The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory

Details The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory

2011-09-23

arxiv.org/abs/1109.5215v1

Physics

Mathematical Physics

40 pages, LaTeX + AMS

Scientific paper

We establish a precise isomorphism between the Schr\"odinger representation and the holomorphic representation in linear and affine field theory. In the linear case this isomorphism is induced by a one-to-one correspondence between complex structures and Schr\"odinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schr\"odinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any K\"ahler polarization.

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