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 July 2010 – September 2010: https://gaugegravity.com/wp-ontent/uploads/2020/09/1-Notebook-7-2010-9-2010-1.pd
Notes on: Gauge Gravity and Teleparallelism, Local Translations, Peirels Bracket, Poisson-Lie Bracket, Weyl, Tanimura, Wong Equations
(p. 02) In lieu of a constant torsion solution, we examine the dS solution described in Pereira’s introduction to teleparallelism.
(p. 09) It is interesting to ask what set of E’s completely form the space of local translations.
- (p. 14) It does seem, although we have not done the computation yet, obvious that the Peirels bracket will satisfy Leibniz II. This is because (A, B) is defined directly in terms of the action and is defined on the space of solutions of the theory defined by S1 and its perturbations. Thus the Feynman bracket seems to be more naturally associated with the Peirels bracket than the Poisson bracket. (Peirels bracket ↔ F- bracket)
- The conceptual view of the origins of General Relativity are that: (as the choice of names suggest)
- i. Special Relativity removed from physics the dependence of Maxwell’s U(1) gauge theory on the choice of inertial reference frame, by a suitable definition of ‘inertial’
- ii. General Relativity removed from physics the notion of inertial frame altogether by declaring that, at a point P, acceleration and gravitation are the same thing to second order.
- The definition of ‘inertial’ given above is associated with Lorentz symmetry. The assertion that gravitation and inertia are the same thing is associated with → here insert a response more careful than the usual one.
- (p 15-18) Efforts to Make Peirels Comprehensible:
- Lie-Poison Algebra is related to Peirels’ bracket.
- This relation must be explored in simple particle mechanics first, then U(1) gauge theory, fluid dynamics … enter equ. for rotating body.
- Reviewing the Poisson bracket formulation of U(1) gauge theory. Unlike in the Poisson-Lie bracket, here we have a symplectic structure.
- Aside: Why are there different classical formulations of GR that seem to be based on completely different physical principles e.g. teleparallelism → it seems to be in some respects non local! (of course so does GR).
- (p. 20) Aside: The fact that the Lorentz and Wong equ. coincide in the case of gravity may be related to the Problem of Motion.
- Aside: The Poisson – Lie formalism allows one to relax the condition of nondegenerat symplectic form 2. Is there an analogous connection that would allow us to relax the condition of nondegenerate tetrad?
- (p. 37- 42) There is a very interesting analogy between the non-linear abelian theory and non-commutative theory.
- Aside: Ross wrote a paper claiming to have proved the analogue of Jacobson’s result fro gauge theory.It would be interesting to examine that proof in the case of gauge theories of local translations.
- The problem is to consistently identify structures in the non-commutative and O – bracket cases
(p. 63) Due to the two types of indices involved and subtleties involving the raising and lowering of indices, we should do a thorough analysis of symmetries of Γ!
- (p. 68 – 70) OK so now I understand everything abut the velocity dependence and ordering.
- Now, because we know instantaneous velocity is finite and measurable from empirical observation, in exactly the same way we assume 𝑚𝑟 is a finite real number.
- In this form these calculations look just like the process of renormalization.
- It is probably advisable that the usual renormalization calculations should be written in a form that emphasizes infinitesimal quantities rather than infinite ones, for then we may use the intuition developed over the centuries regarding the usual calculus, e.g. differential forms.
- The only technical detail left for the production of the complete paper #1 on my (hopefully novel) approach to gravity is the derivation of the geodesic equation, resp. Lorentz force law.
(p. 80) Aside: It is possible that the discovery that diffeomorphism invariance implies a non-separable Hilbert Space is a statement about a direct connection between the Diff. group and the continuum of Complex numbers. See, the non-separable Hilbert Space has a continuum nature about it. Thus their result may mean that considering the Diff. group to be fundamental is, in fact equivalent to considering the continuum to be fundamental.
(p. 89) Now we review the relation of our gravitational theory to Non-Locality
- Leibniz rule #2 is automatically satisfied by the Peirels Bracket.
- Leibniz rule #2 is satisfied for Poisson-Lie Groups and their infinitesimal algebras, the Lie bi-algebras. While Lie Algebras are a rigid structure not allowing for deformation, the Lie bi-algebra allows a deformation (quantization) to a quantum group, which is a kind of non-commutative geometry. … Geometry as a Representation Space.
- It does us credit to remember a completely fundamental difference between YM & GR — the ‘rigidity’ of the solutions. This is basically what people are referring to when they speak about background independence.
(p. 106) Aside: Suppose we are presented with a Lie group … What kind of structure arises from allowing a numerical perturbation of the structure constants … . Is this related to q-deformation of quantum groups?
(p. 107) Aside: Suppose we write the Lie Algebra in terms of its universal enveloping algebra … . Is this related to … ? A review of the classification of Lie Algebras is a good idea to review in this connection.
(p. 111) Identities … in the relation between W & F.
(p. 142) Checking the relation between W & F: From Tanimura …
(p. 152) Because of the peculiarities involved in raising and lowering an index attached to a function one has taken the derivative of, it is highly desirable to do complicated calculations with a metric-compatible derivative. We will do this later, for now we watch our step very carefully and use the ordinary partial derivation. Later, to check the calculations, we will use the metric compatible derivative.
(p. 153) Quite a few important identities were developed yesterday.