Frame transforms, star products and quantum mechanics on phase space

Abstract

Using the notions of frame transform and of square integrable projective representation of a locally compact group G, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group G × G. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase space formulation' of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.2. The image, through the isometry DT , of the product of operators in B 2 (H) is a 'star product of functions' in Ran DT .3. The standard expectation value formula of quantum mechanics -namely,where andρ are, respectively, a bounded selfadjoint operator and a density operator (a positive trace class operator of unit trace) in H -admits, in this framework, a suitable expression in terms of C-valued functions 'on phase space'.The adjoint QT of the isometry DT , like the Weyl map, has a simple integral expression and can be regarded as a 'quantization map'. The paper is organized as follows. In Sect. 2, we discuss the notion of 'frame transform' and its main consequences. In Sect. 3, we briefly review the definition of the Wigner distribution and its relation with projective representations. Next, in Sect. 4, we recall the basic properties of square integrable projective representations, tools that are fundamental for the definition of the (generalized) Wigner transform and of its reverse arrow, the (generalized) Weyl map, see Sect. 5; we will also argue that the generalized Wigner transform is not, in general, a frame transform. Our analysis will culminate in the introduction of the class of frame transforms mentioned before -Sect. 6 -and in the discussion of the main consequences from the point of view of quantum mechanics, see Sect. 7. In Sect. 8, we consider a remarkable example that allows to show the link of our results with the formalism of s-parametrized quasi-distributions developed by Cahill and Glauber [18]. Eventually, in Sect. 9, a few conclusions are drawn.

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

“…This (one-to-one) correspondence extends to all trace class operators [23,24,36,37], yielding a dense subspace LW n of L 2 (R n × R n ), containing the convex cone W n of all functions associated with positive trace class operators. W n contains the convex setW n of all Wigner functions:…”

“…here, the fiducial vector |0 is the ground state of the harmonic oscillator Hamiltonian. The vectors {|z } z∈C form a tight (continuous) frame [16,18,20,23]; i.e., they give rise to an integral resolution of the identity of the form…”

“…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 :…”

“…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],…”

Quantum entropies, Schur concavity and dynamical semigroups

“…In general -dim(H) ≥ 2, no assumption of complete positivity -the class of entropy-nondecreasing dynamical semigroups coincides with the generalized twirling semigroups (Theorem 3), where the semigroups of probability measures of the standard twirling semigroups are replaced with suitable families of signed measures. This is somewhat reminiscent of the formalism of Wigner functions, where the classical probability distributions are replaced with (signed) quantum quasi-distributions [15,26].…”