Snyder noncommutativity and pseudo-Hermitian Hamiltonians from a Jordanian twist

Castro, P. G.; Kullock, R.; Toppan, Francesco

doi:10.1063/1.3602075

Cited by 15 publications

(18 citation statements)

References 40 publications

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“…from modifying standard spacetime to certain noncommutative versions in a specific way. These type of spaces have attracted considerable attention in recent years in the mathematical and physical literature [8,9,10,11,12,13,14,15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

Dey

Fring

2012

Phys. Rev. D

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We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wave packets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets, are clearly identified.

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Section: Introductionmentioning

confidence: 99%

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

Dey

Fring

2012

Phys. Rev. D

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“…Nevertheless, the theory of noncommutativity has evolved from time to time and has shown its usefulness in different areas of modern physics [82][83][84][85][86][87]. Besides, some well-known versions of it, some natural and desirable possibilities may arise when the canonical space-time commutation relation is deformed by allowing general dependence of position and momentum [88][89][90][91][92].…”

Section: A Noncommutative Quantum Mechanicsmentioning

confidence: 99%

Coherent States and Their Applications

2018

Springer Proceedings in Physics

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It was at the dawn of the historical developments of quantum mechanics when Schrödinger, Kennard and Darwin proposed an interesting type of Gaussian wave packets, which do not spread out while evolving in time. Originally, these wave packets are the prototypes of the renowned discovery, which are familiar as "coherent states" today. Coherent states are inevitable in the study of almost all areas of modern science, and the rate of progress of the subject is astonishing nowadays. Nonclassical states constitute one of the distinguished branches of coherent states having applications in various subjects including quantum information processing, quantum optics, quantum superselection principles and mathematical physics. On the other hand, the compelling advancements of non-Hermitian systems and related areas have been appealing, which became popular with the seminal paper by Bender and Boettcher in 1998. The subject of non-Hermitian Hamiltonian systems possessing real eigenvalues are exploding day by day and combining with almost all other subjects rapidly, in particular, in the areas of quantum optics, lasers and condensed matter systems, where one finds ample successful experiments for the proposed theory. For this reason, the study of coherent states for non-Hermitian systems have been very important. In this article, we review the recent developments of coherent and nonclassical states for such systems and discuss their applications and usefulness in different contexts of physics. In addition, since the systems considered here originate from the broader context of the study of minimal uncertainty relations, our review is also of interest to the mathematical physics community.

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“…Some of the work done includes the consideration of different field theories on the Snyder spacetime [25] and its study in connection to quantum gravity phenomenology [26]. Its classical and quantum aspects have been studied from a phenomenological point of view, outside of the noncommutative geometry formalism [27,28,29,30,31,32,33,34,35,36]. The model was also considered in a series of papers [37,38,39,40], and the star product, coproduct and antipodes of its Hopf algebra were calculated.…”

Section: Introductionmentioning

confidence: 99%

Twist for Snyder space

Meljanac

Mignemi

et al. 2018

Eur. Phys. J. C

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We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

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Snyder noncommutativity and pseudo-Hermitian Hamiltonians from a Jordanian twist

Cited by 15 publications

References 40 publications

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

Coherent States and Their Applications

Twist for Snyder space

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