Entropy change during measurement of free particle in the relativistic quantum case.

The relativistic quantum mechanical transition amplitude for a free particle is the propagator of the Gordon-Klein equation and is calculated through multiple methods. This propagator is transformed into an information entropy change by means of a Wick rotation. However, the entropy may also be derived from a variational principle. The Fourier method and path integral approaches produce similar entropy functions with Bessel-function factors, but with key differences in normalization. Various approximations to the path-integral derivation of the relativistic Green's function and related entropy are examined. Uncertainty-like relations are developed for the non-relativistic case, while the relativistic one leads to proper-time quantization. A relation between dimensional time and information-entropy time is developed.