B. J. Hiley

On the intuitive understanding of nonlocality as implied by quantum theory

We bring out the fact that the essential new quality implied by the quantum theory is nonlocality; i.e., that a system cannot be analyzed into parts whose basic properties do not depend on the state of the whole system. This is done in terms of the causal interpretation of the quantum theory, proposed by one of us (D.B.) in 2952, involving the introduction of the “quantum potential.” We show that this approach implies a new universal type of description, in which the standard or canonical form is always supersystem-system-subsystem; and this leads to the radically new notion of unbroken wholeness of the entire universe. Finally, we discuss some of the implications of extending these notions to the relativity domain, and in so doing, we indicate a novel concept of time, in terms of which relativity and quantum theory may eventually be brought together.

A quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells

The time-dependent scattering of one-dimensional Gaussian wave packets of various energies incident on(1) a square potential barrier and(2) a square well is examined numerically, using the quantum potential introduced by Bohm. The time-dependent quantum potential is calculated in each case, and the results displayed on three-dimensional computer plots. The particle trajectories from different initial positions within the wave packet are also shown, giving a detailed description of reflection and tunneling in terms of individual processes. The wider implications of this analysis are also briefly considered.

Non-locality and locality in the stochastic interpretation of quantum mechanics

We review the stochastic interpretation of the quantum theory and show that, like the causal interpretation it necessarily involves non-locality. We compare and contrast our approach with that of Nelson. We then extend the stochastic interpretation to the Pauli equation. This lays the ground for a further extension to the Dirac equation and therefore enables us to discuss this interpretation in a relativistic context. We find that a co consistent treatment of non-locality can be given and that it is indeed possible further to regard this non-locality as a limiting case of a purely local theory in which the transmission of what we have called active information is not restricted to the speed of light. In this case both quantum theory and relativity come out as very good statistical approximations. However, because this basically local theory implies that these latter are not exactly valid, it is possible to propose tests that could in principle distinguish such a theory from the current theories.

Measurement understood through the quantum potential approach

We review briefly the quantum potential approach to quantum theory, and show that it yields a completely consistent account of the measurement process, including especially what has been called the “collapse of the wave function.” This is done with the aid of a new concept of active information, which enables us to describe the evolution of a physical system as a unique actuality, in principle independent of any observer (so that we can, for example, provide a simple and coherent answer to the Schrödinger cat paradox). Finally, we extend this approach to relativistic quantum field theories, and show that it leads to results that are consistent with all the known experimental implications of the theory of relativity, despite the nonlocality which this approach entails.

On a quantum algebraic approach to a generalized phase space

We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of “negative probabilities” by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes.

Imprints of the Quantum World in Classical Mechanics

The imprints left by quantum mechanics in classical (Hamiltonian) mechanics are much more numerous than is usually believed. We show that the Schrödinger equation for a nonrelativistic spinless particle is a classical equation which is equivalent to Hamilton’s equations. Our discussion is quite general, and incorporates time-dependent systems. This gives us the opportunity of discussing the group of Hamiltonian canonical transformations which is a non-linear variant of the usual symplectic group.

The de Broglie Pilot Wave Theory and the Further Development of New Insights Arising Out of It

We briefly review the history of de Broglies notion of the “double solution” and of the ideas which developed from this. We then go on to an extension of these ideas to the many-body system, and bring out the nonlocality implied in such an extension. Finally, we summarize further developments that have stemmed from de Broglies suggestions.

On a new mode of description in physics

We explore the possibilities of a new informal language, applicable to the microdomain, which enables such characteristics as superposition and discreteness to be introduced without recourse to the quantum algorithm. In terms of new notions that are introduced (e.g. ‘potentiation’ and ‘ensemblation’), we show that an experiment need no longer be thought of as a procedure designed to investigate a property of a ‘separately existing system’. Thus, the necessity of a sharp separation between the ‘system under observation’ and the ‘apparatus’ is avoided. Although the new language is very different from that of classical physics, classical notions appear as a special limiting case.This new informal language leads to a mathematical formalism which employs the descriptive terms of a cohomology theory with values in the integers. Thus our theory is not based on the use of a space-time description, continuous or otherwise. In the appropriate limit, the mathematical formalism contains certain features similar to those of classical field theories. It is therefore suggested that all the field equations of physics can be re-expressed in terms of our theory in a way that is independent of their spacetime description. This point is illustrated by Maxwells equations, which are understood in terms of cohomology on a discrete complex. In this description, the electromagnetic four-vector potential and the four-current can be discussed in terms of an ‘ensemblation’ of discontinuous hypersurfaces or varieties. Since the cohomology is defined on the integers the charge is naturally discrete.

The implicate order, algebras, and the spinor

We review some of the essential novel ideas introduced by Bohm through the implicate order and indicate how they can be given mathematical expression in terms of an algebra. We also show how some of the features that are needed in the implicate order were anticipated in the work of Grassmann, Hamilton, and Clifford. By developing these ideas further we are able to show how the spinor itself, when viewed as a geometric object within a geometric algebra, can be given a meaning which transcends the notion of the usual metric geometry in the sense that it must be regarded as an element of a broader and more general pregeometry.

Nonlocality in quantum theory understood in terms of Einstein's nonlinear field approach

We discuss Einsteins ideas on the need for a theory that is both objective and local and also his suggestion for realizing such a theory through nonlinear field equations. We go on to analyze the nonlocality implied by the quantum theory, especially in terms of the experiment of Einstein, Podolsky, and Rosen. We then suggest an objective local field model along Einsteins lines, which might explain quantum nonlocality as a coordination of the properties of pulse-like solutions of the nonlinear equations that would represent particles. Finally, we discuss the implications of our model for Bells inequality.