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

TL;DR

This paper extends Positivstellensatz results to a non-commutative setting for polynomials on complex projective space, characterizing positive elements in the Weyl algebra and their *-representations.

Contribution

It provides a non-commutative Positivstellensatz for the Weyl algebra on CP^n, describing positive elements and classifying *-representations arising from quantization.

Findings

Algebraic description of positive U(1)-invariant polynomials in the non-commutative setting.

Application to classify all *-representations of the quantized polynomial algebra on CP^n.

Extension of Schm"udgen's Positivstellensatz to non-commutative algebras.

Abstract

We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n}, and Schm\"udgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give an algebraic description of the real-valued U(1)-invariant polynomials that are positive in certain *-representations on Hilbert spaces of holomorphic sections of line bundles over CP^n. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive elements. As an application, all *-representations of the quantization of the polynomial *-algebra on CP^n, obtained e.g. through phase space reduction or Berezin--Toeplitz quantization, are determined.