Alberto Levrero

Square-Integrability of Induced Representations of Semidirect Products

We consider a semidirect product G=A×′H, with A abelian, and its unitary representations of the form where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0∩H. We give a new selfcontained proof of the following result: the induced representation is square-integrable if and only if the orbit G[x0] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of .

Group theoretical quantum tomography

The paper is devoted to the mathematical foundation of quantum tomography using the theory of square-integrable representations of unimodular Lie groups.

A Theorem of Ludwig Revisited

Using a recent result of Busch and Gudder, we reconsider a theorem of Ludwig which allows one to identify a class of effect automorphisms as the symmetry transformations in quantum mechanics.

Wavelet transforms and discrete frames associated to semidirect products

We consider a semidirect product G=Rn×′H and its unitary representations U of the form IndG0G(p0m) where Ind is the unitary induction, p0 is in the dual group of Rn, G0 is the stability group of p0, and m is a unitary representation of G0∩H. We give sufficient conditions such that U defines a wavelet transform and a discrete frame.

On discrete frames associated with semidirect products

For groups which are the semidirect product of some vector group with a unimodular group we prove that the existence of a discrete frame obtained from an at-most countable set of vectors through the action of a given unitary representation implies that the representation in use has to be square-integrable.

Phase space observables and isotypic spaces

We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.

Symmetries of the Quantum State Space and Group Representations

The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the ℝ-multipliers of the universal covering G* of G and the characters of G*. As an application, the Poincare group and the Galilei group, both in 3+1 and 2+1 dimensions, are considered.

Frames from imprimitivity systems

Let (P,V) be an irreducible imprimitivity system for a group H based on a dual group  of an Abelian group A and acting on a Hilbert space H. Given ψ∈H, we find necessary and sufficient conditions in order that the set of vectors {∫Â〈â,a〉¯Vh−1 dP(â)ψ : a∈A, h∈H} be a frame in H. Moreover, we apply these results to some examples that are considered in the literature in the context of square-integrability modulo a coset space.

Square-integrable imprimitivity systems

We give a definition of square-integrability for imprimitivity systems and we prove that the square-integrable ones share most of the properties of square-integrable representations of groups.

Integrability of the quantum adiabatic evolution and geometric phases

We show that the cyclic adiabatic evolution of a quantum system is completely integrable as a classical Hamiltonian system. In this context the Berry phases arise naturally as cohomology of the invariant tori.