Dynamic constraints on Feynman's path integral

Pavel Exner

Wataru Ichinose

The Feynman path integral for regularized nonrelativistic quantum electrodynamics is studied rigorously. We begin with the Lagrangian function of the corresponding classical mechanics and construct the Feynman path integral. In the present paper, the electromagnetic potentials are assumed to be periodic with respect to a large box and quantized through their Fourier coefficients with large wave numbers cut off. Firstly, the Feynman path integral with respect to paths on the space of particles and vector potentials is defined rigorously by means of broken line paths under the constraints. Secondly, the Feynman path integral with respect to paths on the space of particles and electromagnetic potentials is also defined rigorously by means of broken line paths and piecewise constant paths without the constraints. This Feynman path integral is stated heuristically in Feynman and Hibbs' book. Thirdly, the vacuum and the state of photons of given momenta and polarizations are expressed concretely as functions of variables consisting of the Fourier coefficients of vector potentials. It is also proved rigorously in terms of distribution theory that the Coulomb potentials between charged particles naturally appear in the above Feynman path integral approach. This shows that the photons give rise to the Coulomb force.

Cécile DeWitt-Morette

Bung-Ning Teng

The Feynman's path integral in space-time is closely related, via classical Lagrangian theory, to Dirac's formulation of quantum mechanics.The essential point of this approach is in making use of the Lagrangian function in space-time and Hamilton's first action principle to construct the probability amplitude in conventional quantum mechanics.This formulation has, as mentioned by Feynman, mathematical and physical incompleteness.We have investigated a part of the physical incompleteness, which can be treated by using the classical technique of eliminating ignorable.coordinates.The corresponding conjugate momenta will then provide constraint relations, which can be used to reduce the Lagrangian function and the path integral to a modified form.The paths are now restricted, via the constraint relations, to limited configuration spaces.A general modified formulation is constructed for paths in a limited space-time region.An important special case is demonstrated.In the special case, Jacobi's action principle is employed to replace Hamilton's action principle, therefore, the paths thus formulated are in space-energy rather than in space-time.

Yoshimasa Nakano

Hamiltonian formalism for systems with both c-numbers and Grassmann numbers developed by Casalbuoni is re-investigated to be pointed out that the invariance properties of the canonical transformation work also in cases of general potential. Following the quantization procedures of Dirac and Faddeev, a form of Feynman path-integral with constraints for boson-fermion systems is presented.

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