Quantum Gravity and String Theory


Physics as Infinite-Dimensional Geometry

Authors: Matti Pitkänen

The topics of this book is a vision about physics as infinite-dimensional Kähler geometry of the "world of classical worlds" (WCW), with "classical world" identified either as light-like 3-D surface X3 of a unique Bohr orbit like 4-surface X4(X3) or X4(X3) itself. The non-determinism of Kähler action defining Kähler function forces to generalize the notion of 3-surface. Zero energy ontology allows to formulate this generalization elegantly using a hierarchy of causal diamonds (CDs) defined as intersections of future and past directed light-cones, and a geometric realization of coupling constant evolution and finite measurement resolution emerges.

The general vision about quantum dynamics is that the basis for WCW spinor fields defines in zero energy ontology unitary U-matrix having as orthogonal rows M-matrices. Given M-matrix is expressible as a product of hermitian square root of density matrix and S-matrix. M-matrices define time-like entanglement coefficients between positive and negative energy parts of zero energy states represented by the modes of WCW spinor fields.

One encounters two challenges.

  1. Provide WCW with Kähler geometry consistent with 4-dimensional general coordinate invariance. Clearly, the definition of metric must assign to given light-like 3-surface X3 a 4-surface X4(X3) as kind of Bohr orbit.
  2. Provide WCW with spinor structure. The idea is to express configuration space gamma matrices using super algebra generators expressible using second quantized fermionic oscillator operators for induced free spinor fields at X4(X3). Isometry generators and contractions of Killing vectors with gamma matrices would generalize Super Kac-Moody algebra.

The condition of mathematical existence poses stringent conditions on the construction.

  1. The experience with loop spaces suggests that a well-defined Riemann connection exists only if this space is union of infinite-dimensional symmetric spaces. Finiteness requires that vacuum Einstein equations are satisfied. The coordinates labeling these symmetric spaces do not contribute to the line element and have interpretation as non-quantum fluctuating classical variables.
  2. The construction of the Kähler structure requires the identification of complex structure. Direct construction of Kähler function as action associated with a preferred extremal for Kähler action leads to a unique result. The group theoretical approach relies on direct guess of isometries of the symmetric spaces involved. Isometry group generalizes Kac-Moody group by replacing finite-dimensional Lie group with the group of symplectic transformations of δ M4+× CP2, where δ M4+ is the boundary of 4-dimensional future light-cone. The generalized conformal symmetries assignable to light-like 3-surfaces and boundaries of causal diamonds bring in stringy aspect and Equivalence Principle can be generalized in terms of generalized coset construction.
  3. Configuration space spinor structure geometrizes fermionic statistics and quantization of spinor fields. Quantum criticality can be formulated in terms of the modified Dirac equation for induced spinor fields allowing a realization of super-conformal symmetries and quantum gravitational holography.
  4. Zero energy ontology combined with the weak form of electric-magnetic duality led to a breakthrough in the understanding of the theory. The boundary conditions at light-like wormhole throats and at space-like 3-surfaces defined by the intersection of the space-time surface with the light-like boundaries of causal diamonds reduce the classical quantization of Kähler electric charge to that for Kähler magnetic charge. The integrability of field equations for the preferred extremals reduces to the condition that the flow lines of various isometry currents define Beltrami fields for which the flow parameter by definition defines a global coordinate. The assumption that isometry currents are proportional to the instanton current for Kähler action reduces Kähler function to a boundary term which by the weak form of electric-magnetic duality reduces to Chern-Simons term. This realizes TGD as almost topological QFT.
  5. There are also number theoretical conjectures about the character of the preferred extremals. The basic idea is that imbedding space allows octonionic structure and that field equations in a given space-time region should reduce to the associativity of the tangent space or normal space so that space-time regions should be quaternionic or co-quaternionic. The first formulation is in terms of the octonionic representation of the imbedding space Clifford algebra and states that the octonionic gamma "matrices" span a quaternionic sub-algebra. Another formulation is in terms of octonion real-analyticity. Octonion real-analytic function f is expressible as f=q1+Iq2, where qi are quaternions and I is an octonionic imaginary unit analogous to the ordinary imaginary unit. q2 (q1) would vanish for quaternionic (co-quaternionic) space-time regions. The local number field structure of octonion real-analytic functions with composition of functions as additional operation would be realized as geometric operations for space-time surfaces. The conjecture is that these two formulations are equivalent.
  6. An important new interpretational element is the identification of the Kähler action from Minkowskian space-time regions as a purely imaginary contribution identified as Morse function making possible quantal interference effects. The contribution from the Euclidian regions interpreted in terms of generalized Feynman graphs is real and identified as Kähler function. These contributions give apart from coefficient identical Chern-Simons terms at wormhole throats and at the space-like ends of space-time surface: it is not clear whether only the contributions these 3-surfaces are present.
  7. Effective 2-dimensionality suggests a reduction of Chern-Simons terms to a sum of real and imaginary terms corresponding to the total areas of of string world sheets from Euclidian and Minkowskian string world sheets and partonic 2-surfaces, which are an essential element of the proposal for what preferred extremals should be. The duality between partonic 2-surfaces and string world sheets suggests that the total area of partonic 2-surfaces is same as that for string world sheets.
  8. The approach leads also to a highly detailed understanding of the Chern-Simons Dirac equation at the wormhole throats and space-like 3-surfaces and Kähler Dirac equation in the interior of the space-time surface. The effective metric defined by the anticommutators of the modified gamma matrices has an attractive interpretation as a geometrization for parameters like sound velocity assigned with condensed matter systems in accordance with effective 2-dimensionality and strong form of holography.

Comments: 480 Pages. Updating

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Submission history

[v1] 3 Aug 2009
[v2] 14 Oct 2009
[v3] 3 Nov 2010
[v4] 2011-12-04 00:33:24
[v5] 2012-01-30 22:35:08
[v6] 2012-07-01 08:25:04
[v7] 2012-08-02 01:03:14
[v8] 2012-08-19 04:20:32
[v9] 2012-09-22 02:34:01
[vA] 2012-10-13 02:50:59
[vB] 2012-11-13 01:10:56
[vC] 2013-01-01 07:52:56
[vD] 2013-02-17 01:33:53
[vE] 2013-09-09 07:06:16
[vF] 2014-09-26 04:42:43

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