Historical research guide · 0777 · 2023
My Personal History With the Quantum Theory of de Sitter Space
Tom Banks
In one minute
Tom Banks wrote this short piece to set the record straight about who proposed what, and when, in the quantum theory of de Sitter space — the geometry our own accelerating universe appears to be heading towards. The physics inside the history is the interesting part. Banks recounts the moment in 1999 when several ideas locked together for him: the entropy Gibbons and Hawking assigned to a de Sitter horizon should be read as the logarithm of the number of quantum states available, which makes empty space a state of maximum entropy holding a finite, countable amount of information, and makes every localized object a reduction of that count. He and Willy Fischler built that into Holographic Space-time, a formalism in which every finite causal diamond carries its own finite Hilbert space. He also revises his own early guess, arguing the horizon is better described by a two-dimensional conformal field theory — and that this version predicts the temperature ripples we already see in the microwave background.
Connection
Chapter 2 treats the vacuum as a real, structured medium rather than an empty stage, and Banks is arguing the strongest version of that from the quantum-gravity side: empty space carries a definite, finite amount of information, and the flat-space vacuum is itself a high-entropy state built from zero-energy modes. His revised horizon model also gives chapter 13 a rare thing — a picture of the vacuum with an observational consequence already on the sky.
From the source
The Gibbons and Hawking entropy of de Sitter space should be read as the logarithm of the dimension of its Hilbert space. That makes empty de Sitter space a maximal entropy state, and it makes any object bound to a geodesic a reduction of the entropy — because from the point of view of any geodesic, anything not bound to it fades into the cosmological horizon in a time of order the de Sitter radius.Section 1, Introduction, paragraphs 1 and 2
Two arguments support a finite number of states. Mathematically, because de Sitter space has no asymptotic space-like or null regions, the de Sitter isometries are all gauge transformations and the whole space consists of gauge copies of a single static patch. Physically, generic asymptotic data extrapolate to a singularity, so specifying both past and future de Sitter asymptotics restricts one to a finite dimensional Hilbert space — a line of reasoning Bousso and Freivogel later proved as a theorem.Section 1, Introduction, the paragraph on the two arguments
Holographic Space-time generalises the de Sitter conjecture to every finite-area causal diamond: a Hilbert bundle over a manifold of time-like trajectories, a finite dimensional Hilbert space for each diamond, discrete time evolution with the Planck time as the interval, and a Quantum Principle of Relativity requiring that the density matrices of any two overlapping diamonds carry identical entanglement spectra.Section 1, the paragraphs introducing Holographic Space-time and the Quantum Principle of Relativity
Banks revises his own earliest ansatz. The claim that the density matrix of empty de Sitter space is maximally uncertain is probably wrong and should be replaced by the Carlip and Solodukhin ansatz, in which the modular Hamiltonian of a diamond is the cut-off Virasoro generator of a two-dimensional conformal field theory living on the stretched horizon, with central charge proportional to the horizon area. A welcome consequence is that the modular Hamiltonian fluctuates in inflationary horizon volumes, which is what produces the observed temperature fluctuations in the cosmic microwave background — a model without them would be ruled out by observation.Section 4, Conclusions, third bullet
Quantum field theory is not a good approximation to everything in a theory of gravity. Inside a causal diamond it is valid at most for states whose entropy is bounded by the diamond area in Planck units raised to the power of one minus one over the space-time dimension — and Cohen, Kaplan and Nelson showed that discarding the excluded states does not spoil the agreement between quantum field theory and any known experiment. Field theory therefore breaks down not only at short distance but in any situation with large entropy.Section 2.1, The Limitations of Quantum Field Theory; Section 4, fourth bullet
The Minkowski vacuum is itself a high entropy state, with the entropy coming from zero energy states — a conclusion Banks reaches by taking flat space as a limit of de Sitter space and, separately, as a limit from negative cosmological constant. He names the nature of the asymptotic Hilbert space, and the existence of a non-perturbative scattering matrix in four dimensions, as the questions that remain open.Section 4, Conclusions, fifth bullet
The way in
https://arxiv.org/abs/2312.10729How to cite it
Tom Banks (2023) My Personal History With the Quantum Theory of de Sitter Space arXiv:2312.10729