Reduced phase space induced decay conditions

TL;DR

This paper develops a method to define decay conditions in gauge field theories by focusing on the reduced phase space, simplifying the analysis of boundary conditions and gauge invariance.

Contribution

It introduces a reduced phase space approach that systematically relates decay conditions of true degrees of freedom to gauge degrees of freedom in field theories.

Findings

Decay conditions are parametrized by true degrees of freedom.

Gauge decay follows from constraints and gauge conditions.

Method simplifies boundary condition analysis in gauge theories.

Abstract

The definition of the phase space of field theories in presence of boundaries of Cauchy surfaces requires a choice of boundary conditions or decay behaviour of those fields. Often these conditions are motivated in part by the decay behaviour of the initial data of known exact solutions. In the case of gauge field theories the initial data are not free but are subject to initial value constraints. Still, the decay behaviour is commonly specified for the kinematical, i.e. unconstrained phase space. This can lead to the following practical problem: The constraints are preferably solved for field variables on which they depend only algebraically, i.e. not involving derivatives, as otherwise one would need to solve partial differential equations. However, the specified decay behaviour may prevent from doing that. On the other hand, a precise specification of decay for all kinematical fields appears unnecessary because the decay of gauge degrees of freedom is not observable. Yet, knowledge of their decay is required as one needs to compute Poisson brackets on the kinematical phase space in order to define what gauge invariance means. Thus the interplay between the constraint structure and the decay properties of the kinematical phase space is complex. In this contribution we develop a reduced phase space induced approach to the decay problem. Upon specifying gauge conditions tailored to the algebraic structure of the constraints, these define a split of the kinematical phase space into gauge and true degrees of freedom. Then the decay conditions of the kinematical phase space is systematically parametrised by a choice of decay for just the true degrees of freedom (i.e. the reduced phase space), the decay of the gauge degrees of freedom then follows unambiguously from solving both the constraints and the gauge conditions.