Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Classical and quantal systems of imprimitivity

N. Giovannini

Published Nov 1, 1981• 6 citations
Publisher

Abstract

We discuss from a group-theoretical point of view, a simple framework in which the classical and the quantal state spaces appear in a unified way. The framework is characterized by the use of (possibly continuous) direct unions of Hilbert spaces. Symmetries are correspondingly represented by families of (anti-) unitary operators. We consider here the associated notions of projective representations as well as the corresponding observables. These observables, classical or quantal, are defined in terms of a slight generalization of the notion of a system of imprimitivity of Mackey.

DOI
10.1063/1.524821

Related papers

Imprimitivity theorem and quaternionic quantum mechanics

Piero Truini, Lawrence C. Biedenharn, Gianni Cassinelli

The study of quantum mechanics in quaternionic Hilbert space is not new in physics; indeed, it goes back to the first attempts in formulating quantum mechanics from an axiomatic point of view, and later, from a more physical viewpoint, to the survey of Finkelstein et al. But recently a new physical motivation for such investigations has been given by the work of Adler in algebraic chromostatics, since it has been pointed out that quaternionic quantum mechanics, as a model of an intrinsically non-Abelian gauge theory, fits the Adler algebraic version of chromostatics. The present paper aims at a critical understanding of quaternionic quantal structure through the examination of the elementary case of a single free particle in non-relativistic space-time. We follow the suggestion by Mackey to exploit the imprimitivity theorem, together with the notion of induced representation, in deriving the theory for the free particle system, starting from the axiomatic standpoint. We show that there exists a unique unitary skew adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic unit.

Hadronic J.; (United States)•Jan 1, 1980•3 citations

Imprimitivity Systems and Quantum Codes

Kalyanapuram Rangachari Parthasarathy

It is shown how classical error correcting codes can be quantized to yield quantum error correcting codes as defined by E. Knill and R. Laflamme [2] by using the notion of an imprimitivity system for a group action in the sense of Frobenius and Mackey [4], [8].

Birkhäuser Basel eBooks•Jan 1, 2003•0 citations

State spaces for classical and quantal, relativistic and nonrelativistic elementary particles

N. Giovannini

We apply a recently developed mathematical formalism to the kinematical classification of elementary physical systems, more precisely to the derivation of the possible state spaces for single (massive) particles and to their corresponding group theoretical interpretation. We show in particular that in all considered cases, relativistic or not, we find in a unified way two solutions, a quantal one and a classical one.

Journal of Mathematical Physics•Nov 1, 1981•2 citations

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

Paolo Aniello

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.

arXiv:1811.00858•Journal of Physics Conference Series•Apr 1, 2019•5 citations

A classification of finite quantum kinematics

J. Tolar

Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view.For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups.This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics.Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N .The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra.The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.

•Aug 17, 2016•4 citations

A classification of finite quantum kinematics

J Tolar

Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.

arXiv:1411.6390•Journal of Physics Conference Series•Oct 10, 2014•1 citation
We discuss from a group-theoretical point of view, a simple framework in which the classical and the quantal state spaces appear in a unified way. The framework is characterized by the use of (possibly continuous) direct unions of Hilbert spaces. Symmetries are correspondingly represented by families of (anti-) unitary operators. We consider here the associated notions of projective representations as well as the corresponding observables. These observables, classical or quantal, are defined in terms of a slight generalization of the notion of a system of imprimitivity of Mackey.
The study of quantum mechanics in quaternionic Hilbert space is not new in physics; indeed, it goes back to the first attempts in formulating quantum mechanics from an axiomatic point of view, and later, from a more physical viewpoint, to the survey of Finkelstein et al. But recently a new physical motivation for such investigations has been given by the work of Adler in algebraic chromostatics, since it has been pointed out that quaternionic quantum mechanics, as a model of an intrinsically non-Abelian gauge theory, fits the Adler algebraic version of chromostatics. The present paper aims at a critical understanding of quaternionic quantal structure through the examination of the elementary case of a single free particle in non-relativistic space-time. We follow the suggestion by Mackey to exploit the imprimitivity theorem, together with the notion of induced representation, in deriving the theory for the free particle system, starting from the axiomatic standpoint. We show that there exists a unique unitary skew adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic unit.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view.For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups.This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics.Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N .The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra.The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.
Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.