A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture)

TL;DR

This paper demonstrates that Euclidean gravitational path integrals obey a fundamental trace inequality analogous to quantum operators, supported by a new positive action conjecture, with proofs in specific gravity models and implications for holography.

Contribution

It establishes a trace inequality for gravitational path integrals and introduces a positive action conjecture, advancing understanding of holographic dualities and quantum gravity.

Findings

Euclidean gravitational path integrals satisfy the trace inequality at all orders.

A positive action conjecture for quantum gravity wavefunctions is proposed and proven for Jackiw-Teitelboim gravity.

The results support the consistency of holographic dualities with quantum operator inequalities.

Abstract

The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators $A,B$ will satisfy the trace inequality $\operatorname{Tr}(AB) \leq \operatorname{Tr}(A) \operatorname{Tr}(B)$. This relation holds on any Hilbert space ${\cal H}$ and is deeply associated with the fact that the algebra $B({\cal H})$ of bounded operators on ${\cal H}$ is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories.