322 research outputs found
Finite dimensional approximations to Wiener measure and path integral formulas on manifolds
Certain natural geometric approximation schemes are developed for Wiener
measure on a compact Riemannian manifold. These approximations closely mimic
the informal path integral formulas used in the physics literature for
representing the heat semi-group on Riemannian manifolds. The path space is
approximated by finite dimensional manifolds consisting of piecewise geodesic
paths adapted to partitions of . The finite dimensional manifolds of
piecewise geodesics carry both an and a type Riemannian
structures . It is proved that as the mesh of the partition tends to
,
where is the energy of the piecewise geodesic path , and
for and , is a ``normalization'' constant, is
the Riemannian volume form relative , and is Wiener measure on
paths on . Here and
where is the scalar curvature of . These results are also shown to
imply the well know integration by parts formula for the Wiener measure.Comment: 48 pages, latex2e using amsart and amssym
Yang-Mills theory and the Segal-Bargmann transform
We use a variant of the classical Segal-Bargmann transform to understand the
canonical quantization of Yang-Mills theory on a space-time cylinder. This
transform gives a rigorous way to make sense of the Hamiltonian on the
gauge-invariant subspace. Our results are a rigorous version of the widely
accepted notion that on the gauge-invariant subspace the Hamiltonian should
reduce to the Laplacian on the compact structure group. We show that the
infinite-dimensional classical Segal-Bargmann transform for the space of
connections, when restricted to the gauge-invariant subspace, becomes the
generalized Segal-Bargmann transform for the the structure group
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