Symmetry Reduction of States I

Abstract

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g . The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a^* a with a \in A , but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A , and the notion of positivity on the reduced algebra A_{red} should be such that states on A_{red} are obtained as reductions of certain states on A . We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold M , which reproduces the coisotropic reduction of M ; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.