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

Publications of the Research Institute for Mathematical Sciences

On the Heisenberg Commutation Relation II

Konrad Schmüdgen

Published Aug 31, 1983• 26 citations
Publisher

Abstract

We study canonical pairs of self-adjoint operators P and Q whose restrictions to a common invariant dense domain Q) of a Hilbert space are essentially self-adjoint and satisfy the Heisenberg commutation relation PQ<p -QP<p-=-i(p for <p^.3).Under some additional assumption, we obtain a classification of such pairs.Moreover, we develop some methods for constructing canonical pairs of this class.

DOI
10.2977/prims/1195182446

Related papers

On pairs of difference operators satisfying [D, X]=Id

A. Z. Górski, Jacek Szmigielski

Dedicated to the memory of Gerda Dorfmeister. Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X]=Id.

arXiv:hep-th/9703015•Journal of Mathematical Physics•Jan 1, 1998•11 citations

Generalizations of the Heisenberg and Schrödinger uncertainty relations

Yan-Ni Dou, Hong-Ke Du

In this paper, releasing the restriction on operators which are self-adjoint, we propose a Heisenberg-type uncertainty relation and a Schrödinger-type uncertainty relation with any pair of operators on a Hilbert space. A generalization of Luo's theorem [S. Luo, “Heisenberg uncertainty relation for mixed states,” Phys. Rev. A 72, 042110 (2005)] is investigated.

Journal of Mathematical Physics•Oct 1, 2013•34 citations

Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians

Eric A. Galapon

We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. Keywords: Time operators; Quantum canonical pairs; Pauli’s theorem 1.

•Nov 10, 2001•64 citations

Self-adjoint symmetry operators connected with the magnetic Heisenberg ring

Bernd Fiedler

Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results:

•Nov 27, 2012•0 citations

Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

Eric Galapon

In single Hilbert space, Pauli's well–known theorem implies that the existence of a self–adjoint time operator canonically conjugate to a given Hamiltonian requires the Hamiltonian to possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self–adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well–known illustrative, implicit counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues surrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.

arXiv:quant-ph/9908033•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Jan 8, 2002•131 citations

On Heisenberg's Uncertainty Principle and the CCR

Reinhard Honegger

Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.

Zeitschrift für Naturforschung A•Mar 1, 1993•7 citations
We study canonical pairs of self-adjoint operators P and Q whose restrictions to a common invariant dense domain Q) of a Hilbert space are essentially self-adjoint and satisfy the Heisenberg commutation relation PQ<p -QP<p-=-i(p for <p^.3).Under some additional assumption, we obtain a classification of such pairs.Moreover, we develop some methods for constructing canonical pairs of this class.
Dedicated to the memory of Gerda Dorfmeister. Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X]=Id.
We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. Keywords: Time operators; Quantum canonical pairs; Pauli’s theorem 1.
Abstract. In [1] we defined symmetry classes, commutation symmetries and symmetry operators in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigated them by means of tools from the representation theory of symmetric groups SN such as decompositions of ideals of the group ring C[SN], idempotents of C[SN], discrete Fourier transforms of SN, Littlewood-Richardson products. In the present paper we consider symmetry operators a ∈ C[SN] on H which fulfil a ∗ = a, where a denotes the complex conjugate of a and the element a ∗ ∈ C[SN] of a = P p app is defined by a ∗ = P p app−1. Such symmetry operators are self-adjoint because of the relation 〈au|v 〉 = 〈u|a ∗ v 〉 which holds true for all a ∈ C[SN] and u, v ∈ H. They yield consequently observables of the quantum mechanical Heisenberg model. We prove the following results:
In single Hilbert space, Pauli's well–known theorem implies that the existence of a self–adjoint time operator canonically conjugate to a given Hamiltonian requires the Hamiltonian to possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self–adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well–known illustrative, implicit counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues surrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.
Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.