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arXiv:1011.4033

The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach

Basil J. Hiley, R. E. Callaghan

Published Nov 17, 2010• 23 citations
arXivHTMLPDF

Abstract

In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.

DOI
10.48550/arxiv.1011.4033

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In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.
In this paper we show how all the quantum properties of Schroedinger and Pauli particles can be described entirely from within a Clifford algebra taken over the reals. There is no need to appeal to any `wave function'. To describe a quantum system, we define the Clifford density element [CDE] as a product of an element of a minimal left ideal and its Clifford conjugate. The properties of the system are then completely specified in terms of bilinear invariants of the first and second kind calculated using the CDE. Thus the quantum properties of a system can be completely described from within the algebra without the need to appeal to any Hilbert space representation. Furthermore we show that the essential bilinear invariants of the second kind are simply the Bohm energy and the Bohm momentum, entities that make their appearance in the Bohm interpretation. We also show how these parameters emerge from standard quantum field theory in the low energy, single particle approximation. There is no need to appeal to classical mechanics at any stage. This clearly shows that the Bohm approach is entirely within the standard quantum formalism. The method has enabled us to lay the foundations of an approach that can be extended to provide a complete relativistic version of Bohm model. In this paper we confine our attention to the details of the non-relativistic case and will present its relativistic extension in a subsequent paper.
Abstract Two key themes of Basil Hiley’s work were the development of David Bohm’s approach to Quantum Mechanics and the importance of Clifford Algebras in fundamental physics. In this paper I will combine these themes by beginning to examine the mathematics of Dirac operators within the Bohm-Hiley approach. I will begin by discussing the geometric interpretation of Dirac operators, and how this is made clearer by the use of Clifford algebras . Next, I will rewrite the Cl(2) Dirac wavefunction in Polar form, and show that new behaviour arises due to topological nonlocality. Finally, I discuss the relationship between the Dirac and Schroedinger equations.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schreodinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
In this paper we explore a different but unitarily equivalent picture to the standard Schrödinger and Heisenberg pictures, namely the Dirac–Bohm picture introduced by Hiley and Dennis. This is not merely another interpretation as it allows us to examine the unfolding quantum process in terms of the algebraic structure of the dynamical variables, which in turn allows us to explore the geometric implications of quantum phenomena. Furthermore this enables us to exploit the double valued nature of the orthogonal and symplectic symmetry groups and show how they profoundly affect the nature of the conjugate momentum by enabling the introduction of classes of vector potentials into Hamiltonian systems. This throws more light on the significance of gauge invariance both in classical and quantum mechanics. The connection with geometric aspects of quantum phenomena allows us to introduce Clifford algebras in a natural way so that it is possible to extend the Dirac–Bohm approach to include spin and relativity, offering new geometric insights into quantum phenomena. In particular the bi-vector aspects allow us to introduce the Gromov non-squeezing theorem in a new way, ultimately offering the possibility of relating our work to the non-commutative geometry of M-theory.
In this paper the quantised version of Newton Second Law is derived assuming merely the existence of de Broglie matter-waves and their basic properties. At the same time we keep an eye towards interpretations of quantum mechanics and will realise that the two most different interpretations (Copenhagen interpretation and the de Broglie-Bohm theory) owe their difference to two fundamentally different approaches to `Harmonisation'. In this regard we shall see that the guiding equation of the de Broglie-Bohm theory currently found in literature is not the most complete equation possible; as a result we answer one of the important questions in interpreting quantum mechanics, namely that `when does the concept of classical path (trajectory) makes sense in quantum mechanics?' Moreover, in light of special-relativistic considerations we shall easily see that in the Number-Division approach, i.e. that of de Broglie-Bohm, the wave operator no longer appears, making it in turn impossible the application of Clifford algebras (Dirac's `square root' of the wave operator).
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.