Path Integral Formulation of Free Propagators and Maximum Entropy
Francesco R. Ruggeri
The Feynman path integral approach to calculating a quantum propagator K(x,x1,t) such that W(x,t) = Integral dx1 K(x,x1,t) W(x1,0) involves a purely classical Lagranian. The form exp(i L(x,dx/dt)) and integration over all possible classical paths between x1,0 and x,t taken as tiny dt slices. For the free propagator, this calculation may be simplified by taking the Fourier transform of K(x,x1,t) which introduces exp(ikx) into the problem. We argue that taking the Fourier transform is not simply a math procedure as p represents physical (classical) momentum. Thus, exp(ipx) appears in this approach as associating physical momentum with a wavelength. Furthermore, it seems that backward forward motion is allowed in each dt slice because constant momentum is a classical solution and any positive or negative momentum may be used with the slice. Thus. “quantum mechanical” features appear already in what seems to be a classical calculation through exp(ipx). In an earlier note, we argued that exp(ipx), (the wavefunction of a free particle) may itself be treated as a conditional probability and obtained by maximizing Shannon’s entropy WlnW subject to the constraint xW i.e. allowing the particle to move back and forth even at time t=0. From exp(ipx), one may desire a Lorentz invariant wavefunction so px becomes px-Et. Then, one may construct the free particle propagator directly from: K(x,x1,t) = Integral dk exp(ik(x-x1)) exp(i kk/2m t) which matches the Feynman approach. In both cases, one uses classical physics and the idea of maximization of entropy. In the second approach, one considers a specific constant momentum p when maximizing W. In the Feynman approach, all momenta are included for a free particle because given that one has particle move form x1,0 to x,t any initial (constant) momentum is possible. Thus, we argue that the Feynman propagator for a free particle follows the same statistical approach as the maximization of a free particle conditional probability (i.e. wavefunction) subject to the constraint xW. This constraint, which seems to represent some forward backward motion directly, appears indirectly in the Feynman approach as exp(ipx) does not appear unless one uses a Fourier transform (which represents a physical wavelength).
•Feb 12, 2021•0 citations