The Dirac picture

Basil J. Hiley, Glen Dennis

We examine Dirac's early algebraic approach which introduces the {\em standard} ket and show that it emerges more clearly from a unitary transformation of the operators based on the action. This establishes a new picture that is unitarily equivalent to both the Schrödinger and Heisenberg pictures. We will call this the Dirac-Bohm picture for the reasons we discuss in the paper. This picture forms the basis of the Feynman path theory and allows us to show that the so-called `Bohm trajectories' are averages of an ensemble of Feynman paths.

Dževad Belkić

In this chapter, the authors show that various pictures of quantum mechanics are interrelated by the unitary transformations. Calling upon the correspondence principle, the quantum-mechanical scattering problem would be quite analogous to the mentioned classical model. The two analysed pictures of quantum mechanics are not the only possible ones. Quite the opposite, all the unitary transformations carried out on the state vectors and the observables of the Schrödinger picture, in fact, define a new formalism of quantum mechanics. There are infinitely many such formalisms but they are all rigorously equivalent. Among all of the possible, alternative pictures of quantum mechanics, of particular importance for scattering theory is the so-called interaction picture, which was first introduced by Dirac. Hence, the alternative name: the Dirac picture.

Philip R. Wallace

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

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.

James L. Pinfold