Jorma Jormakka
I decided to write a short post to explain my main ideas of quantum gravity as apparently there was one person who was interested in it.
Some time ago I looked at General Relativity and did not think it is neither correct nor verified. To me the simple generalization of Newton’s gravitation theory made by Nordstrom (called Nordstrom’s gravitation theory) is much more likely to be correct. The reasons are the following:
` 1. Nordstrom’s theory is a scalar theory and a scalar quantum field can very well be included in a renormalizable quantum field theory. In the very beginning of Supersummetry and Supergravity Wess and Bagger refer to a theorem which shows that in a renormalizable quantum field theory with a mass gap (and I think it assumes also SU(N) symmetry, but you get unitary symmetries from almost any possible symmetry, like rotations, permutations etc., see Heine’s book on Group theory and quantum mechanics) the field can be only of a very limited forms. One of the possible terms is a scalar. Einstein’s GR is not renormalizable, see e.g. Dietmar Ebert’s book Eichentheorien to what happens if you make the Lorentz symmetry into a gauge symmetry. It is possible, but not renormalizable.
2. There should not be energy-stress tensor in quantum distances. There is one in the macroworld, but what causes stress forces? They are caused by molecular level electro-magnetic forces. In a quantum theory electro-magnetic forces are treated by explicit terms, so their effect should not be included as a potential by taking Einstein equations to a quantum field theory. The energy-stress tensor in a quantum field theory should be diagonal. Furthermore, it should give only one equation, not six as in Einstein’s equations. This is because degeneracy results to free parameters (see Heine’s book to compare what happens if in an atom or a molecule there are degenerate energy levels), i.e., a linear combination of wavefunctions. Not individual equations.
3. The Schwarzschild’s solution is a solution for six independent equations in a punctuated vacuum. The solution is not quasiregular. In classical physics almost everything is not only quasiregular, almost everything is conformal. It is very strange to get a deformed ball and when I check the tests of GR assumed to prove that this is the case, I do not verify it. It seems to me that Nordstrom’s theory passes the tests, while GR does not, when correctly calculated. If you do not want a badly deformed ball (not quasiregular), the theory must be Nordstrom’s.
4. There are only two theories where slow mass equals gravitational mass of the universe. One is GR, the other is Nordstrom. It can very well be so that in the macroworld GR gives better predictions (predictions agreeing with experiments) because there are electro-magnetical forces and if they are totally removed, like in Nordstrom, the result cannot agree with the experiment in a real world. GR includes these electro-magnetical forces in the energy-stress tensor. And of course it is so that if I put a book on a table, it stays on the table and does not fall. There is some more stress on the table. But this is because of electrons in the molecules. Assuming there is only gravitation, the book falls through the table quite freely.
5. I really do not like Wess-Bagger Supergravity. It includes Einstein’s equations, but in a 4-dimensional complex space (8 real dimensions). It has to do so, but that is not GR. I do not think there should be these new dimensions.
So, these are the reasons. I could even now write down a quantum gravity theory where gravitation is modelled by Nordstrom’s theory. It goes like this, you take the quantum field theory view where what looks like a wavefunction is actually an operator of creation and destruction operators acting on a state. You put the beginning of Dirac’s equation to the beginning of the Hamiltonian and then add terms what you want, like the gravitation, it will be a classical looking gravitation term. Then you demand some symmetry of some selected gauge group and add terms to get this wanted symmetry. So, like in Balin-Love, Introduction to Gauge Fields. With such a theory you can calculate transition probabilities, but as you usually have to do them from a perturbation series, you see that the series can be made to converge through some rescaling or reorganizing called renormalization. That’s about it. It is not that difficult. You do not need to solve any hard equations. You just write down a Hamiltonian and check that is satisfies what it should satisfy. There is no danger that your theory would be in disagreement with any empirical results from particle accelerators, or in fact with anything we know: gravitation is so weak compared to other forces in elementary particle scale that its effect cannot appear in calculations from perturbation theory. One should think of some black hole or similar situations and deduce what little we could observe. The problem is that many mechanisms might lead to those observed small effects, so this way never proves that your theory is correct.
But I do not write down such a Hamiltonia right now as it has already been made with Nordstrom’s theory as the gravitation theory. I do not know who did it but it most probably works fine. I know it is done as I found a paper mentioning that Nordstrom’s theory can be renormalized – how could one know it unless it was tried? So, it was done, but not accepted. People do not like Nordstrom’s theory because they believe that Einstein showed that Nordstrom’s theory does not match with experiments and GR does match. But maybe it is not quite so. And even if it were so in the macroworld, it means nothing to the quantum world.
This is why I still read the books and try to get something better than this. But so far, I think quantum gravitation can be solved and has been solved long ago, but as always, the solution has been ignored and not published. But there is folklore that such a solution was made (apparently long before I still studied for a doctorate). It looks like the same old thing. Some party is blocking results (no, it is not my results, results of much smarter people, whom we even do not know by name. My results are often not published because they are not that good).