Quantum spin dynamics (QSD): VII. Symplectic structures and continuum lattice formulations of gauge field theories

Interesting nonlinear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator-valued distributions. Therefore, one is usually forced to find a smeared substitute for such a function which corresponds to a regularization. The smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally, one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit. In this paper we begin the investigation of these steps for diffeomorphism-invariant quantum field theories of connections. We introduce a (generalized) projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that a subset of the corresponding projective limit can be identified with the symplectic manifold that one started from. Then we illustrate the programme outlined above by applying it to the Gauss constraint. This paper also complements, as a side result, earlier work by Ashtekar, Corichi and Zapata who observed that certain operators are non-commuting on certain states, although the Poisson brackets between the corresponding classical functions vanish. These authors showed that this is not a contradiction provided that one refrains from a phase space quantization but rather applies a quantization based on the Lie algebra of vector fields on the configuration space of the theory. Here we show that one can provide a phase space quantization, that is, one can find other functions on the classical phase space which give rise to the same operators but whose Poisson algebra precisely mirrors the quantum commutator algebra. The framework developed here is the classical cornerstone on which the semiclassical analysis in a new series of papers called `gauge theory coherent states' is based.