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 ∈ 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 µ-red should be such that states on A µ-red are obtained as reductions of certain states on A. In the special case of the * -algebra of smooth functions on a Poisson manifold M , this reduction scheme reproduces the coisotropic reduction of M , where the reduced manifold M µ-red is just a geometric manifestation of the reduction of the evaluation functionals associated to certain points of M . However, we are mainly interested in applications to non-formal deformation quantization and therefore also discuss the reduction of the Weyl algebra, and of the polynomial algebra whose non-commutative analogs we plan to examine in future projects.

Symmetry Reduction of States II: A non-commutative Positivstellensatz for CPn

“…If Proposition 3.1 applies, then the general reduction scheme from [12] yields the naively expected result for ordered * -algebras of operators. This is completely analogous to the reduction of Poisson manifolds discussed in [12,Sec. 4], with evaluation functionals at points of the µ-levelset being replaced by vector states of µ-eigenvectors.…”

“…In [12], a general reduction scheme was developed for arbitrary "representable Poisson * -algebras", which especially generalizes Marsden-Weinstein reduction of the ordered * -algebra of smooth functions on a symplectic manifold by the action of a commutative Lie group. In the following, we are more interested in the case of * -algebras represented on a pre-Hilbert space:…”

“…For the purpose of this article, it will also be sufficient to only consider the case of a reduction with respect to a 1-dimensional Lie algebra u 1 ∼ = Ê, so that any (linear) "momentum map" J : u 1 → A H is fully determined by one "momentum operator" J (1), and any "momentum" µ ∈ u * 1 by µ (1). By abuse of notation, we will simply write J := J (1) ∈ A H and µ := µ(1) ∈ Ê as in the examples discussed in [12,.…”

“…In this special case, the reduction scheme of [12] reduces to the following: Let J ∈ A H and µ ∈ Ê be given. Denote by…”

“…This article is organized as follows: After recapitulating the necessary preliminaries on ordered * -algebras and quadratic modules in Section 2, Section 3 is devoted to the application of the general reduction procedure for "representable Poisson * -algebras" from [12] to the case of (non-commutative) * -algebras equipped with a Poisson bracket coming from the commutator and equipped with an order obtained from a * -representation on a pre-Hilbert space. A special case of this is the reduction of the Wick star product from 1+n to È n that is covered in Section 4.…”