For clarity, the 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/2019/07/notebook-3-4-2011.pdf
- Note: (**) indicates could not be transcribed.
- (p.04) We can see quite clearly that the teleparallel (1) equation of motion is more general than the formula (4) for the force in Santilli.
- (p. 07) Now we can use the fact that the commutator of a locally translation invariant momentum with a function of x is the translation-covariant derivative of the function.
- (p. 13) Coleman-Mandula may be evaded due to the Groupoid (vs.group) structures. I have a strong feeling that this is correct. What is the irreducibility concept for groupoids/algebroids.
- (p. 20) At this point we cannot put the momentum inside the derivative due to the extra metric factor and the fact that the derivative is not metric-compatible. The general issue is one that has been found before in this project.
- (p. 27) On the deepest level the gauge potential is the function that determines the parallel transport of vectors in the Lie Algebra.
- (p. 28) In other words the torsion acts as a connexion, but it is also a field-strength, being the commutator of covariant derivatives.
- (p. 29) So it appears that in the case of gravity, the field strength plays also the role of connection. This in fact is the reason that the theory looks a bit Abelian: the connection (torsion) is invariant.
- (p. 33) In our case, we have a Lie Algebroid symmetry. We follow the discussion on p. 119 of Weinstein’s Geometric Models paper, and extend the discussion of the Chinese Authors.
- (p. 35) So from yesterday it seems that the cleanest way to proceed in generalizing gauge theory is to promote (**) and (**) to algebroids….Then we might start by assuming that
- 1. (**) is an algebroid over a point
- 2. (**) is an algebroid over a manifold or perhaps a non-commutative space.
- (p. 43) Thus in the absence of Yang-Mills forces we get the geodesic equation.
- (p. 52) The confusion of field strengths. It seems the momenta, (**) , contain everything in the commutator.
- (p. 56) The generalized momentum, even in the translation case, generates to total field strength via commutation…
- (p. 63) Now, we can calculate the variation of (**) under a G.T. and we will find it is invariant.
- (p. 66) The reason for the brief excursion through the bremsstrahlung is to understand the requirement of quantum theory. The universe is not scale invariant, so an ‘Atom’ the size of the solar system is quasi-stable due to the suppression of large accelerations in that system that suppress the bremsstrahlung . As the system shrinks, one requires quantum corrections to stabilize the system.
- What is totally amazing is that the system loses its time dependence. Something happens to the meaning of “t” in the shrinking of the system.
- (p. 70) What is the best way to introduce the translation field strengths. There are dual methods.
- (p. 82) Let’s…consider an expansion of quantum theory in light of Algebroid symmetry.
- (p. 86-87) There has been a substantial shift.
- We previously thought of (**) as ‘living on the spacetime (**) , a conception that is close to Newton’s conception of absolute space. The idea is “well, anything could live there” and thus the spacetime maintains its independent integrity, independent of what happens to live upon it.
- The new view is much closer to a relational understanding. Indeed when we apply a group element to (**) to get (**) we manifest that phase at the new location (**) , and there is an explicit relation (**) that may be thought of as replacing the background manifold. In this picture spacetime is the web of relationships (**) between all objects (**)
- This is quite a satisfactory improvement, from the philosophical side.
- So, now having decided on the superiority of this description, we can go through the usual calculation and carefully notice and (make) changes in the meaning of the relationships deduced.
- (p. 91) New meaning of local gauge invariance.
- (p. 92-93) The notion of parallelism is a physical concept, meaning it must be gauge invariant. I think the commutative diagram picture is quite appropriate for the general case in which there may ultimately be no underlying spacetime…
- (p. 102) At step (9) This is it. (refers to diagram)
- (p. 119) Proof of the Gauge Tranformation rule.