Symmetry Theories, Wigner's Function, Compactification, and Holography

TL;DR

This paper explores how Wigner's function offers a unified way to interpret symmetry data in quantum field theories, especially in complex boundary and entangled states, with applications to string theory and holography.

Contribution

It introduces Wigner's quasi-probabilistic function as a tool to interpret symmetry data in various quantum field theory contexts, including entangled states and boundary conditions.

Findings

Wigner's function provides a physical interpretation of symmetry data.

Application to string compactifications and holographic systems.

Addresses complexity in fixing absolute symmetry data.

Abstract

The global symmetry data of a $D$-dimensional absolute quantum field theory can sometimes be packaged in terms of a $(D+1)$-dimensional bulk system obtained by extending along an interval, with a relative QFT$_D$ at one end and suitable gapped / free boundary conditions at the other end. The partition function of the QFT$_D$ can then be interpreted as a wavefunction depending on background fields. However, in some cases, it is not possible or simply cumbersome to fix an absolute form of the symmetry data. Additionally, it is also of interest to consider entangled and mixed states of relative QFTs as well as entangled and mixed states of gapped / free boundary conditions. We argue that Wigner's quasi-probabilistic function on phase space provides a physical interpretation of the symmetry data in all such situations. We illustrate these considerations in the case of string compactifications and holographic systems.