Tensors vs. Rank 0 Tensors aka Quaternions
Channel: Doug Sweetser
Date: 2019-06-03

This talk was given at the 22nd Eastern Gravity Meeting at UMass Dartmouth, June 1, 2019. Einstein's field equations for gravity can be quantized but not in a way that be used in calculations. Efforts to fix the problem by going up to 11 eleven dimensions then back down to four appear too complicated. By removing all the indices used by the machinery of differential geometry, the result is simpler. This turns out to be less radical than it might appear because all 16 terms of the contraction of two 4-vectors are in a quaternion product, just in different places. One can generate the Klein-Gordon equation, but also create three new equations. I speculate these may help solve riddles that remain in quantum field theory. 

The new (since 2015) approach to gravity uses two symmetries in two 4D spaces. The first is familiar: the Lorentz transformations of Minkowski space-time. The three rotations and three boosts leave the real-valued interval t² - R² unchanged. What is novel is the affine space that I claim is all about energy-momentum. In both classical and quantum mechanics, one needs both where-when a particle is along with the energy-momentum. In the affine space, zero energy is the horizontal line while zero momentum is the vertical line. This time the hyperbolas describe constant imaginary values, tR.

The talk was received with polite indifference. There was one question about whether the proposal was consistent with gravity waves. I have not done a study of the subject, but if the distance R used is a function of time, then that could provide a means for generating a gravity wave. The lowest mode of emission would be a quadrupole which is consistent with observations of energy loss by binary pulsars.

You will also see in the comments section a variety of objections from a professional whose nom de guerre is Purple Penguin. I will make no effort to summarize his objections other that to say they appear to be rooted in a solid understand of how we understand gravity today using general relativity. All I can do is try to understand my work better. The biggest breakthrough I had creating this talk was to discuss together Minkowski space-time and its affine space energy-momentum. There is nothing radical to that approach, just odd (consistent with my style, see the hat).

slides: http://bit.ly/vp-EGM22