  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.
