Quantum Particle on a Quantum Circle

Brzeziński, Tomasz; Rembieliński, Jakub; Smoliński, Kordian A.

doi:10.1142/s0217732393000416

Cited by 22 publications

(33 citation statements)

References 7 publications

(8 reference statements)

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“…Being inspired by the choice of the coherent states for the motion of a particle on a circle [26,27,28,29,30], we define φ …”

Section: Choice Of Coherent Statesmentioning

confidence: 99%

Coherent state quantization of a particle in de Sitter space

Gazeau¹,

Piechocki²

2004

J. Phys. A: Math. Gen.

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We present a coherent state quantization of the dynamics of a relativistic test particle on onesheet hyperboloid embedded in three-dimensional Minkowski space. The group SO 0 (1, 2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO 0 (1, 2) seems to be a new result.

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“…Being inspired by the choice of the coherent states for the motion of a particle on a circle [26,27,28,29,30], we define φ …”

Section: Choice Of Coherent Statesmentioning

confidence: 99%

Coherent state quantization of a particle in de Sitter space

Gazeau¹,

Piechocki²

2004

J. Phys. A: Math. Gen.

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“…R which occurs in the Calogero model [5,6], where R is an exchange operator. With the extended differential operator D ν , we arrive at the deformed Heisenberg algebra with reflection…”

Section: Bosonized Susyqm: General Casementioning

confidence: 99%

“…(4.4) is the massless self-dual equation for a one-component Weyl fermion: (∂ t + ∂ x )ψ = 0 [15]. For this field, the space-time translation and Lorentz transformations have the infinitesimal form 6) and the corresponding theory is Poincaré invariant. So, if one ignores the connection of the theory given by Eq.…”

Section: Linear Differential Equation With Reflectionmentioning

confidence: 99%

“…In Section 2 we find the nonlocal unitary transformation and subsequent reduction procedure relating the conventional form of Witten's SUSYQM with the bosonized form. Here we discuss general properties of the bosonized supersymmetry and arrive naturally at the deformed Heisenberg algebra with reflection emerging in some integrable systems [5,6] and closely related to parabosons and parafermions [7,8]. In Section 3 we show that bosonized supersymmetry can be understood as a supersymmetry of the system of two identical fermions.…”

Section: Introductionmentioning

confidence: 98%

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Three aspects of bosonized supersymmetry and linear differential field equation with reflection

1999

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Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N = 1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to 'up' or 'down' field component gives rise to a linear differential equation with reflection whose 'square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration.

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“…It is important to note that these coherent states are closely related to the coherent states for the motion of massive particle on the circle [5,6,7,8,9].…”

Section: Introductionmentioning

confidence: 99%