General quadratic gauge theory. Constraint structure, symmetries, and physical functions

TL;DR

This paper explores the relationship between Hamiltonian and Lagrangian formulations of quadratic gauge theories, establishing a clear connection between constraints, gauge symmetries, and physical functions, and proving the Dirac conjecture.

Contribution

It introduces superspecial phase-space variables to simplify the Hamiltonian form and provides a rigorous analysis linking constraints, symmetries, and physical functions in quadratic gauge theories.

Findings

Existence of superspecial phase-space variables simplifies the Hamiltonian form.

Derived the general structure of symmetries in quadratic gauge theories.

Proved the Dirac conjecture relating first-class constraints and gauge invariance.

Abstract

How can we relate the constraint structure and constraint dynamics of the general gauge theory in the Hamiltonian formulation with specific features of the theory in the Lagrangian formulation, especially relate the constraint structure with the gauge transformation structure of the Lagrangian action? How can we construct the general expression for the gauge charge if the constraint structure in the Hamiltonian formulation is known? Whether can we identify the physical functions defined as commuting with first-class constraints in the Hamiltonian formulation and the physical functions defined as gauge invariant functions in the Lagrangian formulation? The aim of the present article is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions. To fulfill such a program, we demonstrate the existence of the so-called superspecial phase-space variables in terms of which the quadratic Hamiltonian action takes a simple canonical form. On the basis of such a representation, we analyze a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analyzing a symmetry equation. We then use these results to identify the two definitions of physical functions and thus prove the Dirac conjecture.