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

Abstract

We establish a precise isomorphism between the Schrödinger 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 Schrödinger 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 Schrödinger 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 Kähler polarization.

40 pages, LaTeX + AMS; v2: minor corrections, reference updates

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• The S-matrix in Schrödinger Representation for Curved Spacetimes in General Boundary Quantum Field Theory
• Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution
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• General Boundary Quantum Field Theory in Anti de Sitter Spacetimes
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