November 2010 Wong Equations for Gravitation

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 October – November 2010:

Wong Equations for Gravitation: Work is concurrent with Notebook: September 2010 – February 2011

  • (p. 6-8) A further development in the concept of locality:
    • The Chinese authors develop the concept of a local gauge symmetry and arrive naturally at
      • A) The noncommutative generalization
      • B) The incorporation of extended objects
  • They do not, however, conceive of gravitation clearly in gauge theoretical concepts. They adhere to the untenable view that GR is a special case of gauge theory. This view goes back quite far, but it’s popularity seems clearly due to its championing by Yang himself.
  • The ingredients are
    • i) An algebra of “displacements” This structure is a substiture for the background spacetime.
    • ii) An algebra of “gauge transformations”
    • iii) A Hilbert space upon which i) and ii) act.

(p. 8) We wish here to extend to notion of locality beyond this framework. In fact, we claim that in order to understand Einstein’s theory of gravity one must generalize the structure above–even at the classical level. Before we explain this, we briefly consider the dynamical consequences (types of situations) of constructing a theory invariant under local gauge symmetry.

(p. 9) In order to do this, we do not consider the most general case of W-locality, but rather consider the special situation in which a base space may be clearly identified.

  • The most important feature … for our purposes is
    • *Nontrivial Vacua– The base space can be divided into regions in which, due to the nature of the solutions, the behavior of the theory can be radically different. When quantum effects are included the situation can become remarkably rich.
  • There is considerable debate over whether or not the intuition developed in this situation can be applied when gravity is relevant (e.g. T. Banks on phase transitions in gravitational theories and gravitational instantons). –c.f. Heretics of the false vacuum by T. Banks. See also Banks’ textbook on quantum field theory. I think our extension of the concept of locality may be essential in clarifying this situation.

(p. 12) In plain language, the proposal is simply that in order to consider gravity as a gauge theory we must introduce a further extension of the concept of local symmetry. The behavior of GR can be expressed in terms of local translations

(p. 14) This extension to a more general definition of local symmetry does not have anything a-priori to do with quantum theory, but it would be very interesting to find out otherwise. Perhaps, while not having anything a-priori to do with QM (except perhaps the connection between gauge ideas on QM) it might lead to an extension of QM in which the Berry-Wilczek arguments that connect gauge theory to QM are generalized to allow a new QM based on our new definition of local symmetry.

(p. 77-78) Calculating with eq. 87 of the paper (see the flap of this netbook) Link contains the page found in the flap of the notebook. Now, the approach we take is to integrate each term, and using linearity try to put the resulting relation intothe form in Pereira.

(p. 89) The directions one should go in seems: Gauge theory based on the local symmetry concept of Weyl is highly restrictive, but perhaps our new definition of local symmetry can be more accommodating. It should be based on Groupoids and Algebroids. Have supergroupoids/algebroids been studied?

(p. 103-115) Some thoughts about Lie Groupoids and Algebroids. … Can groupoids reep. algebroids shed some light on the structure of local translation algebra.

  • (p. 116) Division of aspects of our project:
    • Quantization: A separate bibliography needs to be made
    • Gauge Aspects: This would include the majority of what has been done.
    • Background independence
    • Noncommutativity