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arXiv:hep-th/9401009 · Progress of Theoretical Physics

Translations between Quaternion and Complex Quantum Mechanics

S. De Leo, P. Rotelli

Published Nov 1, 1994• 18 citations
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Abstract

While in general there is no one-to-one correspondence between complex and quaternion quantum mechanics (QQM), there exists at least one version of QQM in which a partial set of translations may be made. We define these translations and use the rules to obtain rapid quaternion counterparts (some of which are new) of standard quantum mechanical results.

DOI
10.1143/ptp/92.5.917

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While in general there is no one-to-one correspondence between complex and quaternion quantum mechanics (QQM), there exists at least one version of QQM in which a partial set of translations may be made. We define these translations and use the rules to obtain rapid quaternion counterparts (some of which are new) of standard quantum mechanical results.
We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with complex geometry. In particular we describe the way in which (2n+1)-dimensional complex matrices are reduced to overlapping (n+1)-dimensional quaternionic matrices with generalized quaternionic elements. This step resolves an outstanding difficulty with reduction of purely complex matrix groups within quaternionic QM and avoids anomalous eigenstates. As a result we present a more complete translation from CQM to QQM and vice versa.
Abstract We next discuss certain general results in quaternionic quantum mechanics, which can be formulated independently of the detailed structure of the Hamiltonian H Our emphasis in this chapter will be on topics connected with symmetry generators. We introduce quaternion anti-self-adjoint space translation generators and use them to construct the momentum representation. Corresponding to these generators, we introduce several possible definitions for the self adjoint momentum operators and show that none of them possesses all the properties of the momentum in complex quantum mechanics. A closely analogous discussion is given for the rotation generators and angular momentum.
Solving the Schrödinger equation in quaternionic quantum mechanics (QQM) has been a nut to crack. In complex quantum mechanics (CQM), approximation methods have been used to find the solutions of the Schrödinger equation when exact solutions are difficult to obtain. This study challenges the earlier Eikonal (or WKB) approximation formalism [S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)] developed for an imaginary quaternion potential. In this paper, the correct form of coupled equations, obeying the noncommutative structure and approximated solution for quaternionic time-independent Schrödinger equation (QTISE), has been obtained.
An attempt is made to enlarge the number field underlying quantum mechanics from complex to quaternion. It is not impossible to construct quaternion quantum mechanics in quite a similar manner to the usual complex quantum mechanics, though some limitations are necessary for the position of factors. Further, as a special case, the charge properties of the spin-0 particles are considered. This example may show some new way to generalized the concept of the charge of the elementary particles without adoption of the iso-space.