 00505 Brian C. Hall
 Coherent states and the quantization of 1+1dimensional YangMills
theory
(86K, Latex)
Dec 19, 00

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Abstract. This paper discusses the canonical quantization of 1+1dimensional
YangMills theory on a spacetime cylinder, from the point of view of
coherent states, or equivalently, the SegalBargmann transform. Before
gauge symmetry is imposed, the coherent states are simply ordinary
coherent states labeled by points in an infinitedimensional linear
phase space. Gauge symmetry is imposed by projecting the original
coherent states onto the gaugeinvariant subspace, using a suitable
regularization procedure. We obtain in this way a new family of
"reduced" coherent states labeled by points in the reduced phase space,
which in this case is simply the cotangent bundle of the structure
group K.
The main result explained here, obtained originally in a
joint work of the author with B. Driver, is this: The reduced coherent
states are precisely those associated to the generalized SegalBargmann
transform for K, as introduced by the author from a different point of
view. This result agrees with that of K. Wren, who uses a different
method of implementing the gauge symmetry. The coherent states also
provide a rigorous way of making sense out of the quantum Hamiltonian
for the unreduced system.
Various related issues are discussed, including the complex structure
on the reduced phase space and the question of whether quantization
commutes with reduction.
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