Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on <i>S</i><sup>1</sup>

Tanimura, Shinji

doi:10.1143/ptp.90.271

Cited by 16 publications

(34 citation statements)

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“…[21,22,23]. In the formulation of Ohnuki and Kitakado [24,25] it was shown that there are in fact an infinite number of representations of the algebra of operators which can be understood in terms of a certain gauge field. These representations are classified by the value of a parameter α ∈ [0, 1) which specifies the gauge inequivalent representations and interpolates between the discrete eigenvalues in the operator spectrum.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanics and the generalized uncertainty principle

Bang

Berger

2006

Phys. Rev. D

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Section: Introductionmentioning

confidence: 99%

Quantum mechanics and the generalized uncertainty principle

Bang

Berger

2006

Phys. Rev. D

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“…Finally, we mention an interesting analogy between the failure of the Heisenberg's error-disturbance relations of Ozawa (13) and the subtle behavior of the Kennard-Robertson relation for the periodic boundary condition in box normalization. It is known that complications are associated with the uncertainty relations for periodic systems [17,18,19]. The one-dimensional Schrödinger problem with the periodic boundary condition ψ(−L/2, t) = ψ(L/2, t) in a box [−L/2, L/2] gives the universally valid Kennard-Robertson relation [20]…”

Section: Discussionmentioning

confidence: 99%

Aspects of universally valid Heisenberg uncertainty relation

Fujikawa

Umetsu

2013

Progress of Theoretical and Experimental Physics

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A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart, et al.[ Nature Phys. 8, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs-Kelly uncertainty relation which is also tested by the spinmeasurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs-Kelly uncertainty relation and Heisenberg's error-disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation as is widely accepted.

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“…In fact the inequivalent irreps of Y are parametrized byα ∈ R/Z and are defined by (40) up to equivalences (see [15,16] or e.g. the more recent [12,21]). { e ilx √ 2π } l∈Z ⊂ C ∞ (S 1 ) is an orthonormal basis of L 2 (S 1 ) consisting of eigenvectors of p: ρα(p)e ilx = (l +α)e ilx .…”

Section: Decomposition and Irreducible Representations Of The Observamentioning

confidence: 99%

On Quantum Mechanics with a Magnetic Field on ℝ n and on a Torus $\mathbb{T}^{n}$ , and Their Relation

Fiore¹

2012

Int J Theor Phys

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We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus T n = R n /Λ to the analogous theory on R n constrained by quasiperiodicity under translations in the lattice Λ. The latter theory provides a global description of the former: the quasiperiodic wavefunctions ψ defined on R n play the role of sections of the associated hermitean line bundle E on T n , since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B = dA generate a Lie algebra g Q and together with the periodic functions the algebra of observables O Q . The non-abelian part of g Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g ∈ G Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y Q . We determine the unitary irreducible representations of O Q , Y Q corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. We also clarify how in the n = 2m case a holomorphic structure and Theta functions arise on the associated complex torus.These results apply equally well to the physics of charged scalar particles on R n and on T n in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel'd twists.

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Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S¹

Cited by 16 publications

References 0 publications

Quantum mechanics and the generalized uncertainty principle

Quantum mechanics and the generalized uncertainty principle

Aspects of universally valid Heisenberg uncertainty relation

On Quantum Mechanics with a Magnetic Field on ℝ n and on a Torus $\mathbb{T}^{n}$ , and Their Relation

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Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S1

Cited by 16 publications

References 0 publications

Quantum mechanics and the generalized uncertainty principle

Quantum mechanics and the generalized uncertainty principle

Aspects of universally valid Heisenberg uncertainty relation

On Quantum Mechanics with a Magnetic Field on ℝ n and on a Torus $\mathbb{T}^{n}$ , and Their Relation

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Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S¹