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arXiv:1610.07130 · Quanta

Quantum Trajectories: Dirac, Moyal and Bohm

Basil J. Hiley, Maurice A. de Gosson, Glen Dennis

Published Jun 5, 2019• 3 citations
arXivHTMLPDF

Abstract

We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called quantum trajectories. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the wave properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the algebras originally introduced by Born, Heisenberg and Jordan. This enables us to bring out the geometric aspects of quantum phenomena. Quanta 2019; 8: 11–23.

DOI
10.12743/quanta.v8i1.84

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We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called quantum trajectories. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the wave properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the algebras originally introduced by Born, Heisenberg and Jordan. This enables us to bring out the geometric aspects of quantum phenomena. Quanta 2019; 8: 11–23.
In this paper we show how Dirac, in 1947, anticipated the Bohm approach using an argument based on what is now called the Heisenberg picture. From a detailed examination of these ideas, we show that the role played by the Dirac standard ket is equivalent to the introduction and use of the idempotent in the orthogonal and symplectic Clifford algebras. This formalism is then used to show that the so-called `Bohm trajectories' are the average of an ensemble of individual stochastic Feynman paths. Since the Bohm approach can be simply reduced to classical mechanics, the algebraic formalism presented here provides a natural way to relate quantum mechanics to classical mechanics without the need for decoherence. We show that this approach suggests an underlying fractal space-time of the type discussed by Nottale.
In this paper we discuss the notion of a ‘particle’ and its ‘trajectory’ in the Bohm approach to quantum theory in the context of a non-commutative dynamics which has classical mechanics as a commutative sub-algebra. We study the von Neumann-Moyal algebra and show that the key equations used in the Bohm approach are already contained within this structure. They appear as projections from the non-commutative phase space into space-time. This structure leads us to a more radical view of quantum processes involving individual particles.
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 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.
Henry Stapp has made many significant contributions in quantum physics and its use in trying to understand the mind-matter relationship. I have been influenced by his use of the notion of process to bring more clarity to understand quantum phenomena. In this paper, I want to summarise the latest ideas on the time development of quantum processes that relate the transformation theory of Dirac, Feynman and Schwinger to the notion of weak values which has triggered experimental investigations of the nature of a deeper underlying stochastic structure of quantum processes.
In this paper we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. We show that this algebra contains both the Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.