A method for measuring the real part of the weak (local) value of spin is presented using a variant on the original Stern-Gerlach apparatus. The experiment utilises metastable helium in the 2^3S_1 state. A full simulation using the impulsive approximation has been carried out and it predicts a displacement of the beam by Δ_w = 17 - 33 μ m. This is on the limit of our detector resolution and we will discuss ways of increasing Δ_w. The simulation also indicates how we might observe the imaginary part of the weak value.

In this paper we show how the dynamics of the Schrödinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, \({\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}\), and \({\mathcal{C}}_{0,1}\). Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.

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.

In this paper, we re-examine a series of gedanken welcher Weg (WW) experiments introduced by Scully, Englert and Walther that contain the essential ideas underlying the quantum eraser. For this purpose we use the Bohm model which gives a sharp picture of the behaviour of the atoms involved in these experiments. This model supports the thesis that interference disappears in such WW experiments, even though the centre of mass wave function remains coherent throughout the experiment. It also shows exactly what it means to say ‘that the interference can be restored by manipulating the WW detectors long after the atoms have passed’. It does not support Wheeler’s notion that ‘the past is undefined and undefinable without the observation (in the present)’.

The delayed-choice experiments of the type introduced by Wheeler and extended by Englert, Scully, Süssmann and Walther (ESSW) (1992 Z. Naturf. a 47 1175–86), and others, have formed a rich area for investigating the puzzling behaviour of particles undergoing quantum interference. The surprise provided by the original delayed-choice experiment led Wheeler to the conclusion that ‘no phenomenon is a phenomenon until it is an observed phenomenon’, a radical explanation which implied that ‘the past has no existence except as it is recorded in the present’. However Bohm et al (1985 Nature 315 294–97) have shown that the Bohm interpretation (BI) gives a straightforward account of the behaviour of the particle without resorting to such a radical explanation. The subsequent modifications of this experiment led both Aharonov and Vaidman (1996 Bohmian Mechanics and Quantum Theory: An Appraisal ed J T Cushing, A Fine and S Goldstein (Dordrecht: Kluwer) pp 141–54) and ESSW to conclude that the resulting Bohm-type trajectories in these new situations produce unacceptable properties. For example, if a cavity is placed in one arm of the interferometer, it will be excited by a particle travelling down the other arm. In other words, it is the particle that does not go through the cavity that gives up its energy! If this analysis is correct, this behaviour would be truly bizarre and could only be explained by an extreme non-local transfer of energy that is even stronger than that required in an Einstein, Podolsky and Rosen (EPR)-type process. In this paper, we show that this conclusion is not correct and that if the BI is used correctly, it gives a local explanation, which actually corresponds exactly to the standard quantum mechanics explanation offered by ESSW.

The proposal that the one-parameter solutions of the real part of the Schrodinger equation (quantum Hamilton-Jacobi equation) can be regarded as `quantum particle trajectories' has received considerable attention recently. Opinions as to their significance differ. Some argue that they do play a fundamental role as actual particle trajectories, others regard them as mere metaphysical appendages without any physical significance. Recent work has claimed that in some cases the Bohm approach gives results that disagree with those obtained from standard quantum mechanics and, in consequence, with experiment. Furthermore it is claimed that these trajectories have such unacceptable properties that they can only be considered as `surreal'. We re-examine these questions and show that the specific objections raised by Englert, Scully, Sussmann and Walther cannot be sustained. We also argue that contrary to their negative view, these trajectories can provide a deeper insight into quantum processes.