Friday, May 19, 2006

Roger Penrose vs. holography

First of all, let me mention that Roger Penrose is a great mathematician, a highly influential mathematical physicist, and an original thinker. We mention his name hundreds of times every day in connection with many discoveries. For example, we investigate the Penrose limits of geometries; the Penrose causal diagrams; twistors; many other notions in general relativity (including methods to extract energy from rotating black holes).

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!
Right. Except that we would not say that the gravitational theory defined on M (in the bulk) is a "quantum field theory". Quantum theories of gravity may look like quantum field theories semiclassically but they do not satisfy the usual rules of quantum field theory.
  • 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...
It is the case, modulo the fact that the boundary is a boundary at infinity. Why the S5 does not contribute to the boundary is explained below.
  • 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
I agree that with the third person that Penrose made a small error and the exponent is negative - although its exact value depends on the dimensions.
  • Note that dark energy is /\' > 0 corresponding to a (1 + 4) de Sitter space without an S5 at all. Why the S5?
Because S^5 is a factor in the near-horizon geometry of the 3-branes - the branes whose low-energy effective description (Yang-Mills theory) is equivalent to type IIB string theory on the near-horizon geometry. A 3-brane (with a 3+1-dimensional worldvolume) is surrounded by 6 transverse dimensions. The metric depends on the radial coordinate that becomes the holographic coordinate of AdS_5, and the remaining (angular) 5 coordinates parameterize a sphere. There are other examples of AdS/CFT correspondence where the five-sphere is replaced by a different compact manifold.
  • As Izzi Rabi said of the neutrino "Who ordered that?"
Except that in the case of S^5, we can also answer this question and an intelligent undergraduate student needs 35 seconds to understand this answer.
  • So the observational fact is that /\' > 0 although the theory here has /\' negative
It's because the gravity in AdS5 x S5 - and its negative cosmological constant - has no direct relation with the observed four-dimensional space and its positive (tiny) cosmological constant. String theory on AdS5 x S5 is an equivalent description of a four-dimensional gauge theory without gravity. The backgrounds of string theory that also contain the usual four-dimensional gravity must modify the details of the AdS5 space near the boundary, and the exact AdS5 is just a description that is appropriate for some interval of energies where the strong interactions matter and the four-dimensional gravity does not. Strong interactions are, in this description, equivalent to five-dimensional gravity in the AdS space (with a negative cosmological constant).
  • 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.
No way - one might find it surprising if a great mathematician like Roger Penrose answers this simple mathematical question incorrectly. The conformal boundary of AdS5 x S5 is four-dimensional. It is because the absolute distances between two points near the boundary of AdS5 go to infinity (because the warp factor does), and the conformal scaling that keeps them finite also scales down the size of the S5 (whose size is fixed in the absolute units). Near the boundary, the S5 shrinks to a (zero-dimensional) point relatively to the distances on AdS5, which is why the conformal boundary of AdS5 x S5 is just a four-dimensional Minkowskian manifold and the five-sphere does not contribute any dimensions to the boundary.
  • The Malcadena conjecture supposes that string-theory on AdSxS5 is ... equivalent to a ... supersymmetric Yang-Mills theory on E.
The Maldacena (see my correct spelling) conjecture does not *suppose* the equivalence above; on the contrary, it states it. Moreover, it's known to be true.
  • 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.
This is a standard intuitive expectation of every other beginner who just started to study the subject and who has just made one of the typical elementary newbies' errors. If she's smart, she can be explained what's wrong with that reasoning in 7 minutes.

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} )
at some point of the boundary where F_{ab} is the field strength (and I don't distinguish upper and lower Lorentz indices). This is the graviton with the smallest possible angular momentum around the sphere S^5. Note that the operator resembles the stress-energy tensor which is no coincidence. The gravitons with additional angular momentum are dual to operators like
  • O'_{ac} = Tr ( F_{ab} F_{bc} PHI1 PHI3 PHI6 PHI6 )
where the scalar fields PHI_1...PHI_6 present in the gauge theory are in the adjoint representation (they are matrices), much like everything else. Such (traceless and symmetrized) polynomials in the scalar fields are in one-to-one correspondence with spherical harmonics on S^5. And indeed, the interactions also work as required: the correlators of these operators know about the interactions of gravitons with the angular momentum on S^5.

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).
Indeed, all objects are described by "ordinary" fields in the gauge theory - if the word "ordinary" describes fields that are treated according all the correct and well-known rules of quantum field theory.
  • The only chance of the holographic principle being actually true for these spaces is for the QFTs under consideration to be far from 'ordinary'.
The quantum field theory relevant for the AdS5/CFT4 correspondence are completely standard gauge theories treated according to all rules that have been known in gauge theories for years, and you can be sure that if someone thinks that the duality is not possible, she must be doing a more or less silly mistake (or more than one).
  • 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. ..."
In other words, Prof. Penrose displays his difficulties to understand the very elementary point that dimensions can ever appear or disappear. But they do appear and disappear. In local quantum field theories, the counting of dimensions is absolute because we can always ask what happens if we fill the whole spacetime with points (local operators) whose separation is always between epsilon and 3 epsilon. The number of points will be of order "(L/epsilon)^dimension".

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?
Because it is very important, surprising, and true. Why is it important? It's exactly because old well-known mathematicians have such problems to understand how it could ever be true. It tells us something about the origin of geometry.
  • 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. ....
Today, the support of the correspondence goes well beyond the BPS states. For example, all excited BPS and non-BPS string states can be compared in the special BMN kinematical regime - that we continue to call the Penrose limit. The Penrose limit of the AdS5 x S5 geometry is one of the most complete regimes in which non-BPS agreement between the AdS and the CFT can be obtained. What an irony that the critic who says that the agreement is only at the BPS level is also called Penrose. ;-) Also, branes, black holes, and other things can be successfully compared.

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...
Both semiclassical gravity as well as string theory imply that the Bekenstein-Hawking formula is the correct result for the black hole entropy (and the corrections are negligible) whenever a black hole is much greater than the Planck length. The black holes don't have to be of cosmological size. Assuming the conventional hierarchy of scales, it is enough if they are larger than 10^{-35} m - i.e. ten to minus 35 meters; Prof. Penrose made an error that is 50 orders of magnitude in size. The corrections to the black hole entropy are suppressed by powers of "Plancklength/Radius" and can be neglected for all radii except for the ridiculously small ones (the Planckian ones). It should not be difficult for an open-minded intelligent person to learn how to calculate these order-of-magnitude estimates properly.
  • based on some remarkable agreements between 'entropy calculations' done in different ways, rather than on an actual derivation of the Bekenstein-Hawking expression. ...
This sentence reveals another kind of confusion. In string theory, one can make many superficially different calculations of the entropy of the same black hole. They all lead to the same result in the cases of all black holes where the calculation can be finished and all of them are derivations of the correct expression that we call the Bekenstein-Hawking formula because Bekenstein and Hawking were the first ones to have gotten it, using one of the methods (based on geometric arguments in the bulk). Neither of the string-theoretical derivations is "more actual" than others. The only conceivable interpretation of the words "actual" vs. "non-actual" in the sentence above is that the "non-actual" ones are those that the author has not yet learned.
  • some of the strongest claims can be discounted altogether (such as string theory having provided a complete consistent theory of quantum gravity) ...
String theory is undoubtedly a complete and a consistent theory of quantum gravity: for example, it is a framework to fully calculate a unitary S-matrix for graviton scattering (perturbatively or, using Matrix theory and other approaches, non-perturbatively). What may remain uncertain are two things:
  • 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.
But disputing that it is a complete theory of quantum gravity and that it is a consistent theory of quantum gravity in 2006 is unreasonable. Everyone can discount whatever he wants, but he should not be surprised if his reasoning looks ignorant to other physicists.
  • 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?
That's a good question. In various definitions of "patches" of string theory, some of these dualities can be proved, usually very easily. For example, if we describe M-theory on a K3 surface by a matrix model - namely the six-dimensional (2,0) theory on K3 x S^1 x time - then we can easily prove that M-theory on K3 is equivalent to heterotic string theory on T^3 because both of these theories (and the corresponding spaces) appear as degeneration limits of K3 x S^1 in which either S^1 or K3 is much larger than the other factor. There are dozens of such places where some of the equivalences may be proved, but currently we don't know any framework that would include all of string theory from the very beginning, and that would allow us to prove all of its dualities using the same set of arguments.
  • ... indeed such a reason ... as yet undiscovered, ... still does not reassure us that the string theorists are doing physics." 31.18 p. 926
It's perhaps natural for a scientist who does not know the regime in which the Bekenstein-Hawking formula may be trusted to also be uncertain whether string theory is physics. In this sense, Prof. Penrose made a correct derivation of conclusions from his (flawed) starting point. In this context, Prof. Penrose's misunderstanding of the four points above - and his confusion about dozens of other points in string theory - is undoubtedly primary, and his main incorrect conclusion (about the uncertainty whether string theory is physics) is secondary.


  1. Dear Lubos,

    Quantum 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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