Born rule and Feynman path integral from a phase-consistency condition

Nakka Gurappa

Chung-Ru Lee

Sonia Mazzucchi

John Edward Gough

Abstract It has been argued that the Feynman path integral formalism leads to a quantization rule, and that this is the Born–Jordan rule which gives the unique quantization rule consistent with the correct short-time propagator behavior of the propagator for nonrelativistic systems. We examine this short-time approximation and conclude, contrary to prevailing views, that the asymptotic expansion applies only to Hamiltonian functions that are at most quadratic in the momentum and with constant mass. While the Born–Jordan rule suggests the appropriate quantization of functions in this class, there are other rules which give the same answer, most notably the Weyl quantization scheme.

Jacob Christ

We explore Poincar´e recurrence as a mechanism for generating quantum-likestatistics from recurrent classical dynamics. Modeling the universe as a singlerecurrent, ergodic trajectory on a compact phase space, we show that an embedded observer’s ignorance of their location along the recurrent trajectory definesa natural ensemble. Using the Kolmogorov Extension Theorem, we constructa Kolmogorov-consistent, normalizable measure on the induced path space. Wefurther show that recurrent loop topology fixes the complex phase weighting ofhistories, reconstructing the Feynman path integral. The Born rule emerges fromthe same consistency requirements.