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An explicit model for complementarity in quantum gravity

Restoring the global image out of a fragment of a hologram, QG edition

Since the early 1990s, the black hole complementarity has been a philosophical paradigm capable of solving the information loss paradox. The detailed information may get out of the black hole (in patterns of the Hawking radiation), after all, because the degrees of freedom inside the black hole are scrambled functions of some degrees of freedom outside. But in order for this picture not to contradict some well-known facts, the scrambling must be chaotic enough so that no patterns demonstrating the "equality" of the degrees of freedom in the two regions must be visible in low-energy experiments.

For a long time, this paradigm was rather vague. Numerous papers by Kyriakos Papadodimas and Suvrat Raju – that have been discussed on this blog – could have been viewed as steps towards explicit formulae that show us how the black hole complementarity works. From the beginning, they acknowledge that locality can't be "quite exact". It means that the local fields cannot be taken for granted, either.

Instead, one assumes that there is a fixed Hilbert space of microstates to start with and the algebra of effective local field operators – especially those in the black hole interior – has to be embedded into that Hilbert space. Using the thermal doubling tricks and some other clever ideas, they have showed that there is a way to do so that preserves all the properties of the local operators as long as you don't multiply too many of them.

In simple experiments, physics of the black hole interior looks local. Complicated, very fine experiments are capable of proving the nonlocality – the refusal of the black hole interior to exist "quite independently" from the exterior.

In a new Dutch-Swiss-Indian paper

A toy model of black hole complementarity,
Raju and Papadodimas along with Souvik Banerjee and Jan-Willem Bryan add a few very interesting steps to this whole PR program.

First of all, they are showing that the complementarity doesn't work just in the presence of black holes – e.g. in the presence of "objective" event horizons which are, by definition, the essential ingredients that make a hole black. Instead, they show that the black hole complementarity works even in (macroscopically) empty anti de Sitter space.

I have always found this generalization natural. When one solves black hole information mysteries, he isn't just learning things about "black holes". Black holes are composed of pieces of spacetimes that may exist in the absence of black holes, too. So the lessons from a viable solution to the black hole information puzzle unavoidably teach us something about quantum gravity in general.

Second of all, the four authors construct very explicit expressions for the approximate field operators – check their equation (4.7) as a key result that proves that their treatment is very explicit, indeed.

"A Capella Science" on the LIGO discovery.

I have mentioned that in their toy AdS spacetime, there is no event horizon. Well, they create a "man-made horizon" by demanding that all the field operators are only written as functionals of operators in a "time band" – this is just a phrase denoting the "layer" of the spacetime obeying \[

t_{\rm min}\lt t\lt t_{\rm min}+\Delta t

\] where \(\Delta t\) (measured as proper time in the center of the AdS) is shorter than the AdS radius. With this restriction, a man-made horizon emerges. The time band plays the same role as the near-boundary "black hole exterior" in the normal discussions of the black hole complementarity. But despite the horizon, it's possible to write approximations for the field operators outside the time band. And they may be arbitrarily accurate.

Simple (e.g. single-trace) functions of the operators in the "time band" are relevant for the experiments inside the time band (analogously for the near-boundary region in AdS); complicated (e.g. very long multi-trace) functions of the operators are relevant for the region outside (analogously for the black hole interior).

The general playful fact that the states outside may be approximated arbitrarily well by the action of some restricted operators is known in quantum field theory from the Reeh-Schlieder theorem, a result that they use as well as generalize. An ironic feature of this theorem is that in QFT, the Reeh-Schlieder theorem is often proven by assuming locality; in this picture of quantum gravity, it's used as a clever mechanism that clarifies how nonlocality of quantum gravity works.

In the equation (1.2) we see that what they need are specific polynomials\[

{\mathcal P}_{\alpha, p_c} = \sum_{p=0}^{p_c} (-1)^p \frac{(\alpha H)^p}{p!} \tag{1.2}

\] where \(\alpha=\ln(N)\) and \(p_c=N\ln (N)\). It's a truncated Taylor series for \(\exp(-\alpha H)\). If you first send \(p_c\to\infty\) and then \(\alpha\to\infty\), the operator becomes the projection operator \(P_0\) on the CFT ground state.

Again, the main idea is that if you play clever enough games with some operators restricted to a region of the spacetime, you may approximate physics outside the region arbitrarily accurately, too. Their empty AdS example is a toy model for approximating the field operators in a black hole interior and this approximation is essential for the complete understanding of the black hole information puzzle.

Today, Barry Mazur – a fun guy whom I know well from the Harvard Society of Fellows – and William Stein are releasing their new book about the Riemann Hypothesis and primes. Buy it if you care about this topic. I am going to discuss some related issues now, anyway.

In recent months, I spent more time with the Riemann Hypothesis than with the black hole information puzzle. There is one property of the Riemann zeta function, the universality, that is remarkably similar in spirit to the approximations they use.

Recall that for \({\rm Re}(s)\gt 1\) and especially \({\rm Re}(s)\to\infty\), the function \(\zeta(s)\) basically converges to one (except for a simple pole at \(s=1\)). In the mirror region \({\rm Re}(s)\lt 0\), the behavior of \(\zeta(s)\) is determined from the behavior in \({\rm Re}(s)\gt 1\) by the functional equation [a proof involving the Poisson resummation] that exchanges \(s\leftrightarrow 1-s\).

So the zeta function only shows its "brutally complicated behavior" in the critical strip in between,\[

0\lt {\Re}(s) \lt 1.

\] The zeroes of the zeta function – values of \(s\) for which \(\zeta(s)=0\) – are distributed symmetrically. For a zero \(s\), \(s^*,1-s,1-s^*\) are also zeros. The Riemann Hypothesis conjectures that all the zeroes sit on the critical line i.e. obey \(s=1-s^*\) and we only deal with pairs of mutually complex conjugate values of \(s\) rather than with the quadruplets.

Off-topic: Brian Greene has finally become famous when a pet was named after him, namely the nursery web spider Dolomedes Briangreenei (Czech: lovčík briangreeneový) in Brisbane, Australia.

But back to the point I want to make. The universality says that if you choose any open region \(R\) inside the critical strip and invent any holomorphic function \(f(s)\) that is nonzero everywhere in the region, you may shift the region \(R\) in the vertical direction by some specific, large enough amount \(\Delta t_\epsilon\) and the function \(f(s)\) will agree with \(\zeta(s)\) within an arbitrarily tiny error margin, i.e.\[

\forall \epsilon\gt 0 :\,\exists \Delta t_\epsilon :\, \forall s\in R:\, \\
\qquad\qquad\qquad |f(s)-\zeta(s+i\Delta t_\epsilon)| \lt \epsilon.

\] To fully prove this assertion, one needs to assume the Riemann Hypothesis. But don't forget to be excited about it: when you're moving the zeta function in the vertical direction inside the critical strip, it "probes" all possible holomorphic functions you may imagine, with an arbitrarily good accuracy.

At the end, one shouldn't be shocked that it's the case. The "ultraviolet" behavior of \(\zeta(s)\) in some region of the critical strip is dictated by the locations of the nearby zeroes. And those are distributed as eigenvalues of random matrices (random matrix theory); the normalization of the holomorphic function may be affected by the "infrared" density of the zeroes in a larger nearby region. Because there are infinitely many eigenvalues, an arbitrary configuration of the eigenvalues and their gaps has to be realized somewhere, arbitrarily accurately.

Their Reeh-Schlieder-style constructions are morally similar. They want to approximate local field operators – the analogy of the holomorphic function \(f(s)\) in \(R\) – and they do so making choices in the time band – the analogy of finding \(\Delta t\) for which the several local zeroes of the zeta function have good enough locations.

The analogy may be closer and more quantitative than I suggested so far, especially because the truncation limit \(p_c=N \ln(N)\) resembles expressions that consistently emerge in the discussion of the primes and zeta zeroes, for example, it is an approximation for the \(N\)-th prime.

I can't resist to make another heuristic, popular comment which seems enlightening to me. Holograms – those in the real life – have one cool property, namely that if you only possess one small broken piece of the hologram, you may see the whole 3D object in it (but with a poorer sharpness). Holography in quantum gravity is mathematically analogous to Dennis Gábor's Hungarian (i.e. non-gravitational) holography in wave optics. So this qualitative result must also hold in quantum gravity.

And the paper by Banerjee et al. (discussed above) is doing something similar. Out of the fragment of the hologram – out of the field operators in the thin time band – one can reconstruct everything that can be seen up to the horizon. But they go further and show that there exists a (no longer unique) way to construct functions of the "operators in the fragment of the hologram" that also approximate objects outside the horizon arbitrarily accurately. Regions of spacetime are ready for any possible continuation of the spacetime fields behind the region's boundary.

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