On Dirac's incomplete analysis of gauge transformations

Abstract

Dirac's approach to gauge symmetries is discussed. We follow closely the steps that led him from his conjecture concerning the generators of gauge transformations at a given time -to be contrasted with the common view of gauge transformations as maps from solutions of the equations of motion into other solutions-to his decision to artificially modify the dynamics, substituting the extended Hamiltonian (including all first-class constraints) for the total Hamiltonian (including only the primary first-class constraints). We show in detail that Dirac's analysis was incomplete and, in completing it, we prove that the fulfilment of Dirac's conjecture -in the "non-pathological" cases-does not imply any need to modify the dynamics.We give a couple of simple but significant examples.

Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems

“…This yields the same result. Uniqueness of solutions of the MLCP also follows as an extension of Proposition 5, where one rewrites the second line in (32) as λ 1 (t) − λ 2 (t) ∈ ker(∇h(q)), μ 1 (t) − μ 2 (t) ∈ ker(∇f (q)), and the first line in (33) as ∇h(q)…”

“…The theory of singular Lagrangian and Hamiltonian systems has a long history in Physics, where geometrical, coordinate-free analysis are performed [9,18,26,33]. In particular, equivalence is shown in [9].…”

“…The constraints p n−r = 0 are called the primary constraints in the literature [18,33]. Finally, the classical equality H (q, p, λ) = p Tq − L(q,q, λ) remains valid, and equivalently…”

“…They are, however, tailored to multibody systems where the usefulness of coordinate-free, purely geometrical arguments as in [9,18,26,33] stemming from Dirac's original results is unclear when time comes to study contact complementarity problems.…”

Léon Rosenfeld’s general theory of constrained Hamiltonian dynamics

“…His argumentation gave rise to the widely-held view that "first-class constraints are gauge generators". Aside from the fact that Dirac did not use the term "gauge" anywhere in his lectures, later work on relating the constraints to variational symmetries revealed that a detailed investigation on the full constraint structure of the theory in question is needed; see Pons [66].…”