An Algebraic Approach to Gauge Gravity

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.

An Algebraic Approach to Gravity, Part I: Notebook February 2012 – July 2012:

(p. 1) A consideration of Kleinert’s novel gauge theory.

(p. 11) So there is a fantastically interesting tension between the relativistic philosophy and the Hamiltonian one–If we want a uniform dynamics in which all observables are treated uniformly and evolve according to FODE.

(p. 15) The study I have initiated on the measurability of the electromagnetic field, with the goal of generalizing to the torsion field of teleparallel gravity is quite substantial.

(p. 15-16) Why has the Bohr-Rosenfeld analysis become forgotten? Bohr and Rosenfeld exhibit the basis for the physical interpretation of quantum electrodynamics … Today we start with renewed confidence in the importance of pursuing the Bohr-Rosenfeld analysis. Not only can this lead to insights in teleparallel gravity, but also to the foundations of quantum theory. Bohr reproduces his classic thoughts on non-relativistic quantum theory in the relativistic field theory framework: 1) Application to interpretation of (teleparallel) gauge gravity. 2) Application to formulation of quantum mechanics.

The relationship between the real and reciprocal spaces is absolutely beautiful. It is the physical basis for the tension between the unitarity and covariance requirement of QFT.

(p. 24-25) Relationship between non commutativity and gravity: Szabo et. al. have written a nice paper discussing the fact that Non commutative U(1) gauge theory contains teleparallel gravity. Ref: 01050492v2 25 May 2001. This is quite an old paper but it may be helpful for my concept of gauge gravity.

(p. 24-25) A novel conception of gauge gravity

(p. 26) As for a technical project to begin with now I’d like to extend the Bohr-Rosenfeld analysis to the gravitational field. This should be viable due to the rewriting of the field in its teleparallel form. We really only need the theory of a background field acting upon massive test bodies to replicate the Bohr-Rosenfeld field results. Ideally we should end up with uncertainty relations for the torsion field in exactly the same way that Bohr-Rosenfeld found results for the electrodynamic field strengths.

(p. 81-83) A Review and Summary of Work on Gravity and Quantum Gravity

An Algebraic Approach to Gravity Cont’d: Notebook: April 2012 – May 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.