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TGD: Physics as Infinite-Dimensional Geometry

Note: Newest contributions are at the top!



Year 2009



Octonionic approach to the modified Dirac equation

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?.



What are the basic equations of quantum TGD?

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.

  1. Purely classical equations define the dynamics of space-time sheets as preferred extremals of Kähler action. Preferred extremals are quantum critical in the sense that second variation vanishes for critical deformations. They can be also regarded as hyper-quaternionic surfaces. What these statements precisely mean has become clear during this year.

  2. The purely quantal equations are associated with the representations of various super-conformal algebras and with the modified Dirac equation. The requirement that there are deformations of the space-time surface -actually infinite number of them- giving rise to conserved fermionic charges implies quantum criticality at the level of Kähler action in the sense of critical deformations. The precise form of the modified Dirac equation is not however completely fixed without a further input.

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.

  1. A correlation between 4-D geometry of space-time sheet and quantum numbers is achieved by the identification of exponent of Kähler function as Dirac determinant making possible the entanglement of classical degrees of freedom in the interior of space-time sheet with quantum numbers.

  2. Cartan algebra plays a key role not only in quantum level but also at the level of space-time geometry since quantum critical conserved currents vanish for Cartan algebra of isometries and the measurement interaction terms giving rise to conserved currents are possible only for Cartan algebras. Furthermore, modified Dirac equation makes sense only for eigen states of Cartan algebra generators. The hierarchy of Planck constants realized in terms of the book like structure of the generalized imbedding space assigns to each CD preferred Cartan algebra: in case of Poincare algebra there are two of them corresponding to linear and cylindrical M4 coordinates.

  3. Quantum holography and dimensional reduction hierarchy in which partonic 2-surface defined fermionic sources for 3-D fermionic fields at light-like 3-surfaces Y3l in turn defining fermionic sources for 4-D spinors find an elegant realization. Effective 2-dimensionality is achieved if the replacement of light-like wormhole throat X3l with light-like 3-surface Y3l "parallel" with it in the definition of Dirac determinant corresponds to the U(1) gauge transformation K→ K+f+f* for Kähler function of WCW ("world of classical worlds") so that WCW Kähler metric is not affected. Here is arbitrary holomorphic function of WCW complex coordinates and zero modes.

  4. An elegant description of the interaction between super-conformal representations realized at partonic 2-surfaces and dynamics of space-time surfaces is achieved since the values of Cartan charges are feeded to the 3-D Dirac equation which also receives mass term at the same time. Almost topological QFT at wormhole throats results at the limit when four-momenta vanish: this is in accordance with the original vision.

  5. A detailed view about the physical role of quantum criticality results. Quantum criticality fixes the values of Kähler coupling strength as the analog of critical temperature. Quantum criticality implies that second variation of Kähler action vanishes for critical deformations and the existence of conserved current except in the case of Cartan algebra of isometries. Quantum criticality allows to fix the values of couplings (gravitational coupling, gauge couplings, etc..) appearing in the measurement interaction by using the condition K→ K+f+f*. p-Adic coupling constant evolution can be understood also and corresponds to scale hierarchy for sizes of causal diamonds (CDs).

  6. CP breaking, irreversibility, and the space-time description of dissipation are closely related. What is interesting that dissipation does not make itself visible at the level of configuration space metric since it only induces the gauge transformation K→ K+f+f*. Space-time sheet is however affected. Also the interpretation of preferred extremals of Kähler action in regions where DC-S=0 holds true as asymptotic self organization patterns makes sense. Here DC-S denotes the 3-D modified Dirac operator associated with Chern-Simons action and DC-S,int to the corresponding measurement interaction term expressible as superposition of couplings to various observables to critical conserved currents.

  7. A radically new view about matter antimatter asymmetry based on zero energy ontology emerges and one could understand the experimental absence of antimatter as being due to the fact antimatter corresponds to negative energy states. The identification of bosons as wormhole contacts is the only possible option in this framework.

  8. Almost stringy propagators and a consistency with the identification of wormhole throats as lines of generalized Feynman diagrams is achieved. The notion of bosonic emergence leads to a long sought general master formula for the M-matrix elements. The counterpart for fermionic loop defining bosonic inverse propagator at QFT limit is wormhole contact with fermion and cutoffs in mass squared and hyperbolic angle for loop momenta of fermion and antifermion in the rest system of emitting boson have a precise geometric counterpart in the fundamental theory.

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?.



Handful of problems with a common resolution

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.

  1. The condition that modified Dirac action allows conserved charges leads to the condition that the symmetries in question give rise to vanishing second variations of Kähler action. The interpretation is as quantum criticality and there are good arguments suggesting that the critical symmetries define an infinite-dimensional super-conformal algebra forming an inclusion hierarchy related to a sequence of symmetry breakings closely related to a hierarchy of inclusions of hyper-finite factors of types II1 and III1. This means an enormous generalization of the symmetry breaking patterns of gauge theories.

    There is however a problem. For the translations of M4 the resulting fermionic charges vanish. The trial for the crossword in absence of nothing better would be the following argument. By the abelianity of these charges the vanishing of quantal representation of four-momentum is not a problem and that classical representation for four-momentum or the representation coming from Super-Virasoro representations is enough.

  2. Irrespective of whether the 4-D modified Dirac action or its 3-dimensional dimensional reduction defines the propagator, it seems impossible to obtain a stringy propagator without adding it as a kind of mass insertion. A second trial for a crossword which does not look very convincing. This is certainly a problem at the level of formalism since stringy picture follows in finite measurement resolution from the slicing of space-time sheets with string world sheets.

  3. Quantum classical correspondence requires that the geometry of the space-time sheet should correlate with the quantum numbers characterizing positive (negative) energy part of the quantum state. One could argue that by multiplying WCW spinor field by a suitable phase factor depending on charges of the state, the correspondence follows from stationary phase approximation. Also this crossword looks unsatisfactory.

  4. In quantum measurement theory classical macroscopic variables identified as degrees of freedom assignable to the interior of the space-time sheet correlate with quantum numbers. Stern Gerlach experiment is an excellent example of the situation. The generalization of the imbedding space concept by replacing it with a book like structure implies that imbedding space geometry at given page and for given causal diamond (CD) carries information about the choice of the quantization axes (preferred plane M2 of M4 resp. geodesic sphere of CP2 associated with singular covering/factor space of CD resp. CP2 ). This is a big step but not enough. Modified Dirac action as such does not seem to provide any hint about how to achieve this correspondence. One could even wonder whether dissipative processes characterizing the outcome of quantum jump sequence should have space-time correlate. How to achieve this? There are no guesses for the crosswords here.

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.

  1. The additional term would be essentially the analog of ordinary Dirac action at the imbedding space level. Sint= ΣAQA∫Ψbar gAB j ΓαΨ g1/2d4x ,

    gAB= jAkhkljBl ,

    gABgBCAC ,

    j=jBkhklαhl.

    The gamma matrices in question are modified gamma matrices defined by Kähler action with possible instanton term included. The sum is over isometry charges QA interpreted as quantal charges and jAk denotes the Killing vector field of the isometry. gAB is the inverse of the tensor gAB defined by the local inner products of Killing vectors fields in M4 and CP2. The space-time projections of the Killing vector fields j have interpretation as classical color gauge potentials in the case of SU(3). In M4 degrees of freedom j reduce to the gradients of linear M4 coordinates in case of translations.

  2. An important restriction is that by four-dimensionality of M4 and CP2 the rank of gAB is 4 so that gAB exists only when one considers only four conserved charges. In the case of M4 this is achieved by a restriction to translation generators QA=pA. gAB reduces to Minkowski metric and Killing vector fields are constants. The Cartan sub-algebra could be however replaced by any four commuting charges in the case of Poincare algebra. In the case of SU(3) one must restrict the consideration either to U(2) sub-algebra or its complement. CP2=SU(3)/SU(2) decomposition would suggest the complement as the correct choice. One can indeed build the generators of U(2) as commutators of the charges in the complement.

  3. The added term containing quantal charges must make sense in the modified Dirac equation. This requires that the physical state is an eigenstate of momentum and color charges. This allows only color hyper-charge and color isospin so that there is no hope of obtaining exactly the stringy formula for the propagator. The modified Dirac operator is given by Dtot= D+ Dint= ΓαDα+ ΣAQAgAB jΓα .

    The conserved fermionic isometry currents are

    J= ΣBQBΨbar gBC jCkhkljAlΓαΨ

    =QAΨbar ΓαΨ .

    Here the sum is restricted to a Cartan sub-algebra of Poincare group and color group.

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".

  1. The first guess would be that the added term corresponds to QAγA involving sum over momenta and color charges analogous to pAγA term in super generator G0 and the modified Dirac operator D=ΓαDα corresponds to the analog of super-Kac Moody contribution. Here Γα denotes modified gamma matrix defined by Kähler action. I have considered this option earlier and the detailed analysis shows that the generalized eigenvalues of the 3-D modified Dirac operator should behave like n1/2. This ad hoc assumption does not make this option convincing.

  2. Could one consider a generalization of the additional term to include also charges associated with Super Kac-Moody algebra acting on light-like 3-surfaces? The first problem is that the matrix gAB is invertible only for four vector fields so that one should give up the assumption that charges are conserved. Second problem is that super generators carry fermion number and it seems impossible to define bosonic counterparts for them.

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.

  1. The modified Dirac operator D would take the role of pAγA which looks quite a reasonable generalization and that added term carries information about the momentum and color quantum numbers.

  2. One can avoid the difficulties due to the fact that Gn carry fermion number and just the relevant information about states of Super Virasoro representation is feeded to the modes and spectrum of the modified Dirac equation and to the classical space-time physics defined by the exponent of Kähler action which must receive an additional term coupling it to isometry charges.

  3. The modified Dirac operator D+Dint would annihilate the spinor modes in the interior of the space-time surface expect at the light-like 3-surfaces or partonic 2-surfaces at the ends of light-like 3-surface serving as sources. This gives to the induced spinor field additional terms expressible in terms of the stringy propagator. The propagator would not have exactly stringy character - in particular, only the color hyper charge and isospin appear in it- but there is no absolute need for this. What is essential is that the information about mass and color quantum numbers of the state of super-conformal representation is feeded into the space-time physics.

  4. Dint represents also a mass term in the modified Dirac equation so that particle massivation has a space-time correlate. For instance, the mass calculated by p-adic thermodynamics makes itself visible at the level of classical physics.

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.

  1. The most general option is that the modified gamma matrices appearing in the added term could correspond to a sum of modified gamma matrices assignable to Kähler action and its instanton counterpart.

  2. One can also consider the analog Chern-Simons term with 3-D modified gamma matrices defined by Chern-Simons action and assigned to the light-like wormhole throats at which the induced metric changes its signature from Euclidian to Minkowskian. Wormhole throats define the lines of generalized Feynman diagrams so that the assignment of 3-D stringy propagator with them looks sensible and conforms with quantum holography. Instanton action reduces to Chern-Simons action assignable to wormhole throats but it is not clear whether the instanton term in Dirac action and its counterpart involving coupling to isometry charges are subject to a similar reduction.

    There is support for Chern-Simons option. In the case of Kähler action the dimensional reduction of the modified Dirac operator at wormhole throats is problematic because the determinant of the induced 4-metric vanishes: the dimensional reduction of D to D3 can be defined only through a limiting procedure (this is however nothing unheard-of: in AdS/CFT correspondence similar situation is encountered). For Chern-Simons action situation is different and it defines modified gamma matrices and couplings to isometry charges are well-defined.

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.

  1. The earlier definition was based on the slicing of space-time sheets by 3-D light-like surfaces and dimensional reduction to 3-D Dirac operator D3 with Dirac determinant identified as a product of generalized eigenvalues of D3. This definition generalizes to the recent context and implies that instead of massless particle one has massive particle carrying also other quantum numbers.

  2. The interaction term induced to Kähler action should be consistent with vacuum degeneracy of Kähler action. The interaction term of form

    Lint= C(m2,I3,Y) QAgABj (JαK+iJαI)(g4)1/2

    satisfies this condition. The coefficient C(m2,I3,Y) can depend on mass and color charges. JαK and JαI denote Kähler current and instanton current respectively. 3-D Chern-Simons term is equivalent with instanton term.

    This term is not the most general possible. One can add also couplings to conserved isometry currents as well as to currents whose existence is guaranteed by quantum criticality. For these currents only the covariant divergence vanishes. This would support the interpretation in terms of a measurement interaction feeding information to classical space-time physics about the eigenvalues of the observables of the measured system. The resulting field equations remained second order partial differential equations since the second order partial derivatives appear only linearly in the added terms.

  3. The CP breaking term in the modified Dirac equation means a breaking of time reflection symmetry at the level of fundamental physics. The vision is that the classical non-determinism of Kähler action allows to have space-time correlates for quantum jumps sequences and therefore also for dissipation. This motivates the question whether the CP breaking term could give rise to dissipative effects allowing description in terms of the coupling of the conserved charges to Kähler current and to conserved isometry currents.

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.

  1. The interpretation resolves the problem caused by the fact that the choice of the commuting isometry charges is not unique. Cartan algebra corresponds naturally to the measured observables. For instance, one could choose the Cartan algebra of Poincare group to consist of energy and momentum, angular momentum and boost (velocity) in particular direction as generators of the Cartan algebra of Poincare group. In fact, the choices of a preferred plane M2 subset M4 and geodesic sphere S2 subset CP2 allowing to fix the measurement sub-algebra to a high degree are implied by the replacement of the imbedding space with a book like structure forced by the hierarchy of Planck constants. Therefore the hierarchy of Planck constants seems to be required by quantum measurement theory. One cannot overemphasize the importance of this connection.

  2. What about the space-time correlates of electro-weak charges? The earlier proposal explains this correlation in terms of the properties of quantum states: the coupling of electro-weak charges to Chern-Simons term could give the correlation in stationary phase approximation. It would be however very strange if the coupling of electro-weak charges with the geometry of the space-time sheet would not have the same universal description based on quantum measurement theory as isometry charges have.

    1. The hint as how this description could be achieved comes from a long standing un-answered question motivated by the fact that electro-weak gauge group identifiable as the holonomy group of CP2 can be identified as U(2) subgroup of color group. Could the electro-weak charges be identified as classical color charges? This might make sense since the color charges have also identification as fermionic charges implied by quantum criticality. Could electro-weak charges be only represented as classical color charges by mapping them to classical color currents in the measurement interaction term in the modified Dirac action? At least this question might make sense.

    2. It does not however make sense to couple both electro-weak and color charges to the same fermion current. There are also other fundamental fermion currents which are conserved. All the following currents are conserved.

      Jα=Ψbar OΓαΨ ,

      where O belongs to the set {1,J== JklΣklAB, ΣABJ} .

      Here Jkl is the covariantly constant CP2 Kähler form and ΣAB is the (also covariantly) constant sigma matrix of M4 (flatness is absolutely essential).

    3. Electromagnetic charge can be expressed as a linear combination of currents corresponding to O=1 and O=J and vectorial isospin current corresponds to J. It is natural to couple of electromagnetic charge to the the projection of Killing vector field of color hyper charge and coupling it to the current defined by Oem=a+bJ. This allows to interpret the puzzling finding that electromagnetic charge can be identified as anomalous color hyper-charge for induced spinor fields made already during the first years of TGD. There exist no conserved axial isospin currents in accordance with CVC and PCAC hypothesis which belong to the basic stuff of the hadron physics of old days.

    4. There is also an infinite variety of conserved currents obtained as the quantum critical deformations of the basic fermion currents identified above. This would allow in principle to couple an arbitrary number of observables to the geometry of the space-time sheet by mapping them to Cartan algebras of Poincare and color group for a particular conserved quantum critical current. Quantum criticality would therefore make possible classical space-time correlates of observables necessary for quantum measurement theory.

    5. Note that various coupling constants would appear in the couplings. Quantum criticality should determine the spectrum of these couplings.

  3. Quantum criticality implies fluctuations in long length and time scales and it is not surprising that quantum criticality is needed to produce a correlation between quantal degrees of freedom and macroscopic degrees of freedom. Note that quantum classical correspondence can be regarded as an abstract form of entanglement induced by the entanglement between quantum charges QA and fermion number type charges assignable to zero modes.

  4. Space-time sheets can have several wormhole contacts so that the interpretation in terms of measurement theory coupling short and long length scales suggests that the measurement interaction terms are localizable at the wormhole throats. This would favor Chern-Simons term or possibly instanton term if reducible to Chern-Simons terms. The breaking of CP and T might relate to the fact that state function reductions performed in quantum measurements indeed induce dissipation and breaking of time reversal invariance.

  5. The experimental arrangement quite concretely splits the quantum state to a quantum superposition of space-time sheets such that each eigenstate of the measured observables in the superposition corresponds to different space-time sheet already before the realization of state function reduction. This relates interestingly to the question whether state function reduction really occurs or whether only a branching of wave function defined by WCW spinor field takes place as in multiverse interpretation in which different branches correspond to different observers. TGD inspired theory consciousness requires that state function reduction takes place. Maybe multiversalist might be able to find from this picture support for his own beliefs.

  6. One can argue that "free will" appears not only at the level of quantum jumps but also as the possibility to select the observables appearing in the modified Dirac action dictating in turn the Kähler function defining the Kähler metric of WCW representing the "laws of physics". This need not to be the case. The choice of CD fixes M2 and the geodesic sphere S2: this does not fix completely the choice of the quantization axis but by isometry invariance rotations and color rotations do not affect Kähler function for given CD and for a given type of Cartan algebra. In M4 degrees of freedom the possibility to select the observables in two manners corresponding to linear and spherical Minkowski coordinates could imply that the resulting Kähler functions are different. The corresponding Kähler metrics do not differ if the real parts of the Kähler functions associated with the two choices differ by a term f(Z)+(f(Z))*, where Z denotes complex coordinates of WCW and * complex conjugation, the Kähler metric remains the same. The holomorphic function f can depend also on zero modes. If this is the case then one can allow in given CD superpositions of WCW spinor fields for which the measurement interactions are different. This condition is expected to pose non-trivial constraints on the measurement action and quantize coupling parameters appearing in it.

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.

  1. The term pAαmA contracted with the fermion current is analogous to a gauge potential coupling to fermion number. Since the additional terms in the modified Dirac operator induce stringy propagation, a natural interpretation of the coupling to the induced spinor fields is in terms of gravitation. One might perhaps say that the measurement of four momentum induces gravitational interaction. Besides momentum components also color charges take the role of gravitational charges. As a matter fact, any observable takes this role via coupling to the projections of Killing vector fields of Cartan algebra. The analogy of color interactions with gravitational interactions is indeed one of the oldest ideas in TGD.

  2. One could wonder whether the two terms in the modified Dirac equation be analogous to Einstein tensor and energy momentum tensor in Einstein's equations. Coset construction in which gravitational and inertial four-momenta are replaced by super-symplectic and super Kac-Moody algebras does not support this idea.

  3. The coupling to four-momentum is through fermion number (both quark number and lepton number). For states with a vanishing fermion number isometry charges therefore vanish. In this framework matter antimatter asymmetry would be due to the fact that matter (antimatter) corresponds to positive (negative) energy parts of zero energy states for massive systems so that the contributions to the net gravitational four-momentum are of same sign. Antimatter would be unobservable to us because it resides at negative energy space-time sheets. As a matter fact, I proposed already years ago that gravitational mass is magnitude of the inertial mass but gave up this idea.

  4. Bosons do not couple at all to gravitation if they are purely local bound states of fermion and anti-fermion at the same space-time sheet (say represented by generators of super conformal Kac-Moody algebra). Therefore the only possible identification of gauge bosons is as wormhole contacts. If the fermion and anti-fermion at the opposite throats of the contact correspond to positive and negative energy states the net energy receives a positive contribution from both sheets. If both correspond to positive (negative) energy the contributions to the net four-momentum have opposite signs.

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?.



The recent view about the construction of configuration space spinor structure

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.

  1. One must identify the counterparts of second quantized fermion fields as objects closely related to the configuration space spinor structure. Ramond model has as its basic field the anti-commuting field Gk(x), whose Fourier components are analogous to the gamma matrices of the configuration space and which behaves like a spin 3/2 fermionic field rather than a vector field. This suggests that the complexified gamma matrices of the configuration space are analogous to spin 3/2 fields and therefore expressible in terms of the fermionic oscillator operators so that their anti-commutativity naturally derives from the anti-commutativity of the fermionic oscillator operators.

    As a consequence, configuration space spinor fields can have arbitrary fermion number and there would be hopes of describing the whole physics in terms of configuration space spinor field. Clearly, fermionic oscillator operators would act in degrees of freedom analogous to the spin degrees of freedom of the ordinary spinor and bosonic oscillator operators would act in degrees of freedom analogous to the 'orbital' degrees of freedom of the ordinary spinor field.

  2. The classical theory for the bosonic fields is an essential part of the configuration space geometry. It would be very nice if the classical theory for the spinor fields would be contained in the definition of the configuration space spinor structure somehow. The properties of the modified massless Dirac operator associated with the induced spinor structure are indeed very physical. The modified massless Dirac equation for the induced spinors predicts a separate conservation of baryon and lepton numbers. The differences between quarks and leptons result from the different couplings to the CP2 Kähler potential. In fact, these properties are shared by the solutions of massless Dirac equation of the imbedding space.

  3. Since TGD should have a close relationship to the ordinary quantum field theories it would be highly desirable that the second quantized free induced spinor field would somehow appear in the definition of the configuration space geometry. This is indeed true if the complexified configuration space gamma matrices are linearly related to the oscillator operators associated with the second quantized induced spinor field on the space-time surface and/or its boundaries. There is actually no deep reason forbidding the gamma matrices of the configuration space to be spin half odd-integer objects whereas in the finite-dimensional case this is not possible in general. In fact, in the finite-dimensional case the equivalence of the spinorial and vectorial vielbeins forces the spinor and vector representations of the vielbein group SO(D) to have same dimension and this is possible for D=8-dimensional Euclidian space only. This coincidence might explain the success of 10-dimensional super string models for which the physical degrees of freedom effectively correspond to an 8-dimensional Euclidian space.

  4. It took a long time to realize that the ordinary definition of the gamma matrix algebra in terms of the anti-commutators {gA,gB} = 2gAB must in TGD context be replaced with {gAf,gB} = iJAB\per, where JAB denotes the matrix elements of the Kähler form of the configuration space. The presence of the Hermitian conjugation is necessary because configuration space gamma matrices carry fermion number. This definition is numerically equivalent with the standard one in the complex coordinates. The realization of this delicacy is necessary in order to understand how the square of the configuration space Dirac operator comes out correctly.

  5. The only possible option is that second quantized induced spinor fields are defined at 3-D light-like causal determinants associated with 4-D space-time sheet. The unique partonic dynamics is almost topological QFT defined by Chern-Simons action for the induced Kähler gauge potential and by the modified Dirac action constructed from it by requiring super-conformal symmetry. The resulting theory has all the desired super-conformal symmetries and is exactly solvable at parton level. It is 3-dimensional lightlike 3-surfaces rather than generic 3-surfaces which are the fundamental dynamical objects in this approach.

    The classical dynamics of the interior of space-time surface defines a classical correlate for the partonic quantum dynamics and provides a realization of quantum measurement theory. It is determined by the vacuum functional identified as the Dirac determinant. There are good arguments suggesting that it reduces to an exponent of absolute extremum of Kähler action in each region of the space-time sheet where the Kähler action density has a definite sign.

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.

  1. The Dirac determinant defined by the product of Dirac determinants associated with the light-like partonic 3-surfaces X3l associated with a given space-time sheet X4 is the simplest candidate for vacuum functional identifiable as the exponent of the Kähler function. One can of course worry about the finiteness of the Dirac determinant. p-Adicization requires that the eigenvalues belong to a given algebraic extension of rationals. This restriction would imply a hierarchy of physics corresponding to different extensions and could automatically imply the finiteness and algebraic number property of the Dirac determinants if only finite number of eigenvalues would contribute. The regularization would be performed by physics itself if this were the case.

  2. The basic problem has been how to feed in the information about the preferred extremal of Kähler action to the eigenvalue spectrum of C-S Dirac operator DC-S at light-like 3-surface X3l. The identification of the preferred extremal came possible via boundary conditions at X3l dictated by number theoretical compactification. The basic observation is that the Dirac equation associated with the 4-D Dirac operator DK defined by Kähler action can be seen as a conservation law for a super current. By restricting the super current to flow along X3l by requiring that its normal component vanishes, one obtains a singular solution of 4-D modified Dirac equation restricted to X3l. The ënergy" spectrum for the solutions of DK corresponds to the spectrum of eigenvalues for DC-S and the product of the eigenvalues defines the Dirac determinant in standard manner. Since the eigenmodes are restricted to those localized to regions of non-vanishing induced Kähler form, the number of eigen modes is finite and therefore also Dirac determinant is finite. The eigenvalues can be also algebraic numbers.

  3. It remains to be proven that the product of eigenvalues gives rise to the exponent of Kähler action for the preferred extremal of Kähler action. At this moment the only justification for the conjecture is that this the only thing that one can imagine. The identification of super-symplectic conformal weights as zeros of zeta function defined by the eigenvalues of modified Dirac operator would couple them with the dynamics defined by the Kähler action.

  4. A long-standing conjecture has been that the zeros of Riemann Zeta are somehow relevant for quantum TGD. Rieman zeta is however naturally replaced Dirac zeta defined by the eigenvalues of DC-S and closely related to Riemann Zeta since the spectrum consists essentially for the cyclotron energy spectra for localized solutions region of non-vanishing induced Kähler magnetic field and hence is in good approximation integer valued up to some cutoff integer. In zero energy ontology the Dirac zeta function associated with these eigenvalues defines"square root" of thermodynamics assuming that the energy levels of the system in question are expressible as logarithms of the eigenvalues of the modified Dirac operator defining kind of fundamental constants. Critical points correspond to approximate zeros of Dirac zeta and if Kähler function vanishes at criticality as it ineed should, the thermal energies at critical points are in first order approximation proportional to zeros themselves so that a connection between quantum criticality and approximate zeros of Dirac zeta emerges.

  5. The discretization induced by the number theoretic braids reduces the world of classical worlds to effectively finite-dimensional space and configuration space Clifford algebra reduces to a finite-dimensional algebra. The interpretation is in terms of finite measurement resolution represented in terms of Jones inclusion M subset N of HFFs with M taking the role of complex numbers. The finite-D quantum Clifford algebra spanned by fermionic oscillator operators is identified as a representation for the coset space N/M describing physical states modulo measurement resolution. In the sectors of generalized imbedding space corresponding to non-standard values of Planck constant quantum version of Clifford algebra is in question.

4. Super-conformal symmetries

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.



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