Noncommutative Spacetime; Gravity & Commutators; Noncommutative Pseudo-complex Geometry (3-08–6-08)

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: March 2008-June 2008

(p. 11-21) Study of SU (1,1) with the goal of embedding Rindler space in this manifold. Ref. Perelomov Ch. 5. summary pp. 19-21.

(p. 56) The problem of degenerating representations of the sphere and presumably the hyperboloid (on page 30 (pdf p. 54) might be solved by regularization.

  • (p. 64) The puzzle of space-time noncommutativity in particular Rindler space.
    • S-branes and Tachyons
    • There seems to be a fundamental instability in R-space. It is a patch of flat space and so it seems thee should be no instability, but if observers are present (by observers we mean ~1023 quantum particles stuck together) they destabilize (means: they warp the geometry away from flatness) the space.
    • This analysis goes to the role of observers in quantum gravity. It may be possible to construct a background geometry that is perfectly stable, but when observers are introduced, the space becomes UNstable.
  • (p. 70-75) Just some notes on Schiller’s Born Infeld Kinematics.
    • Q: Does supersymmetric quantum mechanics generate a maximal acceleration?
    • Q: How is pseudo-complexity related to supersymmetry?
  • Pseudo complex geometry provides the natural mathematical framework for “Kinematizing” the dynamical symmetries of Dirac Born Infeld theory.
  • (p. 77-96) Topics:
    • Unitarity; Unitarity and Noncommutative Gauge theories
    • Ward Identities; Ward Identities and Noncommutativity
    • Instability
      • A universe cannot be unstable, it would seem. The instability of a system S, for example a proton electron bound state, is experimentally verified by watching the system evolve in time to another configuration S’ + S”. When this happens, if we restrict attention to S’, we see a decrease in the energy.
    • Conservation
    • Commutators

(p. 110) Do Spacetime Uncertainty Principles Require Strings? (Excerpts from the embed below)

  • …An interesting aspect of SUR’s is that, while arising naturally and possibly universally from string theory, its fundamental physical underpinnings in that context are obscure. In a different approach, which has as its basis a very intuitive and physical thought experiment leading directly to a particular set (examine with care) of SUR’s (which may or may not be precisely equivalent to the string theory version.) Freden Lagen have constructed an accompanying physical theory that precisely reproduces their SUR’s (call them SUR’s). It appears that this formalism suffers from certain deficiencies. Unitarity is called into question in both the scalar field and Gauge Field cases. Different quantization schemes such as Yang-Feldman, Schlesinger, Canonical, PI, etc., which are essentially equivalent in the standard approaches to Quantum Field theory, but may not be where the classical theory to be quantized is written on a noncommutative spacetime. Straightforward application of the calculation rules to the symbols appearing in the NCFT have led to clear violations of unitarity…Further analysis of the consistency of this scheme has produced a variety of difficulties ranging from the forward propagation of (-)(could be a BRST anomaly…?) energy modes to the non-conservation of the BRST current. There are, to my mind two obvious possibilities (and I’m sure some non-obvious ones) for resolution of this situation.
    • 1) SUR’s are inherently stringy despite the fact that the physical considerations used to heuristically describe them do not make direct use of 1-d extended Q.M. entities, but rather only refer to well established classical GR and QM QFT? results.
    • If it turns out that SUR’s are inherently stringy AND the argument of Freden Lagen is very tight this could provide a hint that the string theory quantizization of gravity may be fairly unique…
    • There is a fault in the NCQFT tweaking (or simply its physical interpretation) that needs to be understood…