Tuesday, December 12, 2006

A relativistic quantum theory of gravity

I am happy to announce the release of my new preprint A relativistic quantum theory of gravity. I started this work as a kind of joke: to see whether all relativistic gravitational effects (like the Mercury's perihelion precession and the light bending) can be explained as a consequence of simple two-body potentials acting between particles. Surely, they can. There is no need for curved space-time, general covariance, and other stuff, which manifestly contradicts the ideas of quantum mechanics. So, gravity can be formulated within Relativistic Quantum Dynamics (RQD) similar to electromagnetism. The key ingredient, which enabled this formalism, was the work by Kita who formulated a simple criterion for a two-particle potential to be relativistically invariant.

There was another piece of the puzzle that didn't want to fit and drove me crazy for a week or two. On the one hand, gravity changes the momentum (and energy) of photons. This is clear from the light bending observations. On the other hand, the gravitational time dilation and red shift experimens can be understood only if one assumes that the photon's energy does not change in the gravitational field. I just couldn't understand how these two statements can coexist. Finally, I figured it out. In the red shift experiments one should take into account the full (kinetic + potential) energy of photons, which stays constant in any gravitational field. Indeed, when the photon is absorbed by a detector, all its energy gets absorbed, not just its kinetic energy. Otherwise, the energy conservation law would be violated.
In the light bending situation, the photon's kinetic energy changes together with its momentum. There is not contradiction. Relief.

My next goal is to derive radiative corrections to the Coulomb potential in quantum electrodynamics. The idea is simple. According to RQD, since we know the S-matrix of the renormalized QED, we can fit a dressed particle Hamiltonian to this S-matrix. So, I am going to calculate the S-matrix for the electron-proton scattering up to the one-loop perturbation order (actually, this has been done more than 50 years ago, I just need to reproduce the calculations myself and transfer them to my notation), and then use formulas of RQD to obtain the corresponding electron-proton potential. At the tree-level this is the well-known Breit potentials which describes, in addition to the Coulomb, also magnetic, spin-orbit, and spin-spin interaction. The one-loop correction is supposed to describe the Lamb's shift in the hydrogen atom. So, I need to get this number right. This is going to be the final solution of the bound state problem in quantum field theory. The energies and wave functions of bound states will be obtained from the eigenvalue equation with the dressed particle Hamiltonian, just as in good old quantum mechanics. This should take me a couple of months, if the planets will hold their favorable positions.