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Journal of Physics Conference Series

Structure Process, Weak Values and Local Momentum

Basil J. Hiley

Published Mar 1, 2016• 7 citations
Publisher

Abstract

We explain how weak values and the local momentum can be better understood in terms of Bohm's notion of structure process. The basic ideas of this approach can be expressed in a fully algebraic way, generalising Heisenberg's original matrix mechanics. This approach leads to questions that are now being experimentally investigated by our group at University College London.

DOI
10.1088/1742-6596/701/1/012010

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We explain how weak values and the local momentum can be better understood in terms of Bohm's notion of structure process. The basic ideas of this approach can be expressed in a fully algebraic way, generalising Heisenberg's original matrix mechanics. This approach leads to questions that are now being experimentally investigated by our group at University College London.
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.
There has been a recent revival of interest in the notion of a 'trajectory' of a quantum particle. In this paper, we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm 'trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis et al.
In this chapter we show how the weak values, are related to the T 0 µ ( x, t ) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T 0 µ ( x, t ) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al . (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties. Introduction The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al ., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems. One of the new features that we will concentrate on in this chapter is the possible measurement of the T 0 µ ( x, t ) components of the energy–momentum tensor.
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.