A Gauge Fixing Procedure for Causal Fermion Systems

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.

arXiv comment30 pages, LaTeX, introduction and preliminaries expanded (published version)

References in corpus (13)

• A Lorentzian Quantum Geometry
• Local U(2,2) Symmetry in Relativistic Quantum Mechanics
• Causal Fermion Systems as a Candidate for a Unified Physical Theory
• Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
• Causal Fermion Systems: A Primer for Lorentzian Geometers
• Perturbative Description of the Fermionic Projector: Normalization, Causality and Furry's Theorem
• Causal Variational Principles on Measure Spaces
• Light-Cone Expansion of the Dirac Sea in the Presence of Chiral and Scalar Potentials
• The Causal Action in Minkowski Space and Surface Layer Integrals
• Perturbation Theory for Critical Points of Causal Variational Principles
• Spinors on Singular Spaces and the Topology of Causal Fermion Systems
• Light-Cone Expansion of the Dirac Sea to First Order in the External Potential
• Derivation of Local Gauge Freedom from a Measurement Principle

Cited by in corpus (10)

• Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems
• Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
• Two-Dimensional Area and Matter Flux in the Theory of Causal Fermion Systems
• The Linear Dynamics of Wave Functions in Causal Fermion Systems
• Modified Measures as an Effective Theory for Causal Fermion Systems
• Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles
• Linear Bosonic Quantum Field Theories Arising from Causal Variational Principles
• A Positive Quasilocal Mass for Causal Variational Principles
• Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
• A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems