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Topological Geometrodynamics: an Overview
Note: Newest contributions are at the top!
Massive particles are the basic problem of the twistor program. The twistorialization of massive particles does not seem to be a problem in TGD framework thanks to the possibility to interpret them as massless particles in 8-D sense but the situation has been unsatisfactory for virtual particles (see this).
The ideas possibly allowing to circumvent this problem emerged from a totally unexpected direction (the finding of Martin Grusenick described in previous posting). The TGD inspired explanation of the results together with quantum classical correspondence (see this) led to the proposal that Einstein tensor in space-time regions where the energy density vanishes describes the presence of virtual particles assignable to the exchange of momentum between massive objects. This identification is consistent with Einstein's equations in zero energy ontology and leads to a proposal for a concrete geometric identification of virtual particles in TGD framework.
The basic idea is following. In TGD gauge bosons are identified as wormhole contacts with positive energy fermion and negative energy antifermion at the opposite light-like throats of the contact. The observation is that even if the fermions are on mass shell particles, the sum of their momenta has a continuum of values so that interpretation as a virtual boson makes sense. Free fermions do not correspond to single light-like throat but if the interaction region involves a pair of parallel space-time sheets with distance of order CP2 length of order 104 Planck lengths, the CP2 extremal describing fermion has a high probably to touch the second space-time sheet so that topological sum giving rise to second light-like throat is formed. This throat has spherical topology and carries purely bosonic quantum numbers. If on mass shell state with opposite energy is in question, the situation reduces to that for bosons. One can also formulate conditions guaranteing that no new momentum degrees of freedom appear in loops integrals. Since only on mass shell throats are involved one can apply TGD based twistorial description also to virtual particles without any further assumptions.
I have updated the chapter about massless particle and particle massivation to correspond to the recent progress in the understanding of the basic TGD. I glued below the abstract.
In this chapter the goal is to summarize the recent theoretical understanding of the spectrum of massless particles and particle massivation in TGD framework. After a summary of the recent phenomenological picture behind particle massivation the notions of number theoretical compactification and number theoretical braid are introduced and the construction of quantum TGD at parton level in terms of second quantization of modified Dirac action is described. The recent understanding of super-conformal symmetries are analyzed in detail. TGD color differs in several respect from QCD color and a detailed analysis of color partial waves associated with quark and lepton chiralities of imbedding space spinors fields is carried out with a special emphasis given to the contribution of color partial wave to mass squared of the fermion. The last sections are devoted to p-adic thermodynamics and to a model providing a formula for the modular contribution to mass squared.
Although the basic predictions of p-adic mass calculations were known almost 15 years ago, the justification of the basic assumptions from basic principles of TGD (and also the discovery of these principles!) has taken a considerable time. Particle massivation can be regarded as a generation of thermal conformal weight identified as mass squared and due to a thermal mixing of a state with vanishing conformal weight with those having higher conformal weights. The observed mass squared is not p-adic thermal expectation of mass squared but that of conformal weight so that there are no problems with Lorentz invariance.
One can imagine several microscopic mechanisms of massivation. The following proposal is the winner in the fight for survival between several competing scenarios.
An important question concerns the justification of p-adic thermodynamics.
p-Adic thermodynamics is what gives to this approach its predictive power.
The predictions of the general theory are consistent with the earliest mass calculations, and the earlier ad hoc parameters disappear. In particular, optimal lowest order predictions for the charged lepton masses are obtained and photon, gluon and graviton appear as essentially massless particles.
For the updated version of the chapter see Massless particles and Particle Massivation.
During the last five years both the mathematical and physical understanding of quantum TGD has developed dramatically. Some ideas have died and large number of conjectures have turned to be un-necessary strong, un-necessary, or simply wrong. The outcome is that the books about basic TGD do not correspond the actual situation in the theory. Therefore I decided to perform a major cleaning operation throwing away the obsolete stuff and making good arguments more precise. Good household is not my only motivation: this kind of process, although it challenges the ego, is always extremely fruitful. The basic goal has been to replace the perspective as it was for five years ago with the one which is outcome of the development of visions and concepts like fundamental description of quantum TGD as almost topological QFT in terms of modified Dirac action for fermions at light-like 3-surfaces identified as the basic objects of the theory, zero energy ontology, finite measurement resolution as a fundamental physical principle realized in terms of Jones inclusions and having number theoretic braids as space-time correlate, generalization of S-matrix to M-matrix, number theoretical universality and number theoretical compactification reducing standard model symmetries to number theory and allowing to solve some basic problems of quantum TGD, realization of the hierarchy of Planck constants in terms of the generalization of imbedding space concept, discovery of a hierarchy of symplectic fusion algebras provided concrete understanding of the super-symplectic conformal invariance, and so on.
I started the cleaning up process from the chapter Configuration Space Spinor Structure and I glue below the abstract.
Quantum TGD should be reducible to the classical spinor geometry of the configuration space. In particular, physical states should correspond to the modes of the configuration space spinor fields. The immediate consequence is that configuration space spinor fields cannot, as one might naively expect, be carriers of a definite spin and unit fermion number. Concerning the construction of the configuration space spinor structure there are some important clues.
1. Geometrization of fermionic statistics in terms of configuration space spinor structure
The great vision has been that the second quantization of the induced spinor fields can be understood geometrically in terms of the configuration space spinor structure in the sense that the anti-commutation relations for configuration space gamma matrices require anti-commutation relations for the oscillator operators for free second quantized induced spinor fields.
2. Modified Dirac equation for induced classical spinor fields
The identification of the light-like partonic 3-surfaces as carriers of elementary particle quantum numbers inspired by the TGD based quantum measurement theory forces the identification of the modified Dirac action as that associated with the Chern-Simons action for the induced Kähler gauge potential. At the fundamental level TGD would be almost-topological super-conformal QFT in the sense that only the light-likeness condition for the partonic 3-surfaces would involve the induced metric. Chern-Simons dynamics would thus involve the induced metric only via the generalized eigenvalue equation for the modified Dirac operator involving the light-like normal of X3l subset X4. N=4 super-conformal symmetry emerges as a maximal Super-Kac Moody symmetry for this option. The application of D to any generalized eigen-mode gives a zero mode and zero modes and generalized eigen-modes define a cohomology.
The basic idea is that Dirac determinant defined by eigenvalues of DC-S can be identified as the exponent of Kähler action for a preferred extremal. There are however two problems. Without further conditions the eigenvalues of DC-S are functions of the transversal coordinates of X3l and the standard definition of Dirac determinant fails. Second problem is how to feed the information about preferred extremal to the eigenvalue spectrum. The solution of these problems is discussed below.
The eigen modes of the modified Dirac equation are interpreted as generators of exact N=4 super-conformal symmetries in both quark and lepton sectors. These super-symmetries correspond to pure super gauge transformations and no spartners of ordinary particles are predicted: in particular N=2 space-time super-symmetry is generated by the righthanded neutrino is absent contrary to the earliest beliefs. There is no need to emphasize the experimental implications of this finding.
An essential difference with respect to standard super-conformal symmetries is that Majorana condition is not satisfied, the super generators carry quark or lepton number, and the usual super-space formalism does not apply. The situation is saved by the fact that super generators of super-conformal algebras anticommute to Hamiltonians of symplectic transformations rather than vector fields representing the transformations.
Configuration space gamma matrices identified as super generators of super-symplectic or super Kac-Moody algebras (depending on CH coordinates used) are expressible in terms of the oscillator operators associated with the eigen modes of the modified Dirac operator. The number of generalized eigen modes turns out to be finite so that standard canonical quantization does not work unless one restricts the set of points involved defined as intersection of number theoretic braid with the partonic 2-surface. The interpretation is in terms of finite measurement resolution and the surprising thing is that this notion is implied by the vacuum degeneracy of Kähler action.
3. The exponent of Kähler function as Dirac determinant for the modified Dirac action
Although quantum criticality in principle predicts the possible values of Kähler coupling strength, one might hope that there exists even more fundamental approach involving no coupling constants and predicting even quantum criticality and realizing quantum gravitational holography.
The almost topological QFT property of partonic formulation based on Chern-Simons action and corresponding modified Dirac action allows a rich structure of N=4 super-conformal symmetries. In particular, the generalized Kac-Moody symmetries leave corresponding X3-local isometries respecting the light-likeness condition. A rather detailed view about various aspects of super-conformal symmetries emerge leading to identification of fermionic anti-commutation relations and explicit expressions for configuration space gamma matrices and Kähler metric. This picture is consistent with the conditions posed by p-adic mass calculations.
Number theoretical considerations play a key role and lead to the picture in which effective discretization occurs so that partonic two-surface is effectively replaced by a discrete set of algebraic points belonging to the intersection of the real partonic 2-surface and its p-adic counterpart obeying the same algebraic equations. This implies effective discretization of super-conformal field theory giving N-point functions defining vertices via discrete versions of stringy formulas.
For the updated version of the chapter see Configuration Space Spinor Structure.
The quantization of Planck constant has been the basic them of TGD since 2005 and the perspective in the earlier version of this chapter reflected the situation for about year and one half after the basic idea stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by hbargr = GM1M2/v0, v0 ≈ 2-11 for the inner planets. The general form of hbargr is dictated by Equivalence Principle. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales.
The second crucial empirical input were the anomalies associated with living matter. Mention only the effects of ELF radiation at EEG frequencies on vertebrate brain and anomalous behavior of the ionic currents through cell membrane. If the value of Planck constant is large, the energy of EEG photons is above thermal energy and one can understand the effects on both physiology and behavior. If ionic currents through cell membrane have large Planck constant the scale of quantum coherence is large and one can understand the observed low dissipation in terms of quantum coherence.
1. The evolution of mathematical ideas
From the beginning the basic challenge -besides the need to deduce a general formula for the quantized Planck constant- was to understand how the quantization of Planck constant is mathematically possible. From the beginning it was clear that since particles with different values of Planck constant cannot appear in the same vertex, a generalization of space-time concept is needed to achieve this.
During last five years or so many deep ideas -both physical and mathematical- related to the construction of quantum TGD have emerged and this has led to a profound change of perspective in this and also other chapters. The overall view about TGD is described briefly here.
The evolution of physical ideas related to the hierarchy of Planck constants and dark matter as a hierarchy of phases of matter with non-standard value of Planck constants was much faster than the evolution of mathematical ideas and quite a number of applications have been developed during last five years.
3. Brief summary about the generalization of the imbedding space concept
A brief summary of the basic vision in order might help reader to assimilate the more detailed representation about the generalization of imbedding space.
For the updated version of the chapter see Does TGD Predict a Spectrum of Planck Constants?.