Reformulation of gauge theories in terms of gauge invariant fields

Abstract

We present a reformulation of gauge theories in terms of gauge invariant fields. Focusing on Abelian theories, we show that the gauge and matter covariant fields can be recombined to introduce new gauge invariant degrees of freedom. Starting from the $(1+1)$ dimensional case on the lattice, with both periodic and open boundary conditions, we then generalize to higher dimensions and to the continuum limit. To show explicit and physically relevant examples of the reformulation, we apply it to the Hamiltonian of a single particle in a (static) magnetic field, to pure abelian lattice gauge theories, to the Lagrangian of quantum electrodynamics in $(3+1)$ dimensions and to the Hamiltonian of the $2d$ and $3d$ Hofstadter model. In the latter, we show that the particular construction used to eliminate the the gauge covariant fields enters the definition of the magnetic Brillouin zone. Finally, we briefly comment on relevance of the presented reformulation to the study of interacting gauge theories.

20 pages, 2 figures. v2: improved manuscript with the correct implementation of periodic boundary conditions. v3: published version with the new Section IX

References in corpus (7)

• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Loop, String, and Hadron Dynamics in SU(2) Hamiltonian Lattice Gauge Theories
• Confinement and lack of thermalization after quenches in the bosonic Schwinger model
• Confined phases of one-dimensional spinless fermions coupled to $Z_2$ gauge theory
• A gauge redundancy-free formulation of compact QED with dynamical matter for quantum and classical computations
• Deconfining disordered phase in two-dimensional quantum link models
• Exact diagonalization of cubic lattice models in commensurate Abelian magnetic fluxes and translational invariant non-Abelian potentials