Building a frame and gauge free formulation of quantum mechanics

TL;DR

This paper develops a frame and gauge free formulation of quantum mechanics using absolute quantities, providing clearer physical interpretations, extending the theory's scope, and offering new insights into phenomena like the Aharonov-Bohm effect.

Contribution

It introduces an absolute, frame and gauge invariant reformulation of quantum mechanics, replacing wave functions with physically interpretable absolute quantities and equations.

Findings

Sharper uncertainty relations derived.

Extended validity of Ehrenfest's theorem.

Potential explanation of Aharonov-Bohm effect through electromagnetic field strength.

Abstract

The wave function of quantum mechanics is not a boost invariant and gauge invariant quantity. Correspondingly, reference frame dependence and gauge dependence are inherited to most of the elements of the usual formulation of quantum mechanics (including operators, states and events). If a frame dependent and gauge dependent formalism is called, in short, a relative formalism, then the aim of the paper is to establish an absolute, i.e., frame and gauge free, reformulation of quantum mechanics. To fulfil this aim, we develop absolute quantities and the corresponding equations instead of the wave function and the Schr\"odinger equation. The absolute quantities have a more direct physical interpretation than the wave function has, and the corresponding equations express explicitly the independent physical aspects of the system which are contained in the Schr\"odinger equation in a mixed and more hidden form. Based on the absolute quantities and equations, events, states and physical quantities are introduced also in an absolute way. The formalism makes it possible to obtain some sharper versions of the uncertainty relation and to extend the validity of Ehrenfest's theorem. The absolute formulation allows wide extensions of quantum mechanics. To give examples, we discuss two known nonlinear extensions and, in close detail, a dissipative system. An argument is provided that the absolute formalism may lead to an explanation of the Aharonov-Bohm effect purely in terms of the electromagnetic field strength tensor. At last, on special relativistic and curved spacetimes, absolute quantities and equations instead of the Klein-Gordon wave function and equation are given, and their nonrelativistic limit is derived.