This chapter examines the assumptions behind Noether’s theorem connecting symmetries and conservation laws, taking an algebraic approach to compare classical and quantum versions of this theorem. 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 the square root of ?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. This expresses a relation between quantum and statistical mechanics, closely connected to the principle that ‘inverse temperature is imaginary time’.