# An Algebraic Approach to Gauge Gravity: A Concise Overview

Below an excerpt from an email George Karatheodoris sent to Roger Penrose explaining his novel approach to the gauge gravity problem, dated April 2, 2012.

“The approach I am taking to the quantum gravity problem is somewhat conservative in that it appears to adhere to the quantum theory, but as you will see it does introduce novel elements (even as regards to quantum theory) that should have significant effects. Basically, I feel that the approaches to “gauge gravity”, come in two forms: 1) the results from string theory that show that gravitational behavior of a kind is implicit in Yang-Mills theory provided that it is radically reinterpreted and provided that the rules for calculating observables are given by formulae proposed by Witten et. al.  2) results on the level of the classical action pioneered by Utiyama in the 1950’s and carried through by many authors.  The second approach is an attempt to understand gravity as a traditional gauge theory with the gauge group being something like the Poincare group–it is not accompanied by the non-local interpretation used in the stringy approach.  It is the second approach the I am working in, but I disagree with much of the literature on it.

The first result that I have found is that I can interpret the geodesic equation as the Wong equation for a generalized gauge theory.  I will explain exactly what this means.  First of all the Wong equation for a Yang-Mills theory coupled to matter is the equation of motion for that coupled system (matter, gauge field) in the approximation in which one treats it exactly like classical electrodynamics.  The gauge fields are treated as classical fields and the matter fields are approximated by classical charged particles, just as in classical electrodynamics.  The equations look just like the equations of classical electrodynamics but with the Yang-Mills field strength, the color vector charge in the current, and the covariant derivative replacing their Abelian counterparts.

One exciting feature that arises is a precise notion of background independence.  In string theory one does not have background independence because there is a group associated with the spacetime asymptotically, and if this asymptotic symmetry is changed one is talking about a new model within the string theory framework that is dynamically unrelated to the original model.  In the generalized gauge theory that I am looking at one has true background independence because the structure functions appearing in the 4- momentum bracket are dynamically determined, not input parameters.  In fact choosing a dynamics for the structure functions is introducing dynamics for the gravitational field itself.  It seems that the mathematical structure that appears naturally is well known, it is called an algebroid “symmetry”. So, one way of stating the proposal is to say that the gauge “algebra” of gravity is actually a Lie algebroid.  The attached note does not elaborate on this aspect, but future work will emphasize it.

So now what about the word “generalized” in “generalized gauge theory”?  Here I have to make an innovation.  In the literature it is often claimed that the correct choice of fibre when constructing a gravitational gauge theory is the tangent space or something very like it.  I do not believe that this is correct.  More details are provided in the attached paper, but briefly what seems necessary in order to describe gravity is to take Weyl’s local gauge principle one logical step further.  Weyl suggested that one should require gauge theories to be invariant under transformations of a group that can vary from point to point in spacetime.  I propose that gravity is what results when you remove the restriction that the symmetry group is the same at all points.  This may seem crazy, but it can be made precise: the bracket of the 4-momenta has structure function on its right-hand side, and in this way the structure “constants” of the symmetry vary from point to point.

The mathematics seems to be quite beautiful and I have told you a small fraction of the results and ideas…Again, I thank you so much and I hope that you find something of interest here.”

Cheers,
George Stephen Karatheodoris.