Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 050, 31 pages      arXiv:1009.5615      https://doi.org/10.3842/SIGMA.2012.050

Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation

Robert Oeckl
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, C.P. 58190, Morelia, Michoacán, Mexico

Received April 27, 2012, in final form August 03, 2012; Published online August 09, 2012

Abstract
We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein-Gordon theory.

Key words: geometric quantization; topological quantum field theory; coherent states; foundations of quantum theory; quantum field theory.

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