Remarks on Dirac-Bergmann algorithm, Dirac's conjecture and the extended Hamiltonian

TL;DR

This paper critically examines the Dirac-Bergmann algorithm's application to constrained systems, highlighting issues with the extended Hamiltonian and gauge invariance, and clarifies Dirac's conjecture with pedagogical examples.

Contribution

It clarifies the correct treatment of the extended Hamiltonian and gauge invariants, and refines the understanding of Dirac's conjecture in constrained Hamiltonian systems.

Findings

Incorrect results arise from naive treatment of the extended Hamiltonian.

Redefinition of gauge invariants aligns with the Stueckelberg trick.

Dirac's conjecture holds only for the extended Hamiltonian.

Abstract

The Dirac-Bergmann algorithm for the Hamiltonian analysis of constrained systems is a nice and powerful tool, widely used for quantization and non-perturbative counting of degrees of freedom. However, certain aspects of its application to systems with first-class constraints are often overlooked in the literature, which is unfortunate, as a naive treatment leads to incorrect results. In particular, when transitioning from the total to the extended Hamiltonian, the physical information encoded in the constrained modes is lost unless a suitable redefinition of gauge invariant quantities is made. An example of this is electrodynamics, in which the electric field gets an additional contribution to its longitudinal component in the form of the gradient of an arbitrary Lagrange multiplier. Moreover, Dirac's conjecture, the common claim that all first-class constraints are independent generators of gauge transformations, is somewhat misleading in the standard notion of gauge symmetry used in field theories. At the level of the total Hamiltonian, the true gauge generator is a specific combination of primary and secondary first-class constraints; in general, Dirac's conjecture holds only in the case of the extended Hamiltonian. The aim of the paper is primarily pedagogical. We review these issues, providing examples and general arguments. Also, we show that the aforementioned redefinition of gauge invariants within the extended Hamiltonian approach is equivalent to a form of the Stueckelberg trick applied to variables that are second-class with respect to the primary constraints.