t care about the regular piece) in the entanglement entropy across this -dimensional surface may be calculated directly from the coefficient above. To this order in the expansion, the relevant part of is

where we have introduced the UV-cutoff (which appears as in the lower limit of integration, to make the dimensions of the auxiliary time variable work-out; this is perhaps clearest by examining eq. (1.12) of [2]) and the area of the surface . Substituting this expression for the effective action into eq. (8) for the entropy, we obtain

which is eq. (81) in [1]. As explained therein, the reason this matches the flat space result — that is, the case in which is a flat plane — is because even in curved spacetime, any surface can be locally approximated by flat Minkowski space. In particular, we’ll see that this result remains the leading-order correction to the black hole entropy, because the near-horizon region is approximately Rindler (flat). In other words, this result is exact for flat space (hence the equality), but provides only the leading-order divergence for more general, curved geometries.

For concreteness, we’ll limit ourselves to four dimensions henceforth, in which the flat space result above is

In the presence of a black hole, there will be higher-order corrections to this expression. In particular, in we also have a log divergence from the term:

where in the last step, we’ve dropped higher-order terms as well as the log IR divergence, since here we’re only interested in the UV part. From (33), we then see that the term results in an expression for the UV divergent part of the black hole entanglement entropy of the form

cf. (82) [1]. Specifying to a particular black hole solution then requires working out the projections of the Ricci and Riemann tensors on the subspace orthogonal to ( and , respectively). For the simplest case of a massless, minimally coupled scalar field () on a Schwarzschild black hole background, the above yields

where is the horizon radius; see section 3.9.1 of [1] for more details. Note that since the Ricci scalar , the logarithmic term represents a purely topological correction to the flat space entropy (36) (in contrast to flat space, the Euler number for a black hole geometry is non-zero). Curvature corrections can still show up in UV-finite terms, of course, but that’s not what we’re seeing here: in this sense the log term is universal.

Note that we’ve specifically labeled the entropy in (39) with the subscript “ent” to denote that this is the entanglement entropy due to quantum fields on the classical background. We now come to the confusion alluded in the opening paragraph of this post, namely, what is the relation between the entanglement entropy of the black hole and either the thermodynamic or gravitational entropies, if any?

Recall that in classical systems, the thermodynamic entropy and the information-theoretic entropy coincide: not merely formally, but ontologically as well. The reason is that the correct probability mass function will be as broadly distributed as possible subject to the physical constraints on the system (equivalently, in the case of statistical inference, whatever partial information we have available). If only the average energy is fixed, then this corresponds to the Boltzmann distribution. Note that this same logic extends to the entanglement entropy as well (modulo certain skeptical reservations to which I alluded before), which is the underlying physical reason why the Shannon and quantum mechanical von Neumann entropies take the same form as well. In simple quantum systems therefore, the entanglement entropy of the fields coincides with this thermodynamic (equivalently, information-theoretic/Shannon) entropy.

More generally however, “thermodynamic entropy” is a statement about the internal microstates of the system, and is quantified by the total change in the free energy w.r.t. temperature :

where the total effective action . Crucially, observe that here, in contrast to our partition function (22) for fields on a fixed background, the total Euclidean path integral that prepares the state also includes an integration over the metrics:

where represents the gravitational part of the action (e.g., the Einstein-Hilbert term), and the contribution from the matter fields. (For reference, we’re following the reasoning and notation in section 4.1 of [1] here).

In the information-theoretic context above, introducing a black hole amounts to imposing additional conditions on (i.e., more information about) the system. Specifically, it imposes constraints on the class of metrics in the Euclidean path integral: the existence of a fixed point of the isometry generated by the Killing vector in the highly symmetric case of Schwarzschild, and suitable asymptotic behaviour at large radius. Hence in computing this path integral, one first performs the integration over matter fields on backgrounds with a conical singularity at :

where on the r.h.s., represents the effective action for the fields described above; note that the contribution from entanglement entropy is entirely encoded in this portion. The path integral (41) now looks like this:

where on the far right is the semiclassical effective action obtained from the saddle-point approximation. That is, the metric is the solution to

at fixed . Since the saddle-point returns an on-shell action, is a regular metric (i.e., without conical singularities). One can think of this as the equilibrium geometry at fixed around which the metric fluctuates due to the quantum corrections represented by the second term . Note that the latter represents an off-shell contribution, since it is computed on singular backgrounds which do not satisfy the equations of motion for the metric (44).

To compute the thermodynamic entropy of the black hole, we now plug this total effective action into (40). A priori, this expression involves a total derivative w.r.t. , so that we write

except that due to the equilibrium condition (44), the second term in parentheses vanishes anyway, and the thermodynamic entropy is given by

This, then, is the precise relationship between the thermodynamic, gravitational, and entanglement entropies (at least for the Schwarzschild black hole). The thermodynamic entropy is a statement about the possible internal microstates at equilibrium — meaning, states which satisfy the quantum-corrected Einstein equations (44) — which therefore includes the entropy from possible (regular) metric configurations as well as the quantum corrections given by (39).

Note that the famous Bekenstein-Hawking entropy refers only to the gravitational entropy, i.e., . This is sometimes referred to as the “classical” part, because it represents the tree-level contribution to the path integral result. That is, if we were to restore Planck’s constant, we’d find that comes with a prefactor, while is order . Accordingly, is often called the first/one-loop quantum correction to the Bekenstein-Hawking entropy. (Confusion warning: the heat kernel method allowed us to compute itself to one-loop in the expansion of the matter action, i.e., quadratic order in the source , but the entire contribution appears at one-loop order in the -expansion.)

However, despite the long-standing effort to elucidate this entropy, it’s still woefully unclear what microscopic (read: quantum-gravitational) degrees of freedom are truly being counted. The path integral over metrics makes it tempting to interpret the gravitational entropy as accounting for all possible geometrical configurations that satisfy the prescribed boundary conditions. And indeed, as we have remarked before, this is how one would interpret the Bekenstein entropy in the absence of Hawking’s famous calculation. But insofar as entropy is a physical quantity, the interpretation of as owing to the equivalence class of geometries is rather unsatisfactory, since in this case the entropy must be associated to a single black hole, whose geometry certainly does not appear to be in any sort of quantum superposition. The situation is even less clear once one takes normalization into account, whereby — in most situations at least — the gravitational and matter couplings are harmoniously renormalized in such a way that one cannot help but question the fundamental distinction between the two; see [1] for an overview of renormalization in this context.

Furthermore, this entire calculational method relies crucially on the presence of an abelian isometry w.r.t. the Killing vector in Euclidean time, i.e., the rotational symmetry mentioned above. But the presence of such a Killing isometry is by no means a physical requisite for a given system/region to have a meaningful entropy; things are simply (fiendishly!) more difficult to calculate in systems without such high degrees of symmetry. However, this means that cases in which all three of these entropies can be so cleanly assigned to the black hole horizon may be quite rare. Evaporating black holes are perhaps the most topical example of a non-static geometry that causes headaches. As a specific example in this vein, Netta Engelhardt and collaborators have compellingly argued that it is the apparent horizon, rather than the event horizon, to which one can meaningfully associated a coarse-grained entropy à la Shannon [4,5]. Thus, while we’re making exciting progress, even the partial interpretation for which we’ve labored above should be taken with caution. There is more work to be done!

You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.

— John von Neumann to Claude Shannon, as to what the latter should call his uncertainty function.

I recently read the classic article [1] by Jaynes, which elucidates the relationship between subjective and objective notions of entropy—that is, between the information-theoretic notion of entropy as a reflection of our ignorance, and the thermodynamic notion of entropy as a measure of the statistical microstates of a system. These two notions are formally identical, and can be written

where is the probability of the outcome/microstate. However, as Jaynes rightfully stresses, the fact that these concepts share the same mathematical form does not necessarily imply any relation between them. Nonetheless, a physicist cannot resist the intuitive sense that there must be some deeper relationship lurking beneath the surface. And indeed, Jaynes paper [1] makes it beautifully clear that in fact these distinct notions are referencing the same underlying concept—though I may differ from Jaynes in explaining precisely how the subjective/objective divide is crossed.

(As an aside, readers without a theoretical physics background may be confused by the fact that the thermodynamic entropy is typically written with a prefactor , known as Boltzmann’s constant. This is merely a compensatory factor, to account for humans’ rather arbitrary decision to measure temperatures relative to the freezing point of water at atmospheric pressure. Being civilized people, we shall instead work in natural units, in which .)

As a prelude, let us first explain what entropy has to do with subjective uncertainty. That is, why is (1) a sensible/meaningful/well-defined measure of the information we have available? Enter Claude Shannon, whose landmark paper [2] proved, perhaps surprisingly, that (1) is in fact the unique, unambiguous quantification of the “amount of uncertainty” represented by a discrete probability distribution! Jaynes offers a more intuitive proof in his appendix A [1], which proceeds as follows. Suppose we have a variable , which assumes any of the discrete values . These could represent, e.g., particular outcomes or measurement values. To each , we associated the corresponding probability , with . The question is then whether one can find a function which uniquely quantifies the amount of uncertainty represented by this probability distribution. Remarkably, Shannon showed [2] that this can indeed be done, provided satisfies the following three properties:

1. is a continuous function of .
2. If all are equal, then (since probabilities sum to 1), and is a monotonically increasing function of .
3. If we decompose an event into sub-events, then the original value of must be the weighted sum of the individual values of for each composite event.

The first requirement simply ensures that is well-defined. The second encodes the fact that if all events are equally likely, then the uncertainty increases monotonically with the number of events. The third property is referred to as the composition law by Jaynes, and is easily explained with the help of figure 6 in Shannon’s paper [2], which I’ve reproduced here:On the left, we have three probabilities , , and , so the uncertainty for this distribution is given by . On the right, we’ve coarse-grained the system such that at the highest level (that is, the left-most branch), we have two equally likely outcomes, and hence uncertainty . However, this represents a different state of knowledge than the original , insofar as it disregards the fine-grained information in the lower branch. Obviously, our uncertainty function should not change simply because we’ve chosen to group certain clusters of events together. Thus in order for to be invariant under this composition, we must have

where the prefactor is the probabilistic weight associated with this composition of events.

To generalize this, it is convenient to represent the probabilities as rational fractions,

(The fact that is continuous enables us to make this assumption without loss of generality). This enables us to fix the form of by applying the composition law to the case in which we coarse-grain equally likely alternatives into clusters of size , which results in the requirement

To see this, consider the example given by Jaynes, in which we take , and let . If all outcomes are equally likely, then the uncertainty, given by the left-hand side of (4), is . As in the previous example, we want this uncertainty to be preserved under coarse-graining, where in this case we group the possible outcomes into clusters of size , as illustrated in the figure below:

Thus we see that at the highest level, the uncertainty — had we thrown away the fine-grained information in each cluster — is . The second term on the right-hand side of (4) is then the weighted value of for each cluster, as in the simpler example above.

The final trick is to consider the case where all , in which case , and (4) reduces to , the solution to which is , where is some constant (note that by the second condition, ). Substituting this into (4), we obtain

where we’ve used the fact that probabilities sum to 1 to write . QED.

Thus, on the subjective side, we see that entropy (1) is the least biased representation of our state of knowledge. And this fact underlies a form of statistical inference known as maximum entropy or max-ent estimation. The basic problem is how to make inferences — that is, assign probabilities — based on incomplete information. Obviously, we want our decisions to be as unbiased as possible, and therefore we must use the probability distribution which has the maximum entropy subject to known constraints—in other words, the distribution which is maximally non-committal with respect to missing information. To use any other distribution is tantamount to imposing additional, arbitrary constraints, thereby biasing our decisions. In this sense, max-ent estimation may be regarded as a mathematization of rationality, insofar as it represents the absolute best (i.e., most rational) guess we could possibly make given the information at hand.

To make contact with the objective notion of entropy in thermodynamics, let’s examine the max-ent procedure in more detail. Suppose, as in the examples above, that we have a variable which assumes discrete values with associated probabilities . In a typical statistical inference problem, we don’t know what these probabilities are, but we know the expectation value of some function ,

Now the question is, based on this information, what is the expectation value of some other function ? As it stands, the problem is insoluble: we need to know values to compute , but the normalization

and (6) collectively provide only 2 constraints. Max-ent is the additional principle which not only makes this problem tractable, but ensures that our distribution will be as broad as possible (i.e., that we don’t concentrate our probability mass any more narrowly than the given information allows).

As the name suggest, the idea is to maximize the entropy (1) subject to the constraints (6) and (7). We therefore introduce the Lagrange multipliers and , and minimize the Lagrange function

Imposing then returns the two constraint equations, along with

We can then redefine to absorb the 1, whereupon we obtain the more elegant result

To solve for the Lagrange multipliers, we simply substitute this into the contraints (7) and (6), respectively:

Now observe that the sum in these expressions is none other than the partition function for the canonical ensemble at inverse temperature :

We therefore have

i.e., is the cumulant generating function (in QFT, this would be the generator of connected Feynman diagrams; see effective action), corresponds to the thermodynamic energy, and (10) is none other than the Boltzmann distribution. Of course, this naturally extends to multiple functions , which may generically depend on other parameters ; see [1] for details—Jaynes paper is a gem, and well-worth reading.

This essentially completes the connection with the thermodynamic concept of entropy. If only the average energy of the system is fixed, then the microstates will be described by the Boltzmann distribution (10). This is an objective fact about the system, reflecting the lack of any additional physical constraints. Thus the subjective and objective notions of entropy both reference the same underlying concept, namely, that the correct probability mass function is dispersed as widely as possible given the constraints on the system (or one’s knowledge thereof). Indeed, a priori, one may go so far as to define the thermodynamic entropy as the quantity which is maximized at thermal equilibrium (cf. the fundamental assumption of statistical mechanics).

Having said all that, one must exercise greater care in extending this analogy to continuous variables, and I’m unsure how much of this intuition survives in the quantum mechanical case (though Jaynes has a follow-up paper [3] in which he extends these ideas to density matrices). And that’s not to mention quantum field theory, where the von Neumann entropy is notoriously divergent. Nonetheless, I suspect a similar unity is at play. The connection between statistical mechanics and Euclidean field theory is certainly well-known, though I still find thermal Green functions rather remarkable. Relative entropy is another example, insofar as it provides a well-defined link between von Neumann entropy and the modular hamiltonian, and the latter appears to encode certain thermodynamic properties of the vacuum state. This is particularly interesting in the context of holography (as an admittedly biased reference, see [4]), and is a topic to which I hope to return soon.

References

1. E. T. Jaynes, “Information Theory and Statistical Mechanics,” The Physical Review 106 no. 4, (1957).
2. C. E. Shannon, “A Mathematical Theory of Communication,” The Bell System Technical Journal 27, (1948).
3. E. T. Jaynes, “Information Theory and Statistical Mechanics II,” The Physical Review 108 no. 2, (1957).
4. R. Jefferson, “Comments on black hole interiors and modular inclusions,” arXiv:1811.08900.

The issue of boundary conditions in AdS/CFT has confused me for years; not because it’s intrinsically complicated, but because most of the literature simply regurgitates a superficial explanation for the standard prescription which collapses at the first inquiry. Typically, the story is presented as follows. Since we’ll be interested in the near-boundary behaviour of bulk fields, it is convenient to work in the Poincaré patch, in which the metric for Euclidean AdS reads

where parametrizes the transverse directions. Now consider the action for a bulk scalar , dual to some primary operator in the CFT:

where . The on-shell equations of motion yield the familiar Klein-Gordon equation,

This is a second-order differential equation with two independent solutions, whose leading behaviour as one approaches the boundary at scale like , where are given by

For compactness, we shall henceforth set the AdS radius . The general solution to the Klein-Gordon equation in the limit may therefore be expressed as a linear combination of these solutions, with coefficients and :

The subleading behaviour plays no role in the following, so we’ll suppress it henceforth.

Note that the condition imposes a limit on the allowed mass of the bulk field, namely that . This is known as the Breitenlohner-Freedman (BF) bound. (Unlike in flat space, a negative mass squared doesn’t necessarily imply that the theory is unstable; intuitively, the negative curvature of AdS compensates for the backreaction, provided the excitations are sufficiently small). Furthermore, if one integrates the action (2) from to the cutoff at , the result will be finite for , i.e., . Thus is a solution for all masses above the BF bound.

For , the situation is slightly more subtle. One can show that the boundary term obtained by integrating (2) by parts is non-zero only if ; i.e., for , there exists a second, inequivalent action (read: theory). And in this case, the bulk integral up to the cutoff is finite only for , which corresponds to the restricted mass range

The upper limit is called the unitarity bound, since the limit corresponds to the constraint, from unitarity, on the representation of conformal field theories. Hence for masses below the unitarity bound (but above the BF bound, of course), one may choose either or , which implies two different bulk theories for the same CFT. More on this below. (Slightly more technical details can be found in section 5.3 of [1]).

With the above in hand, the vast majority of the literature simply recites the “standard prescription”, in which one identifies as the source of the boundary operator , which is in turn identified with . This appears to be based on the observation that, since by definition, the second term dominates in the limit, which requires that one set in order that the solution be normalizable—though as remarked above, this is clearly not the full story in the mass range (6). Sometimes, an attempt is made to justify this identification by the observation that one can multiply through by , whereupon one sees that

In accordance with the extrapolate dictionary, this indeed suggests that sources the bulk field . But it does not explain why should be identified as the boundary dual of this bulk field; that is, it does not explain why and should be related in the CFT action as

since a priori these are independent coefficients appearing in the solution to the differential equation above. Furthermore, it does not follow from the fact that the bulk coefficient has the correct conformal dimension as the boundary source field (and that both will be set to zero on-shell) that the two must be identified: this may be aesthetically suggestive, but it is by no means logically necessary.

Some clarification is found in the classic paper by Klebanov and Witten [2]. They observe that the extrapolate dictionary — or what one would identify, in more modern language, as the HKLL prescription for bulk reconstruction — can be written

where is the bulk-boundary propagator with conformal dimension ,

and is the boundary limit of the bulk field , which serves as the source for the dual operator . Choosing then implies that in order to be consistent with (5), we must identify , and

which one obtains by taking the limit of the integral above. We then recognize the position-space Green function , whence

which is the statement that is the source for the one-point function . That is, recall that the generating function can be written in terms of the Feynman propagator as

whereupon the functional derivative with respect to the source yields the -point function

Hence, taking a single derivative, we have

which is precisely the form of (12), with and , thus explaining the relation (8). (Note that we’re ignoring a constant factor here; the exact relation between and the vev is fixed in [2] to ). Of course, setting causes the one-point function to vanish on-shell. The fact that we set the sources to zero resolves the apparent tension between setting for normalizability, and simultaneously having it appear in (12): the expansion (5) was obtained by solving the equations of motion, so we only have on-shell.

Thus we see that the bulk-boundary correspondence in the form (9) provides the extra constraint needed to relate the coefficients and in the expansion (5) as (vev of) operator and source, respectively, and that choosing the alternative boundary condition simply interchanges these roles. It is an interesting and under-appreciated fact that any holographic CFT therefore admits two different bulk duals, i.e., two different quantum field theories are encoded in the same CFT Lagrangian!

The above essentially summarizes the basic story for the usual case, i.e., a CFT action of the form

However, the question I encountered during my research that actually motivated this post is what happens to these boundary conditions in the presence of a double-trace deformation,

To that end, suppose we start with a CFT at some UV fixed point. If the coupling is irrelevant (), then the deformation does not change the effective field theory, since it only plays a role in high-energy correlators. However, if the coupling is relevant (), then the perturbation induces a flow into the IR. And it turns out that the boundary conditions change from (in the UV) to (in the IR) as a result.

First, let’s understand why we need to start with the “alternative boundary condition” in the UV. The paper [3] motivates this by examining the Green function in momentum space, which scales like , where . Thus in the UV, we choose so that the Green function converges. This explanation isn’t entirely satisfactory though, since it’s not clear why this should converge with in the IR.

In this sense, a better explanation for choosing in the UV is obtained simply by examining the mass dimensions in the coupling term (17): in order for to be dimensionless, we must have . The condition that be a relevant coupling then implies that . Since can only satisfy this condition for masses below the Breitenlohner-Freedman bound, we are constrained to choose . Note that from this perspective, the choice is equally valid, but it corresponds to an irrelevant perturbation instead, so nothing interesting happens.

Understanding why the theory flows to at the new IR fixed point is more complicated, and requires computing the beta functions for the couplings. This is done in [4], where they compute the evolution equation for the coupling (i.e., the beta function for ), which enables them to identify the fixed points which correspond to the alternative and standard quantizations in the UV and IR, respectively.

It is then natural to ask what happens at intermediate points along the flow, since the above applies only to the UV/IR fixed points, at which a single choice (either or ) is determined. This leads to the notion of mixed boundary conditions, as introduced by Witten in [5]. In order to understand his proposal, let’s examine the relationship between and above from another angle. The term in the effective action corresponding to the single-trace insertion is

Taking functional derivatives with respect to the source then yields the operator expectation value,

which of course simply restates the relationship (15). As explained above, the standard quantization identifies and , which amounts to a trivial relabeling of the variables in this expression. However, the alternative quantization, in which and , is slightly more subtle: a similar relabeling would work for the present case, but not for a multi-trace deformation. Instead, consider varying the effective action with respect to the operator :

Note that unlike (19), this is an operator statement; it may seem like an odd thing to write down, but in fact it’s formally the same expression that arises in the derivation of Ward identities. Specifically, if one considers an infinitesimal shift of the field variable , then invariance of the generating functional requires

where we’ve taken the action (16) with in place of . The resemblance to Noether’s theorem is intriguing, but I’m not sure whether there’s a deeper relation to symmetries in QFT at work here. In any case, Witten’s proposal is that in the alternative quantization, one should identify

which simply states that . Strictly speaking, we can’t identify in this expression, since it doesn’t tell us the vev of the operator . Rather, to find the expectation value — in particular, that it should be identified with — requires us to solve the bulk equations of motion with this boundary condition, substitute the result into the bulk action, and then take variational derivatives with respect to the source as usual. In the end of course, one recovers the simple identification , hence Witten actually writes (22) directly with in place of , with the understanding that the expression is purely formal.

Now consider a double-trace deformation,

where we’ve rescaled the coupling for consistency with the cited literature (and the convenient factor of 2). According to (22), this leads to the mixed boundary condition

As observed in [3], this neatly interpolates between the standard and alternative prescriptions above. To see this, observe that since the deformation is explicitly chosen to be relevant, the dimensionless coupling grows as we move into the IR, i.e., as the energy scale . Since , (5) requires that we identify , with the standard choice . Conversely, the coupling shrinks to zero in the UV, where , and hence we identify as the vev of the operator instead, with the alternative choice .

Witten’s prescription extends to higher-trace deformations as well, though I won’t discuss those here. The point, for the purposes of the present post, is that it provides a unified treatment of boundary conditions in holography that smoothly interpolates between the UV/IR, and hence allows one to naturally treat the bulk dual all along the flow.

I am grateful to my former officemate, Diptarka Das, for several enlightening discussions on this topic, and for graciously allowing me to pester him across several afternoons until I was satisfied.

References

Rindler space provides a convenient example to elucidate some basic properties of AQFT — specifically Tomita-Takesaki theory — in what is arguably the case of greatest interest to high-energy theorists. We shall begin by introducing a few fundamental objects in the theory, and then apply them to Rindler space, loosely following the excellent review by Witten [1], as well as the work of Papadodimas [2] and Papadodimas & Raju [3].

We begin with the -algebra of observables in some spacetime region acting on the Hilbert space of some quantum field theory (accordingly, we will be concerned with Type III algebras throughout this post). In fact, we shall be slightly more restrictive and assume that is a von Neumann algebra. Recall that a -algebra is the set of bounded operators which is closed both in the norm topology and under the adjoint operation. The weak closure of a -algebra is a von Neumann algebra, which further implies by von Neumann’s double commutant theorem. Additionally, we take to be equipped with the cyclic and separating vector . Cyclic means that states , are dense in ; separating means that iff . (Note that, glossing over various representational subtleties, the vector therefore serves as the vacuum state of ). For concreteness, as well as later convenience, we shall take to be the right Rindler wedge , and denote . The commutant is then the algebra in the left Rindler wedge , . (The very last equality is sometimes called Haag duality, and applies when and are causal complements).

Given a von Neumann algebra , Tomita-Takesaki theory admits a canonical construction of the commutant , which we will use to derive some relationships between the left and right Rindler wedges. The starting point is the antilinear map , defined by

where recall that an antilinear map between complex vector spaces satisfies and . Note that is a state-dependent operator, in that it is defined in terms of its action on the cyclic and separating state . To minimize clutter, we shall omit the subscript on henceforth, but one should bear in mind that most of the analysis below does not hold for general (excited) states, which fail the cyclic/separating condition.

Case in point: the fact that is separating is necessary for to be well-defined, otherwise there would exist an annihilation operator such that but , which contradicts the definition (1). The fact that is cyclic then ensures that it maps to a dense set of states on . Additionally, it follows from the definition of the action of on an arbitrary state that, as an operator, we must have . Note that by the square of the operator, we do not mean ; rather, this implies that is its own inverse, . It turns out that , the corresponding operator on the commutant ; see [1] for an elementary proof. Lastly, simply taking to be the identity operator in (1) shows that leaves the vacuum invariant, .

Since is invertible, it admits a unique polar decomposition,

where is an antiunitary operator (meaning ) called the modular conjugation, and is a self-adjoint operator called the modular operator. The decomposition (2), combined with the fact that , implies that

which is sometimes taken as its definition. Additionally, since is a positive Hermitian operator, we may define the modular Hamiltonian such that

(N.b., some authors — e.g., Witten [1] — define with an additional factor of such that . Our notation here is consistent with [2,3]). Since both and leave the vacuum state invariant, this property extends to the modular operator as well, . It it important to note that does not have support purely within the right Rindler wedge , i.e., on the algebra , but acts on the full Hilbert space defined over both the left and right wedges. This is immediate from (4), and the fact that the domain of is . We shall return to this point below. (By extension, , but this does not contradict the fact that is separating since is not localized to any subregion).

For the last of the preliminaries, we note the following useful identities:

These essentially follow from the fact that in conjunction with the polar decomposition. Combining (5) with the aforementioned fact that leads to and . Finally, expressing in terms of and via (2) enables one to show that it likewise leaves the vacuum invariant; i.e., we have

Now, the fundamental result of Tomita-Takesaki theory is comprised of the following two facts: first, the modular operator defines a 1-parameter family of modular automorphisms

In other words, the algebra is invariant under modular flow. Second, the modular conjugation induces an isomorphism between the algebra and its commutant ,

i.e., , defines an element such that . The isomorphism (8) is rather remarkable: it allows one to map operators between the left and right Rindler wedges, and features crucially in the works of Papadodimas and Raju on reconstructing black hole interiors [2,3].

To demonstrate how this works, let be a unitary operator with support in . This creates a state which is indistinguishable from the vacuum state for all observers (i.e., all operators ) in :

where in the last step, we used the fact that (note that this wouldn’t have worked if were non-unitary). Now consider the state

Perhaps surprisingly, this state is indistinguishable from the vacuum for an observer in , i.e., it represents an excitation localized entirely outside ! (One has to be careful in saying that it’s localized in , since under suitable time evolution it may evolve through the left Rindler horizon). This is straightforward to show, using the properties of the modular conjugation above:

where is an operator in the commutant. Indistinguishably of from the vacuum for observers in then follows by the same logic as (9). However, while it is true that the state is (initially) localized entirely in , the same is not true for the operator ! This is due to the fact, mentioned above, that has support in both and , and hence the operator . This example highlights the subtle yet crucial distinction between localized states vs. localized operators. Another way to emphasize this is that, as operators,

(since these exist in different algebras), even though as states,

as shown in (11). [I am grateful to Kyriakos Papadodimas for clarifying this distinction to me].

Note the similarity to the Reeh-Schlieder theorem at play here: it appears that we’ve created an excitation in by acting with a unitary operator in the spacelike separated region ; but this isn’t quite true, since we relied on the modular operator to do so, which lives in both and . To accomplish this feat purely with operators whose support lies entirely in would require these operators to be non-unitary, as explained in the eponymous post. This suggests a deep relationship between locality and unitarity embedded within this framework.

Now, let’s make the above formalism a bit more concrete. In the case of Rindler space specifically, it turns out that the modular Hamiltonian is proportional to the generator of Lorentz boosts. Witten [1] offers an intuitive path integral explanation of this fact. The derivation is non-rigorous, since it assumes a factorized Hilbert space , but in the course of doing so it provides some insight into where and why this factorization fails, and highlights which operators do and do not exist—the reduced density matrix for either wedge being a prime example of the latter.

First, consider the Euclidean path integral that prepares the vacuum state . This is illustrated in figure (a) of the image below, which we’ve taken from [1]. The shaded region represents an integral over the half-plane which produces the vacuum state on the boundary. Similarly, integrating over the upper half-plane produces the bra . Stitching these together by integrating over all boundary values at yields the density matrix . As we shall shortly see, this is related to the modular operator .

Now pretend that we could factorize the Hilbert space as , where the fields at are respectively supported in , . Strictly speaking, this defines and as open subregions of , but taking the weak closure to construct the associated von Neumann algebras entails including the bifurcation point , as well as the lightlike Rindler horizons themselves. In some sense, this is the source of the problem, since we need to regulate these boundary regions when performing the path integral over either subspace. Specifically, consider the reduced density matrix for the right Rindler wedge,

which is illustrated in figure (b). The boundary condition indicates that we identify the fields in the upper and lower half-plane when computing the partial trace, while the cut along indicates that are kept free. The regularization issue mentioned above arises from the fact that local operators at the boundary point are not well-defined, but must be treated as smeared distributions over some finite region. In typical calculations of entanglement entropy, one excises some small region around the endpoint of the cut, which acts as a UV cutoff that regulates the divergent result. [I am grateful to Jamie Sully for discussions on this issue]. But we can’t strictly do this while preserving the (von Neumann) algebraic structure above, which is one reason this procedure doesn’t rigorously hold.

Now consider figure (c), which represents the Euclidean wedge with opening angle . Leaving both boundary conditions unfixed implies that this computes some operator, which acts as a rotation in the Euclidean plane:

where in the second expression we’ve gone to Lorentzian time . But the latter is precisely a Lorentz boost of the real plane with boost parameter , and we know that the generator of such a boost can be expressed as

Quite independent of the above analysis, generates a Lorentz boost by a real parameter , i.e., the operator . Now if the factorization of the Hilbert space were legitimate, we could split (with domains of integration restricted to the associated wedges), such that the path integral over the wedge in figure (c) yields the operator , where . Such an operator propagates degrees of freedom from the upper boundary of the wedge at , to the lower boundary. By setting , one sees that we recover the path integral in figure (b), and therefore

Of course, the same analysis holds for the left wedge, with generating . We then recover the modular operator by appealing to the tensor factor structure, which implies , and hence

where the promised proportionality between the modular Hamiltonian and the generator of Lorentz boosts is (note that many authors absorb the factor of into the definition of ).

The formulation in terms of the wedge operator provides a cute illustration of how maps states in to states in and vice-versa, cf. (13). The interpretation is that removes a wedge of angle from the slice in as depicted in figure (c), and adds an equal wedge in (in the positive half-plane; i.e., it rotates the slice). Since we started with vacuum in , one can think of this as rotating excitations in by an angle ; setting thus moves an operator insertion from to , which is precisely the action of in equation (13) above.

As mentioned above, this path integral picture is not strictly valid since the Hilbert space of Type III factors does not admit a tensor product structure. One way in which this manifests is that cannot be split into in such a way as to yield operators whose support lies solely in . This implies that the reduced density matrices do not really exist! Given how much of our understanding of entanglement in quantum field theory is based on the reduced density matrix — and more recently, at least in certain holographic applications, the modular Hamiltonian itself — this is in principle a rather serious problem. It remains to be seen how much of our ideas about entanglement must accordingly be revised, or whether this is analagous to Haag’s theorem about the non-existence of the interaction picture: theoretically damning, but practically irrelevant.

To see how one might get around this issue, consider two arbitrary, well-defined states . The non-existence of the operator manifests in a universal UV divergence in at , as explained above. Nonetheless, the matrix elements are still well-defined. An intuitive argument for this is that in the energy eigenbasis, correlation functions between well-defined (read: finite) states simply don’t see the problematic UV modes. [I am grateful to Ben Freivogel for this explanation]. Of course, one could certainly inquire as to the ontological status of matrix elements of an operator which does not technically exist. But, at least at an epistemic level, this offers a potential back-door to entanglement entropy. That is, taking the trace of the reduced density matrix as is typically done is not a rigorous method for computing entanglement, since as discussed above is not a valid operator in the theory. However, insofar as entanglement is essentially a measure of correlation functions, one could imagine performing the sum over all possible correlators instead. While the resulting sum may still be divergent, the individual terms are well-defined, and this may lend some support to the practical validity of many existing results despite the false foundation.

References

1. E. Witten, “Notes on Some Entanglement Properties of Quantum Field Theory,”, arXiv:1803.04993
2. K. Papadodimas, “A class of non-equilibrium states and the black hole interior,” arXiv:1708.06328
3. K. Papadodimas and S. Raju, “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity,” arXiv:1310.6335