April 2012-May 2012

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.

Notebook April – May 2012: An Algebraic Approach to Gauge Gravity (cont’d) https://gaugegravity.com/2-notebook-4-5-2012/

Work is concurrent with Notebook: February 2012 – July 2012

(p. 9) The project … should be to precisely (geometrically) quantize the theory of particles on a curved background in the teleparallel formalism. BRST and geometric quantization.

  • Review the traditional approach e.g. page 127 of Woodhouse’s book on Geometric Quantization.
  • Reformulate in teleparallel formulation
  • Write an analogue to Bohr-Rosenfeld on the physics of the gravitational field.

(p. 11) Spin– It is possible to construct classical phase spaces for particles with spin from co-adjoint orbits of the Poincare group. By quantizing them we find the empirically correct quantum phase space. Geometric techniques are essentials here because the interval degrees of freedom cannot be separated into configuration and momentum variables.

(p. 16) Ideas for metric on the space of displacements. Want to encode an idea of how ‘far away’ two displacements are from each other, following the lead from ordinary geometry.

(p. 19) What can we understand about the dynamics of the field (Gravitational or Electromagnetic) from the dynamics of test particles.

(p. 29-34) Algebroids: What exactly does the anchor map and its induced map do?

  • (p. 35) Theory of Algebroids and Groupoids
    • Simple Definition
    • Categorical Definition
    • The grout and tiles example–concrete e.g. of transformation groupoid
    • Notes for brief history
    • Pseudo-groups and Lie and Cartan’s old and original ideas
    • Bibliography for groupoids and algebroids
    • Simplest examples and concrete cases for each example
    • Global and Local issues

(p. 43-48) Regarding a possible review of generalized gauge theory of gauge gravity. We wish here to propose a generalization of gauge theory of the kind that was instituted by Weyl and carried to fruition by Yang and Mills. Some logical generalizations are provided in the Chinese book.