Many people know the Penrose triangle or Penrose tilings. Our colleagues outside the mainstream also like the spin networks that he invented decades before "loop quantum gravity" was constructed.

Two weeks ago I was asked to comment on Roger Penrose's remarks about holography. His text will be written in italics and my answer in Roman fonts. Because it is based on a fast e-mail, the text below won't be perfect.

To summarize, Roger Penrose believes that holography must be wrong. Because some of his errors are easy to catch, let me make this comment public.

Roger Penrose: 31.16 The Road to Reality

What is the "holographic conjecture"?

- the states of a quantum field theory defined on some spacetime M can be put in direct 1-1 correspondence with the states of another quantum field theory, where the second quantum field theory is defined on another spacetime E of lower dimension!

- Often, E is presented as though it were a (timelike) boundary of M, or at least some conformally smooth timelike submanifold of M. However, this is not the case in the usual example...

- The holographic principle is ... analogous to a hologram, where a 3-dimensional image is perceived when a (basically) 2-dimensional surface is viewed.

- The most familiar form of this 'holographic principle', ... is sometimes referred to as .. the AdS/CFT conjecture. Here M is ... a (1 + 9) dimensional product AdS5xS5, where AdS5 is the ('unwrapped') (1 + 4) dimensional anti-de Sitter space ... here there are four space dimensions.

- "The S5 is spacelike 5-sphere whose radius is of cosmological dimension, equal to (-/\')^1/2, where /\' is the (negative) cosmological constant of AdS5." I think that is a typo in the text and it is (-/\')^-1/2 because [/\] = 1/AREA = CURVATURE

- Note that dark energy is /\' > 0 corresponding to a (1 + 4) de Sitter space without an S5 at all. Why the S5?

- As Izzi Rabi said of the neutrino "Who ordered that?"

- So the observational fact is that /\' > 0 although the theory here has /\' negative

- The smaller space E is to be the 4-dimensional ... conformal infinity of AdS5. We note that E, being 4-dimensional, is certainly not the boundary of M... since M = AdS5xS5 is 10 dimensional. Instead the boundary of M can be thought of (but not conformally) as ExS5.

- The Malcadena conjecture supposes that string-theory on AdSxS5 is ... equivalent to a ... supersymmetric Yang-Mills theory on E.

- Here there is no chance of appealing to the type of 'quantum energy' argument (in 31.10) for explaining away the gross discrepancy between the functional freedom of an ordinary field on M, namely M^9, and an ordinary field on E, namely E^3. Since the extra dimensions of M are in no way 'small' - being of cosmological scale - the flood of additional degrees of freedom, from the fields' dependence on the S5 part of M, would spoil any possibility of an agreement between the two field theories.

On the other hand, someone else can publish this misunderstanding in a book because he has no one who would tell him why the reasoning is flawed. Instead, everyone tells him: Yes, Prof. Penrose, you are such a genius who can disprove the AdS/CFT in such an elegant way! ;-) Such an uncritical approach of the environment may be helpful whenever the person is doing all things perfectly, but it becomes counterproductive if the person starts to make serious and elementary errors.

OK, so what's the correct answer to Prof. Penrose's questions?

It is true that the radius of S^5 is equal to the radius of AdS_5 but it does not imply that the dual theory is more than four-dimensional. Indeed, all spherical harmonics of fields on the S^5 can be easily found in the four-dimensional gauge theory. I must start with some basic relevant mathematical background for the holographic duality.

In the correspondence, the modes of perturbative particles in the AdS bulk - such as the gravitons - are in one-to-one correspondence with local operators on the boundary. Gravitons can be found in many "states of motion" around the sphere S^5, and this motion corresponds to additional scalar factors inserted as factors inside the full operators.

The states of motion around the S^5 can be decomposed into energy eigenstates and I will use the term "spherical harmonics" known from the case of the two-sphere.

The operator dual to the graviton looks schematically like

- O_{ac} = Tr ( F_{ab} F_{bc} )

- O'_{ac} = Tr ( F_{ab} F_{bc} PHI1 PHI3 PHI6 PHI6 )

Also, we can ignore all modes with non-minimal values of the angular momentum around S^5, which means that we can consider a limiting theory of AdS5 x S5 that only depends on AdS5, despite the large radius of S5. Such a limiting theory will describe all modes of supergravity whose energy is too small to excite higher spherical harmonics on S^5. This regime exists because AdS5 is noncompact and the energy in it can approach zero; the S5 is compact and therefore it has a gap. This difference in the compactness makes it consistent to forget about S^5 in a certain limit that still keeps AdS_5.

- The same would apply to ordinary QFTs on M and E, since one-particle states are themselves described simply by 'ordinary fields' (26.2).

- The only chance of the holographic principle being actually true for these spaces is for the QFTs under consideration to be far from 'ordinary'.

- Penrose elaborates on this with more details. "It is my opinion that the the importance this kind of discrepancy in functional freedom has been profoundly underrated. ..."

But in gravity, such a counting is not possible because there are no exactly local operators and distances below Lplanck are meaningless. The accelerators that try to probe them by high-energy particles create larger and larger black holes instead. We cannot fill a region with too many accurate measuring sticks because accurate sticks are heavy and too many of them would collapse into a black hole. The real number of degrees of freedom that gravity allows does not scale as the volume.

The only objective and physically meaningful measure of the phrase "number of degrees of freedom" - a phrase that is vague in general - is given by counting the actual number of physical states in the Hilbert space in some energy regime. That's encoded in the entropy. Indeed, an entropy of a gravitational system - such as the black hole - does not scale with the volume but only with the surface area. That's why the counting of degrees of freedom, if done properly including the correct treatment of quantum mechanics, implies that the "magnitude" of a gravitational theory in D noncompact dimensions is equivalent to the number of degrees of freedom of a non-gravitational theory in D-1 dimensions: that's what holography is all about.

- Why is the AdS/CFT conjecture taken so seriously?

- The support for it seems to come from a correspondence between BPS states on the two sides, that had been noted by Maldacena and from a number of other correspondences ... there are also some additional 'coincidences' that seem to need explaining. ....

In the last two years, the reasoning has went beyond the Penrose limit, using the methods of spin chains, integrable systems, and so forth. Everything that has been easy enough to understand has been checked, it's OK, and the possible discrepancies in the most complicated calculations appear at the three-loop level contributions to the gauge theory quantities. People differ whether we should be calculating it to ever higher orders because no one would believe that a real discrepancy exists.

- The AdS/CFT conjecture arose as another way of looking at the 'string' derivation of the Bekenstein-Hawking black-hole entropy formula... This would only be of relevance to cosmological-size black holes...

- based on some remarkable agreements between 'entropy calculations' done in different ways, rather than on an actual derivation of the Bekenstein-Hawking expression. ...

- some of the strongest claims can be discounted altogether (such as string theory having provided a complete consistent theory of quantum gravity) ...

- what is the most general set of quantities that string theory can calculate and how it should be done (especially in the cosmological context);
- and whether string theory is the correct theory of the particular world we observe around or just a theory that agrees approximately with everything we know but that will reveal discrepancies.

- The strength of string's theory's case appears to rest on a number of remarkable mathematical relationships between seemingly different 'physical situations'... Are these relationships 'coincidence', or is there some deeper reason behind them?

- ... indeed such a reason ... as yet undiscovered, ... still does not reassure us that the string theorists are doing physics." 31.18 p. 926

Dear Lubos,

ReplyDeleteQuantum information of the quantum black hole (=spacetime inside out) must be smaller than 10^123 qbits due to the holographical principle. Please what is a demonstration topological string computation ?

Horizon (=entanglement entropy) converges to the empty space. Please what is interrelation between nonlocal dynamics of the black hole horizon and a drift of the cosmic horizon towards vacuum ?

Wavelength of the inflaton (=dilaton) is comparable to the horizon radius. That's why the entanglement entropy increases. Please what is rule of proportion of the CY-fold entropy to the 10^123 Planckian surfaces ?

Thank you very much!

Best, Planckeon

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