Star products: a group-theoretical point of view

Abstract

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product explicit formulae can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Grönewold-Moyal star product. In the second example, the link with wavelet analysis is clarified.

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

“…The square integrable representations of a semidirect product, with an abelian normal factor, can be classified [12]. The classical example is the one-dimensional affine group, which gives rise to the standard wavelet transform [12,14,16,24]. In this case, the analyzing vector ψ, or mother wavelet, must belong to the domain of the (unbounded) Duflo-Moore operator.…”

“…Let B 2 (H) be the Hilbert space of Hilbert-Schmidt operators in H, and B 1 (H) ⊂ B 2 (H) the Banach space of trace class operators. Given a square integrable projective representation U : G → U (H) (with G unimodular ), we call dequantization map the linear isometry determined by [23,24] D :…”

“…For functions living in Ran(D) this operation can be regarded as the dequantized product of operators. The pair (L 2 (G), ⋆) is a H * -algebra [24,35], whose annihilator ideal is Ran(D) ⊥ . For this group-theoretical star product we have, among others [24,34], the following result:…”

“…In the case of the group of translations on phase space -G = R n × R n -H = L 2 (R n ), U is the Weyl system, Ran(D) = L 2 (G) = L 2 (R n × R n , (2π) −n d n q d n p; C) (i.e., the annihilator ideal is trivial) and d U = 1. Taking into account the fact that γ(q, p ; q ′ , p ′ ) = exp(i(q · p ′ − p · q ′ )/2), the γ-twisted convolution is nothing but the classical twisted convolution [15,24]. Note however that the function (Dρ)(q, p) = tr(U (q, p) * ρ ) associated with a quantum state -ρ ∈ B 1 (H),ρ ≥ 0, tr(ρ) = 1 -is not its Wigner distribution ρ [15,31], but rather the corresponding quantum characteristic function ρ.…”

“…Thus, the convex setP n ⊂ P n of normalized functions of classical positive type coincides with the set of characteristic functions. Passing to the quantum setting, recall that in the phase-space approach to quantum mechanics a pure stateρ ψ = |ψ ψ| is replaced with its Wigner function ρ ψ ∈ L 2 (R n × R n ) [15,17,23,24,31],…”