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. S_int= Σ_AQ_A∫Ψbar g^AB j_Bα Γ^αΨ g^1/2d⁴x ,

   g_AB= j_A^kh_klj_B^l ,

   g^ABg_BC=δ^A_C ,

   j_Bα=j_B^kh_kl∂_αh^l.

   The gamma matrices in question are modified gamma matrices defined by Kähler action with possible instanton term included. The sum is over isometry charges Q_A interpreted as quantal charges and j^Ak denotes the Killing vector field of the isometry. g^AB is the inverse of the tensor g_AB defined by the local inner products of Killing vectors fields in M⁴ and CP₂. The space-time projections of the Killing vector fields j_Bα have interpretation as classical color gauge potentials in the case of SU(3). In M⁴ degrees of freedom j_Bα 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 g_AB is 4 so that g^AB exists only when one considers only four conserved charges. In the case of M⁴ this is achieved by a restriction to translation generators Q_A=p_A. g_AB 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 D_tot= D+ D_int= Γ^αD_α+ Σ_AQ_Ag^AB j_BαΓ^α .

   The conserved fermionic isometry currents are

   J^Aα= Σ_BQ_BΨbar g^BC j_C^kh_klj_A^lΓ^αΨ

   =Q_AΨbar Γ^αΨ .

   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 Q_Aγ^A involving sum over momenta and color charges analogous to p_Aγ^A term in super generator G₀ and the modified Dirac operator D=Γ^αD_α corresponds to the analog of super-Kac Moody contribution. Here Γ^α 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 n^<1/a. 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 g_AB 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 p_A 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 p_Aγ^A 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 G_n 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+D_int 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. D_int 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

   L_int= C(m²,I₃,Y) Q_Ag^ABj_Bα (J^α_K+iJ^α_I)(g₄)^1/2

   satisfies this condition. The coefficient C(m²,I₃,Y) can depend on mass and color charges. J^α_K and J^α_I 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.

      J^α=Ψbar OΓ^αΨ ,

      where O belongs to the set {1,J== J_klΣ^kl,Σ_AB, Σ_ABJ} .

      Here J_kl is the covariantly constant CP₂ Kähler form and Σ_AB 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 O_em=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 Q_A 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, the Kähler metric remains the same. The 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 p_A∂_αm^A 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 chapter Construction of Quantum Theory: Symmetries of the book "Towards M-matrix".