Mathematical Physics


Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X2 at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM+4 + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM+4 x CP2 localized with respect to X2. H is identified as Kac-Moody type group associated with isometries of H = M+4 x CP2 acting on light-like 3-surfaces and thus on X2. An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.

Comments: 26 Pages.

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Submission history

[v1] 16 Jun 2010
[v2] 2012-01-30 22:03:21

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