I dare claim that this endless cleaning is not a mere exotic form of cleaning neurosis. My working style is that of a light-hearted jazz musician and this produces a lot of stuff which does not represent eternal truths, and it is better to throw away this stuff away in order to not totally confuse the potential reader (in the beginning of the cleaning operation and seeing what has happened in my household I really hope that no such reader exists! When everything shines again I hope that that my friend might exist after all!.
The progress has been especially jazzy during last five years as several new visions about what TGD might be have seen the daylight. Mention only zero energy ontology, the notion of finite measurement resolution, the role of hyper-finite factors of type II1, the hierarchy of Planck constants, the construction of configuration space geometry in terms of second quantized induced spinor fields, number theoretic compactification,...). These ideas are now converging to an overall view in which various approaches to quantum TGD (physics as infinite dimensional geometry, physics as generalized number theory, physics from number theoretical universality, physics from finite measurement resolution implying effective discretization, TGD as almost topological QFT) neatly fuse together to single coherent overall view.
What is so fine in this cleaning up process that it forces to read all the stuff written during years and critically estimate the internal consistency or lack of it. I will never get rid of the feeling of deep shame than an age old archeological remnant which should have been destroyed for aeons ago creates in me. It is difficult to tolerate the childish enthusiasm of those older copies of me talking what now seems to me total non-sense.
But there is also a reward from all of this pain and trouble. New beautiful connections emerge and arguments and concepts become more precise. It is also wonderful to feel that you really might have something to give to the human kind. It might not be comparable to the Fantasie Impromptu of Chopin, but it is not totally worthless. May be it reduces just to the message that I did my best.
I also learn how incredibly tortuous the path to truth is and that it is good for ego to learn how fragile the most convincing looking argument is, how many different variants it can evolve to depending on what one means with basic concepts, and that the most difficult part in science is finding the correct interpretation. Without it you cannot write the rules. Some of us have a really good luck, and are able to do it during their lifetime and become heros. They can be however be sure that practically no one bothers to go through the same difficult path to really understand the origin of the rules.
In any case, all this work has not been in vain. I feel that I have good justifications for saying that Quantum TGD is a wonderful child full of vigor and energy and exists more and more intensively also as a mathematical theory. In the following I try to sum up some highlights about what has happened during last months. I hope that I find time to write something also the What's New sections of seven books about quantum TGD.
Number theoretical compactification as a bottle neck notion
The detailed formulation of the notion of "number theoretic compactification" (or M8H duality) stating that TGD allows equivalent formulations in terms of 4-surfaces of 8-D Minkowski space M8 (hyper-octonions) and H=M4× CP2 is responsible for everything else that has taken place during last months.
Number theoretical compactification makes strong predictions about the structure of preferred extremals of Kähler action consistent with the known extremals. The slicing of preferred extremals by stringy world sheets and their partonic duals is the basic prediction so that dimensional reduction gives string model type theory. A related prediction is a slicing by light-like 3-surfaces parallel to the fundamental light-like 3-surface X3l at which the signature of the induced metric changes: X3l carries elementary particle quantum numbers.
Finite measurement resolution replacing effectively light-like 3-surfaces with braids replaces space-time surfaces with collections of string world sheets. Note that the strings connecting braid points at partonic 2-surfaces are like strings connecting branes. The string model in question however differs in many respects from string model and the string tension -essentially density of Kähler action per unit length - does not equal to the inverse of gravitational constant.
Construction of configuration space geometry and spinor structure in terms of second quantized induced spinor fields
Thanks to the input from number theoretical compactification, the construction of the configuration space geometry and spinor structure in terms of second quantized induced spinor fields is now relatively well understood. Second quantization and configuration space geometry are in very intimate relationship and explicit formulas for configuration space Kähler function can be written. Even an explicit formula for Kähler coupling strength revealing its number theoretic anatomy is possible.
- The key idea is that the Dirac determinant for the modified Dirac operator defined assignable to some action defining the dynamics of space-time surface or perhaps 3-surface. The replacement of induced gamma matrices with modified gamma matrices guarantees super-conformal symmetry. The basic question is "Which action?". For five years ago I would have answered "Kähler action" without hesitation but one of the basic blunders of the last years was the attempt to reduce the entire physics to Chern-Simons action for induced Kähler gauge potential. The motivation was TGD as an almost topological QFT formulated in terms of modified Dirac action associated with C-S action and localized to the light-like 3-surfaces. Step by step I realized that the correct formulation must involve the modified Dirac operator associated with Kähler action, which indeed allows also the almost topological QFT formulation in terms of holography for preferred extremals.
- For a moment I thought that Kähler action is enough. It however seems that it must be complexified by adding imaginary instanton term for a preferred extremal and defines exponent of Kähler function with a phase defined by instanton term added. By its topological character instanton term does not induce imaginary part to the Kähler metric but induces Chern-Simons action at light-like 3-surfaces representing particles. This would realize the long sought CP breaking at the fundamental level explaing matter antimatter asymmetry and hadronic CP breaking in TGD Universe. Also time reversal asymmetry is implied and becomes quite explicit in the sketched generalized Feynman rules.
- If instanton term is absent there is only finite number of eigenmodes and the physical interpretation of the situation is very transparent: fermions in electroweak magnetic fields with regions in which induced Kähler form is non-vanishing forming natural units allowing finite number of analogs of cyclotron stats. Second quantization allows to satisfy anticommutation at only finite number of points of partonic 2-surface so that the notion of braid as a correlate for finite measurement resolution would emerge automatically. Dirac determinant reduces to a product of finite number of generalized eigenvalues and everything is nice. This picture is especially attractive from the point of view of number theoretical universality.
- If instanton term is allowed, infinite number of conformal excitations assignable to the strings connecting braid points are possible. In this case the Dirac determinant can be defined by standard zeta function regularization reducing to that for Riemann Zeta but it is questionable whether this option is number theoretically universal. It is not yet clear whether one must allow conformal excitations in the definition of Dirac determinant or not or whether the two definitions might give rise to same configuration space metric (but not same Kähler function since a real part of a holomorphic function of configuration space coordinates can distinguish between them!). More generally, the independence on conformal cutoff would have interpretation as renormalization group invariance of the configuration space metric.
- The generalized eigenvalues of the modified Dirac operator relate closely to Higgs mechanism. It however turned out that Higgs vacuum expectation does not cause massivation of gauge bosons. Rather the Higgs expectation value in boson state is expressible in terms of this kind of eigenvalues for gauge boson giving directly the ground state contribution to the mass of fermion (gauge boson is bound state of fermion and antifermion at the opposite throats of a wormhole contact). The ground state contribution to fermion mass would be small since p-adic thermodynamic contribution from the conformal excitations would dominate over the small ground state contribution. This also leads to a formula of Weinberg angle in terms of the generalized eigenvalues. Quite generally, the view about what causes what in particle massivation is drastically modified.
Space-time correlate of quantum criticality and the identification of preferred extremals
The geometric properties of preferred extremals are fixed to a high degree by number theoretic compactification. This is not quite enough. A good candidate for the additional field equations satisfied by the preferred extremals of Kähler action is revealed by the study of the modified Dirac equation (this result could have been deduced for more than decade ago!).
- The Noether currents associated with Kähler action involve modified gamma matrices, which are contractions of the vector field associated with the first variation of Kähler action with ordinary gamma matrices. These currents are conserved only if the second variation of Kähler action vanishes. This is quite a strong condition. It is satisfied trivially by vacuum extremals but might be too strong for general extremals.
- A weaker condition is that only the second variations associated with the dynamical symmetries vanish. This would give a hierarchy of criticalities beginning from that for vacuum extremals in which case all second variations vanish identically. Thus field equations alone would imply the basic vision that TGD Universe is a Universe at the edge: it would not be needed as an additional postulate. A generalization of Thom's catastrophe theory would be in question: systems would live only at the edges of catastrophe graph defined by the V shaped boundary of cusp in the simplest situation.
- This at-the-edge property has also several other aspects. There would be also criticality with respect to phase transitions changing Planck constant very important in TGD inspired quantum biology. As also the number theoretical criticality with respect to quantum jumps transforming p-adic and real space-time sheets to each other and assigned with the formation of cognitive representations and realization of intentional actions in TGD inspired theory of consciousness. Number theoretical would be distinct from number theoretical universality. Only those surfaces whose mathematical representations can be interpret both in terms of real and p-adic numbers would be analogous to rationals common to all number fields and would represent number theoretical criticality.
Finite measurement resolution and number theoretic braids
Finite measurement resolution has the notion of number theoretic braid as a space-time correlate. This concept is now rather well-understood.
- The basic assumption is that the braids must be definable in purely physical terms: one cannot pick up braid points just randomly as a mathematician armed with selection axiom would do. For instance, braid points could be identified as points of partonic two-surface at which induced Kähler field strength has extremum, as intersections of M4 and CP2 projections with with 2-D critical manifolds associated with the criticality with respect to the phase transition changing Planck constant. There is also a much more general definition inspired by the hierarchy of symplectic triangulations which can be realized in terms of quantization of Kähler magnetic fluxes and extrema of induced Kähler field strength. What is the precise rule characterizing allowed rules defining braids is not quite clear yet. This definition would allow an infinite hierarchy of conformal cutoffs in terms of symplectic triangulations with the resulting cutoff conformal algebras realized in terms of finite number of fermionic oscillator operators assignable to the braid points.
- Finite measurement resolution reduces the light-like 3-surfaces to braids and space-time surfaces to strings and the infinite-dimensional world of classical worlds reduces to a finite-dimensional space (δ M4+/-× CP2)n/Sn consisting of n braid points at partonic 2-surface. In the similar manner the space of configuration space spinor fields modulo finite measurement resolution reduces to a finite-dimensional space. This means enormous simplification at the mathematical level. There is a strong temptation to believe that the Clifford algebra in question can be regarded as a coset space of infinite-dimensional hyper-finite factors of type II1 N and M, where N subset M defines the measurement resolution, and that this algebra could be regarded as a quantum Clifford algebra for the nonstandard values of Planck constant.
Super-conformal symmetries and the structure of the world of classical worlds
The understanding of super-conformal symmetries is now much more detailed than before and I have deleted an impressive collection of wrong and not-even-wrong statements.
- It seems now clear that the coset construction for super-symplectic Virasoro algebra and Kac-Moody algebra realizes Equivalence Principle at quantum level. The space-time correlate for Equivalence Principle follows from the stringy picture. General Relativistic form of Equivalence Principle holds only in long length scales, not for a cosmic string like objects for instance. This resolves the basic poorly understood issues which have plagued the understanding of GRT-TGD correspondence and allows to throw away a lot of trash.
- The understanding of the detailed structure of the configuration space has improved considerably. Configuration space is union of symmetric spaces over zero modes identified as coset spaces and the challenge is to understand what this statement might mean.
- The values of the induced Kähler field strength for the partonic 2-surface defines the most important zero modes meaning that dynamics of induced Kähler field is completely classical. Coset construction has its counterpart at the level of configuration space as a union of coset spaces. The symmetric space associated with a given induced Kähler form correspond to the orbit of a symplectic group.
- Symplectic group can be made local with respect to the partonic 2-surface - or rather with the coordinate defined by the value of induced Kähler field strength taking the role of complex coordinate in conformal field theories. Kac-Moody sub-algebra defined at light-like 3-surface, whose elements vanish at the partonic 2-surfaces defining its ends, acts as a gauge algebra defining further zero modes.
- Quantum fluctuating degrees of freedom correspond to the coset space defined by the symplectic algebra and by the sub-Kac-Moody algebra. Note that the entire Kac-Moody algebra appears in the coset construction and p-adic mass calculations whereas only the sub-algebra appeas in the coset space construction.
- The identification of induced Kähler form of X2 as purely classical field means that configuration space functional integral is only over the fluctuations contributing to the induced metric metric of X2. Therefore at the configuration space level the only quantum fluctuating degrees of freedom are purely gravitational. Besides this present are fermionic degrees of freedom, modular degrees of freedom, other zero modes (Kac-Moody algebra), and topological degrees of freedom.
About the construction of M-matrix
The toughest challenge of TGD has been the construction of TGD counterpart of S-matrix - M-matrix as I call it. The understanding of the generalized Feynman rules is now rather detailed and the notion of finite measurement resolution gives excellent hopes about calculational rules making possible practical calculations.
- The first fundamental element is zero energy ontology allowing to identify M-matrix as time-like entanglement coefficients between positive and negative energy parts of zero energy state (counterpart of physical event) assignable to the light-like boundaries of causal diamond identified as intersection of future and past directed light-cones defining the basic piece of the world of classical worlds. There is entire hierarchy of CDs within CDs and this allow to understand p-adic coupling constant evolution in terms of finite measurement resolution defined by the size of smallest CD included.
- Second basic notion is generalized Feynman diagram identified as light-like 3-surface or equivalently as region of space-time with Euclidian signature of metric accompanying the light-like 3-surface. Euclidian regions would represent particles and Minkowskian regions classical fields. The conformal symmetries and stringy picture implied by the finite measurement resolution suggest strongly stringy Feynman rules.
- A very powerful form of General Coordinate Invariance would be the condition that one can deduce configuration space metric by using any light-like 3-surface in the slicing of space-time surface to light-like 3-surfaces parallel to the surface X3l at which the signature of the induced metric changes. Invariance would not mean invariance of Kähler function but only that of Kähler metric. This condition should pose extremely powerful constraints on the form of various expressions appearing in generalized Feynman diagrammatics.
- Vertices correspond to partonic 2-surfaces and n-points functions of an N=4 conformal field theory in which second quantized induced spinor fields are the fundamental fields. The TGD based interpretation of N=4 for algebra is now well-understood and reflects directly the basic symmetries of TGD. Discretization implied by the number theoretic braids implies a huge simplification of the situation and mean stringy theory at space-time level.
- Propagators assigned with light-like 3-surfaces connecting vertices should be stringy. The problem is how to obtain conformal excitations propagating along strings connecting braid points as zero modes of the modified Dirac operator. These excitations with non-vanishing conformal weight necessarily break the effective 2-dimensionality of 3-surfaces and thus holography. In the proposed - and yet admittedly speculative - picture about the properties of preferred extremals the only possible manner to obtain this breaking seems to be complexification of Kähler action by adding to it as imaginary part the CP breaking instanton action. The "only" in the preceding sentence should be taken with a grain of salt since the implications of number theoretical compactification for the geometry of preferred extremals are not completely understood.
- Besides implying CP breaking and the breaking of time reversal symmetry, the instanton term would break the effective 2-dimensionality of 3-surfaces (holography) and would give rise to stringy propagation of fermions whereas at the space-time level effective 2-dimensionality seem to prevail apart from the non-determinism of Kähler action. One can speak of a radiative generation of kinetic and mass terms in stringy propagator. The classical non-determinism of Kähler action would be responsible for generating the analogs of self energy vertices and break the effective 2-dimensionality of 3-surfaces. This conforms with what one might expect. Note that only the conformal excitations of induced spinor field would break the exact holography.