Bill Zajc sent me a link to the following fresh CERN talk by my (former) PhD adviser Tom Banks:
his and Willy Fischler's paper about firewalls.
But otherwise the results are presented as corollaries of some much deeper wisdom that I've been exposed to since 1999. Although I added another hour to the exposure, the beef in the hypothetical deeper wisdom – Tom's axiomatic holography – remains utterly incomprehensible to me.
When I was listening to this talk, it made me organize my reasons why it seems impossible to me to make sense of these things. What are the reasons?
Tom claims to have a completely new starting point to understand quantum gravity, one that is more fundamental and more accurate than any alternative that is already known to us. In this setup, Hilbert spaces and operators may be associated with finite regions of spacetime, with causal diamonds – essentially intersections of the (filled) future light cone of one point and the (filled) past light cone of another one.
This seems like a much more ambitious framework than the formulations of string theory we know. Those work with infinite spacetimes (flat Minkowski space or another space in perturbative string theory; flat Minkowski space times a simple compactification manifold in Matrix theory; AdS space times something in AdS/CFT) and the infinite extent seems essential for string theory's consistency. The boundary conditions at infinity are the players that allow us to attach the fields to their preferred values, choose the superselection sector, make something like the S-matrix scattering problem well-defined, and circumvent the fact that all local operators refuse to be gauge-invariant in a diffeomorphism-invariant theory (because the diffeomorphisms, i.e. gauge redundancies, change the coordinate location of such operators).
So I am surely not as confused about his statements as some people in the audience who think that Matrix theory "must" be a special case of Tom's axiomatic approach (they think so perhaps because Matrix theory is the insight that is most often associated with his name). It's surely not because Matrix theory is dealing with the whole infinite space but Tom's axiomatic approach is all about our ability to describe finite causal diamonds separately.
But I am still confused about almost everything else. ;-)
The main problem is that despite Tom's repeated promises that he proposes a "new formalism", there doesn't seem to be any formalism at all. The papers don't really have any mathematical structures or equations or expressions of a new kind (and in most cases, not even those of the old kinds) that could allow a mathematically inclined reader to "smell" where the scent of the new perspective is coming from. At least I've never "smelled" this scent. A new formalism is exactly what seems to be completely absent!
A compatible observation with this absence of the formalism is that major conclusions – whether they seem "probably correct" or "probably wrong" – are never derived through mathematical steps from comprehensible, well-defined axioms or assumptions. Instead, when it comes to the claims that should be the interesting conclusions, we're usually told that "Tom agrees with someone".
Now, the picture seems to involve many assumptions that I consider correct but some assumptions that I consider incorrect. But what is most confusing about the collection is that we're never told why the particular combination of the assumptions is picked and what can be derived out of this collection that would convince us that Tom sees something that others don't. In other words, we're never shown – I think – that/why the set of assumptions is the "complete set" needed to achieve something interesting.
If I mention the starting points of Tom that are correct, he is surely a believer in the proper Copenhagen-like, intrinsically probabilistic interpretation of quantum mechanics. Much like your humble correspondent, he thinks that Niels Bohr and friends had known pretty much everything they should have known (including the "conceptual framework of decoherence" even though decoherence was only articulated explicitly in recent decades) and he knows why the "alternative interpretations" of quantum mechanics such as GRW and Bohm-like "realist" interpretations (and perhaps the many worlds of the realist varieties) are wrong. He also understands holography as a novel feature of quantum gravity that forces us to qualitatively deviate from some wisdoms we got used to in the context of the effective quantum field theories. I have surely learned some things from him so some of the conclusions I am making might be secretly influenced by Tom's leadership.
Those things are great but they aren't new and the agreement seems to stop here.
It's partly because many technical assumptions that are being added to the mix for reasons that seem totally unclear. For example, Hilbert spaces are being associated with particular diamonds, regions of spacetime, or even particular world lines in the spacetime. I don't think it's really possible because this association seems to assume exactly the kind of locality – well-definedness of locations in the spacetime and even well-definedness of exact world lines – that the holographic nature of the new framework (and sometimes even the quantum fuzziness of the old type) should reject.
Moreover, the world lines of "all conceivable observers" seem like an even less well-defined collection of concepts because these world lines should also include non-differentiable, almost everywhere time-like trajectories (because they do contribute to the path integral) and if you look at these trajectories with too high a resolution, you must run into the conflict with the quantum gravity's ban on "probing too short distances". Quite generally, his picture seems to make the metric tensor degrees of freedom play too classical a role; it disagrees with the dictum "spacetime is doomed" that I surely believe to be right.
So for these and other reasons, I don't really believe that there's a way within quantum gravity to exactly associate a Hilbert space with a finite causal diamond and/or a world line of an observer in this causal diamond (despite the fact that I do think that something could or should become more well-defined or simpler for regions with null boundaries). The previous statement should be viewed as a no-go theorem (well, a hypothesis) about possible formulations of string/M-theory or another hypothetical consistent theory of quantum gravity. It's a non-existence statement I believe to hold although I can't prove it. Tom believes the opposite one but I don't think he is proving it, either. But even if I decided to join his faith system, it is still not clear to me why it would be interesting, i.e. what interesting conclusions or insights I could derive out of this membership in the "church". ;-)
In my opinion, the association of the Hilbert spaces with finite causal diamonds – when we take it as an exact statement about a quantum gravitational theory – probably violates the holographic character of the underlying dynamics. Or using a weaker statement, if this approach works, it must come with a mechanism that erases the bulk degrees of freedom (here I mean some degrees of freedom inside the intersections of several causal diamonds or on their shared boundaries) so that only the surface degrees of freedom remain physical. I don't see any structure in Tom's "atlas of diamonds" that would fulfill this task so this "atlas of diamonds" seems effectively local in the sense of a local effective QFT.
Another technicality that seems to be an artifact of misleading intuition is the spinor bundle and the "qubits" composition of the Hilbert space. There doesn't exist a reason why the Hilbert space dimension should be a power of two; it's mostly the incorrect paradigm believed by those who understand our ordinary computers (that just happen to be based on binary numbers, but even this thing could be done differently – and in fact, it has been) more than they understand physics (which doesn't include Tom, as the glitches with the laptop during the talk show). In fact, in the examples of interesting systems in quantum gravity we may describe, the Hilbert space dimension is never a power of two (think about the microstates of a Strominger-Vafa black hole with some charges). The entropy is never an integer multiple of \(\ln 2\). After all, it has no reasons to be.
Again, even if we joined Tom (and others, in the case of this particular assumption) in his belief that these special Hilbert spaces constructed out of qubits play a special fundamental role in quantum gravity, what of it? What encouraging conclusions (either unifying conclusions or those that nontrivially agree with some facts of Nature we know either from experiments or from "mathematical experiments" we did we particular vacua of string/M-theory) could we derive out of this assumption besides the assumption itself? In other words, why? If there's no answer to these "why" question, is it surprising that most people don't buy these proposals?
More generally, Tom seems to view the dimensionality of the Hilbert spaces associated with the diamonds to be entries on his (never completely specified) list of axioms. This very ambition seems very bizarre for a fundamental theory of spacetime because the dimension should be derived from a more elementary starting point: it's a thermodynamic property of the spacetime and its subsets. It's not clear to me how a precise i.e. microscopic theory of the spacetime could ever result from some properties of the physical system (spacetime) in the macroscopic regime, i.e. from the thermodynamic limit, which – by the basic character of thermodynamics – only contains a very small information about the microscopic theory. Quite typically, when we have a well-defined framework, the dimensions are the dimensions of representations of an algebra of operators and we must know the whole algebra, not only the numerical values of the dimensions.
In a later part of the talk, Tom is suggesting that his/their insights are at least compatible if not (partly?) equivalent to some cute proposals by Ted Jacobson. Now, I would positively interpret Jacobson's work as a nice way to "localize" some Bekenstein-Hawking-led efforts to relate general relativity with thermal phenomena. Bekenstein and Hawking would only tell you things like \(S=A/4G\) and \(T=a/2\pi\) (gravitational acceleration at the event horizon: I wrote it for the Unruh effect here) for some particular static global solutions etc. Jacobson is associating temperatures and entropy densities with small places in the spacetime and reinterpreting Einstein's equations as some equations of thermodynamics applied to the situation in which the physical system of interest is the dynamical spacetime.
I think it's cute, much of it is right, but many of the extra words that Jacobson has said about those things are self-evidently wrong, too. Many statements in related papers seem to be extremely loosely connected with the actual calculations that do work. Effectively, Jacobson is telling you "if you liked this calculation, then you must also agree with this and that because it's being said by the same Jacobson". Sorry, that's not how a rational careful person increases the collection of things he believes to hold. Tom's proselytizing strategy seems to be a bit similar. Too many statements just don't follow from any arguments or calculations.
A particular feature of Tom's approach is that quantum gravity is presented as some kind of "quantized hydrodynamics". Andy Strominger co-led the program to relate Einstein's equations near the event horizon with the equations of hydrodynamics (Navier-Stokes equations in various forms). I sort of understand what's going on over here although some questions remain. But it's clear that there's a mathematically nontrivial yet verifiable map between two a priori distinct systems of equations.
Tom's picture seems to be different. The calculations seem to be missing. On the other hand, Tom seems to extract much more far-reaching conceptual conclusions out of this (hypothetical) map. But if there's some evidence for these connections, I fail to see it again.
Well, the main problem here is that we really shouldn't quantize hydrodynamics because hydrodynamics is an effective theory of a high-entropy system. It ignores lots of degrees of freedom whose existence is needed for the high entropy. If I am a bit more explicit, what I mean is that we know that "a liter of [static] water at room temperature" doesn't correspond to a unique quantum state in the Hilbert space because all the water molecules are still very complicated and chaotic, stupid. However, in hydrodynamics, this object effectively does correspond to a unique classical configuration (vanishing speed, constant temperature etc.). That's really why hydrodynamics isn't and can't be a detailed microscopic, fully quantum theory of the liquid.
If you look what is the main feature that allows an effective theory to be both useful and intrinsically non-fundamental, you will realize that it's the incorporation of "local temperature" among its degrees of freedom. According to statistical physics, the temperature is only a meaningful quantity that describes a physical system once we start to look at the ensembles of microstates – many microstates in a set that we effectively refuse to distinguish. And because we don't distinguish them, we're inevitably overlooking some microscopic degrees of freedom. If we're not overlooking anything, we simply can't associate the temperature to generic states because a fixed temperature comes with very special probability distributions for the microstates and the most generic microstates just can't agree with the thermal distributions for any \(T\).
Once you understand and endorse the previous paragraph, you should agree with me that all effective theories that use the concept of a temperature are inevitably non-fundamental, effective theories for high-entropy systems that overlook some degrees of freedom. Bekenstein, Hawking, and perhaps Jacobson have told us how to construct such a description of black holes and quantum gravity. But it's an inseparable part of the picture that this framework simply can't be microscopically exact, it can't tell us about any details how the entropy is actually encoded. Tom – and perhaps Jacobson – seem to disagree with the argument above (it's supposed to be a full proof, not just an invitation for you to believe or not believe something) and I think that they're manifestly wrong about this point.
Also, Tom says a mostly correct thing that the Hawking evaporation results from the interactions of the quantum fields around the black hole – which kind of admit a local description – with the degrees of freedom of the black hole or its event horizon that are inherently non-field-theoretical. I agree with that. It's just an interaction between a system whose microscopic structure (local fields: they're OK outside the black hole) seems to be kind of understood; and degrees of freedom (inside the black hole, those that carry the bulk of the black hole entropy) that seem to be mysterious.
While I stay positive, I should mention that this paradigm explains why it's so hard to uniquely answer the question "Where is the entropy of a black hole located?". There can't be a canonical, good answer because the black hole doesn't respect the rules of locality, as we know them from the effective quantum field theories, so it doesn't have to organize the degrees of freedom by their location.
However, I would still be more cautious when I am saying similar things. I would say that the black hole dynamics and theories describing the black hole evaporation may work even if they say nothing about the location of the degrees of freedom, even if they reject the quantum-field-theory-like paradigms for the bulk of the black hole entropy. However, there may still exist some "dual descriptions" of the set of black hole microstates that effectively does such a thing (black hole fuzzball solutions are a major example) and I don't see anything wrong with their hypothetical existence. Even if they exist, they don't hurt; they're just a new "dual" way to look at the Hilbert space of the microstates. So I don't understand how it could ever be helpful for learning something if we assumed that such a local description of the degrees of freedom mustn't exist. Again, it looks like another assumption by Tom that seems completely useless because nothing useful may ever come out of it.
A few paragraphs above, I mentioned that descriptions with the temperature are inevitably non-fundamental. Jacobson effectively relates curvature tensors with local temperatures of a sort so this description with curvatures – general relativity – isn't a fundamental description of the microstates, either. Indeed, any particular vacuum of string theory is producing lots of degrees of freedom that go well beyond the general theory of relativity. The metric tensor is just the ground state in an infinite tower of excited closed string states, if I mention the most transparent perturbative string theory example.
Tom seems to say that his axiomatic framework can say lots of things about the fundamental microscopic laws of physics, perhaps the spectrum of particles etc. It seems very obvious to me that this can't be the case. Because he incorporates the temperature (by identifying it with the spacetime curvature, according to some of those Jacobson's recipes), it is inevitably just an effective, thermodynamic description of some properties of the quantum gravitational system.
I think that Tom has never derived anything about the spectrum of fields or particles out of his starting point. It could be just because I have overlooked something he has said or written but the previous paragraph contains an argument that tells me that such a derivation can't really exist. In other words, Tom is working with some effective description – or some vague properties of it – and assumes that it must contain the deepest definition of "a theory of everything" even though it's strikingly obvious to me that it can't contain such a thing.
Again, Tom differs from the people whose proposals drive me up the wall because they are ignorant about most of the established science – he knows the physics up to some relatively recent point of departure very well and he also knows how to present it. But despite this "promising DNA" and even "welcome conclusions", the newly proposed ideas seem frustratingly illogical to me.
Also, Tom meaningfully concludes that there are no firewalls but his arguments seem to be incomprehensible to me, too. And they seem to have a very little overlap with the reasons why your humble correspondent or e.g. Raju and Papadodimas and others think that the AMPS proof is wrong. I can't get rid of the feeling that Tom hasn't attacked the detailed microscopic steps in the AMPS argument at all – he is rejecting everything based on his opposition to the framework of the effective field theory. But even though the effective field theory is misleading about many points, it must still be very accurate and nontrivially sensible when it comes to many other points because this success has been verified experimentally. So Tom's refusal of the effective field theory even as a "standard that a deeper theory must emulate when it comes to the everyday life predictions" seems like a case of the baby thrown out with the bath water and it's hard to believe anything based on this "very revolutionary" approach.