On the role of constraints and degrees of freedom in the Hamiltonian formalism

TL;DR

This paper clarifies the role of constraints and degrees of freedom in Hamiltonian formalism, emphasizing proper interpretation of gauge symmetries and primary constraints in degenerate systems.

Contribution

It provides a pedagogical explanation of degenerate Hamiltonian systems, correcting misconceptions about gauge symmetries and physical degrees of freedom in field theories.

Findings

Clarifies the relationship between primary constraints and gauge symmetries.

Explains why gauge freedom produces twice as many first-class constraints.

Highlights the importance of primary constraints in the total Hamiltonian.

Abstract

Unfortunately, the Hamiltonian mechanics of degenerate Lagrangian systems is usually presented as a mere recipe of Dirac, with no explanation as to how it works. Then it comes to discussing conjectures of whether all primary constraints correspond to gauge symmetries, and it goes all the way to absolutely wrong claims such as the statement that electrodynamics or gravity have only two physical components each, with others being spurious. One has to be very careful because non-dynamical, or constrained, does not mean unphysical. I give a pedagogical introduction to the degenerate Hamiltonian systems, showing both very simple mechanical examples and general arguments about how it works. For the familiar field theory models, I explain why the gauge freedom there "hits twice" in the sense of producing twice as many first-class constraints as gauge symmetries, and why primary, and only primary, constraints should be put into the total Hamiltonian.