Louis de Broglie was a critical link between the work of Albert Einstein and Max Born's statistical interpretation of quantum mechanics. But a quarter-century later, de Broglie's ideas were revived and re-interpreted by David Bohm as faster-than-light potentials and forces at a deeper level of reality than indeterministic quantum mechanics. Bohm saw these superluminal forces as instantaneous connections between all the particles of a "holistic" and deterministic universe.

It was de Broglie who first argued that if light, which was thought to consist of continuous waves, is actually discrete particles (which Einstein called light quanta in 1905, later called photons), then matter, which is known to consist of discrete particles, might also have a continuous wave nature. Einstein enthusiastically endorsed d Broglie's view.

The fundamental idea of [my 1924 thesis] was the following: The fact that, following Einstein's introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated... My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons."

De Broglie called the light wave "transporting" the photon a "pilot wave," where Einstein had called it a "ghost field" or "guiding field," without attributing any controlling energy, force, or impulse to the field.

Einstein said that the light wave at some position is a measure of the probability of finding a light particle there, that is, the intensity of the light wave is proportional to the number of photons there. It may have been implicit in his 1905 light quantum hypothesis, as de Broglie seems to think.

Although Einstein had described "ghost" and "guiding" fields to colleagues as early as 1921, we don't have specific quotes from Einstein until 1927 at the fifth Solvay conference, where he explains it in terms of the absolute square of Erwin Schrödinger's new wave function ψ.

|ψ|² expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.

(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. pp.440-442)

Schrödinger violently disagreed with Einstein's probabilities and his statistical interpretation of the wave function, which became a part of Niels Bohr and Werner Heisenberg Copenhagen interpretation of quantum mechanics. For Schrödinger, the light wave was distributed energy, and for matter particles like the electron, his wave function was distributed matter and charge. Einstein too became disillusioned with his own discovery of chance and the statistical basis of quantum mechanics, the most famous critic of quantum mechanics for the rest of hia life.

Over a year earlier than the Solvay conference, in July of 1926, Max Born used de Broglie's matter waves, as described by Schrödinger's wave equation, to quantify the interpretation of the wave as the probability of finding an electron going off in a specific collision direction as proportional to the square of the probability amplitude wave function in that direction. Born gave full credit to Einstein, de Broglie, and Schrödinger for the idea, although the "statistical interpretation" and the role of chance itself is pure Einstein.

Born wrote in 1926...

Collision processes not only yield the most convincing experimental proof of the basic assumptions of quantum theory, but also seem suitable for explaining the physical meaning of the formal laws of the so-called “quantum mechanics.”

A year before the introduction of Werner Heisenberg's uncertainty principle and the "orthodox" Copenhagen Interpretation, Born already sees there are multiple interpretations of quantum mechanics

Indeed, as it seems, it always produces the correct term values of the stationary states and the correct amplitudes for the oscillations that are radiated by the transitions, but opinions are divided regarding the physical interpretation of the formulas. The matrix form of quantum mechanics that was founded by Heisenberg and developed by him and the author of this article starts from the thought that an exact representation of processes in space and time is quite impossible and that one must then content oneself with presenting the relations between the observed quantities, which can only be interpreted as properties of the motions in the limiting classical cases. On the other hand, Schrödinger (3) seems to have ascribed a reality of the same kind that light waves possessed to the waves that he regards as the carriers of atomic processes by using the de Broglie procedure; he attempts “to construct wave packets that have relatively small dimensions in all directions,” and which can obviously represent the moving corpuscle directly.

Here Born offers a third, "statistical" interpretation of quantum mechanics, and he gives credit to Einstein for the relation between waves and particles.

Einstein had described the intensity of a continuous light wave as proportional to the probability of discrete light quanta being found there. He called the light wave a "ghost field" (Gespensterfeld) or a "guiding field (Führungfeld).

Neither of these viewpoints seems satisfactory to me. Here, I would like to try to give a third interpretation and probe its utility in collision processes.

I shall recall a remark that Einstein made about the behavior of the wave field and light quanta. He said that perhaps the waves only have to be wherever one needs to know the path of the corpuscular light quanta, and in that sense, he spoke of a “ghost field.” It determines the probability that a light quantum - viz., the carrier of energy and impulse – follows a certain path; however, the field itself is ascribed no energy and no impulse.

One would do better to postpone these thoughts, when coupled directly to quantum mechanics, until the place of the electromagnetic field in the formalism has been established. However, from the complete analogy between light quanta and electrons, one might consider formulating the laws of electron motion in a similar manner. This is closely related to regarding the de Broglie-Schrödinger waves as “ghost fields,” or better yet, “guiding fields.”

I would then like to pursue the following idea heuristically: The guiding field, which is represented by a scalar function ψ of the coordinates of all particles that are involved and time, propagates according to Schrödinger’s differential equation. However, impulse and energy will be carried along as when corpuscles (i.e., electrons) are actually flying around. The paths of these corpuscles are determined only to the extent that they are constrained by the law of energy and impulse; moreover, only a probability that a certain path will be followed will be determined by the function ψ. One can perhaps summarize this, somewhat paradoxically, as: The motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws.

(Quantum mechanics of collision processes (Quantenmechanik der Stoßvorgänge), Zeitschrift für Physik, 38 (1926), 803-827)

That same year, Bohm wrote his classic book Causality and Chance in Modern Physics, and de Broglie wrote a preface, in which he hoped Bohm could discover a deeper level deterministic physics, which would explain and replace the probabilistic and statistical properties of quantum mechanics seen by Einstein, Heisenberg, Born, and Dirac.

De Broglie wrote...

A long time ago in an article in the Journal de Physique of May 1927 I put forward a causal explanation of wave mechanics which I called the "theory of double solutions" but I abandoned it, discouraged by criticisms which this attempt roused. In his 1952 paper Professor Bohm has taken up certain ideas from this article and commenting and enlarging on them in a most interesting way he has successfully developed important arguments in favour of a causal reinterpretation of quantum physics. Professor Bohm's paper has led me to take my old concepts up again, and with my young colleagues at the Institute Henri Poincaré, and in particular M. Jean-Pierre Vigier, we have been able to obtain certain encouraging results. M. Vigier working with Professor Bohm himself has developed an interesting interpretation of the statistical significance of |ψ|2 in wave mechanics...

Professor Bohm...has shrewdly and carefully analyzed the idea of chance and has shown that it comes in at each stage in the progress of our knowledge, when we are not aware that we are at the brink of a deeper level of reality, which still eludes us. Convinced that theoretical physics has always led, and will always led, to the discovery of deeper and deeper levels of the physical world, and that this process will continue without any limit, he has concluded that quantum physics has no right to consider its present concepts definitive, and that it cannot stop researchers imagining deeper domains of reality than those which it has already explored.

(Causality and Chance in Modern Physics, p.x)