Interior operators in AdS/CFT

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]:

PR_blackhole

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].

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