Quantized Space-Time

Hellund, E. J.; Tanaka, Katsumi

doi:10.1103/physrev.94.192

Cited by 38 publications

(16 citation statements)

References 6 publications

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“…Later on, he studied how to deal with the electomagnetic field in such a context [2]. Then, during several decades, there were only a few works on this subject [3,4,5,6].…”

Section: Introductionmentioning

confidence: 99%

Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator

Quesne¹,

Tkachuk²

2006

J. Phys. A: Math. Gen.

121

122

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The D-dimensional (β, β ′ )-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a (D + 1)-dimensional quantized spacetime. In the D = 3 and β = 0 case, the latter reproduces Snyder algebra. The deformed Poincaré transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a (1 + 1)-dimensional spacetime described by such an algebra is studied in the case where β ′ = 0. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for β < 1/(m 2 c 2 ). A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.

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“…Later on, he studied how to deal with the electomagnetic field in such a context [2]. Then, during several decades, there were only a few works on this subject [3,4,5,6].…”

Section: Introductionmentioning

confidence: 99%

Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator

Quesne¹,

Tkachuk²

2006

J. Phys. A: Math. Gen.

121

122

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“…Usually they are given in momentum representation. In [9,10] it was realized by operatorŝand̂defined aŝ…”

Section: The Representation Imentioning

confidence: 99%

Quantum Mechanics on a Curved Snyder Space

Mignemi

Štrajn

2016

Advances in High Energy Physics

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We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.

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“…La construction d'opkrateurs physiques dans un espace-temps quantifit a fait l'objet de plusieurs recherches; dans ces thtories les coordonnees spatiotemporelles ne commutent pas: les relations de commutation usuelles de la mtcanique quantique diffkrent par un terme qui est proportionnel au carrt de la longueur tltmentaire (6)(7)(8)(9).…”

Section: Introductionunclassified

Sur l'existence d'une longueur élémentaire et d'un intervalle de temps élémentaire

Lacroix¹

1978

Can. J. Phys.

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R e~u le 27janvier 1978 Nous passons brievement en revue divers travaux qui ont 6te faits sur I'introduction d'une longueur ClCmentaire 1, et d'un intervalle de temps klkmentaire to dans les theories physiques. Nous dkcrivons des arguments que nous avons trouves, argumentsformules pard'autres auteurs, etqui militenten faveurdes hypothttsessurl'existence de1,etde to. Une equation auxdifferences finies est proposee et les solutions d e quelques problemes du mouvement en une dimension sont donntes.We have briefly examined several studies which have been madeconcerning the introduction of an elementary length 1, and an elementary time interval to into physical theories. We have discussed the arguments which we have found, arguments formulated by other authors, and which support the hypothesesconcerning the existence oflo and of to. A finite difference equation is proposed and the solutions of some problems of movement in one dimension are given.Can. J. Phys ,56, 1109(1978) 1. Introduction L'idte d'introduire une longueur tltmentaire lo dans les thtories physiques, longueur qui serait une constante universelle, est due a Heisenberg (1, 2). Partant de l'hypothkse de I'existence d'une longueur tltrnentaire, diverses tentatives ont t t t faites pour tlaborer une thtorie d'un espace-temps discret qui serait invariant pour des transformations appartenant a un sous-groupe de Lorentz (3-5).La construction d'opkrateurs physiques dans un espace-temps quantifit a fait l'objet de plusieurs recherches; dans ces thtories les coordonnees spatiotemporelles ne commutent pas: les relations de commutation usuelles de la mtcanique quantique diffkrent par un terme qui est proportionnel au carrt de la longueur tltmentaire (6-9).Das (lo), Cole (1 1) et Hasebe (5) ont formu16 une thtorie des champs dans un espace-temps discret, thtorie qui a de profondes similitudes avec la th6orie quantique des champs. Dans ces theories les tquations difftrentielles partielles de la thtorie quantique des champs sont remplactes par des tquations aux differences finies correspondantes, et les optrateurs de dtrivtes partielles sont des combinaisons lintaires d'optrateurs de difftrences finies. Rtcemment, Welch (12) a prouvt que les optrateurs physiques, tel par exemple celui de la quantite de mouvement, ne sont pas des combinaisons lintaires d'optrateurs de difftrences finies bien connus de l'analyse numtrique (ce qui contredisait les assertions de Cole), et il a propost une tquation qui jouerait le meme r61e dans un espace-temps discret que celui jout par les tquations de Hamilton en

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Quantized Space-Time

Cited by 38 publications

References 6 publications

Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator

Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator

Quantum Mechanics on a Curved Snyder Space

Sur l'existence d'une longueur élémentaire et d'un intervalle de temps élémentaire

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