The EUP Dirac Oscillator: A Path Integral Approach
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A. Merad · M. Aouachria · H. Benzair
The EUP Dirac Oscillator: A Path Integral Approach
Received: 21 June 2020 / Accepted: 23 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The Green function for Dirac oscillator in (1+1) dimension in the context of the extended uncertainty principle (EUP) is calculated exactly via the path integral formalism. The spectrum energy is determined, the corresponding wave functions suitably normalized are derived and they are expressed by the Gegenbauer’s polynomials. Special cases are considered. 1 Introduction It is well known that the spin is a fundamental physical quantity in quantum physics and plays a significant role in various areas of physics, in particular, in the explanation of the mesoscopic phenomena. In the relativistic case, the exact analytical solutions of physical models much required, enables us to explore at the same time relativistic and spin effects, and the relativistic principles require that space-time must be described in the unified manner. Indeed, Feynman’s path integral formulation for systems with spin has not yet been definitively achieved due to the discrete nature of the spin and the requirements of relativistic invariance. In fact, path integral uses classical and continuous concepts such as trajectories whereas the spin is irreducibly of a discrete nature, without classical equivalent, and to satisfy the relativistic invariance requirements on the other hand. To overcome this difficulty within this framework, some models were presented for this purpose. For example, the Feynman attempt for the free Dirac electron using the Poisson stochastic process [1], the Schulman description of the spin of a nonrelativistic particle by the top model using the three Euler angles [2], and its extension to the relativistic case [3], the Barut–Zanghi theory for the classical spinning electron related to zitterbewegung [4], the bosonic and fermionic Schwinger model in the related coherent state space [5–7] and the supersymmetric model using the Grassmann variables for the spin evolution with many developments [8– 11]. Recently, the applicability of this Feynman formulation for the spin system has undergone notable development in various domains of physics with different topologies modeled by deformed algebras. For example, effects of the gravitational field in quantum mechanics in presence of the generalized uncertainty principle (GUP) [12–14], and on the noncommutative geometry in quantum system [15,16]. Consequently, in this regard, a significant number of papers have been published. Citing for instance, within the GUP framework the A. Merad (B) PRIMALAB, Département de Physique, Faculté des Sciences de la matière, Université de Batna 1, 05000 Batna, Algeria E-mail: [email protected] M. Aouachria LPEA, Département de Physique, Faculté des Sciences de la Matière, Université de Batna 1, Batna, Algeria H. Benzair Laboratoire LRPPS, Faculté des Sciences et de la Technologie et des Sciences de la Matière, Université Kasdi Merbah
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