Symmetry Reduction of States I
| Autor: | Schmitt, Philipp, Schötz, Matthias |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 38 pages, comments welcome! |
| Databáze: | arXiv |
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