A brief history of the discovery of the foundations of Quantum Mechanics

Understanding of Planck’s constant nature is the key that can open the door to foundations of quantum mechanics. Nature of the Planck constant is a key to build a complete quantum theory outside the framework of the currently dominating axiomatic approach. It is well know, that the Schrödinger, Klein – Gordon – Fock and Dirac equations were postulated and have not yet been derived from the first principles. There is reason to believe that discovering the nature of Planck’s constant is a direct, and perhaps the only, way to understand the foundations of quantum theory. This statement is true, since only understanding where Planck’s constant comes from, will allow us to derive from the principle of least action, the equations that describe quantum physics.

Several years ago (2014–2018), three independent methods for calculating Planck’s constant from first principles (i.e., from the geometry of the Universe) were proposed. The resulting understanding of the nature of Planck’s constant made it possible to write down the equations of full electrodynamics, free from the known limitations of Maxwell’s equations. It should be emphasized that the resulting equations also describe quantum phenomena.

Before moving on, let me briefly mention here some known limitations of Maxwell’s electrodynamics:

(i) It should be noted that the observed cosmological redshift for distant galaxies does not follow from Maxwell’s theory in any way, and we must introduce it artificially, similar to the Doppler effect, which is not entirely correct. In fact, the redshift caused by the slow change in the metric tensor should appear in the equations of electrodynamics in an expanding universe. This is due to the continuous change in the spatial metric, which leads to a continuous removal of energy from the transverse electromagnetic field at every point in space.

(ii) Furthermore, Maxwell’s electrodynamics does not work at the microscale (it does not describe quantum objects), although the electromagnetic field itself is quantized, as proved by Einstein (his 1905 Nobel Prize-winning work on the photoelectric effect) and Debye (1910).

(iii) Moreover, there are a number of well-known paradoxes inherent in Maxwell’s electrodynamics (see, for example, V. Ginzburg’s book “Additional Chapters of Electrodynamics”). For example, it is well known that a stationary isolated charge will self-accelerate in accordance with Maxwell’s equations. All these facts indicate that Maxwell’s electrodynamics is incomplete and needs to be generalized to the case of a changing space metric.

It should be noted that the above-mentioned problems are not independent, as they might seem at first glance, and, given the non-stationary spatial metric, all of them can be solved simultaneously. The problem with Maxwell’s equations of electrodynamics is that they were written for a static universe (the metric tensor is independent of time), which led to the aforementioned problems. This is why Maxwell’s electrodynamics does not take the cosmological redshift into account in the equations themselves. Moreover, it is also unsuitable for describing phenomena on the microscale (quantum mechanics), when the energy loss of the system is associated with the expansion of the universe (i.e., with small adiabatic changes in the metric tensor). In this case, the energy loss (due to the electromagnetic field) becomes comparable in magnitude to the energy of the quantum system itself and must be correctly taken into account. Clearly, in the case of a changing metric tensor, the physical system ceases to be closed, and its energy is not conserved (see cosmological redshift). What is conserved in this case? The adiabatic invariant of the transverse electromagnetic field (Planck’s constant) is conserved.

Interestingly, a similar problem occurred here as with dark matter (when the consideration of an incomplete system necessitated the postulation of dark matter). The key difference is that physicists did not apply Maxwell’s electrodynamics beyond its scope (since it obviously didn’t work for quantum objects). Physicists were forced to develop a new, axiom-based computational method called quantum mechanics, while astrophysicists decided to correct Newtonian mechanics by postulating dark matter.

Here in this paper: Journal of Applied Mathematics and Physics (2017) 5, 582 (or https://www.scirp.org/pdf/JAMP_2017030814454579.pdf) the Planck’s constant was calculated from first principles, and the equations of the complete electrodynamics on an adiabatically varying Finsler manifold were obtained in linear approximation. By Finsler geometry I mean here a broad class of non-Riemannian geometries characterized by (at least) non-symmetric connections (asymmetric Christoffel symbols). By the way, only in Finsler geometry does the cosmological constant arise naturally from metric. In Riemannian geometry, it does not arise naturally and must be introduced artificially, which is unsatisfactory.

As expected, the explanation of the redshift follows automatically from the equations. However, a complete surprise for me was the explanation of the Aharonov-Bohm (A-B) effect, which follows directly from the resulting equations! I must admit, I didn’t expect this, as I thought it (the A-B effect) was hidden much deeper. Thus, the resulting theory (generalized electrodynamics on an adiabatically modified Finsler manifold) describes both the macro- and microworlds, does not require the introduction of wave functions, perfectly explains why quantization occurs, and can be used to solve a wide range of problems. This approach not only made it possible to derive the equations of motion for any quantum system from the variational principle, but also resolved the problem of wave function collapse (the EPR paradox). As a result, it became possible to resolve the problem of the huge discrepancy between the observed value of the cosmological constant and the calculated value of the vacuum energy density (vacuum catastrophe), also called “the worst theoretical prediction in all of physics.“

At the same time, an article was written proposing a third (simplified) version of the calculation of Planck’s constant on a Riemannian manifold, with the goal of popularizing this idea among a wider audience of fellow physicists. The article was published in the journal Mod.Phys.Lett.A. in 2018:

https://www.worldscientific.com/doi/10.1142/S0217732319503152

For your convenience, I will briefly outline the essence of the main part of this work.
As you know, at one of the Solvay conferences in the early 20th century, the following problem was proposed and immediately solved: Suppose we have a pendulum whose thread slowly (adiabatically) changes with time. Question: What is the conserved quantity in this case? Clearly, the energy E is not conserved (since we are changing the length of the thread). I note here that the adiabatic change in the pendulum thread length is a complete analogue of the adiabatic change in the resonator volume of a propagating transverse electromagnetic field, as a result of which the photon energy decreases (the cosmological redshift).

Right there, at the congress, this question was immediately answered. Participants concluded that the approximate conserved quantity (adiabatic invariant) would be ET = const, where E is the pendulum’s energy and T is its period.

It is well known that the electromagnetic field is an oscillator (a two-dimensional oscillator, since the Faraday tensor has rank 2). If a free electromagnetic field propagates along an adiabatically varying manifold (in this simplified work, an adiabatically varying Riemannian manifold was considered), then it is obvious that the photon energy will not be conserved (recall the cosmological redshift). Then one can calculate the value of the adiabatic invariant ET = const, and I did this in the mentioned article in the simplest way. Such a calculation can be done using the first volume of Landau and Lifshitz’s course on theoretical physics (the section on “adiabatic invariants”) or otherwise, as was done in my article mentioned above. As expected, we obtain ET = h (I note once again that the cosmological constant does not arise in the Riemannian world, but arises naturally from torsion only in Finsler geometry). For this reason, I did not introduce the cosmological constant in this article (since I did not consider it possible to postulate it as some kind of “external” parameter of the theory), and the resulting value of Planck’s constant differs from that calculated for the Finsler manifold by exactly the value of the cosmological constant (i.e. by factor 1.5). Thus, it can be argued that a photon propagating along a geodesic also serves as a probe for measuring the change in the metric tensor over time. In this case the disregarding by cosmological constant corresponds to the propagating photon “not feeling” the part of the metric (torsion) responsible for the emergence of the cosmological constant.

Let’s summarize the above.

1) Planck’s constant is an adiabatic invariant of a free electromagnetic field propagating along an adiabatically (slowly) varying Finsler manifold. In this case, so-called quantization is directly derived from the requirement that the boundary conditions for a transverse EM field propagating in an inhomogeneous medium be satisfied. We emphasize that the inhomogeneity is due to the fact that the metric tensor varies over time (incidentally, the Aharonov-Bohm effect is a direct consequence of this inhomogeneity; see the article).

2) The wave functions are functions of the corresponding Störm-Liouville problem, in terms of which the photon functions were expanded.

3) In this work, general equations of complete electrodynamics were obtained that uniformly describe any electromagnetically interacting systems: from relativistic (including GR) to quantum systems. The fact that these equations explain the Aharonov-Bohm effect, naturally account for the cosmological redshift, and resolve the paradoxes of quantum mechanics is a strong argument in favor of the resulting equations.

These results not only provide a profound understanding of the origin and cause of quantization, but also seamlessly unify GR and QM.

Furthermore, as it later turned out, these equations are the equations of a “Grand Unified Theory.” This was predictable, since the same value of Planck’s constant appears not only in the case of electromagnetic interaction but also in the cases of weak and strong interactions, but we will discuss this later.