[Paper Review] Gauging spacetime inversions in quantum gravity

Daniel Harlow, Tokiro Numasawa|arXiv (Cornell University)|Nov 16, 2023

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper argues that in quantum gravity, spacetime inversion symmetries like $ΓΓΓ$ must be gauged due to the absence of global symmetries. It shows that the Euclidean path integral automatically implements $ΓΓΓ$ symmetry, enforces a real Hilbert space in closed universes, and requires inclusion of non-time-orientable Lorentzian manifolds—demonstrating that such geometries are essential for consistency in AdS/CFT.

ABSTRACT

Spacetime inversion symmetries such as parity and time reversal play a central role in physics, but they are usually treated as global symmetries. In quantum gravity there are no global symmetries, so any spacetime inversion symmetries must be gauge symmetries. In particular this includes $\mathcal{CRT}$ symmetry (in even dimensions usually combined with a rotation to become $\mathcal{CPT}$), which in quantum field theory is always a symmetry and seems likely to be a symmetry of quantum gravity as well. In this article we discuss what it means to gauge a spacetime inversion symmetry, and we explain some of the more unusual consequences of doing this. In particular we argue that the gauging of $\mathcal{CRT}$ is automatically implemented by the sum over topologies in the Euclidean gravity path integral, that in a closed universe the Hilbert space of quantum gravity must be a real vector space, and that in Lorentzian signature manifolds which are not time-orientable must be included as valid configurations of the theory. In particular we give an example of an asymptotically-AdS time-unorientable geometry which must be included to reproduce computable results in the dual CFT.

Motivation & Objective

• To understand the implications of gauging spacetime inversion symmetries—particularly $ΓΓΓ$—in quantum gravity.
• To resolve the tension between $ΓΓΓ$ being a symmetry in quantum field theories and the absence of global symmetries in quantum gravity.
• To establish that $ΓΓΓ$ symmetry is automatically realized via the sum over topologies in the Euclidean path integral.
• To show that in a closed universe, the Hilbert space of quantum gravity must be a real vector space due to antiunitary time-reversal symmetry.
• To argue that non-time-orientable Lorentzian manifolds must be included as physical configurations in the quantum gravity path integral.

Proposed method

• Analyzing the analytic continuation of $ΓΓΓ$ from Euclidean to Lorentzian signature, where it corresponds to a $π$-rotation in the plane of one spatial direction and time.
• Using the path integral formulation of Euclidean quantum gravity to argue that the sum over topologies naturally implements $ΓΓΓ$ as a gauge symmetry.
• Applying the argument that antiunitary time-reversal operators in closed universes project the Hilbert space onto a real vector space.
• Constructing an explicit example of an asymptotically-AdS, non-time-orientable spacetime that must be included to reproduce CFT correlation functions.
• Employing conformal frame transitions and holomorphic transition functions to derive a topological formula linking winding numbers to the Euler characteristic.
• Using the Gauss-Bonnet theorem and triangulation to relate the Euler density to the winding of conformal frames across patch overlaps.

Experimental results

Research questions

• RQ1 How can spacetime inversion symmetries like $ΓΓΓ$ be consistently gauged in quantum gravity, given the absence of global symmetries?
• RQ2 What are the physical consequences of gauging an antiunitary symmetry such as time reversal in a closed universe?
• RQ3 Why must non-time-orientable Lorentzian manifolds be included in the quantum gravity path integral?
• RQ4 How does the sum over topologies in the Euclidean path integral enforce $ΓΓΓ$ symmetry?
• RQ5 What role do conformal frame winding numbers play in determining the Euler characteristic of a 2D surface in quantum gravity?

Key findings

• The $ΓΓΓ$ symmetry is automatically implemented in the sum over topologies in the Euclidean gravity path integral, making it a fundamental feature of the quantum gravity path integral.
• In a closed universe, the Hilbert space of quantum gravity must be a real vector space due to the antiunitary nature of the $ΓΓΓ$ operator, which implies that physical states are invariant under complex conjugation and time reversal.
• Non-time-orientable Lorentzian manifolds must be included as valid configurations in the quantum gravity path integral, as demonstrated by a concrete example in asymptotically-AdS spacetime.
• The inclusion of a non-time-orientable geometry in the path integral is necessary to reproduce computable results in the dual CFT, showing its physical necessity.
• The winding of conformal frames across patch overlaps is directly related to the Euler characteristic via a topological formula derived from the Gauss-Bonnet theorem.
• The formula $\chi(\Sigma) = \sum_{\alpha<\beta<\gamma} \int_{V_{\alpha\beta\gamma}} w_{\alpha\beta\gamma} $ shows that the Euler characteristic counts the total winding number of conformal frames across triple overlaps.