Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

We show that the Ashtekar-Isham extension A/G of the configuration space of Yang-Mills theories A/G is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.These results are then used to prove that A/G is contained in a zero measure subset of A/G with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on A/G. Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the "extended" configuration space A/G.

The Segal Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group G with its normalized Haar measure + H , the Hall transform is an isometric isomorphism from L 2

We study the dynamics of flat Friedmann-Robertson-Walker (FRW) cosmologies in the presence of a triplet of massive vector fields with SO(3) global symmetry. We find an E 3 -symmetric ansatz for the vector fields that is compatible with the E 3 -invariant FRW metric and propose a method to make invariant ansätze for more general cosmological models. We use techniques of dynamical systems to study qualitatively the behaviour of the model and find, in particular, that the effective equation of state of the system changes gradually from a radiation-dominated to a matter-dominated form and that the scale of the transition depends on the mass of the gauge fields. † Work supported in part by a JNICT graduate scholarship BD/138/90-RM and by STRIDE/FEDER Project JN.91.02.‡ Work supported by a GTAE (Grupo Teórico de Altas Energias) grant.

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