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

TL;DR

The paper examines subtle aspects of the Dirac–Bergmann algorithm for constrained Hamiltonian systems, showing that naive use of the extended Hamiltonian can obscure physical content carried by constrained modes. Through a pedagogical progression from a simple toy model to Yang–Mills theory and General Relativity, it analyzes the correct gauge generators via Castellani’s construction and contrasts them with Dirac’s conjecture, clarifying how gauge invariance should act on observables. A central message is that, to maintain equivalence between total and extended formulations, one must redefine gauge-invariant quantities (akin to a Stueckelberg trick) so they commute with all first-class constraints; this preserves the physical content, including static configurations, while allowing extra gauge freedom. The work provides a unified view of gauge generators, extended Hamiltonians, and observable definitions across canonical mechanical systems, YM theory, and GR, with implications for consistent quantization and interpretation of gauge theories.

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.