A statistical derivation of non-relativistic quantum theory

U. Klein

Published 2011-09-28, updated 2014-12-22Version 2

A previous derivation of the single-particle Schr\"odinger equation from statistical assumptions is generalized to an arbitrary number $N$ of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It is found that the same statistical assumptions that imply Schr\"odinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. For a single particle, spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. For a $N$-particle system, spin is introduced as a consequence of gauge coupling.The classical limit $\hbar \to 0$ of Schr\"odinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.