Relations between path integrals and the variational principles of Hamilton

Karl Zeile

John J. Hegseth

Many quantization schemes rely on analogs of classical mechanics where the connections with classical mechanics are indirect. In this work I propose a new and direct connection between classical mechanics and quantum mechanics where the quantum mechanical propagator is derived from a variational principle. I identify this variational principle as a generalized form of Hamilton's principle. This proposed variational principle is unusual because the physical system is allowed to have imperfect information, i.e., there is incomplete knowledge of the physical state. Two distribution functionals over possible generalized momentum paths a[p(t)] and generalized coordinates paths b[q(t)] are defined. A generalized action is defined that corresponds to a contraction of a[p(t)], b[q(t)], and a matrix of the action evaluated at all possible p and q paths. Hamilton's principle is the extremization of the generalized action over all possible distributions. The normalization of the two distributions allows their values to be negative and they are shown to be the real and imaginary parts of the complex amplitude. The amplitude in the Feynman path integral is shown to be an optimal vector that extremizes the generalized action. This formulation is also shown to be directly applicable to statistical mechanics and I show how irreversible behavior and the micro-canonical ensemble follows immediately.

Ashok Kumar Kapoor

Martin C. Gutzwiller

M. Z. Shaharir

The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)= 1 / 2 p j g jk (q)p k + 1 / 2 (A j (q),p j )+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.

Xiang-Yao Wu, Ben-Shan Wu, Hong Li +1 more

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.