Spaces of solutions of relativistic field theory with constraints

Abstract

In this paper I shall explain how the reduction results of Marsden and Weinstein [38] can be used to study the space of solutions of relativistic field theories. Two of the main examples that will be discussed are the Einstein equations and the Yang-Mills equations. The basic paper on spaces of solutions is that of Segal [49]. That paper deals with unconstrained systems and is primarily motivated by semilinear wave equations. We are mainly concerned here with systems with constraints in the sense of Dirac. Roughly speaking, these are systems whose four dimensional Euler-Lagrange equations are not all hyperbolic but rather split into hyperbolic evolution equations and elliptic constraint equations.