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

arXiv:math-ph/0009035 · Modern Physics Letters B

QUANTUM GROUPS AND VON NEUMANN THEOREM

Alfredo Iorio, Giuseppe Vitiello

Published Feb 20, 1994• 17 citations
arXivHTMLPDF

Abstract

We discuss the q-deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in quantum mechanics. We show that the q-deformation parameter labels the Weyl systems in quantum mechanics and the unitarily inequivalent representations of the canonical commutation relations in quantum field theory.
DOI
10.1142/s0217984994000285

Related papers

Quantum Groups and Von Neumann Theorem

Alfredo Iorio

We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory

•Jan 1, 2000•14 citations

On the uniqueness of unitary representations of the non-commutative Heisenberg–Weyl algebra

Laure Gouba, Frederik G. Scholtz

In this paper, we discuss the uniqueness of the unitary representations of the noncommutative Heisenberg–Weyl algebra. We show that, apart from a critical line for the noncommutative position and momentum parameters, the Stone–von Neumann theorem still holds, which implies uniqueness of the unitary representation of the Heisenberg–Weyl algebra.

arXiv:0903.3816•Canadian Journal of Physics•Sep 1, 2009•3 citations

Von Neumann’s uniqueness theorem in theories with nonphysical particles

K. V. Antipin, Yu. S. Vernov, M. N. Mnatsakanova

Von Neumann’s uniqueness theorem is extended to the class of anti-Fock representations of canonical commutation relations.

arXiv:1301.7712•Physics of Particles and Nuclei Letters•Mar 1, 2015•0 citations

A ( p,q )-Deformation of Quantum Mechanics

Gong Ren-shan

On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p,q)-derivative operator is discussed by means of (p,q)-integration. Then a (p,q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisenberg–Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p,q)-deformation discussed here is identical with q-deformation given by Truong, the (p,q)-deformed Schrodinger picture is in fact different from the q-deformed one.

Communications in Theoretical Physics•Mar 15, 1998•2 citations

Q-deformation of bosonic paraquantum trilinear commutation relations

M. H. Traikia, N. Mebarki

Using a weak deformation of the Weyl-Heisenberg algebra of a paraquantization of order 2. After defining a generalized commutation and anti-commutation relations, modified triliniar commutation relations are derived.

AIP conference proceedings•Jan 1, 2012•0 citations

The q-deformation of quantum mechanics of one degree of freedom

V. Maximov, Anatol Odzijewicz

The q-quantum mechanics of the one degree of freedom is studied. Among others the holomorphic representation of q-deformed Heisenberg–Weyl algebra and its realization by covariant Berezin symbols is described.

Journal of Mathematical Physics•Apr 1, 1995•14 citations
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory
On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p,q)-derivative operator is discussed by means of (p,q)-integration. Then a (p,q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisenberg–Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p,q)-deformation discussed here is identical with q-deformation given by Truong, the (p,q)-deformed Schrodinger picture is in fact different from the q-deformed one.