Symmetry Reduction of States I

TL;DR

This paper develops a general framework for symmetry reduction of states on *-algebras with Poisson structures, emphasizing a flexible notion of positivity that depends on the context, and illustrates it with three detailed examples.

Contribution

It introduces a broad theory of symmetry reduction for *-algebras, incorporating a context-dependent positivity concept and providing concrete examples.

Findings

Unified approach to symmetry reduction of states on *-algebras.

Highlights importance of context-specific positivity notions.

Demonstrates the theory with three detailed examples.

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, for which positive elements are 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 $M$, reduction of the Weyl algebra with respect to translation symmetry, and reduction of the polynomial algebra with respect to a $U(1)$-action.