April 2010 – July 2010: Issues in Gauge Gravity, Wong Equations

Author Commentary Below: A few notations from this notebook, to indicate topics covered. Author’s page numbering is ignored, any page numbers referenced below are from the scanned PDF documents.**Most helpful if notebook is read with Two Page Scrolling in Adobe, under “View”.

Notebook April 2010 – July 2010 https://gaugegravity.com/wp-content/uploads/2020/10/1.-Notebook-4-2010-7-2010.pdf Notes on the connection between Noncommutativity and Generalized Gauge Theory tucked at the end of this notebook: https://gaugegravity.com/wp-content/uploads/2021/01/Vol.-18a-added-pages.pdf

Below: July 2010, Author’s Brief Summary

I was still working on some fine points of the Wong equations for gravity during this whole notebook but also exploring lots of different issues in gauge gravity generally, including lattice gravity, gauge transformations , generalized gauge structures as laid out by the Chinese authors, the question of observables in quantum gravity, and noncommutativity. This is the notebook in which I made the discovery that noncommutativity and non-locality are inherent in gauge gravity. I also studied the Seiberg-Witten image of local translations and supersymmetry in the Wong equations. The work was discussed with Sergio.

Table of Contents

  1. On April 23-26, I did calculations on the Wong equations.
  2. On page 5, some notes on the Hamiltonian and action principle for a particle in a background gauge gravity field
  3. On page 6, it is necessary to have

On page 6, I also realized that the bracket of two velocities is not the field strength but rather is an ww .

  • On page 7 and 8, more computations on the Wong equations
  • On page 8, the Bianchi identity is calculated.
  • Page 9 was an effort to put gauge gravity in the framework of the Chinese authors.
  • Some observations
    • May be tempted to conclude that gravitation is the theory for which the displacement algebra and the gauge algebra are isomorphic
    • Classically, the refinement of composite fiber bundles seems necessary. It is unclear how this refinement changes the Chinese triple structure for noncommutative gauge theory. (Author notes end here. Continued below, from (pdf) pages 20-21)

(p. 38) …(I)t is relevant to consider, very carefully, what ‘quantum corrections’ to gravity appear like, in the context of gauge theory of gravity.

(p. 44) Is it just a coincidence that Steinacker and Yang both get gravity coming from the u(1) sector (the abelian sector) of some noncommutative theory. Is this related to the teleparallel formulation? Here is a list of some quantum effects in abelian gauge theory…

(p. 54) The solid state physicist knows how to go from QM to QFT. This entails a change in the DOF, but in distinction to the effective field theory philosophy, this change is due to a change in the number of degrees of freedom, N, not the energy, E. Also this use of field theory has nothing to do with particle creation/annihilation. C.f. Weinberg’s discussion of the Cluster Decomposition theorem.

(p. 58- 59) Still the big issue is connecting precisely the Chinese and Connes’ approaches.It’s a good idea to solve some explicit examples in teleparallel gravity to see the distinction between the diffeo and local translation approaches in the context of specific examples.

(p. 61) Here I want to explore the precise way in which noncommutative geometry arises in the consideration of quantum gauge theories. The Chinese authors have argued convincingly that noncommutativity arises naturally from gauge theory considerations and the attempt to tighten concepts.

(p. 69) Does the concept of Wigner Quantum System (WQS) enter? Is the Feynman system really a WQS?

  • (p. 69 -70) First let’s consider the unity of space and time.
    • The relativistic unity
    • My operational considerations that suggest a minimal space in which to store information about time measurements….
  • The reason we need 2 models , one classical and one quantum, is in order to examine whether the difference in quantum and classical measurement makes a difference in the measurement of time.
  • It may not, according to QM (at least the time in the Schrodinger equation) time passes between quantum measurements, but there is no quantum measurement of time. This is in stark contrast with space (conceived as places where quantum particles might manifest)
  • This is all a bit mysterious since the inability to tell WHEN a particular radioactive atom will decay was was used as a quintessential result of quantum mechanical uncertainty.

Aside: It’s starting to look like gauge theory is fundamental to QM itself. In this case, as I have already anticipated, “quantized spacetime” or noncommutative geometry should have fundamental gauge theory input; as in the Chinese approach. (This note is after reading Wilczek; Zee paper and Chapter in Nair on geometric quantization)

(p. 76 – 77) Some notes on Observables in Quantum Gravity

(p. 78 – 110) Gauge Transformations in Feynman’s Formalism (Below, solution on p. 55 in notebook, p. 110 in the PDF doc.)

So, modulo factors, the prescription for gauge transformations I invented for the F-Algebra reproduces Eq. 1.2.8 in the Chinese book. So now we can turn our attention to other things e.g. let’s look at Lattice gauge Translation Theory.

(p. 112 – 114) How is a tangent space represented in Lattice Theory? Here we examine Gauge Transformations in the case of Gravity, following as closely as possible the procedure introduced for Yang-Mills.

(p. 119 – 121) On the inherent noncommutativity (resp. nonlocality) of gravity.

(p. 134) Making a Lie algebra out of a soft algebra. Let’s begin with an example … The goal is to reformulate this system in a Lie algebra style, i.e. increase the number of generators and eliminate the dependence of the structure constants on the generators.

  • (p. 187) We can turn this soft algebra into countably infinite many hard i.e. Lie Algebras.
  • Let’s also find a bunch of solutions to teleparallel gravity and find the specific Lie algebras that they produce.
  • Revisit the Tiemblo approach and take perhaps WZW in 4D model for the Godstone consideration.

End Note: Loose notes added after the last page: “…to elucidate the connection between noncommutativity and Generalized Gauge Theory. “ https://gaugegravity.com/wp-content/uploads/2021/01/Vol.-18a-added-pages.pdf