Abstract
We study Hamiltonian form of unfree gauge symmetry where the gauge parameters have to obey differential equations. We consider the general case such that the Dirac-Bergmann algorithm does not necessarily terminate at secondary constraints, and tertiary and higher order constraints may arise. Given the involution relations for the first-class constraints of all generations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form, including the differential equations constraining gauge parameters. All the field theories with unfree gauge symmetry share the common feature: they admit sort of "global constants of motion" such that do not depend on the local degrees of freedom. The simplest example is the cosmological constant in the unimodular gravity. We consider these constants as modular parameters rather than conserved quantities. We provide a systematic way of identifying all the modular parameters. We demonstrate that the modular parameters contribute to the Hamiltonian constraints, while they are not explicitly involved in the action. The Hamiltonian analysis of the unfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, including explicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRST Hamiltonian quantization method for the case of unfree gauge symmetry. The main distinction is in the content of the non-minimal sector and gauge fixing procedure. The general formalism is exemplified by traceless tensor fields of irreducible spin $s$ with the gauge symmetry parameters obeying transversality equations.
Highlights
Gauge symmetry is usually understood as a set of the infinitesimal transformations of the fields such that leaves the action intact, while the transformation parameters are the functions of space-time
We work out constrained Hamiltonian formalism corresponding to the unfree gauge symmetry with gauge parameters constrained by differential equations
In the Hamiltonian form, the phenomenon of the unfree gauge symmetry has been clarified from viewpoint of involution relations between Hamiltonian and constraints
Summary
Gauge symmetry is usually understood as a set of the infinitesimal transformations of the fields such that leaves the action intact, while the transformation parameters are the functions of space-time. All the field theories with unfree gauge symmetry share the common feature: they admit the “global constants of motion” such that do not depend on the local degrees of freedom, with Λ of the UG being the simplest example This general fact is explained from various viewpoints in the recent papers [8,9,10]. The new phenomenon here is that the modular parameters, being connected to the nontrivial asymptotics of the fields, can make the constraints explicitly depending on the space-time point x, even though the original Lagrangian is x independent. This phenomenon has previously unnoticed analogue in Lagrangian formalism. We provide a convenient formula for the degree of freedom counting in Lagrangian formalism in the case of unfree gauge symmetry
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