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arXiv:hep-th/9406191

Quantum Dynamics in Regions of Quaternionic Curvature

Steven P. Brumby, Gayatri Joshi

Published Jun 29, 1994• 0 citations
arXivHTMLPDF

Abstract

The complex unit appearing in the equations of quantum mechanics is generalised to a quaternionic structure on spacetime, leading to the consideration of complex quantum mechanical particles whose dynamical behaviour is governed by inhomogeneous Dirac and Schrödinger equations. Mixing of hyper-complex components of wavefunctions occurs through their interaction with potentials dissipative into the extra quaternionic degrees of freedom. An interferometric experiment is analysed to illustrate the effect.

DOI
10.48550/arxiv.hep-th/9406191

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The complex unit appearing in the equations of quantum mechanics is generalised to a quaternionic structure on spacetime, leading to the consideration of complex quantum mechanical particles whose dynamical behaviour is governed by inhomogeneous Dirac and Schrödinger equations. Mixing of hyper-complex components of wavefunctions occurs through their interaction with potentials dissipative into the extra quaternionic degrees of freedom. An interferometric experiment is analysed to illustrate the effect.
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.
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.
A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The role played by the new imaginaries is studied. The principal conceptual difficulty concerns the theory of composite systems where the ordinary tensor product fails due to noncommutativity. It is shown that the natural resolution of this difficulty introduces new degrees of freedom similar to isospin and hypercharge. The problem of the Schrödinger equation, ``which i should appear?'' is studied and a generalization of Stone's theorem is used to resolve this problem.
Having shown in our previous works that the real-valued Schrodinger wave equation can be used to find mathematical functions to construct spacetime structures of quantum particles, in this work, we will discuss the possibility to formulate a real-valued Dirac equation in which all physical objects and all differential operators that are used to describe the dynamics of a particle are real quantities and, furthermore, since solutions to the Dirac equation are wavefunctions that have four components, it is possible to suggest that solutions to the real-valued Dirac equation should be interpreted as a parameterisation of 3-dimensional differentiable manifolds which are embedded submanifolds of the Euclidean space R^4.