October 2011-December 2011

Author’s page numbering is ignored; any page numbers referenced below are those from the scanned PDF documents.

Notebook: https://gaugegravity.com/wp-content/uploads/2019/11/Notebook-10-12-2011.pdf

  • Non Generated Diffeomorphisms (continued)

Author Commentary:

(p. 1-14) We make an inductive hypothesis.

(p. 15-16) So, since I’ve made the interpolating function, it would seem that a sufficient constraint on so that it lies on a 1-parameter group containing the identity is that . If this constraint is relaxed we must deal with a more complicated computation. Already at the level of m+n=3, there is an addition to the differential equation that determines . It does, however, seem that only the are non-zero. So, the solution is:

(p. 17) It seems that one must retain products of 𝒷’s. We can check whether the approximation was sensible by checking the group law for the final function. In addition it is not too hard to find the exact solution, all we have in the RHS of the differential equation is sums of exponentials..

(p. 19) Now we set to zero any quantity quadratic in the 𝒷’s . That means that 𝓹 must be equal to 1.

(p. 24) We have discovered an interpolating function in the case that we can ignore all quadratic terms in the perturbation. the question is: does this give us the right to ignore the perturbation for all 𝓽?

(p. 25) More on the Unitarity problem of non-commutative QFT and the finding of a good set of uncertainty relations.

(p. 28-30) Small perturbation of a small rotation on a 1-parameter subgroup.