A Deformation Theory of Self-Dual Einstein Spaces

The self-dual Einstein equations on a compact Riemannian 4-manifold can be expressed as a quadratic condition on the curvature of an $SU(2)$ (spin) connection which is a covariant generalization of the self-dual Yang-Mills equations. Local properties of the moduli space of self-dual Einstein connections are described in the context of an elliptic complex which arises in the linearization of the quadratic equations on the $SU(2)$ curvature. In particular, it is shown that the moduli space is discrete when the cosmological constant is positive; when the cosmological constant is negative the moduli space can be a manifold the dimension of which is controlled by the Atiyah-Singer index theorem.Comment: 13 page

Covariant Phase Space Formulation of Parametrized Field Theories

Parametrized field theories, which are generally covariant versions of ordinary field theories, are studied from the point of view of the covariant phase space: the space of solutions of the field equations equipped with a canonical (pre)symplectic structure. Motivated by issues arising in general relativity, we focus on: phase space representations of the spacetime diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces.Comment: 22 page

Natural Symmetries of the Yang-Mills Equations

We define a natural generalized symmetry of the Yang-Mills equations as an infinitesimal transformation of the Yang-Mills field, built in a local, gauge invariant, and Poincar\'e invariant fashion from the Yang-Mills field strength and its derivatives to any order, which maps solutions of the field equations to other solutions. On the jet bundle of Yang-Mills connections we introduce a spinorial coordinate system that is adapted to the solution subspace defined by the Yang-Mills equations. In terms of this coordinate system the complete classification of natural symmetries is carried out in a straightforward manner. We find that all natural symmetries of the Yang-Mills equations stem from the gauge transformations admitted by the equations.Comment: 23 pages, plain Te

The Problems of Time and Observables: Some Recent Mathematical Results

We present 2 recent results on the problems of time and observables in canonical gravity. (1) We cannot use parametrized field theory to solve the problem of time because, strictly speaking, general relativity is not a parametrized field theory. (2) We show that there are essentially no local observables for vacuum spacetimes.Comment: Talk presented at the Lanczos Centenary Conference 3 pages, plain Te

Schrodinger representation for the polarized Gowdy model

The polarized ${\bf T}^3$ Gowdy model is, in a standard gauge, characterized by a point particle degree of freedom and a scalar field degree of freedom obeying a linear field equation on ${\bf R}\times{\bf S}^1$. The Fock representation of the scalar field has been well-studied. Here we construct the Schrodinger representation for the scalar field at a fixed value of the Gowdy time in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure. We show that ``typical'' field configurations are slightly more singular than square-integrable functions on the circle. For each time the corresponding Schrodinger representation is unitarily equivalent to the Fock representation, and hence all the Schrodinger representations are equivalent. However, the failure of unitary implementability of time evolution in this model manifests itself in the mutual singularity of the Gaussian measures at different times.Comment: 13 page