See Bohm and Hiley: The Undivided Universe, and,, and references therein.

Bohm and Hiley have stated that they found their own choice of terms of an " interpretation in terms of hidden variables " to be too restrictive.

This stage covers work by Bohm and in collaboration with Jean-Pierre Vigier and Basil Hiley.

Bohm is clear that this theory is non-deterministic ( the work with Hiley includes a stochastic theory ).

Taking reference to the work of Bohm and Peat Science, Order and Creativity, Arleta Griffor – noted by Paavo Pylkkänen for her “ deep and extensive knowledge of Bohm's philosophy ” and member of the research group of Bohm's co-worker Basil Hiley – underlines the importance of the kind of listening involved in the Bohm dialogue and points to Bohm's statement that

David Bohm, his co-worker Basil Hiley and other physicists of Birkbeck college, University of London, worked towards representing the implicate order in form of an appropriate algebra or other pregeometry.

( See also: Work by Bohm and Hiley on implicate orders, pre-space and algebraic structures )

* Quantum Implications: Essays in Honour of David Bohm, by F. David Peat ( Editor ) and Basil Hiley ( Editor ), Routledge & Kegan Paul Ltd, London & New York, 1987

Hiley and F. David Peat, ( eds ) Quantum Implications: Essays in Honour of David Bohm, Routledge, 1987 ISBN 0-415-06960-2

On the other hand the Bohm interpretation of quantum mechanics keeps counter-factual definiteness while introducing a conjectured non-local mechanism in form of the ' quantum potential ', defined as one of the terms of the Schrödinger equation.

In the resulting representation, also called the de Broglie – Bohm theory or Bohmian mechanics ,, the wave – particle duality is not a property of matter itself, but an appearance generated by the particle's motion subject to a guiding equation or quantum potential.

In the formulation of the De Broglie – Bohm theory, there is only a wave function for the entire universe ( which always evolves by the Schrödinger equation ).

For a de Broglie – Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.

The de Broglie – Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles ; the orientation of the apparatus affects the wavefunction.

: The Hamilton – Jacobi equation is the equation derived from a Newtonian system with potential and velocity field The potential is the classical potential that appears in Schrödinger's equation and the other term involving is the quantum potential, terminology introduced by Bohm.

Around this time Erwin Madelung also developed a hydrodynamic version of Schrödinger's equation which is incorrectly considered as a basis for the density current derivation of the de Broglie – Bohm theory.

While de Broglie – Bohm theory has Lagrangians and Hamilton-Jacobi equations as a primary focus and backdrop, with the icon of the quantum potential, Bohmian mechanics considers the continuity equation as primary and has the guiding equation as its icon.

Adler's Trace Dynamics has been discussed in relation to the differential space theory of quantum systems by Norbert Wiener and Amand Siegel, to its variant by David Bohm and Jeffrey Bub, and to modifications of the Schrödinger equation by additional terms such as the quantum potential term or stochastic terms, and to hidden variable theories.

Commenting on this, other writers ( such as John von Neumann and David Bohm ) have suggested that consequently there would have to be ' hidden ' variables responsible for random measurement results, something which was not expressly claimed in the original paper.

It is in that qualified sense that Born rule is, for the de Broglie – Bohm theory, a theorem rather than ( as in ordinary quantum theory ) an additional postulate.

In Dürr et al., the authors describe an extension of de Broglie – Bohm theory for handling creation and annihilation operators, which they refer to as " Bell-type quantum field theories ".

Thus, for the de Broglie – Bohm theory, the particle's spin is not an intrinsic property of the particle — instead spin is, so to speak, in the wave function of the particle in relation to the particular device being used to measure the spin.

In particular, the usual operators-as-observables formalism is, for de Broglie – Bohm theory, a theorem.

The motivation for de Broglie – Bohm theory is to describe a system of particles.

The de Broglie – Bohm theory makes the same ( empirically correct ) predictions for the Bell test experiments as ordinary quantum mechanics.

Bohm's formulation of de Broglie – Bohm theory in terms of a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952.

In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the luminiferous ether was found to be unnecessary in special relativity.

This stage is known as the de Broglie – Bohm Theory in Bell's work 1987 and is the basis for ' The Quantum Theory of Motion ' 1993.

The Bohm criterion for ions entering the magnetic sheath applies to the motion along the field, while at the entrance to the Debye sheath it applies to the motion normal to the surface.

Decoherence provides an explanatory mechanism for the appearance of wavefunction collapse and was first developed by David Bohm in 1952 who applied it to Louis DeBroglie's pilot wave theory, producing Bohmian mechanics, the first successful hidden variables interpretation of quantum mechanics.

In practical terms, a Bohm dialogue, twenty to forty participants sit in a circle for a few hours during regular meetings, or for a few days in a workshop environment.

A separate " molecular " Aharonov – Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is " neither nonlocal nor topological ", depending only on local quantities along the nuclear path.

The magnetic Aharonov – Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential A forms part.

( 1985 ) demonstrated Aharonov – Bohm oscillations in ordinary, non-superconducting metallic rings ; for a discussion, see Schwarzschild ( 1986 ) and Imry & Webb ( 1989 ).

By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov – Bohm interference phenomenon from the phase shift has been predicted ; again, the absence of an electric field means that, classically, there would be no effect.

The Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions.

* de Broglie – Bohm – Bell pilot wave formulation of quantum mechanics

Non-locality, however, soon became established as an integral feature of quantum theory ( see EPR paradox ), and David Bohm extended de Broglie's model to explicitly include it.

The pilot wave model, originally developed by Louis de Broglie and further developed by David Bohm into the hidden variable theory proposes that there is no duality, but rather a system exhibits both particle properties and wave properties simultaneously, and particles are guided, in a deterministic fashion, by the pilot wave ( or its " quantum potential ") which will direct them to areas of constructive interference in preference to areas of destructive interference.

An example is the Bohm interpretation of quantum mechanics.

In his later career, de Broglie worked to develop a causal explanation of wave mechanics, in opposition to the wholly probabilistic models which dominate quantum mechanical theory ; it was refined by David Bohm in the 1950s.

The de Broglie – Bohm theory, also called the pilot-wave theory, Bohmian mechanics, and the causal interpretation, is an interpretation of quantum theory.

In Bohm's original papers 1952, he discusses how de Broglie – Bohm theory results in the usual measurement results of quantum mechanics.

Similarly, in the de Broglie – Bohm theory, there are anomalous initial conditions which would produce measurement statistics in violation of the Born rule ( i. e., in conflict with the predictions of standard quantum theory ).

However, while standard quantum mechanics is limited to discussing the results of ' measurements ', de Broglie – Bohm theory is a theory which governs the dynamics of a system without the intervention of outside observers ( p. 117 in Bell ).

De Broglie – Bohm theory gives the same results as quantum mechanics.

Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived ( in the epistemic sense mentioned above ), on the de Broglie – Bohm theory.

Decades later John Bell proved Bell's theorem ( see p. 14 in Bell ), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such " hidden-variable " completions of quantum mechanics must either be nonlocal ( as the Bohm interpretation is ) or give up the assumption that experiments produce unique results ( see counterfactual definiteness and many-worlds interpretation ).