  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".
