This page provides a space for additional discussion of workshop topics

Holography

The holography challenge -- Steve Giddings 6/23/15

Early in the workshop there was significant discussion on the problem of "decoding the hologram." There are some different views about how/whether a sharp "decoding map" can be defined, and what is necessary to be able to sharply resolve and describe all the relevant bulk physics, particularly on scales small as compared to the AdS radius.

At the final workshop discussion, I stated the following "holography challenge:"

Suppose we assume that H_bulk = H_bdy. Provide a precise formlation, purely in terms of CFT data, of bulk observables that reduce to observables of local quantum field theory on scales <<R_ADS in a controlled approximation for small but nonzero g_s.

Some of those present indicated that they thought that the HKLL construction meets the challenge, and none objected to the statement that it does, at least on an AdS background. Of course, as has been discussed, there are additional issues about other backgrounds, such as black holes.

Please share your thoughts whether you think HKLL does indeed meet the challenge -- or whether you see limitations to or problems with the statement that it does. (In particular, there are questions about a systematic expansion in g_s, which I raised in my talk during the "hologram week;" resolution of these seem to require certain properties of smearing functions. There is also the set of questions which resulted in the discussion about code subspaces, or the role of gauge invariance.)

Inflationary cosmology

IR safe cosmo observables -- Martin Sloth 4/28/15

I just want to reply here to Richard's comments directed at me during the discussion from last friday, since I could not be there myself.

I guess that the point is not that I/we don’t want to discuss how to formulate the prescription of defining IR safe observables of [1] at a full quantum operator level, but rather that it shouldn’t be necessary given that we have proven/conjectured that all the infrared effects can be captured by classical perturbations of the geometry (see [1] and [2]).

Of course any macroscopic classical object can ultimately be described by quantum mechanics, but it is just not very practical (and if done correct it should give the classical result exponentially precise).

In the same way the dependence of the IR safe observables on the IR resolution scale should be understandable from a full fundamental quantum field theory treatment, but it is just not very practical, and it should agree exponentially precisely with our treatment.

Since we are utilising that the IR part of the qft correlation functions can be understood classically, the only caveat would be if perturbations are not becoming fully classical on super-horizon scales. But there are several papers addressing this issue in a full quantum treatment -- maybe [3] is a good recent reference in the present context.

First, my thanks to Martin for his reply. I wish we could have discussed this in person at KITP, and I hope we will be able to do so in the near future. Second, while I have great admiration for the progress Martin and others have made, the word proof'' has a very strong meaning and I don't believe it applies to what has been done. In particular, there has not been any precise quantification of how good is the approximation of representing IR loop effects by simply distorting tree order factors of k^2 to k^i k^j contracted into a classical, long wavelength metric'' and then averaging this separately from the other operators. I would like to compare exact one loop results with the approximation. It ought to be possible to see what parts of the graviton and zeta propagators this recovers and what parts it does not. Right now the only thing I am sure about is that that approximation correctly recovers the dependence on the IR cutoff on the initial value surface.

I know from my own work that there is a distinction between the IR cutoff and secular growth caused by modes which experience horizon crossing at whatever is the current time. It seems to me that such growth effects are not exponentially suppressed, and may not be correctly described by the approximation. In arXiv:1204.1784, Shun-Pei Miao and I consider a simple example of this procedure in the context of a coincident propagator, see equations (41-43) on page 9. As expected, the technique correctly reproduces the cutoff dependence but it gives a false result for secular growth when the exact result shows none.

I am also worried about assumptions about the classicality of quantum IR effects from gravitons. This is probably true at some level, but it isn't clear in what sense. For scalar potential models on nondynamcial de Sitter background this can be made rigorous through proving that Starobinsky's stochastic formalism really does exactly capture the leading IR logarithms at each order in the loop expansion. Nothing of the kind exists yet for the GR + inflaton system. Further, naive extrapolations of Starobinsky's formalism have been shown not to correctly reproduce the one loop IR logarithms acquired by the wave function of a massless fermion as it propagates through the ensemble of gravitons created by de Sitter inflation. This really is a ``proof'' that these naive extrapolations are wrong because they are being compared against exact one loop results. The detailed, explicit and fully renormalized computations are described in arXiv:0803.2377, and Shun-Pei Miao spoke on this paper at KITP on April 28. In particular, each of the following plausible-sounding procedures fails to produce correct results:
(1) Employing IR truncated mode sums for the fields;
(2) Removing the dimensional regularization; and
(3) Ignoring differentiated gravitons.

Finally, the various ``Ward identities'' being invoked seem to contain hidden assumptions about analyticity which may not correct. As stated, the results are only true for attaching legs with k=0, which are not observable. To extend them to give observable results one must assume that the relations remain approximately true even for nonzero k. The problem with this sort of extrapolation should be clear from the fact that it could be used to fix the topology of space by making local measurements. For example, there are no extra coordinate symmetries, and presumably no related Ward identities, if the spatial manifold is T^3, rather than R^3. (In fact the strictly constant modes of T^3 are NOT pure gauge.) Yet we should not be able to use local measurements to tell the difference between R^3 and T^3 if the T^3 radii are super-horizon.

\epsilon suppression in IR safe correlators -- Yuko Urakawa, 5/7/15

I’d like to make a comment on 1/\epsilon enhancement Richard mentioned in the working group discussion on Apr. 24th, since I was not there unfortunately.

Richard mentioned that the IR safe correlators get the correction like \zeta(t, x) \zeta(t, y) [1 + \zeta + \zeta^2 ....] (#) and additional contributions in the square brackets, which appear after the coordinate change, can give P^2 ~ (G H^2/\epsilon)^2 which is 1/\epsilon bigger than a typical contribution from the loop correction. Here, P is the amplitude of zeta (multiplied by k^3).

I think that the additional contributions are not of order P^2, but they are also multiplied by slow-roll parameters and become same order as the contribution from the loop corrections. (If they are not of the same order, the additional piece in the IR safe correlator cannot cancel the IR divergence from the loop corrections.)

Here, let me explain how the "missing" slow-roll parameters appears.

The IR safe correlator is, in the Fourier space, given by <\zeta(t, e^{\zeta_L} k) \zeta(t, e^{\zeta_L} k')> where k and k' are wavenumbers and \zeta_L is an IR mode much softer than k and k'. (Here, for illustrative purposes, let me take the Fourier space. But basically the same argument follows also in the position space.)

Skipping the detail of the computation, the IR safe correlator can be given by a compact expression as <\zeta(t, e^{\zeta_L} k) \zeta(t, e^{\zeta_L} k')> ~ k^{-3} [1+ 1/2<\zeta_L^2> \partial^2_{ln k} + ... .] k^3 <\zeta(t, k) \zeta(t, k')>. (##) At first glance, one may think that the additional piece, the second term, is of order P^2, but since \partial^2_{ln k} operates on k^3<\zeta(t, k) \zeta(t, k')> ~ P, the second term is, (n_s-1)^2 <\zeta_L^2> <\zeta(t, k) \zeta(t, k’)>, (###) which is now multiplied by (n_s-1)^2, which is typically of order \epsilon^2. (In (##), k^3 appears when we correctly take into account the volume effect.)

This shows that a naive counting of \epsilon can tell us a wrong answer.

After some manipulation, the one loop corrections can be given by the terms of order \epsilon^2 \times P^2 and IR regular terms. Then, the additional contribution, given in (###), can cancel the IR divergence of the loop corrections.

My thanks to Yuko for a clear explanation, with supporting references, of how an IR safe observable can preserve the pattern of \epsilon suppression of the naive observables at lowest order. However, there are some problems:
(1) The quantity (n_s - 1) involves the 2nd slow roll parameter, in addition to the first one. The 2nd slow roll parameter does not need to be small and it is not, in any case, identical to the factors of 1/\epsilon which \zeta propagators carry.
(2) It isn't clear that the next order corrections have any suppression at all.
(3) The proposed observable is wildly nonlocal through its dependence upon \zeta_L.
(4) What is long wavelength'' presumably changes with time, and it's not clear why the uncorrected Fourier modes should be used to determine what is long wavelength'' when this is not trusted for the original correlator.
(5) The proposed observable is not invariant, whereas all true invariants I know exhibit the problem with \epsilon-suppression.

The last point is especially relevant in view of Yuko's recent work with Tanaka (arXiv:1402.2076) on which she spoke at KITP on April 13. If I understood this correctly, they claim the absence of secular growth effects ONLY for invariant operators in the presence of a special state.

I would also like to use this opportunity to raise with Yuko (and everyone else) an issue which has been bothering me for some time about gauge fixing. She and Tanaka have argued that the factors of e^{\zeta_L} must be added to the naive correlator because requiring the graviton to be transverse (\partial_i \gamma_{ij}(t,x) = 0) fails to completely fix the spatial gauge. Suppose the spatial manifold is T^3, rather than R^3. Then I think transversality really does fix the spatial gauge. Does this mean there is no need for the factors of e^{\zeta_L} of T^3? But without those factors, loop corrections to the \zeta-\zeta correlator on T^3 depend upon the coordinate radius of T^3, which is just another sort of IR cutoff that one would think should not matter.

If I correctly understood the May 6 KITP talk by Andy Strominger, it is wrong in flat space to consider the ``extra symmetries'' of R^3 as gauge transformations. Why would the same not be true for inflation? In which case we do not need to gauge fix them, or make observables invariant under these symmetries, and the justification for the factors of e^{\zeta_L} must be sought elsewhere.

Reply toRichard Woodard--Yuko Urakawa6/7/15 Thanks for your reply, Richard. (I wrote the following in another place, but let me copy and paste the same here.)

About (1)
Ok. Remember that in case the second slow-roll parameter \eta is not very small, this should be taken into account also in the loop computations. And this can change the result. I expect that also in such case the cancellation between the IR divergence from loops and the contributions due to the change of the distance measure to the invariant distance takes place, when we evaluate the (local) gauge invariant quantity.

About (2)
The computation will become messy, but I expect that as far as the slow-roll approximation holds, the cancellation can be checked also at higher orders.

In the above refs, the cancellation was checked including the quadratic terms of the slow-roll parameters, assuming that all the slow-roll parameters are small. The interpretation of the result is different, but at the algebraic level, the same result was derived by Steve and Martin in http://arxiv.org/abs/1005.1056
Our purpose in 1009.2947 was to derive the condition on the vacuum state to cancel the IR divergence and we partially kept the sub Hubble contributions. Because of that, our argument gets a bit complicated. Just to see the IR cancelation, I recommend the papers by Martin&Steve or Brynes et al.

About (3)
I agree. There is a remaining issue to be solved about how to define the (local) gauge invariant quantity.

About (4)
Yes, I agree. Because of that, we evaluated the fluctuation in the configuration space. Then, I think what you wrote does not matter. In the above message, for illustrative purpose, I discussed in the momentum space, but the cancellation can be shown also in the configuration space.

About (5)
Let me understand your point more clearly. If loop corrections have a different slow-roll parameter dependence from the contributions due to the distance change, the cancellation of the divergence between them never happens. Are you claiming that even for the (local) gauge invariants, the IR divergences and the secular growths are not canceled away? And actually I’m not sure what do you mean by “all true invariants”?

Suppose the spatial manifold is T^3, rather than R^3. Then I think transversality really does fix the spatial gauge.

I suppose that the same issue can occur also for T^3, as far as T^3 has an unobservable region, which is not causally connected to us. If we could impose the transverse gauge condition all over on T^3, we can completely fix the gauge as you said. But, in employing the gauge condition, we are implicitly assuming that we can observe the whole of T^3. Once we accept that we can impose a gauge condition only in the observable region, there inevitably appear residual freedoms associated with the gauge dofs outside of the causally connected region. The topology is not crucial there, I think.

If I correctly understood the May 6 KITP talk by Andy Strominger, it is wrong in flat space to consider the ``extra symmetries'' of R^3 as gauge transformations.

I expect that the frequency of the soft mode addressed in his talk, \omega, is much lower frequency than other modes, but not exactly 0. In an actual measurement, there is a limitation on the time scale, since the graviton propagation and the measurement should take place in a finite time scale. I don’t expect that much lower frequency modes than the limitation can leave detectable imprints. Unfortunately, I was not in his talk, so I might be missing the point. So, please correct me if I’m wrong. If what I wrote here is correct, I think that the situation is similar to what happens in cosmology. The soft modes can be gauged away as far as they are far super Hubble, but when the modes enter our Hubble patch, it can leave an observable imprint. To evaluate such effect, we should compute the sub Hubble dynamics, and the imprint may not be the simple log effect.

Holographic spacetime

Informal talks -- Tom Banks 3/26/15

I'm happy to give a series of informal talks on the Holographic Space-time formalism. I'm giving a quick overview on April 1 as part of the official program of the workshop. These talks will fill in the details and point out the many open problems.

[editor's note -- Tom's informal lectures are now available online, under recorded talks]

Some answers and apologies -- Tom Banks 4/2/15

Yesterday, at the end of the talk, my mouth got disconnected from my brain and I said some stupid things. First I want to apologize to Eva . She's completely right. The best way to visualize the N^2 DOF that contribute to black hole entropy is to go out on the Coulomb branch. This doesn't happen at low K_0 + P_0 energy because of the coupling of the gauge theory moduli to the curvature of the sphere, but the Hawking Page transition in this example is precisely the point where the entropy of all these Coulomb branch excitations dominates the energetics. The stuff I said about low HP temperature implying low energy is just wrong, as Kyriakos reminded me. I was thinking of low energy in a completely different sense : according to the time dependent Hamiltonian of a given time-like geodesic, over times short compared to the AdS radius, which is the appropriate description of local physics, according to HST. This discussion gave me some new ideas about how to define that Hamiltonian in the gauge theory, and I'll report on them if they lead anywhere.

A better answer to Steve's question about single particles is, I think, that HST doesn't have single particle states. The generic state described by a small block of the matrix is a superposition of multiparticle states restricted by a given angular location and total radial momentum according to a particular timelike geodesic. When we try to follow this into a small causal diamond we have two problems: we may not have enough angular momentum modes to localize the state on the small sphere. AND we have the problem with constraints that I talked about. So, on the one hand the thing is fuzzed out over the sphere and on the other hand all of its states get mixed together by the action of the Hamiltonian. However, by conservation of momentum (which in HST is imposed by saying that for a geodesic at rest w.r.t to the standard one, but far away from it all of this takes place in a large causal diamond) when we look to the future of this small diamond the state must still be concentrated in a small angle on the bigger sphere, we see that the state describes the undisturbed flow of momentum, but with the coefficients in the superpositions of multiparticle states changed by a "mini-S matrix". I think anyone who wants to get a better field theory understanding of this than I have will have to master the Fadeev Kulish papers.

Question for Tom about black hole entropy in HST - Ted Jacobson, 6/1/15

I'm puzzled about the role of the background Minkowski metric. As I understood it, HST is constructed so the dimension of Hilbert space attributable to a causal diamond goes like the area of the bounding sphere, where the area is measured with respect to the fixed, Minkowski metric. Now suppose we have a black hole. We know the entropy should go like the area of the horizon measured with respect to the emergent, "hydrodynamic" metric. But at the underlying HST level, a different area would presumably be assigned to this surface by the Minkowski metric. This suggests that the dimension of the underlying Hilbert space has no particular relation to the black hole entropy. But if that's the case, then what is the motivation for the HST axioms in the first place? Probably I'm misunderstanding something basic about HST...

Reply to Ted's question - Tom Banks, 6/12/15

Let me start from the assumption that black holes evaporate in a unitary manner in models of quantum gravity in asymptotically flat spacetime. It follows that in the remote past and future there is a complete set of Minkowski geodesics. The HST formalism works with a set of time-like trajectories, which asymptote to these geodesics. We assign a quantum system to each such trajectory with a time parameter which is asymptotically the proper time along those geodesics.

Now consider a process in which a black hole is created in the collision of two jets of gravitons. I argued in my penultimate lecture that our matrix model could model that situation because of the definition of particle jets in terms of constrained states in the matrix model Hilbert space. When the number of constraints defining the incoming state at time - T in the subfactor of the Hilbert space describing a particular interval of proper time 2T , becomes of order T^2 then it is highly improbable that the final state will be anything other than an generic unconstrained state in the Hilbert space. This description was carried out using a trajectory which was at rest in the c.m. of the collision and was placed right at the center of the black hole. My first controversial contention is that one can extend the description of the physics along this trajectory to infinite proper time after the black hole is formed and that it becomes a Minkowski geodesic in the remote future, after the black hole has evaporated. The classical black hole interior metric is geodesically incomplete, so denies this possibility. I would claim that this is not a signal that the quantum mechanics stops, but simply that a finite proper time after horizon crossing, the quantum system is in an equilibrium state associated geometrically with the stretched horizon . The stretched horizon of a black hole formed in collision and evaporating into particles is a timelike cylinder of finite extent. The entropy associated with this cylinder (number of fermionic quantum degrees of freedom necessary to describe it) changes slowly with time, eventually going to zero. In the matrix model, this is modeled by saying that a certain subfactor of the entire Hilbert space is in a generic time dependent state of T^2 degrees of freedom, typically not satisfying of order ET constraints. The probability of finding this subsystem in such a constrained state is of order e^{- ET} (note T is a time, not a temperature) and this is interpreted as Hawking radiation of a jet of energy E.

Now we come to the crucial question: what should we consider to be the degrees of freedom causally connected to this "trajectory inside the black hole" at proper times larger than a Schwarzschild time after horizon crossing. Classical GR does not give an answer to this question, because it doesn't allow us to think about proper times that long. For a while, up through our first paper on firewalls, Willy and I considered modeling this by a singular QM evolution which stops at a finite time (this just means that as one approaches that time the system goes through every unitary in the unitary group of its Hilbert space an infinite number of times), but that makes no sense because the black hole evaporates. The proper answer is to think about the QM of H_{in} as describing the evolution of the equilibrated stretched horizon and its interaction with the external world. Since at each proper time after local equilibrium sets in, there is SOME probability that the black hole will emit massless Hawking quanta, we have to take the causal diamond of the trajectory inside the black hole to grow as it would in Minkowski space. That is to say, the QM MUST contain states, which are included in the factor of the Hilbert space acted on by $H_{in} (T + t)$. This Hamiltonian is a function of (T + t)^2 fermions and we want to think about t as being much larger than T. The state of "a black hole with a jet of Hawking quanta of energy E" satisfies of order T (T + t) constraints such that the matrix elements between two blocks one of size E and the other of size (T - E) vanish, as well as all of those matrix elements connecting those two blocks to a larger block of size t . The latter constraints are "inherited from the initial state at time - (T + t) while those between the two small blocks are created by a fluctuation in the equilibrium dynamics of the subsystem of T^2 fermions. Geometrically one can think of this causal structure as describing the causal connections between some time like trajectory on the stretched horizon, but if I pick any given trajectory I break the spherical symmetry. So we should think of this as the causal structure of a trajectory in the dead center of the black hole, but the quantum system associated with that trajectory doesn't have any interior particles to talk about and just describes the dynamics of DOF on the stretched horizon. A geometrical slogan that captures what's going on is that the quantum dynamics of H_{in} (\tau ) along trajectories inside the black hole takes place in a Hilbert space describing the domain of dependence of the stretched horizon. This Hilbert space grows with increasing proper time just like the corresponding Minkowski surface (all of this is in the limit of large black hole mass) . At proper times $\tau$ large compared to T, the black hole itself is described by a small subset of the DOF inside the diamond.

I won't repeat what I said in my last lecture about how this framework allows for the description of a system falling into the black hole long before evaporation but long after formation, and how the consistency condition between the description along trajectories which enter the black hole along with the in-falling system, and those which are a few Schwarzschild radii away allows one to describe "a system in the black hole interior" for a period of time of order the Schwarzschild radius.

In Ben's lecture you complained that the causal diamond picture didn't make sense "because we were doing field theory". The point Willy and I have been trying to make is that we are not doing field theory. All we know from string theory and AdS/CFT is that the computation of a certain subset of all boundary correlators (S matrix in the Minkowski case) in a restricted kinematic regime are approximately reproduced by field theory. Many people have taken this as license to use the full field theory Hilbert space apparatus, with some UV cutoff, to describe what's going on in local regions. I believe we should have known for a long time that this is wrong. The restrictions imposed on the Hilbert space by black hole physics cannot be accounted for by a UV cutoff. Field theory IMHO is almost as conceptually remote from the correct theory as QM is from CM. Willy and I have also discovered something somewhat more surprising. In our formalism, most of the DOF associated with a causal diamond, cannot be localized in the bulk of that diamond. The best I can come to a field theoretic description of what's going on is to say that the only field theory states in a causal diamond, which are really there in the theory of quantum gravity, are modes that are pure gauge everywhere except on the boundary.

## Online discussion

This page provides a space for additional discussion of workshop topicsHolographyThe holography challenge -- Steve Giddings 6/23/15Early in the workshop there was significant discussion on the problem of "decoding the hologram." There are some different views about how/whether a sharp "decoding map" can be defined, and what is necessary to be able to sharply resolve and describe all the relevant bulk physics, particularly on scales small as compared to the AdS radius.

At the final workshop discussion, I stated the following "holography challenge:"

Suppose we assume that H_bulk = H_bdy. Provide a precise formlation, purely in terms of CFT data, of bulk observables that reduce to observables of local quantum field theory on scales <<R_ADS in a controlled approximation for small but nonzero g_s.Some of those present indicated that they thought that the HKLL construction meets the challenge, and none objected to the statement that it does, at least on an AdS background. Of course, as has been discussed, there are additional issues about other backgrounds, such as black holes.

Please share your thoughts whether you think HKLL does indeed meet the challenge -- or whether you see limitations to or problems with the statement that it does. (In particular, there are questions about a systematic expansion in g_s, which I raised in my talk during the "hologram week;" resolution of these seem to require certain properties of smearing functions. There is also the set of questions which resulted in the discussion about code subspaces, or the role of gauge invariance.)

Inflationary cosmology## IR safe cosmo observables -- Martin Sloth 4/28/15

I just want to reply here to Richard's comments directed at me during the discussion from last friday, since I could not be there myself.

I guess that the point is not that I/we don’t want to discuss how to formulate the prescription of defining IR safe observables of [1] at a full quantum operator level, but rather that it shouldn’t be necessary given that we have proven/conjectured that all the infrared effects can be captured by classical perturbations of the geometry (see [1] and [2]).

Of course any macroscopic classical object can ultimately be described by quantum mechanics, but it is just not very practical (and if done correct it should give the classical result exponentially precise).

In the same way the dependence of the IR safe observables on the IR resolution scale should be understandable from a full fundamental quantum field theory treatment, but it is just not very practical, and it should agree exponentially precisely with our treatment.

Since we are utilising that the IR part of the qft correlation functions can be understood classically, the only caveat would be if perturbations are not becoming fully classical on super-horizon scales. But there are several papers addressing this issue in a full quantum treatment -- maybe [3] is a good recent reference in the present context.

[1] http://arxiv.org/abs/1104.0002

[2] http://arxiv.org/abs/1005.1056

[3] http://arxiv.org/abs/1210.7792

Reply to Martin Sloth -- Richard Woodard 6/1/15First, my thanks to Martin for his reply. I wish we could have discussed this in person at KITP, and I hope we will be able to do so in the near future. Second, while I have great admiration for the progress Martin and others have made, the word proof'' has a very strong meaning and I don't believe it applies to what has been done. In particular, there has not been any precise quantification of how good is the approximation of representing IR loop effects by simply distorting tree order factors of k^2 to k^i k^j contracted into a classical, long wavelength metric'' and then averaging this separately from the other operators. I would like to compare exact one loop results with the approximation. It ought to be possible to see what parts of the graviton and zeta propagators this recovers and what parts it does not. Right now the only thing I am sure about is that that approximation correctly recovers the dependence on the IR cutoff on the initial value surface.

I know from my own work that there is a distinction between the IR cutoff and secular growth caused by modes which experience horizon crossing at whatever is the current time. It seems to me that such growth effects are not exponentially suppressed, and may not be correctly described by the approximation. In arXiv:1204.1784, Shun-Pei Miao and I consider a simple example of this procedure in the context of a coincident propagator, see equations (41-43) on page 9. As expected, the technique correctly reproduces the cutoff dependence but it gives a false result for secular growth when the exact result shows none.

I am also worried about assumptions about the classicality of quantum IR effects from gravitons. This is probably true at some level, but it isn't clear in what sense. For scalar potential models on nondynamcial de Sitter background this can be made rigorous through proving that Starobinsky's stochastic formalism really does exactly capture the leading IR logarithms at each order in the loop expansion. Nothing of the kind exists yet for the GR + inflaton system. Further, naive extrapolations of Starobinsky's formalism have been shown not to correctly reproduce the one loop IR logarithms acquired by the wave function of a massless fermion as it propagates through the ensemble of gravitons created by de Sitter inflation. This really is a ``proof'' that these naive extrapolations are wrong because they are being compared against exact one loop results. The detailed, explicit and fully renormalized computations are described in arXiv:0803.2377, and Shun-Pei Miao spoke on this paper at KITP on April 28. In particular, each of the following plausible-sounding procedures fails to produce correct results:

(1) Employing IR truncated mode sums for the fields;

(2) Removing the dimensional regularization; and

(3) Ignoring differentiated gravitons.

Finally, the various ``Ward identities'' being invoked seem to contain hidden assumptions about analyticity which may not correct. As stated, the results are only true for attaching legs with k=0, which are not observable. To extend them to give observable results one must assume that the relations remain approximately true even for nonzero k. The problem with this sort of extrapolation should be clear from the fact that it could be used to fix the topology of space by making local measurements. For example, there are no extra coordinate symmetries, and presumably no related Ward identities, if the spatial manifold is T^3, rather than R^3. (In fact the strictly constant modes of T^3 are NOT pure gauge.) Yet we should not be able to use local measurements to tell the difference between R^3 and T^3 if the T^3 radii are super-horizon.

## \epsilon suppression in IR safe correlators -- Yuko Urakawa, 5/7/15

I’d like to make a comment on 1/\epsilon enhancement Richard mentioned in the working group discussion on Apr. 24th, since I was not there unfortunately.

Richard mentioned that the IR safe correlators get the correction like

\zeta(t, x) \zeta(t, y) [1 + \zeta + \zeta^2 ....] (#)

and additional contributions in the square brackets, which appear after the coordinate change, can give

P^2 ~ (G H^2/\epsilon)^2

which is 1/\epsilon bigger than a typical contribution from the loop correction. Here, P is the amplitude of zeta (multiplied by k^3).

I think that the additional contributions are not of order P^2, but they are also multiplied by slow-roll parameters and become same order as the contribution from the loop corrections. (If they are not of the same order, the additional piece in the IR safe correlator cannot cancel the IR divergence from the loop corrections.)

Here, let me explain how the "missing" slow-roll parameters appears.

The IR safe correlator is, in the Fourier space, given by

<\zeta(t, e^{\zeta_L} k) \zeta(t, e^{\zeta_L} k')>

where k and k' are wavenumbers and \zeta_L is an IR mode much softer than k and k'. (Here, for illustrative purposes, let me take the Fourier space. But basically the same argument follows also in the position space.)

Skipping the detail of the computation, the IR safe correlator can be given by a compact expression as

<\zeta(t, e^{\zeta_L} k) \zeta(t, e^{\zeta_L} k')>

~ k^{-3} [1+ 1/2<\zeta_L^2> \partial^2_{ln k} + ... .] k^3 <\zeta(t, k) \zeta(t, k')>. (##)

At first glance, one may think that the additional piece, the second term, is of order P^2, but since \partial^2_{ln k} operates on

k^3<\zeta(t, k) \zeta(t, k')> ~ P,

the second term is,

(n_s-1)^2 <\zeta_L^2> <\zeta(t, k) \zeta(t, k’)>, (###)

which is now multiplied by (n_s-1)^2, which is typically of order \epsilon^2. (In (##), k^3 appears when we correctly take into account the volume effect.)

This shows that a naive counting of \epsilon can tell us a wrong answer.

After some manipulation, the one loop corrections can be given by the terms of order \epsilon^2 \times P^2 and IR regular terms. Then, the additional contribution, given in (###), can cancel the IR divergence of the loop corrections.

Here I only gave a schematic explanation. A more detailed discussion can be found in our paper

http://arxiv.org/pdf/1009.2947.pdf

This is based on QFT computation. A similar result is derived by Byrnes et al. based on delta N formalism in

http://arxiv.org/pdf/1005.3307v3.pdf

Reply to Yuko Urakawa -- Richard Woodard 6/1/15My thanks to Yuko for a clear explanation, with supporting references, of how an IR safe observable can preserve the pattern of \epsilon suppression of the naive observables at lowest order. However, there are some problems:

(1) The quantity (n_s - 1) involves the 2nd slow roll parameter, in addition to the first one. The 2nd slow roll parameter does not need to be small and it is not, in any case, identical to the factors of 1/\epsilon which \zeta propagators carry.

(2) It isn't clear that the next order corrections have any suppression at all.

(3) The proposed observable is wildly nonlocal through its dependence upon \zeta_L.

(4) What is long wavelength'' presumably changes with time, and it's not clear why the uncorrected Fourier modes should be used to determine what is long wavelength'' when this is not trusted for the original correlator.

(5) The proposed observable is not invariant, whereas all true invariants I know exhibit the problem with \epsilon-suppression.

The last point is especially relevant in view of Yuko's recent work with Tanaka (arXiv:1402.2076) on which she spoke at KITP on April 13. If I understood this correctly, they claim the absence of secular growth effects ONLY for invariant operators in the presence of a special state.

I would also like to use this opportunity to raise with Yuko (and everyone else) an issue which has been bothering me for some time about gauge fixing. She and Tanaka have argued that the factors of e^{\zeta_L} must be added to the naive correlator because requiring the graviton to be transverse (\partial_i \gamma_{ij}(t,x) = 0) fails to completely fix the spatial gauge. Suppose the spatial manifold is T^3, rather than R^3. Then I think transversality really does fix the spatial gauge. Does this mean there is no need for the factors of e^{\zeta_L} of T^3? But without those factors, loop corrections to the \zeta-\zeta correlator on T^3 depend upon the coordinate radius of T^3, which is just another sort of IR cutoff that one would think should not matter.

If I correctly understood the May 6 KITP talk by Andy Strominger, it is wrong in flat space to consider the ``extra symmetries'' of R^3 as gauge transformations. Why would the same not be true for inflation? In which case we do not need to gauge fix them, or make observables invariant under these symmetries, and the justification for the factors of e^{\zeta_L} must be sought elsewhere.

Reply toRichard Woodard--Yuko Urakawa6/7/15Thanks for your reply, Richard. (I wrote the following in another place, but let me copy and paste the same here.)

About (1)

Ok. Remember that in case the second slow-roll parameter \eta is not very small, this should be taken into account also in the loop computations. And this can change the result. I expect that also in such case the cancellation between the IR divergence from loops and the contributions due to the change of the distance measure to the invariant distance takes place, when we evaluate the (local) gauge invariant quantity.

About (2)

The computation will become messy, but I expect that as far as the slow-roll approximation holds, the cancellation can be checked also at higher orders.

In the above refs, the cancellation was checked including the quadratic terms of the slow-roll parameters, assuming that all the slow-roll parameters are small. The interpretation of the result is different, but at the algebraic level, the same result was derived by Steve and Martin in

http://arxiv.org/abs/1005.1056

Our purpose in 1009.2947 was to derive the condition on the vacuum state to cancel the IR divergence and we partially kept the sub Hubble contributions. Because of that, our argument gets a bit complicated. Just to see the IR cancelation, I recommend the papers by Martin&Steve or Brynes et al.

About (3)

I agree. There is a remaining issue to be solved about how to define the (local) gauge invariant quantity.

About (4)

Yes, I agree. Because of that, we evaluated the fluctuation in the configuration space. Then, I think what you wrote does not matter. In the above message, for illustrative purpose, I discussed in the momentum space, but the cancellation can be shown also in the configuration space.

About (5)

Let me understand your point more clearly. If loop corrections have a different slow-roll parameter dependence from the contributions due to the distance change, the cancellation of the divergence between them never happens. Are you claiming that even for the (local) gauge invariants, the IR divergences and the secular growths are not canceled away? And actually I’m not sure what do you mean by “all true invariants”?

- Suppose the spatial manifold is T^3, rather than R^3. Then I think transversality really does fix the spatial gauge.

I suppose that the same issue can occur also for T^3, as far as T^3 has an unobservable region, which is not causally connected to us. If we could impose the transverse gauge condition all over on T^3, we can completely fix the gauge as you said. But, in employing the gauge condition, we are implicitly assuming that we can observe the whole of T^3. Once we accept that we can impose a gauge condition only in the observable region, there inevitably appear residual freedoms associated with the gauge dofs outside of the causally connected region. The topology is not crucial there, I think.- If I correctly understood the May 6 KITP talk by Andy Strominger, it is wrong in flat space to consider the ``extra symmetries'' of R^3 as gauge transformations.

I expect that the frequency of the soft mode addressed in his talk, \omega, is much lower frequency than other modes, but not exactly 0. In an actual measurement, there is a limitation on the time scale, since the graviton propagation and the measurement should take place in a finite time scale. I don’t expect that much lower frequency modes than the limitation can leave detectable imprints. Unfortunately, I was not in his talk, so I might be missing the point. So, please correct me if I’m wrong.If what I wrote here is correct, I think that the situation is similar to what happens in cosmology. The soft modes can be gauged away as far as they are far super Hubble, but when the modes enter our Hubble patch, it can leave an observable imprint. To evaluate such effect, we should compute the sub Hubble dynamics, and the imprint may not be the simple log effect.

Holographic spacetimeInformal talks -- Tom Banks 3/26/15I'm happy to give a series of informal talks on the Holographic Space-time formalism. I'm giving a quick overview on April 1 as part of the official program of the workshop. These talks will fill in the details and point out the many open problems.

[editor's note -- Tom's informal lectures are now available online, under recorded talks]

## Some answers and apologies -- Tom Banks 4/2/15

Yesterday, at the end of the talk, my mouth got disconnected from my brain and I said some stupid things. First I want to apologize to Eva . She's completely right. The best way to visualize the N^2 DOF that contribute to black hole entropy is to go out on the Coulomb branch. This doesn't happen at low K_0 + P_0 energy because of the coupling of the gauge theory moduli to the curvature of the sphere, but the Hawking Page transition in this example is precisely the point where the entropy of all these Coulomb branch excitations dominates the energetics. The stuff I said about low HP temperature implying low energy is just wrong, as Kyriakos reminded me.

I was thinking of low energy in a completely different sense : according to the time dependent Hamiltonian of a given time-like geodesic, over times short compared to the AdS radius, which is the appropriate description of local physics, according to HST. This discussion gave me some new ideas about how to define that Hamiltonian in the gauge theory, and I'll report on them if they lead anywhere.

A better answer to Steve's question about single particles is, I think, that HST doesn't have single particle states. The generic state described by a small block of the matrix is a superposition of multiparticle states restricted by a given angular location and total radial momentum according to a particular timelike geodesic. When we try to follow this into a small causal diamond we have two problems: we may not have enough angular momentum modes to localize the state on the small sphere. AND we have the problem with constraints that I talked about. So, on the one hand the thing is fuzzed out over the sphere and on the other hand all of its states get mixed together by the action of the Hamiltonian. However, by conservation of momentum (which in HST is imposed by saying that for a geodesic at rest w.r.t to the standard one, but far away from it all of this takes place in a large causal diamond) when we look to the future of this small diamond the state must still be concentrated in a small angle on the bigger sphere, we see that the state describes the undisturbed flow of momentum, but with the coefficients in the superpositions of multiparticle states changed by a "mini-S matrix". I think anyone who wants to get a better field theory understanding of this than I have will have to master the Fadeev Kulish papers.

Question for Tom about black hole entropy in HST - Ted Jacobson, 6/1/15I'm puzzled about the role of the background Minkowski metric. As I understood it, HST is constructed so the dimension of Hilbert space attributable to a causal diamond goes like the area of the bounding sphere, where the area is measured with respect to the fixed, Minkowski metric. Now suppose we have a black hole. We know the entropy should go like the area of the horizon measured with respect to the emergent, "hydrodynamic" metric. But at the underlying HST level, a different area would presumably be assigned to this surface by the Minkowski metric. This suggests that the dimension of the underlying Hilbert space has no particular relation to the black hole entropy. But if that's the case, then what is the motivation for the HST axioms in the first place? Probably I'm misunderstanding something basic about HST...

Reply to Ted's question - Tom Banks,6/12/15Let me start from the assumption that black holes evaporate in a unitary manner in models of quantum gravity in asymptotically flat spacetime. It follows that in the remote past and future there is a complete set of Minkowski geodesics. The HST formalism works with a set of time-like trajectories, which asymptote to these geodesics. We assign a quantum system to each such trajectory with a time parameter which is asymptotically the proper time along those geodesics.

Now consider a process in which a black hole is created in the collision of two jets of gravitons. I argued in my penultimate lecture that our matrix model could model that situation because of the definition of particle jets in terms of constrained states in the matrix model Hilbert space. When the number of constraints defining the incoming state at time - T in the subfactor of the Hilbert space describing a particular interval of proper time 2T , becomes of order T^2 then it is highly improbable that the final state will be anything other than an generic unconstrained state in the Hilbert space. This description was carried out using a trajectory which was at rest in the c.m. of the collision and was placed right at the center of the black hole. My first controversial contention is that one can extend the description of the physics along this trajectory to infinite proper time after the black hole is formed and that it becomes a Minkowski geodesic in the remote future, after the black hole has evaporated. The classical black hole interior metric is geodesically incomplete, so denies this possibility. I would claim that this is not a signal that the quantum mechanics stops, but simply that a finite proper time after horizon crossing, the quantum system is in an equilibrium state associated geometrically with the stretched horizon . The stretched horizon of a black hole formed in collision and evaporating into particles is a timelike cylinder of finite extent. The entropy associated with this cylinder (number of fermionic quantum degrees of freedom necessary to describe it) changes slowly with time, eventually going to zero. In the matrix model, this is modeled by saying that a certain subfactor of the entire Hilbert space is in a generic time dependent state of T^2 degrees of freedom, typically not satisfying of order ET constraints. The probability of finding this subsystem in such a constrained state is of order

e^{- ET} (note T is a time, not a temperature) and this is interpreted as Hawking radiation of a jet of energy E.

Now we come to the crucial question: what should we consider to be the degrees of freedom causally connected to this "trajectory inside the black hole" at proper times larger than a Schwarzschild time after horizon crossing. Classical GR does not give an answer to this question, because it doesn't allow us to think about proper times that long. For a while, up through our first paper on firewalls, Willy and I considered modeling this by a singular QM evolution which stops at a finite time (this just means that as one approaches that time the system goes through every unitary in the unitary group of its Hilbert space an infinite number of times), but that makes no sense because the black hole evaporates. The proper answer is to think about the QM of H_{in} as describing the evolution of the equilibrated stretched horizon and its interaction with the external world. Since at each proper time after local equilibrium sets in, there is SOME probability that the black hole will emit massless Hawking quanta, we have to take the causal diamond of the trajectory inside the black hole to grow as it would in Minkowski space. That is to say, the QM MUST contain states, which are included in the factor of the Hilbert space acted on by $H_{in} (T + t)$. This Hamiltonian is a function of (T + t)^2 fermions and we want to think about t as being much larger than T. The state of "a black hole with a jet of Hawking quanta of energy E" satisfies of order T (T + t) constraints such that the matrix elements between two blocks one of size E and the other of size (T - E) vanish, as well as all of those matrix elements connecting those two blocks to a larger block of size t . The latter constraints are "inherited from the initial state at time - (T + t) while those between the two small blocks are created by a fluctuation in the equilibrium dynamics of the subsystem of T^2 fermions. Geometrically one can think of this causal structure as describing the causal connections between some time like trajectory on the stretched horizon, but if I pick any given trajectory I break the spherical symmetry. So we should think of this as the causal structure of a trajectory in the dead center of the black hole, but the quantum system associated with that trajectory doesn't have any interior particles to talk about and just describes the dynamics of DOF on the stretched horizon. A geometrical slogan that captures what's going on is that the quantum dynamics of H_{in} (\tau ) along trajectories inside the black hole takes place in a Hilbert space describing the domain of dependence of the stretched horizon. This Hilbert space grows with increasing proper time just like the corresponding Minkowski surface (all of this is in the limit of large black hole mass) . At proper times $\tau$ large compared to T, the black hole itself is described by a small subset of the DOF inside the diamond.

I won't repeat what I said in my last lecture about how this framework allows for the description of a system falling into the black hole long before evaporation but long after formation, and how the consistency condition between the description along trajectories which enter the black hole along with the in-falling system, and those which are a few Schwarzschild radii away allows one to describe "a system in the black hole interior" for a period of time of order the Schwarzschild radius.

In Ben's lecture you complained that the causal diamond picture didn't make sense "because we were doing field theory". The point Willy and I have been trying to make is that we are not doing field theory. All we know from string theory and AdS/CFT is that the computation of a certain subset of all boundary correlators (S matrix in the Minkowski case) in a restricted kinematic regime are approximately reproduced by field theory. Many people have taken this as license to use the full field theory Hilbert space apparatus, with some UV cutoff, to describe what's going on in local regions. I believe we should have known for a long time that this is wrong. The restrictions imposed on the Hilbert space by black hole physics cannot be accounted for by a UV cutoff. Field theory IMHO is almost as conceptually remote from the correct theory as QM is from CM. Willy and I have also discovered something somewhat more surprising. In our formalism, most of the DOF associated with a causal diamond, cannot be localized in the bulk of that diamond. The best I can come to a field theoretic description of what's going on is to say that the only field theory states in a causal diamond, which are really there in the theory of quantum gravity, are modes that are pure gauge everywhere except on the boundary.

I hope this helps.