A Gauge Fixing Procedure for Causal Fermion Systems

TL;DR

This paper develops a gauge fixing procedure for causal fermion systems, utilizing spectral and polar decompositions, and introduces coordinate systems to fix gauge freedom, with applications to Dirac sea configurations.

Contribution

It presents a detailed method to fix local gauge freedom in causal fermion systems using spectral decompositions and Riemannian geometry, providing explicit coordinate constructions.

Findings

Established distinguished gauges fixing local gauge freedom

Constructed Gaussian and wave coordinate systems for causal fermion systems

Applied methods to Dirac sea configurations in finite and infinite volumes

Abstract

Causal fermion systems incorporate local gauge symmetry in the sense that the Lagrangian and all inherent structures are invariant under local phase transformations of the physical wave functions. In the present paper it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give rise to distinguished gauges where the local gauge freedom is fixed completely up to global gauge transformations. The main method is to use spectral and polar decompositions of operators on Hilbert spaces and on indefinite inner product spaces. We also introduce and make use of a Riemannian metric which is induced on the manifold of all regular correlation operators by the Hilbert-Schmidt scalar product. Gaussian coordinate systems corresponding to this Riemannian metric are constructed. Moreover, we work with so-called wave charts where the physical wave functions are used as coordinates. Our constructions and results are illustrated in the example of Dirac sea configurations in finite and infinite spatial volume.