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 Tr(AB) ≤ Tr(A)Tr(B). This relation holds on any Hilbert space \( \mathcal{H} \) and is deeply associated with the fact that the algebra B(\( \mathcal{H} \)) of bounded operators on \( \mathcal{H} \) 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.
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Acknowledgments
DM thanks Clifford Johnson for discussion of JT gravity, Jorma Louko for conversations related to positive action conjecture for wavefunctions, and Gary Horowitz for discussions of existing positive action theorems. He also thanks the Perimeter Institute for its hospitality during the final stages of this work. EC thanks Alexey Milekhin for conversations related to the trace inequality and positivity of entropy. The work of DM and ZW was supported by NSF grant PHY-2107939, and by funds from the University of California. ECs participation in this project was made possible by a DeBenectis Postdoctoral Fellowship and through the support of the ID# 62312 grant from the John Templeton Foundation, as part of the ‘The Quantum Information Structure of Spacetime’ Project (QISS). The opinions expressed in this project/publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
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Colafranceschi, E., Marolf, D. & Wang, Z. A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture). J. High Energ. Phys. 2024, 140 (2024). https://doi.org/10.1007/JHEP04(2024)140
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DOI: https://doi.org/10.1007/JHEP04(2024)140