[Paper Review] A Paradox and its Resolution Illustrate Principles of de Sitter Holography

Leonard Susskind|arXiv (Cornell University)|Apr 2, 2023

Cosmology and Gravitation Theories6 citations

TL;DR

This paper resolves a paradox in de Sitter holography where semiclassical correlation functions exhibit imaginary parts, contradicting the holographic principle’s prediction of purely real correlation functions due to a maximally mixed density matrix. The resolution lies in recognizing time-reversal as a gauge symmetry and introducing quantum reference frames (QRFs), which allow gauge-fixing to a preferred time direction, restoring consistency between semiclassical and holographic descriptions by making the imaginary part non-zero only in the gauge-fixed, observer-dependent formulation.

ABSTRACT

Semiclassical gravity and the holographic description of the static patch of de Sitter space appear to disagree about properties of correlation functions. Certain holographic correlation functions are necessarily real whereas their semiclassical counterparts have both real and imaginary parts. The resolution of this apparent contradiction involves the fact that time-reversal is a gauge symmetry in de Sitter space -- a point made by Harlow and Ooguri -- and the need for an observer (or quantum reference frame) as advocated by Chandrasekaran, Longo, Penington, and Witten.

Motivation & Objective

• To resolve a contradiction between semiclassical gravity and holographic principles in de Sitter space, where correlation functions have imaginary parts in the former but not in the latter.
• To clarify the role of time-reversal symmetry in de Sitter space, showing it acts as a gauge symmetry—implying all physical states must be invariant under it.
• To demonstrate that the absence of imaginary parts in holographic correlation functions arises from gauge invariance, not a flaw in the formalism.
• To establish that observers and quantum reference frames (QRFs) are essential for recovering the semiclassical limit in de Sitter holography.
• To show that gauge-fixing to a preferred time direction (e.g., forward-moving clocks) restores non-zero imaginary parts in correlation functions, resolving the paradox.

Proposed method

• Apply the holographic principle to the static patch of de Sitter space, positing that bulk physics is encoded in degrees of freedom on the stretched horizon.
• Use the assumption of a maximally mixed density matrix (Tr[O] = ⟨O⟩) to enforce that all expectation values are trace-normalized, implying no preferred state.
• Invoke Harlow and Ooguri’s insight that time-reversal is a gauge symmetry in de Sitter space, requiring all physical states and observables to be time-reversal invariant.
• Analyze the commutator [A(t₁), A(t₂)] as a time-reversal-odd operator, which must vanish in the gauge-invariant formulation due to symmetry.
• Introduce quantum reference frames (QRFs), particularly clocks, as physical subsystems that break time-reversal symmetry through gauge fixing.
• Fix the gauge by selecting a specific time direction (e.g., forward-moving clocks), allowing non-zero imaginary parts in correlation functions, consistent with semiclassical field theory.

Figure 1: Penrose diagram for de Sitter space. The dashed black curve is the stretched horizon of the right-side static patch. The dashed red curve shows a particle being emitted and absorbed by the stretched horizon.

Experimental results

Research questions

• RQ1 Why do semiclassical correlation functions in de Sitter space have non-zero imaginary parts, while holographic correlation functions computed with a maximally mixed density matrix appear to have only real parts?
• RQ2 How can time-reversal symmetry, which is a gauge symmetry in de Sitter space, be reconciled with the appearance of imaginary parts in correlation functions?
• RQ3 What is the role of observers and quantum reference frames (QRFs) in resolving the apparent conflict between semiclassical and holographic descriptions?
• RQ4 Why does the trace of a commutator vanish in the holographic description, and how does this relate to gauge invariance and the absence of time-reversal-odd operators?
• RQ5 How does gauge-fixing via a physical clock restore the semiclassical prediction of non-zero imaginary parts in correlation functions?

Key findings

• The imaginary part of the correlation function ⟨A(t₁)A(t₂)⟩ vanishes in the gauge-invariant holographic description because it is proportional to the trace of a commutator, which is zero.
• Time-reversal is a gauge symmetry in de Sitter space, implying that all physical states and observables must be invariant under time reversal, and thus time-reversal-odd operators like [A(t₁), A(t₂)] must vanish.
• The paradox is resolved by recognizing that the semiclassical approximation breaks time-reversal symmetry, which is only allowed after gauge-fixing via a physical observer or quantum reference frame.
• The existence of an observer—a fluctuation that includes a clock—is necessary to define a preferred time direction and to break the time-reversal gauge symmetry.
• In the gauge-fixed description (e.g., with a forward-going clock), the imaginary part of the correlation function becomes non-zero, matching the semiclassical result.
• The gauge-fixing procedure is analogous to Coulomb gauge fixing in QED, where gauge-invariant states have zero expectation value for certain operators, but the gauge-fixed theory allows non-zero values.