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arXiv:hep-th/9509059 · Modern Physics Letters A

THE QUATERNIONIC DIRAC LAGRANGIAN

Stefano De Leo, PIETRO ROTELLI

Published Feb 20, 1996• 38 citations
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Abstract

We discuss the use of the variational principle within quaternionic quantum mechanics. This is nontrivial because of the noncommutative nature of quaternions. We derive the Dirac-Lagrangian density corresponding to the two-component Dirac equation. This Lagrangian is complex projected as anticipated in previous articles and this feature is necessary even for a classical real Lagrangian.

DOI
10.1142/s0217732396000400

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We discuss the use of the variational principle within quaternionic quantum mechanics. This is nontrivial because of the noncommutative nature of quaternions. We derive the Dirac-Lagrangian density corresponding to the two-component Dirac equation. This Lagrangian is complex projected as anticipated in previous articles and this feature is necessary even for a classical real Lagrangian.
The Dirac equation provides a description of spin 1/2 particles, consistent with both the principles of quantum mechanics and of special relativity. Often its presentation to students is based on mathematical propositions that may hide the physical meaning of its contents. Here we show that Dirac spinors provide the quantum description of two unit classical vectors: one whose components are the speed of an elementary particle and the rate of change of its proper time and a second vector which fixes the velocity direction. In this context both the spin degree of freedom and antiparticles can be understood from the rotation symmetry of these unit vectors. Within this approach the Dirac Lagrangian acquires a direct physical meaning as the quantum operator describing the total time-derivative.
In this paper, we solve the quaternionic Dirac equation [Formula: see text] in the real Hilbert space, and we ascertain that their free particle solutions set comprises eight elements in the case of a massive particle, and a four elements solutions set in the case of a massless particle, a richer situation when compared to the four elements solutions set of the usual complex Dirac equation [Formula: see text]. These free particle solutions were unknown in the previous solutions of anti-Hermitian quaternionic quantum mechanics, and constitute an essential element in order to build a quaternionic quantum field theory [Formula: see text].
The quaternionic generalization of the Dirac equation is investigated. From elementary considerations of unitarity and Lorentz invariance it is demonstrated that potentials with quaternionic parts are not consistent with representation independence. This result leads to the conclusion that either quaternionic quantum mechanics singles out a special class of Dirac representations, or that allowed potentials appearing in the problem have no j or k components. If the latter alternative is the case, consideration of nonrelativistic wave mechanics casts doubt on the existence of simple experimental tests of quaternionic quantum mechanics.
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.
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.