The variable Planck’s constant due to imaginary gravitational temperature

Abstract

In a talk in PIRT-2015 we described how the Tolman-Thorne temperature of a general relativistic spinning star can be used to define a variable Planck-type constant where the temperature becomes imaginary and how this variable has a negligible value except in some region near a stationary gravitating particle of given mass and spin. In the present talk we recover the Lamb shift of the proton-electron system at thousands’ place in MHz from the spread of this region. While this may suggest that the variable Planck’s constant could be used with linear Fourier analysis to approximate gravitational effects in coupling gravity with non-gravitational interactions quite effectively, we also consider Einstein-Yang-Mills coupled equations for possible solutions that could be stable because the variable coupling implied by a variable Planck’s constant could stop the solutions from blowing up. While no new significant result for these coupled equations has been found yet, we provide a progress report of an ongoing program in which eventually we would like to explain at least qualitatively every valid result in QM and QFT with the help of the variable Planck’s constant suggested by GR (by which we mean here Einstein equations) and without using renormalization. We elaborate the first problem in more details. The remarkable coincidence in the earlier talk gave us a variable Planck’s constant hg of gravitational origin roughly at 0.2 Bohr radius for a gravitating particle having the mass and spin of a proton. In this talk we visit Welton’s argument for the Lamb shift to see how far the Lamb shift can be explained with this hg. The fluctuations of the quantized EM field used in Welton’s argument would take place only in the region where hg is meaningful. If we consider the Fourier transform of a real-valued function of a single variable having nonzero value 1 only on an interval of the positive axis, we see that the outer endpoint is roughly inversely proportional to the lowest significant wave number and inner endpoint is roughly inversely proportional to the highest significant wave number. The omitted frequencies produce the Gibbs’ phenomenon. Thus the ratio of the cutoff frequencies used in Welton’s argument becomes in our discussion . If we take rmin to be the distance from the center of mass of the proton-electron system where the gravitational temperature becomes imaginary and rmax to be the distance where hg ≈ h (hg decreasing as inverse square of r thereafter), we get values of our Lamb shift in the range 1019 to 1249 MHz. We do not expect to get the experimental value of 1057 MHz following our crude qualitative arguments. The discussion on the variation of values based on Welton’s argument namely 667 to 1394 MHz (see for example the textbook of F. Schwabl) makes one appreciative of our values and hence of our suspicion that gravity is behind the significant part of the Lamb shift. The next task is to look closely wherever Planck’s constant is used in the equations of quantum physics. Just naively replacing this Planck’s constant by hg will not give the full or a consistent story. In particular one would like to know how the Dirac equation and Einstein-Yang-Mills coupled equations get modified with hg. A variable Planck’s constant will make the coupling constant of the Einstein-Yang-Mills coupled system a varying function and the scale invariance of the system will be lost. We have done some initial calculation for the case of static Einstein-Yang-Mills equations. Spinning has not been added yet to keep the field equations manageable and so the justification behind using hg is somewhat missing. However the indications are that we cannot have SU(2) solutions, and we cannot have spherically symmetric solutions unless possibly when the gauge group is more complicated. Otherwise the coupling is forced to become constant and we get the known unstable solutions. A sufficiently complicated gauge group gives more equations. Looking at the phase space of the known solutions we then expect that the varying coupling field may stop the blowing up of some otherwise singular solutions. The global solution is expected to be a stable solution because its constant coupling analog is not the unstable separator of the classes of YM-potentials going to ±∞. Some related issues and how our results can be improved will be discussed.