Symmetry Reduction of States I

Philipp Schmitt, Matthias Schötz

Published 2021-07-10Version 1

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 $A$. 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 $M_{red}$ 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.