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- Title
Symmetry reduction of states I.
- Authors
Schmitt, Philipp; Schötz, Matthias
- 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∈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 Ared should be such that states on Ared 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.
- Subjects
COMMUTATIVE algebra; POISSON brackets; SYMMETRY; LIE algebras; SMOOTHNESS of functions; HERMITIAN forms; POISSON'S equation
- Publication
Journal of Noncommutative Geometry, 2024, Vol 18, Issue 2, p501
- ISSN
1661-6952
- Publication type
Article
- DOI
10.4171/JNCG/534