On the Space-Time Formulation of Non-Relativistic Quantum Mechanics

For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space-time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrödinger equation, the transformation function K(x,t′;y,t′) in the space representation on the basis of the composition rule K(x, t″; y, t′) = ∫K(x, t″; z, t)dzK(z, t; y, t′) (1) coupled with the supposition that it is approximated to zeroth order in the quantum of action h by the so-called semi-classical kernel Ko(x, t″; y, t′) = (i/h)∂2S/∂x∂y1/2 exp (i/ℏ) S(x,t″; y, t′ (2) written in terms of the classical action S(x, t″; y, t′) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t″ − t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.