Spectral Projections for Density Matrices in Quantum Field Theories

TL;DR

This paper explores spectral projections of density matrices in quantum field theories, revealing universal divergences, constructing new states, and applying to non-Hermitian operators, with implications for holography and quantum geometry.

Contribution

It introduces a method using spectral projections to analyze density matrices, including regularization, eigenvalue density computation, and applications to holography and non-Hermitian operators.

Findings

Universal divergent terms in stress energy tensor expectation values

Construction of new quantum states via spectral projections

Application to non-Hermitian operators like transition matrices

Abstract

In this paper, we investigate the spectral projection of density matrices in quantum field theory. With appropriate regularization, the spectral projectors of density matrices are expected to be well-defined. These projectors can be obtained using the Riesz projection formula, which allows us to compute both the density of eigenvalues and the expectation values of local operators in the projected states. We find that there are universal divergent terms in the expectation value of the stress energy tensor, where the coefficients depend universally on the density of eigenvalues and a function that describes the dependence of eigenvalues on boundary location. Using projection states, we can construct a series of new states in quantum field theories and discuss their general properties, focusing on the holographic aspects. We observe that quantum fluctuations are suppressed in the semiclassical limit. We also demonstrate that the fixed area state, previously constructed using gravitational path integrals, can be constructed by suitably superposition of appromiate amount of projection states. Additionally, we apply spectral projection to non-Hermitian operators, such as transition matrices, to obtain their eigenvalues and densities. Finally, we highlight potential applications of spectral projections, including the construction of new density and transition matrices and the understanding of superpositions of geometric states.