Symmetries and conservation laws in histories-based generalized quantum mechanics

TL;DR

This paper develops a framework for symmetries and conservation laws in histories-based generalized quantum mechanics, extending traditional concepts to more general theories with quasitemporal structures and logic for single-time histories.

Contribution

It introduces a classification of symmetries in histories-based quantum mechanics, establishes criteria for physical equivalence, and proves a Noether-type theorem linking symmetries and conservation laws.

Findings

Classified symmetries into orthochronous and non-orthochronous types.

Established a criterion for physical equivalence of histories.

Proved a Noether-type theorem relating symmetries to conservation laws.

Abstract

Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved.