Abstract
This paper is the review on theory of infinite-dimensional pseudo-differential operators(PDO) and their application to quantization of systems with the infinite number of degrees of freedom. There are considered various approaches to the theory of infinite-dimensional PDO (e.g., Berezin’s approach based on polynomial operators). There is presented in details the calculus of PDO based on the theory of distributions on infinite-dimensional spaces (general locally convex spaces). This calculus is based on the “Feynman measure” on the phase space (introduced by Smolyanov in 80th). Symbols of the most important PDO in the representation of second quantization are calculated.
This paper was partly supported by EU-Network “QP and Applications” and Nat. Sc. Found., grant N PHY99-07949 at KITP, Santa-Barbara, and visiting professor fellowship at Russian State Humanitarian University.
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Khrennikov, A. (2006). Distributions and Pseudo-Differential Operators on Infinite-Dimensional Spaces with Applications in Quantum Physics. In: Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 164. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7514-0_12
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