Maslov's complex germ and the Weyl--Moyal algebra in quantum mechanics and in quantum field theory

TL;DR

This paper surveys the use of the Weyl algebra and Moyal product to simplify the Maslov--Shvedov method in quantum mechanics and quantum field theory, offering new insights and definitions.

Contribution

It introduces a unified algebraic framework using Weyl algebra and Moyal product, simplifying complex germ methods and perturbative quantum field theory calculations.

Findings

Simplified the Maslov--Shvedov method using Weyl algebra

Provided a new simple definition of the Maslov index modulo 4

Presented a self-consistent approach to perturbative quantum field theory without infinities

Abstract

The paper is a survey of some author's results related with the Maslov--Shvedov method of complex germ and with quantum field theory. The main idea is that many results of the method of complex germ and of perturbative quantum field theory can be made more simple and natural if instead of the algebra of (pseudo)differential operators one uses the Weyl algebra (operators with Weyl symbols) with the Moyal *-product. Section 1, devoted to quantum mechanics, contains a closed mathematical description of the Maslov--Shvedov method in the theory of Schrodinger equation, including the method of canonical operator. In particular, it contains a new simple definition of the Maslov index modulo 4. Section 2, devoted to quantum field theory, contains a logically self-consistent exposition of the main results of perturbative quantum field theory not using the subtraction of infinities from the quantum Hamiltonian of free field and normal ordering of operators. It also contains a result (dynamical evolution in quantum field theory in quasiclassical approximation) close to the Maslov--Shvedov quantum field theory complex germ.