A novel interpretation of the Klein-Gordon equation

TL;DR

This paper proposes a new interpretation of the Klein-Gordon equation, suggesting solutions are influenced by both initial and final boundary conditions, leading to potential deviations from standard quantum mechanics.

Contribution

It introduces a boundary condition-based interpretation of the Klein-Gordon equation, offering a framework that nearly recovers quantum probabilities and predicts deviations under certain conditions.

Findings

Standard quantum probabilities are nearly recovered in the non-relativistic limit.

Deviations from quantum mechanics occur when energy constraints are near hbar/T.

The approach provides a new perspective on relativistic quantum equations.

Abstract

The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrodinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision near hbar/T (where T is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted.