How do you interpret Planck constant? What's the (physical) meaning of it? And should it have one?

Dear Kimmo, thank you for discussion and your question.

Planck constant is just adiabatic invariant of free electromagnetic (EM) field, propagated in our Universe (characterized by slowly changed curvature). For EM field it should be unique.

Ok, obviously we work based on different set of axioms. In my model, Planck constant is realized to be photon's mass.

I was working within the concept of the Einstein-Cartan-Shroedinger model (Riemann-Cartan geometry).

Besides that the main idea was: try to reduce the number of axioms (we need to construct the Quantum Theory) to zero. So, as it was shown, to construct the Quantum theory we need the same axioms which we use to formulate General Relativity, not more. We do not need the axiom of “wave function” existence.

A.:

I offer a possible such unification. Perhaps you will look at sections 1.1, 1.2, and 1.3 in the attached extract from "Physics Math Correlations." This part of the work provides a possible model for quantum gravity.

An unification of three forces at the level of relativistic quantum field theory implies equality of three gauge couplings   α1=α2=α3. The three displayed couplings are those corresponding to U(1) hypercharge, SU(2) weak force and SU(3) strong force. As the name suggests strong coupling is larger than weak coupling that is  α3 > α2, therefore forces are not unified.

The physicist's dream, however, is to find a single description for all forces. To this end, one uses the idea of renormalization group evolution of gauge couplings. Couplings constants are not really constants but they vary with the energy scale, therefore the correct equality would be  α1(μ)=α2(μ)=α3(μ), and one needs to solve for the energy scale μ.

It turns out that the extrapolation depends on the particle content of the theory. For example a SU(5) particle content is different from that of SO(10), specially in the scalar sector. In supersymmetric cases particle content is doubled as all fermions need a scalar partner whereas all scalars need a fermionic partner. Therefore the energy scale μ where three couplings unify (say Mx) is model dependent. But roughly it is about 1015-1016 GeV range, where they are approximately equal.

Next question, which one may ask is that is the disagreement at Mx scale is due to gravitational effects? This has been addressed in the literature. A few references are; 

Christopher T. Hill, Phys. Lett. B135 (1984) 47-51,

Thomas G. Rizzo, Phys. Lett. B142 (1984) 163-167,

Q. Shafi , C. Wetterich, Phys. Rev .Lett. 52 (1984) 875.

Essentially in these papers the unification condition has been modified to the following; α1(1+ε1)=α2(1+ε2)=α3(1+ε3) where the epsilon terms parametrize effects of gravity induced operators on the unification of couplings.

This approach naturally extends the framework of unified field theory and includes the effects of gravitation via non-renormalizable effective operators.

I am deeply convinced that it is impossible to combine Quantum Theory (QT) with General Relativity (GR) without understanding what the quantization is, where it comes from and what it does mean.

Trying to juggle the terms and methods, we remind the student that knowing only integers trying to pick up a response to the teacher's task: "What is the square root of two". It is clear that this was not 2 and not 1, but if we do not introduce the irrational numbers and not determine the operation of taking root, the pupil will not find an answer to this question.

More than 100 years we are trying to combine two incomplete (approximate) models (even not theories), which are the QT and GR. They cannot be unified without understanding what their incompleteness is.

Thus, the main questions to be answered are:

What is quantization? To which entity it corresponds physically? From where, why and how it occurs, why Planck's constant is 6.62x10 ^ (- 27) rather than 3.1415926, what is the true geometry of our world, etc ...

So far not found the answers to these questions - our quest for unification will be reminded of the behavior of the student mentioned above. And in answer to a simple question SQRT (2) =? We'll get the 1, then 2, then 18 (unreal zero-point oscillations of the Fock vectors, which are in incredible discrepancy with the observations of the cosmological constant with gap more than 100 orders of magnitude, unrenormalizable theories, the singularities in GR, the problem of confinement on the brane, and more and more and more… )

If a same set of equations can explain the rotation curves of galaxies as well as the Hydrogen spectra, then that would be sign of unification. By replacing Schwarzschild metric of general relativity with a new metric, it is possible to explain Hydrogen spectra as well as the rotation curves of galaxies using Einstein's field equations.

Article Hydrogen spectra using Einstein's field equations