dorsal/arxiv
View SchemaCoherent states and the quantization of 1+1-dimensional Yang-Mills theory
| Authors | Brian C. Hall |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0012050 |
| URL | https://arxiv.org/abs/quant-ph/0012050 |
| DOI | 10.1142/S0129055X0100096X |
| Journal | Rev. Math. Phys. 13 (2001) 1281-1306 |
Abstract
This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills theory on a spacetime cylinder, from the point of view of coherent states, or equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant 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 Segal-Bargmann 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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"abstract": "This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills\ntheory on a spacetime cylinder, from the point of view of coherent states, or\nequivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,\nthe coherent states are simply ordinary coherent states labeled by points in an\ninfinite-dimensional linear phase space. Gauge symmetry is imposed by\nprojecting the original coherent states onto the gauge-invariant subspace,\nusing a suitable regularization procedure. We obtain in this way a new family\nof \"reduced\" coherent states labeled by points in the reduced phase space,\nwhich in this case is simply the cotangent bundle of the structure group K.\n The main result explained here, obtained originally in a joint work of the\nauthor with B. Driver, is this: The reduced coherent states are precisely those\nassociated to the generalized Segal-Bargmann transform for K, as introduced by\nthe author from a different point of view. This result agrees with that of K.\nWren, who uses a different method of implementing the gauge symmetry. The\ncoherent states also provide a rigorous way of making sense out of the quantum\nHamiltonian for the unreduced system.\n Various related issues are discussed, including the complex structure on the\nreduced phase space and the question of whether quantization commutes with\nreduction.",
"arxiv_id": "quant-ph/0012050",
"authors": [
"Brian C. Hall"
],
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"doi": "10.1142/S0129055X0100096X",
"journal_ref": "Rev. Math. Phys. 13 (2001) 1281-1306",
"title": "Coherent states and the quantization of 1+1-dimensional Yang-Mills theory",
"url": "https://arxiv.org/abs/quant-ph/0012050"
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