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Chapter 1
The Geometry of the Quantum Potential in Different Contexts
1.1Bohm’s Original 1952 Approach on the Quantum Potential
The probabilistic interpretation of the wavefunction developed by the physicists of the Copenhagen and Gottinga schools seems to be in agreement with experimental facts regarding the microscopic world. However, indeed it is not forced by these experimental results: the outcomes of experiments on atomic and subatomic processes seem to indicate only that it is consistent to consider the wavefunction as a parameter which contains information on probability but do not exclude the possibility that the wavefunction may possess other properties. On the other hand, as even Heisenberg observed at the dawning of quantum theory, the standard interpretation is radically a-causal and atomic processes cannot therefore be incorporated into a space-time picture. Since in the standard interpretation of quantum mechanics quantum phenomena cannot be explained as events happening in space-time, namely the dogma of formulation of physics in terms of motion in space–time must be abandoned, the standard quantum mechanics cannot be considered satisfactory if one wants to develop a coherent geometrodynamic picture of the quantum world.
The purely probabilistic interpretation of the wavefunction characteristic of standard quantum mechanics can be considered coherent only if one wishes to reduce physics to a kind of algorithm which is efficient to correlate the statistical results of experiments. If one wishes to do more, and attempts to understand the experimental results regarding the microscopic world in terms of a causally connected series of individual processes and thus to develop a real geometrodynamic picture of the quantum world, then it becomes natural and plausible to search for possible further significances of the wavefunction (beyond its probabilistic aspect), and to introduce other elements in addition to the wavefunction.
This is what Bohm’s version of quantum mechanics allows us to realize: to suggest a formulation of quantum mechanics where probability only enters as a subsidiary condition on a causal theory of the motion of individual events. In Bohmian quantum mechanics the wavefunction turns out to have a direct physical significance in each individual process, its statistical meaning is only a secondary property. Besides the wavefunction, this approach introduces an additional element in order to obtain a geometrodynamic description of subatomic processes, namely a particle, conceived in the classical sense as pursuing a definite continuous trajectory in space and time.
Bohmian quantum mechanics, known in the literature also as de Broglie-Bohm pilot wave theory, can be considered as the most significant and satisfactory hidden variables theory predictably equivalent to quantum mechanics, able to give a causal completion to quantum mechanics. It can be inserted inside that important research stream directed to complete standard quantum theory in a deterministic sense.
This approach was originally proposed by Louis de Broglie in 1927 at the Solvay Conference. As regards the non-relativistic problem, de Broglie proposed that the wavefunction of each one-body physical system is associated with a set of identical particles which have different positions and are distributed in space according to the usual quantum formula, given by
. But he recognized a dual role for the wavefunction: on one hand, it carries information about the probable position of the particle (just like in the standard interpretation), on the other hand, it influences the position by exerting a force on the orbit. According to the de Broglie view, the
wavefunction would act like a sort of pilot wave which guides the particles in regions where such wavefunction is more intense [
27]. In the context of his proposal, de Broglie applied his guidance formula to compute the orbits for the hydrogen atom stationary states. The approach met however with a general lack of enthusiasm at the Solvay Conference. The unfavourable climate, presumably generated by Heisenberg’s discovery of the ‘uncertainty’ relations (and the interpretation of these relations provided by Heisenberg himself, which implies a crucial role of the observer), eventually led him to abandon this research programme. De Broglie returned to research in this field 25 years later when David Bohm rediscovered the approach and developed it to the level of a fully fledged physical theory.
In 1952 David Bohm published in Physical Review two fundamental papers entitled A suggested interpretation of the quantum theory in terms of hidden variables [30, 31]. In these two classic works Bohm was able to extend de Broglie’s approach in a coherent way also to many-body systems, ...
