Abstract

The structure constants of the “algebra” of constraints of a parametrized field theory are derived by a simple geometrical argument based exclusively on the path independence of the dynamical evolution; the change in the canonical variables during the evolution from a given initial surface to a given final surface must be independent of the particular sequence of intermediate surface used in the actual evaluation of this change. The requirement of path independence also implies that the theory will propagate consistently only initial data such that the Hamiltonian vanishes. The vanishing of the Hamiltonian arises because the metric of the surface is a canonical variable rather than a c-number. It is not assumed the constraints can be solved to express four of the momenta in terms of the remaining canonical variables. It is shown that the signature of spacetime can be read off from the commutator of two Hamiltonian constraints at different points. The analysis applies equally well irrespective of whether the spacetime is a prescribed Riemannian background or whether it is determined by the theory itself as in general relativity. In the former case the structure of the commutators imposes consistency conditions for a theory in which states are defined on arbitrary spacelike surfaces; whereas, in the later case it provides the conditions for the existence of spacetime— “embeddability” conditions which ensure that the evolution of a three-geometry can be viewed as the “motion” of a three-dimensional cut in a four-dimensional spacetime of hyperbolic signature.