Super G-spaces

We review the basic theory of super $G$-spaces. We prove a theorem relating the action of a super Harish-Chandra pair $(G_0, \mathfrak{g})$ on a supermanifold to the action of the corresponding super Lie group $G$. The theorem was stated in [DM99] without proof. The proof given here does not use Frobenius theorem but relies on Koszul realization of the structure sheaf of a super Lie group (see [Kosz83]). We prove the representability of the stability subgroup functor

Covariant localizations in the torus and the phase observables

We describe all the localization observables of a quantum particle in a one-dimensional box in terms of sequences of unit vectors in a Hilbert space. An alternative representation in terms of positive semidefinite complex matrices is furnished and the commutative localizations are singled out. As a consequence, we also get a vector sequence characterization of the covariant phase observables.Comment: 16 pages, no figure, Latex2

The Logic of Quantum Mechanics

none2Enyclopedia of mathematics and its applications: Vol 15BELTRAMETTI E. G.; G. CASSINELLIBeltrametti, ENRICO GIOVANNI; Cassinelli, Giovann

Generalized orthogonality relations and SU(1,1)-quantum tomography

none3We present a mathematically precise derivation of some generalized orthogonality relations for the discrete series representations of SU(1; 1). These orthogonality relations are applied to derive tomographical reconstruction formulas. Their physical interpretation is also discussed.C. Carmeli; G. Cassinelli; F. ZizziCarmeli, Claudio; Cassinelli, Giovanni; F., Zizz

The theory of symmetry actions in quantum mechanics: with an application to the Galilei group

This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher level and it is written with an attempt to be concise, to respect conceptual clarity and mathematical rigor. The basic structures of quantum mechanics are used to identify the automorphism group of quantum mechanics. The main concept of a symmetry action is defined as a group homomorphism from a given group, the group of symmetries, to the automorphism group of quantum mechanics. The structure of symmetry actions is determined under the assumption that the symmetry group is a Lie group. The Galilei invariance is used to illustrate the general theory by giving a systematic presentation of a Galilei invariant elementary particle. A brief description of the Galilei invariant wave equations is also given

Frames from imprimitivity systems

Let ( P , V ) be an irreducible imprimitivity system for a group H based on a dualgroup A\u2c6 of an Abelian group A and acting on a Hilbert spaceH. Given H, we\ufb01nd necessary and suf\ufb01cient conditions in order that the set of vectors be a frame inH. Moreover, we apply theseresults to some examples that are considered in the literature in the context ofsquare-integrability modulo a coset space