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Maslov's complex germ and the Weyl--Moyal algebra in quantum mechanics and in quantum field theory
Mathematical Physics · Submitted 28 Feb 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The paper is a survey of some author's results related with the Maslov--Shvedov method of complex germ and with quantum field theory. | contribution 0.58 | 0.60 | 0.06 | asserted 0.98 |
| s1 | 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. | contribution 0.67 | 0.56 | 0.88 | asserted 0.75 |
| s2 | 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. | contribution 0.82 | 0.65 | 0.04 | asserted 1.00 |
| s3 | In particular, it contains a new simple definition of the Maslov index modulo 4. | result 0.59 | 0.87 | 0.04 | asserted 1.00 |
| s4 | 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. | contribution 0.88 | 0.79 | 0.08 | asserted 1.00 |
| s5 | It also contains a result (dynamical evolution in quantum field theory in quasiclassical approximation) close to the Maslov--Shvedov quantum field theory complex germ. | result 0.88 | 0.79 | 0.07 | asserted 0.80 |
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Original 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.
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1 arXiv paper cite this in 11 sentences
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Cites it as background 1 paper
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In particular, we consider the anisotropic Gaussian wave packet, following the independent works of Heller, Huber, Littlejohn (isotropic case) [ref] and Littlejohn [ref] , Maslov, Shvedov [ref] (anisotropic case) and more recently Robert [ref] , Nazaikinskii, Schulze, Sternin [ref] , and Stoyanovsky [ref] .
Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation · Anisotropic Wave Packet Dynamics · 0.91
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