Ustin Chashchikhin - Quantum theory of gravitation

In general relativity, the source of the gravitational field is the energy-momentum tensor, a measure of the energy and momentum density in a volume:

(2) General relativity equations

Therefore, to find the quantum of the gravitational field – the assumed graviton – we need to find the minimal, elementary values of energy and momentum.

Yes, energy and momentum are quantized:

(3)

And usually physicists believe that the fact of quantization of energy and momentum (and hence the energy-momentum tensor in general relativity) is an obvious argument in favor of the graviton and the quantum nature of the gravitational field. But any attempt to create a quantum theory of gravitation faces insurmountable difficulties – all these models are non-renormalizable.

So this is a dead end? What is the problem?

The fact is that the question of quantization of energy and momentum is actually much more complicated.

Planck’s constant, unlike electric charge, is neither an elementary energy nor an elementary momentum.

And the frequency \nu 1 1 Since this editing platform has not LaTeX converter and Greek alphabet, then I have to use LaTeX notation and graphics instead of Greek alphabet. in expression (3) can take a variety of values. Yes, electromagnetic waves are quantized, but a quantum of infrared light has less energy than UV, and a quantum of UV light has less energy than an X-ray quantum.

Therefore, the quantum of electromagnetic radiation (3) is not yet an elementary minimum energy, since in nature there are smaller values of radiation quanta of other wavelengths.

Therefore, the question of the existence of elementary energy depends not only on the Planck constant, but also on the question of the existence of an elementary frequency.

This is the question I want to focus the reader’s attention on. This is the question that must be solved in order to create a quantum theory of gravitation.

Therefore, my approach is the following: just as elementary electric charge is necessary to construct QED, so also elementary minimum values of mass, energy, momentum, and frequency are necessary to construct quantum theory of gravitation.

For a more complete understanding, let’s look at the graphs of the dependence of electric charge, energy, and momentum.

The electric charge is quantized. Therefore electromagnetic field is quantized.

Energy and momentum are quantized in n, but they are not quantized in frequency \nu. Therefore gravitational field is not quantized.

Quantized energy spectrum in n and continuous energy spectrum in frequency \nu. Expression “matter is quantized” means quantization of energy and momentum at n.

The lightest particles with a rest mass are the electron and positron. Is the mass of an electron the elementary mass – the minimum mass that all other masses are multiples of it? Is here the same situation as in the case of an electric charge?

To answer this question, it is enough to calculate the ratio of the masses of all elementary particles to the mass of the electron. If all these relations are equal to natural numbers, then we can say that the mass of the electron is an elementary mass.

However, after making these simple calculations, we can see that these ratios are not natural numbers, for example, the ratio of the muon and electron masses is ~206.768…, the ratio of the proton and electron masses is ~1836.1527…, and so on.

You can double-check these calculations for all particles with a rest mass. As a result of these simple calculations, it is easy to see that, unlike the electric charge, the masses of elementary particles are not proportional to the mass of the electron.

(4)

What conclusions can be drawn from these facts?

Can we say that the mass of an electron is an elementary mass based on these facts? Isn’t it that the opposite conclusion follows from these facts?

This means that we can no longer describe the gravitational interaction between an electron and a proton as an exchange of one virtual graviton. A proton is heavier than an electron and its gravitational field is stronger than that of an electron. And stronger in a non-integer number of times ~1836.1527…

Are we reasoning correctly?

Maybe there is a mass of neutrinos and it is the minimum and elementary mass? Or maybe there is a certain elementary particle – the carrier of the minimum elementary mass? Then it is logical to assume that all other particles with a rest mass must be constructed from neutrinos or from such a particle – carrier of elementary mass. If this were true, then this particle with a mass less than the mass of the electron would appear at particle collisions in colliders. However, experiments rather refute than confirm this line of thought [17—19].

Is it possible to assert the presence of an elementary mass on the basis of these data? No, I think.

Even if we found an elementary mass in a non-relativistic theory, in relativity, attempts to quantize mass are complicated by the fact that in it the mass depends on the velocity:

Therefore, in the theory of relativity, the question of the existence of an elementary mass depends on the existence of an elementary velocity.

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Since this editing platform has not LaTeX converter and Greek alphabet, then I have to use LaTeX notation and graphics instead of Greek alphabet.