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Title: A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture)
A<sc>bstract</sc> 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 operatorsA, Bwill satisfy the trace inequality Tr(AB) ≤ Tr(A)Tr(B). This relation holds on any Hilbert space$$ \mathcal{H} $$ H and is deeply associated with the fact that the algebraB($$ \mathcal{H} $$ H ) of bounded operators on$$ \mathcal{H} $$ 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.  more » « less
Award ID(s):
2107939
PAR ID:
10511714
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer
Date Published:
Journal Name:
Journal of High Energy Physics
Volume:
2024
Issue:
4
ISSN:
1029-8479
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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