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Czechoslovak Journal of Physics

Feynman quantization: An operator approach to path integration over commuting and anticommuting variables

V. Janiš

Published Aug 1, 1990• 1 citation
Publisher
DOI
10.1007/bf01597956

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The Lagrangian formulation of quantum dynamics in terms of path-integrals due to Feynman describes systems for which the Hamiltonian is classical in form and quantization is carried out in terms of commutators rather than anticommutators. The difficulty with the Feynman method is the actual evaluation of the pathintegral itself. An explicit evaluation is given for classical wave motion in one dimension. This requires an extension of the Feynman method which was introduced by Tobocman and studied in detail by Davies. The work of Corson on the question of a unified formulation of dynamics is discussed. (Author)
The concept of a Fock–Stueckelberg space of quantum states and a procedure of an operator quantization using only Lagrangians (kinematical quantization) are introduced. A propagator operator 𝒦, matrix elements of which are Green’s functions, is used, and an equation of motion for it is derived. We prove that kinematical quantization is an operator (coordinate-free) form of the Feynman quantization technique. The Feynman path integral (FPI) is obtained as a spectral representation of the operator 𝒦 in a coordinate basis. The connection of a representation of commutation relations in this scheme, the domain of integration in FPI, and causality is mentioned.