Getting to the Bottom of Noether's Theorem

TL;DR

This paper explores the assumptions behind Noether's theorem in classical and quantum mechanics using algebraic structures, revealing a deep connection between quantum symmetries and statistical mechanics principles.

Contribution

It clarifies the conditions under which Noether's theorem applies in algebraic frameworks and links these conditions to fundamental principles in quantum and statistical mechanics.

Findings

Mapping observables to generators is key for Noether's theorem in algebraic settings

The second condition on dynamical correspondence relates quantum mechanics to statistical mechanics

Identifies the algebraic structures underlying symmetries in quantum systems

Abstract

We examine the assumptions behind Noether's theorem connecting symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, observables are naturally elements of a Jordan algebra, while generators of one-parameter groups of transformations are naturally elements of a Lie algebra. Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one-parameter group that preserves itself. In ordinary complex quantum mechanics this mapping is multiplication by $\sqrt{-1}$. In the more general framework of unital JB-algebras, Alfsen and Shultz call such a mapping a "dynamical correspondence", and show its presence allows us to identify the unital JB-algebra with the self-adjoint part of a complex C*-algebra. However, to prove their result, they impose a second, more obscure, condition on the dynamical correspondence. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time".