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

SIGMA 11 (2015), 071, 12 pages      arXiv:1507.02201      https://doi.org/10.3842/SIGMA.2015.071

Path Integrals on Euclidean Space Forms

Guillermo Capobianco and Walter Reartes
Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca, Buenos Aires, Argentina

Received July 09, 2015, in final form August 31, 2015; Published online September 03, 2015

Abstract
In this paper we develop a quantization method for flat compact manifolds based on path integrals. In this method the Hilbert space of holomorphic functions in the complexification of the manifold is used. This space is a reproducing kernel Hilbert space. A definition of the Feynman propagator, based on the reproducing property of this space, is proposed. In the $\mathbb{R}^n$ case the obtained results coincide with the known expressions.

Key words: path integrals; holomorphic quantization; space forms; reproducing kernel Hilbert spaces.

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