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The European Physical Journal C

Relativistic scattering of fermions in quaternionic quantum mechanics

HASSAN HASSANABADI, Hadi Sobhani, A. Banerjee

Published Sep 1, 2017• 30 citations
Publisher

Abstract

In this article, we propose a quaternionic version of the Dirac equation in the presence of scalar and vector potentials. It has been shown that in complex limit of such an equation, the complex version of this equation can be covered. After setting a quaternionic form for the Dirac delta potential, scattering due to the considered interaction has been studied. Wave functions and discontinuity conditions of the problem considered have been derived in detail. Using the continuity equation, we have found a constraint implying the conservation law of the probability current.

DOI
10.1140/epjc/s10052-017-5154-5

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In this article, we propose a quaternionic version of the Dirac equation in the presence of scalar and vector potentials. It has been shown that in complex limit of such an equation, the complex version of this equation can be covered. After setting a quaternionic form for the Dirac delta potential, scattering due to the considered interaction has been studied. Wave functions and discontinuity conditions of the problem considered have been derived in detail. Using the continuity equation, we have found a constraint implying the conservation law of the probability current.
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The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the possibility of self-interacting physical systems. The self-interaction precludes confining states, and also imposes non-stationary quantum states, both of them representing novel situations that cannot be observed in terms of quantum wave functions. These results remark the differences between quaternionic quantum mechanics ($\mathbbm H$QM) and complex quantum mechanics ($\mathbbm C$QM), and also establish a method of solving the wave equation that may be applied to a variety of different cases.
Abstract In this paper, by introducing the q -deformed delta function, the Dirac equation for a fermion in the presence of Dirac delta scalar and vector potential is developed. Then, considering the boundary condition and the effect of the q -deformed Dirac delta potential, the scattering state is investigated. Further, the reflection and transmission coefficients are determined. Moreover, the effect of the q -parameter and the potential coefficient on the reflection and transmission coefficients is reported.
We perform a one-dimensional complexified quaternionic version of the Dirac equation based on iii-complex geometry. The problem of the missing complex parameters in Quaternionic Quantum Mechanics with iii-complex geometry is overcome by a nice ``trick'' which allows to avoid the Dirac algebra constraints in formulating our relativistic equation. A brief comparison with other quaternionic formulations is also presented.