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General Framework of Quaternionic Quantum Mechanics

Stephen L. Adler

Published Apr 27, 1995• 12 citations
Publisher

Abstract

Abstract We proceed now to give the basic kinematic and dynamical framework of quaternionic quantum mechanics. In much of what follows there is a close analogy with the familiar framework of complex quantum mechanics, but there are a number of characteristic features of the quaternionic case that play a significant role in the sequel, and to which we alert the reader.

DOI
10.1093/oso/9780195066432.003.0002

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Abstract In this chapter we discuss a number of methods for handling stationary state problems in quaternionic quantum mechanics and then consider more generally methods in which the phase of the wave function plays a primary role. We first describe exact methods for reducing the anti Hermitian quaternionic Hamiltonian H to a complex Hermitian Hamiltonian H for an equivalent complex quantum mechanics problem. We then develop stationary state perturbation theory, and as an application compute the leading approximation to the Hamiltonian phase operator IH. We also give a quaternionic analog of the Rayleigh-Ritz variational principle. We proceed next to an extended discussion of the geometric phase in quaternionic quantum mechanics, in both the adiabatic and the nonadiabatic cases. Finally, we give the quaternionic generalization of the eikonal or WKB approximation.

•Apr 27, 1995•0 citations
Abstract We proceed now to give the basic kinematic and dynamical framework of quaternionic quantum mechanics. In much of what follows there is a close analogy with the familiar framework of complex quantum mechanics, but there are a number of characteristic features of the quaternionic case that play a significant role in the sequel, and to which we alert the reader.
Abstract It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex Hilbert space. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Authored by a world-renowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum mechanics for the purpose of determining whether quaternionic Hilbert space is the appropriate arena for the long sought-after unification of the standard model forces with gravitation. Significant results from earlier literature, together with many new results obtained by the author, are integrated to give a coherent picture of the subject. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic Hilbert space. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still challenge the field. This well-written treatise is a very significant contribution to theoretical physics. It will be eagerly read by a wide range of physicists.
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.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
Abstract In this chapter we discuss a number of methods for handling stationary state problems in quaternionic quantum mechanics and then consider more generally methods in which the phase of the wave function plays a primary role. We first describe exact methods for reducing the anti Hermitian quaternionic Hamiltonian H to a complex Hermitian Hamiltonian H for an equivalent complex quantum mechanics problem. We then develop stationary state perturbation theory, and as an application compute the leading approximation to the Hamiltonian phase operator IH. We also give a quaternionic analog of the Rayleigh-Ritz variational principle. We proceed next to an extended discussion of the geometric phase in quaternionic quantum mechanics, in both the adiabatic and the nonadiabatic cases. Finally, we give the quaternionic generalization of the eikonal or WKB approximation.