Point transformations and non-linear systems in quantum theory

Abstract

The quantization of the non-linear mechanical system L(t) = 12Q̇igij(Q) Q̇j) − V(Q) (i, j = 1, …, n), is considered in detail. It is shown that the quantum action principle is satisfied when allowable q-number variations are used. When the above system is formulated as a first-order system in phase space with the Lagrangian L′ = PiQi − H, it is shown that c-number variations are allowable and that the action principle has its conventional practical form. From the action principle with a variation of the coupling constant, the S-matrix is calculated within perturbation theory. In its Hamiltonian form the well-known additional h̷2 potential is found, and in its Lagrangian form we find, apart from the log term, a different h̷2 potential only recently given in the literature. The relations of these results to the Feynman path integral formalism is shown by means of explicit calculations.