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TGD: Physics as Infinite-Dimensional Geometry
Note: Newest contributions are at the top!
The recent progress in the understanding of the modified Dirac equation defining quantum TGD at fundamental level (see this and this) stimulated a further progress. I managed to find a general ansatz for the modified Dirac equation solving it with very general assumptions about the preferred extremal of Kähler action.
A key role in the ansatz is played by the assumption that modified Dirac equation can be formulated using an octonionic representation of imbedding space gamma matrices. Associativity requires that the space-time surface is associative in the sense that the modified gamma matrices expressible in terms of octonionic gamma matrices of H span quaternionic sub-algebra at each point of space-time surface. Also octonionic spinors at given point of space-time surface must be associative: that is they span same quaternionic subspace of octonions as gamma matrices do. Besides this the 4-D modified Dirac operator defined by Kähler action and the 3-D Dirac operator defined by Chern-Simons action and corresponding measurement interaction term must commute: this condition must hold true in any case. The point is that associativity conditions fix the solution ansatz highly uniquely since the action of various operators in Dirac equation is not allowed to lead out from the quaternionic sub-space and the resulting ansatz makes sense also for ordinary gamma matrices.
It must be emphasized that octonionization is far from a trivial process. The mapping of sigma matrices of imbedding space to their octonionic counterparts means projection of the vielbein group SO(7,1) to G2 acting as automorphism group of octonions and only the right handed parts of electroweak gauge potentials survive so that only neutral Abelian part of classical electroweak gauge field defined in terms of CP2 remains. More over, electroweak holonomy group is mapped to rotation group so that electroweak interactions transform to gravitational interactions in the octonionic context! If octonionic and ordinary representations of gamma matrices are physically equivalent this represents kind of number theoretical variant for the possiblity to represent gauge interactions as gravitational interactions. This effective reduction to electrodynamics is absolutely essential for the associativity and simplifies the situation enormously. The conjecture is that the resulting solutions as such define also solutions of the modified Dirac equation for ordinary gamma matrices.
The additional outcome is a nice formulation for the notion of octo-twistor using the fact that octonion units define a natural analog of Pauli spin matrices having interpretation as quaternions. Associativity condition reduces the octo-twistors locally to quaternionic twistors which are more or less equivalent with the ordinary twistors and their construction recipe might work almost as such. It must be however emphasized that this notion of twistor is local unlike the standard notion of twistor since projections of momentum and color charge vector to space-time surface are considered. The two spinors defining the octo-twistor correspond to quark and lepton like spinors having different chirality as 8-D spinors.
The basic motivation for octo-twistors is that they might allow to overcome the problems caused by the massivation in the case of ordinary twistors. One might think that 4-D massive particles correspond to 8-D massless particles. A more refined idea emerges from modified Dirac equation. The space-time vector field obtained by contracting the space-time projections of four-momentum and the vector defined by Cartan color charges might be light-like with respect to the effective metric defined by the anticommutators of the modified gamma matrices. Whether this additional condition is consistent with field equations for the preferred extremals of Kähler action remains to be seen. Note that the geometry of the space-time sheet depends on momentum and color quantum numbers in accordance with quantum classical correspondence: this is what makes possible entanglement of classical and quantum degrees of freedom essential for quantum measurement theory.
Since it not much point in typing the detailed equations I give a link to a ten page pdf file Octo-twistors and modified Dirac equation representing the calculations. For details and background see the chapter Does the modified Dirac action define the fundamental action principle?.
After 32 years of hard work it is finally possible to proudly present the basic equations of quantum TGD. There are two kinds of equations.
Quantum classical correspondence requires a coupling between quantum and classical and this coupling should also give rise to a generalization of quantum measurement theory. The big question mark is how to realize this coupling. Few weeks ago I realized that the addition of a measurement interaction term to the modified Dirac action does the job.
In the previous posting about how the addition of measurement interaction term to the modified Dirac actions solves a handful of problems of quantum TGD I was not yet able to decide the precise form of the measurement interaction. There is however a long list of arguments supporting the identification of the measurement interaction as the one defined by 3-D Chern-Simons term assignable with wormhole throats so that the dynamics in the interior of space-time sheet is not affected. This means that 3-D light-like wormhole throats carry induced spinor field which can be regarded as independent degrees of freedom having the spinors fields at partonic 2-surfaces as sources and acting as 3-D sources for the 4-D induced spinor field. The most general measurement interaction would involve the corresponding coupling also for Kähler action but is not physically motivated. Here are the arguments.
My overall feeling is that TGD is finally a mature physical theory with a clear physical interpretation and precise equations. As I started this business my optimistic belief was that it would be a matter of few years to write the Feynman rules. The continual trial and error process made it soon obvious that standard recipes fail and that deep conceptual problems must be solved before one can even dream about defining S-matrix in TGD framework. This forced a construction of TGD inspired theory of consciousness and vision about quantum biology as a by-product. During last half decade (zero energy ontology, the notion of finite measurement resolution, the hierarchy of Planck constants, bosonic emergence,...) it has become clear how dramatic a generalization of existing ontology and epistemology of physics is needed before it is possible to write the generalized Feynman rules. But it seems that they can be written now!
For details see the new chapter Does modified Dirac action defined the fundamental variational principle?.
Theory building could be compared to pattern recognition or to a solving a crossword puzzle. It is essential to make trials, even if one is aware that they are probably wrong. When stares long enough to the letters which do not quite fit, one suddenly realizes what one particular crossword must actually be and it is soon clear what those other crosswords are. In the following I describe an example in which this analogy is rather concrete. Let us begin by listing the problems.
Each of these problems makes one suspect that something is lacking from the modified Dirac action: there should be a manner to feed information about quantum numbers of the state to the modified Dirac action in turn determining vacuum functional as an exponent Kähler function identified as Kähler action for the preferred extremal assumed to be dictated by by quantum criticality and equivalently by hyper-quaternionicity.
This observation leads to what might be the correct question. Could a general coordinate invariant and Poincare invariant modification of the modified Dirac action consistent with the vacuum degeneracy of Kähler action allow to achieve this information flow somehow? This seems to be possible. In the following I proceed step by step by improving the trial to get the final result.
1. The first guess
The idea is simple: add to the modified Dirac action a source term which is analogous to the Dirac action in M4×CP2.
2. Does one obtain stringy propagator?
Before trying to answer to the question whether one really obtains stringy propagator one must define what one means with "stringy propagator".
The next question is "What do we really need?". Only the information about quantum numbers of quantum state in super-conformal representation at partonic 2-surface must be feeded to the propagator. The minimum of this kind is information about isometry charges: that is conserved four-momentum and color quantum numbers. This observation inspires the third guess. All that is needed is that the eigenvalue of pA belongs to the mass shell defined by Super Virasoro conditions at partonic 2-surface. Same applies to the eigenvalues of color hypercharge and isospin. Let us forget for a moment electro-weak quantum numbers and look what this gives.
3. Should one assume that the source term is almost topological?
Kähler function contains besides real part also imaginary part which does not however contribute to the configuration space metric since it is induced by instanton term assignable to Kähler action and corresponding modified Dirac action. The CP breaking term is unavoidable in the previous scenario and is expected to relate to the small CP breaking of particle physics and to the generation of matter antimatter asymmetry. It is not completely clear what the situation is in the recent case.
A careful consideration of the CP breaking effects predicted by various options should make it possible to make a unique choice.
4. The definition of Dirac determinant and the additional term in Kähler action
The modification forces also to reconsider the definition of the Dirac determinant.
5. A connection with quantum measurement theory
It is encouraging that isometry charges and also other charges could make themselves visible in the geometry of space-time surface as they should by quantum classical correspondence. This suggests the interpretation in terms of quantum measurement theory.
6. New view about gravitational mass and matter antimatter asymmetry
The physical interpretation of the additional term in modified Dirac action forces quite a radical revision of the ideas about matter and antimatter.
For background and more reader friendly formulas see the section "Handful of problems with a common solution" of the new chapter Does the modified Dirac action define the fundamental variational principle?.
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.