Foundations of Quantum Mechanics 1

It is known that QM (as well as QED and QFT) are based on a number of axioms. To understand the meaning of QM, we must attempt to remove the axioms from the foundations of QM. To do this, we need to find the missing pieces that were replaced by the axioms when QM was created.

Let’s begin our journey with indisputable facts that can serve as a basis for our reasoning. What do we have?

Fact 1. The observed cosmological redshift indicates the expansion of the Universe. This suggests that the metric tensor changes adiabatically with time.

Corollary 1: The photon is not a closed system; its energy is not an integral of motion (see cosmological redshift).

Corollary 2: The photon has an approximately conserved quantity—the adiabatic invariant. The theory of adiabatic invariants has been well known since the Solvay Conference (where the corresponding problem for a harmonic oscillator was considered) and is well covered in standard textbooks on classical mechanics.

Corollary 3. A photon propagates in an inhomogeneous medium. Indeed, if the metric tensor changes with time, then the curvature of space at the initial point of the photon’s oscillation is not equal to the curvature that will arise at the final point of the oscillation after a period of oscillation. We obtain anisotropy for a moving object (the curvature of space behind is greater than in front). How small, and how significant, is this difference? This question is addressed in a paper published in the Journal of Applied Mathematics and Physics (2017) 5, 582. Calculations show that the magnitude of the anisotropy is significant, and it is this anisotropy that leads to quantization of the transverse EM field (anisotropy leads to the emergence of an adiabatic invariant of the field). The calculated value of the adiabatic invariant for a photon propagating in expanding space equipped with the Finsler metric is (in the vicinity of our galaxy cluster) precisely h = 6.6 x 10^-27 erg s. [1].

The calculations in the cited paper were performed in a general form for the curvature, and then (to obtain a numerical estimate for a terrestrial laboratory) the parameters (the Hubble constant and the cosmological constant) characterizing the immediate vicinity of our galaxy cluster were substituted. It turned out that the calculated value of the adiabatic invariant (Planck’s constant) coincides with the experimental value to within two decimal places, i.e., within the measurement errors of the Hubble constant and the cosmological constant!

Simplified calculations of Planck’s constant were also performed for the case of a (pseudo-)Riemannian geometry of the Universe [2]. However, in this case, the cosmological constant is not geometric in nature and therefore cannot influence the propagation of light.

Fact 2. The transverse EM field is quantized regardless of whether charges are nearby. This fact was confirmed by the explanation of the photoelectric effect (Einstein’s 1905 work on the photoelectric effect, for which he was awarded the Nobel Prize). However, now the value of Planck’s constant is not only an experimental fact but also follows from the aforementioned calculations. In this work, it was shown that the transverse EM field is quantized due to the adiabatic change in the spatial metric, and the adiabatic invariant of the field is h = 6.6 x 10^-27 erg s.

Fact 3. All experimentally (directly) observed interactions are electromagnetic. For now, we will discuss only the electromagnetic multipole interaction. That is, the interaction of particles that are electric monopoles (e.g., protons, electrons), dipoles (e.g., molecules), quadrupoles (e.g., neutrons), etc.

Corollary 4. From facts 2 and 3, it logically follows that the quantization of the motion of charged particles is due to the quantization of the transverse electromagnetic field (note that the longitudinal central electric field is not quantized, since it does not change with a change in the metric). This is also indicated by the fact that Planck’s constant (which is an adiabatic invariant of the transverse EM field) is included in the equations of QM.

Fact 4. Bohm’s pilot wave. A harmonically moving charge (for example, an electron in an atom) generates a harmonic transverse EM field. In the absence of radiation, this is a standing transverse EM field. In V. Ginzburg’s book “Additional Chapters of Electrodynamics,” this field is not very appropriately called “virtual.” It is better to call it the bound transverse EM field, or the Bohm pilot wave.

Fact 5. QM, QED, and QFT are constructed for the case of exclusively inertial frames of reference.

Corollary 5. QM, QED, and QFT, built on the axiom that the superposition principle holds, cannot be unified with GR in any way (since within GR, the superposition principle is obviously not satisfied).

Corollary 6. Facts 1, 3, and 5 imply that both curvature and the change in the metric over time are not taken into account in the equations of QM, QED, and QFT. The entire Standard Model is constructed within the Minkowski metric. Thus, strictly speaking, in this sense, QM and QFT consider an incomplete system, which is compensated for by the introduced axioms. In reality (taking into account the change in the metric), one should consider the complete system (charges, transverse and longitudinal EM fields, change in the metric), and only then will it be possible to escape the axioms underlying QM.

Fact 6. Maxwell’s electrodynamics is incomplete because it 1) does not take into account the expansion of the Universe (the change in the metric tensor), 2) as a consequence of the first point, it does not describe the quantization of the transverse EM field, and 3) suffers from a number of paradoxes, for example, such as the spontaneous acceleration of a free charge (see V. Ginzburg’s book “Additional Chapters of Electrodynamics”).

Corollary 7. Maxwell’s electrodynamics must be reformulated on an adiabatically changing manifold describing the expansion of the Universe.

What should we do?
From Corollaries 5 and 6, we can see that the solution to the problem of unifying quantum physics and GR should not be sought within the framework of QM or QFT.

From Corollaries 1, 2, 3, 4, and 7, it follows that the solution to the problem of unifying Quantum Physics and GR can be obtained by constructing complete electrodynamics on an adiabatically variable manifold. As a result of this step, we obtain 1) a quantized transverse EM field, 2) the value of Planck’s constant h = 6.6 x 10^-27 erg s, 3) the equations of motion for charged particles, which will describe quantum systems and naturally unify with GR, and 3) the QM equations in the limit of the Minkowski metric.
It is precisely this program that was implemented in [1], the results of which we will discuss in the next post.