Kirill Russkov | Searcharxiv

arXiv2026

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

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.

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arXiv2025

Revisiting the relaxation of constraints in gauge theories

Recently, there were works claiming that path integral quantisation of gauge theories necessarily requires relaxation of Lagrangian constraints. As has also been noted in the literature, it is of course wrong since there perfectly exist gauge field quantisations respecting the constraints, and at the same time the very idea of changing the classical theory in this way has many times appeared in other works. On the other hand, what was done in the path integral approach is fixing a gauge in terms of zero-momentum variables. We would like to show that this relaxation is what normally happens when one fixes such a gauge at the level of action principle in a Lagrangian theory. Moreover, there is an interesting analogy to be drawn. Namely, one of the ways to quantise a gauge theory is to build an extended Hamiltonian and then add new conditions by hand such as to make it a second class system. The constraints' relaxation occurs when one does the same at the level of the total Hamiltonian, i.e. a second class system with the primary constraints only.

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