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 Y³_l 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 X³_l with light-like 3-surface Y³_l "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 as asymptotic self organization patterns makes sense. Here D_C-S denotes the 3-D modified Dirac operator associated with Chern-Simons action and D_C-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 section "Does the modified Dirac action define the fundamental action principle?" of the chapter Construction of Quantum Theory: Symmetries of the book "Towards M-matrix".

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".

The latest vision about the role of hyperfinite factors in TGD

I realized of the importance of von Neumann algebras known as hyper-finite factors for more than half decade ago. The mathematics involved is extremely heavy technically and for a physicist at my age and with my brain the only reasonable goal is to understand this notion conceptually and see whether it relates naturally to own visions. Fermionic Fock space finding geometrization in quantum TGD is indeed a canonical representation for HFFs of II₁ having very close relations to quantum groups, topological quantum field theories, statistical mechanics, etc.. so that there are excellent motivations for taking HFFs of various types seriously.

It is clear that at least the hyper-finite factors of type II₁ assignable to WCW (world of classical worlds) spinors must have a profound role in TGD. Whether also HFFS of type III₁ appearing in relativistic quantum field theories emerge when WCW spinors are replaced with spinor fields in WCW is not completely clear. I have proposed several ideas about the role of hyper-finite factors in TGD framework. In particular, inclusions of factors and Connes tensor product provide an excellent candidate for defining the notion of measurement resolution.

The perspective provided by zero energy ontology, the recent advances in the understanding of M-matrix at QFT limit using the notion of bosonic emergence, as well as the more mature view about what these mysterious factors are, motivate a fresh discussion of the subject. There are several question to be considered but before posing these questions it is good to give some references.

1. Basic notions related to factors

It would take too much text to explain the basic ideas and facts of factors so that I give links to references that I have used with comments.

1. There is Wikipedia article about von Neumann algebras. Hyper-finite factors of type II₁, II_∞ and type III₁ are of special importance from TGD point of view. HFFs of type III₁ are encountered also in quantum field theories.

2. The slides of Longo with title Operator Algebras and Index Theorems in Quantum Field Theory should be useful. In particular, it explains Tomita-Takesaki formula, which turned out to have very nice interpretation in zero energy ontology. The ordinary fermionic Fock space does not satisfy the conditions of the theorem (existence of cyclic and separable state) but the space formed by zero energy states naturally does so. Positive and negative energy parts of zero energy states for which vacua are annihilated by creation resp. annihilation operators corresponds to the decomposition of the space of bounded operators of Hilbert space to a "vee" product of factors and its commutant generalizing the tensor product in the description of mutually non-interacting systems.

3. The article Von Neumann algebra automorphisms and time- thermodynamics relation in general covariant quantum theories of Connes and Rovelli should give ideas about physical interpretation. The insights given by article allowed to realize that the hope about the identification of the modular automorphism Δ^it as M-matrix for a complex value of t is not realistic.

4. The article The role of Type III Factors in Quantum Field Theory by Yngvason gives a good idea about their role in relativistic QFT and helped to realize that WCW local Clifford algebra and WCW spinor fields from fixed causal diamond define naturally HFF of type II_∞ since isotropies correspond to group SO(3) leaving the tips of CD invariant. The extension to the union of WCWs associated with Lorentz boosts of CD would give naturally rise to HFF of type III₁.

5. Crossed product is an important concept which I have not applied earlier in TGD framework. Essentially semi-direct product of algebra with group is in question and under some assumptions gives rise to factor. Now the algebra is that of WCW spinor fields in CD and group Lorentz group transforming CDs. Crossed product construction leads to an understanding of factors in quantum TGD and it seems that the relevant factors in TGD framework are HFFs of type III₁. The minimal choices for the group in question is Lorentz group and its non-compactness implies HFF of type III₁ as a result. Translations acting on either tip of CD induce Lorentz transformation so that they induce also as modular automorphisms.

6. KMS The notion of KMS state is important and gives a connection between thermodynamics and factors.

7. A strongly TGD colored brief summary about basics of von Neumann algebras and the recent view about M-matrix contra HFFS can be found in the section "The latest vision about the role of HFFs in TGD" of the chapter Construction of Quantum Theory: M-matrix of "Towards S-matrix".

2. Conceptual problems

It is safest to start from the conceptual problems and take a role of skeptic.

1. Under what conditions the assumptions of Tomita-Takesaki formula stating the existence of modular automorphism and isomorphy of the factor and its commutant hold true? What is the physical interpretation of the formula M^¢=JMJ relating factor and its commutant in TGD framework? The answer turned out to be that positive and zero energy parts of the zero energy state relate in this manner and the transformation involves also time reflection with respect to the center of CD in rest frame of CD inducing transformation of light-like 3-surfaces and space-time surfaces to the time-mirrored counterparts.

2. Is the identification M=Δ^it sensible is quantum TGD and zero energy ontology, where M-matrix is "complex square root" of exponent of Hamiltonian defining thermodynamical state and the notion of unitary time evolution is given up? The notion of state ω leading to Δ is essentially thermodynamical and one can wonder whether one should take also a "complex square root" of ω to get M-matrix giving rise to a genuine quantum theory.

3. TGD based quantum measurement theory involves both quantum fluctuating degrees of freedom assignable to light-like 3-surfaces and zero modes identifiable as classical degrees of freedom assignable to interior of the space-time sheet. Zero modes have also fermionic counterparts. State preparation should generate entanglement between the quantal and classical states. What this means at the level of von Neumann algebras?

4. What is the TGD counterpart for causal disjointness. At space-time level different space-time sheets could correspond to such regions whereas at imbedding space level causally disjoint CDs would represent such regions.

3. Technical problems

There are also more technical questions.

1. What is the von Neumann algebra needed in TGD framework? Does one have a a direct integral over factors (at least direct integral over zero modes)? Which factors appear in it? Can one construct the factor as a crossed product of some group G with direct physical interpretation (say Lorentz group affecting CD) and of naturally appearing factor A? Is A a HFF of type II_∞ assignable to a fixed CD? What is the natural Hilbert space H in which A acts?

2. What are the geometric transformations induced modular automorphisms of II_∞ inducing the scaling down of the trace? Is the action of G induced by the boosts in Lorentz group. Could also translations and scalings induce the action? What is the factor associated with the union of Poincare transforms of CD? log(Δ) is Hermitian algebraically: what does the non-unitarity of exp(log(Δ)it) mean physically?

3. Could ω correspond to a vacuum which in conformal degrees of freedom depends on the choice of the sphere S² defining the radial coordinate playing the role of complex variable in the case of the radial conformal algebra. Does ^*-operation in M correspond to Hermitian conjugation for fermionic oscillator operators and change of sign of super conformal weights?

The exponent of the modified Dirac action gives rise to the exponent of Kähler function as Dirac determinant and fermionic inner product defined by fermionic Feynman rules. It is implausible that this exponent could as such correspond to ω or Δ^it having conceptual roots in thermodynamics rather than QFT. If one assumes that the exponent of the modified Dirac action defines a "complex square root" of ω the situation changes. This raises technical questions relating to the notion of square root of ω.

1. Does the complex square root of ω have a polar decomposition to a product of positive definite matrix (square root of the density matrix) and unitary matrix and does ω^1/2 correspond to the modulus in the decomposition? Does the square root of Δ have similar decomposition with modulus equal equal to Δ^1/2 in standard picture so that modular automorphism, which is inherent property of von Neumann algebra, would not be affected?

2. Δ^it or rather its generalization is defined modulo a unitary operator defined by some Hamiltonian and is therefore highly non-unique as such. This non-uniqueness applies also to |Δ|. Could this non-uniqueness correspond to the thermodynamical degrees of freedom?

4. Cautious conclusions

The cautious conclusions are following.

1. The notion of state as it appears in the theory of factors is not enough for the purposes of quantum TGD. The reason is that state in this sense is essentially the counterpart of thermodynamical state.

2. The construction of M-matrix might be understood in the framework of factors if one replaces state with its "complex square root" natural if quantum theory is regarded as a "complex square root" of thermodynamics. The replacement of exponent of Hamiltonian with imaginary exponent of action is the counterpart for this generalization in QFT framework.

3. The identification of Δ^it as M-matrix is not consistent with zero energy ontology and causal diamond as basic building block of WCW. It might be that the varying thermodynamical part of M-matrix -say that giving rise to p-adic thermodynamics- involves Δ^t.

4. Also the idea that Connes tensor product could fix M-matrix is too optimistic but an elegant formulation in terms of partial trace for the notion of M-matrix modulo measurement resolution exists and Connes tensor product allows interpretation as entanglement between sub-spaces consisting of states not distinguishable in the measurement resolution used. The partial trace also gives rise to non-pure states naturally. M-matrix as a complex square root of ω could be determined uniquely from the condition of universality and the requirement that it decomposes to a tensor product of M-matrices associated with factors space M/N and N for any reasonable choices of N.

The challenge is to show that the identification of M-matrix as a "complex square root" of state ω is consistent with the definition of M-matrix using only the modified Dirac action for second quantized induced spinor fields identifiable as square root" of Kähler action emerges. The key idea is the notion of bosonic emergence used as a guideline in the construction of QFT limit of TGD and meaning that bosonic propagators and vertices emerge radiatively from fermionic loops.

For details see either the section "The latest vision about the role of HFFs in TGD" of the chapter Construction of Quantum Theory: M-matrix of the book "Towards S-matrix" or of the chapter Was von Neumann Right After All of the book "Overall View about TGD".

Comments about M-matrix and Connes tensor product

I have proposed that the identification of M-matrix as Connes tensor product defined by finite measurement resolution could lead to a universal definition of dynamics. This hypothesis is fascinating but - mainly due to my poor understanding of HFFS of type II₁ - has remained just an interesting hypothesis. In the following I represent a formulation of this idea which is more precise than the earlier formulations and take the role of skeptic and reconsider also hyper-finite factors of type III₁ appearing in quantum field theories. I also consider the possibility that M-matrices could relate to a quantum variant of so called 2-vector space formulated by John Baez and collaborators.

1. Finite measurement resolution and M-matrix

Consider first the formulation of finite measurement resolution in terms of inclusions of HFFs of type II₁.

1. Finite measurement resolution means that M-matrix elements are defined by integrating- that is taking a trace- over the degrees of freedom defined by the sub-factor N subset M in the sense that states created by N from a given state do not differ from it in given measurement resolution. As a consequence,N takes the role of complex numbers in the ordinary quantum theory.
2. This requires that the tensor product decomposition M= (M/N)⊗ N .

   exists in some sense. The factor space M/N could be seen as an analog of or even identical with a tensor product of quantum spinors with different quantum phases q=exp(i2π/n), n > 3, and would have a fractal quantum dimension fixed by the inclusion. Quantum spinors are finite-dimensional.

3. Consider M-matrix elements M_{(r,n),(r1,n1)}, where r labels the states of M/N. Transition probabilities with a finite measurement resolution are defined by taking trace over N meaning summation over both n and n₁. One obtains

   p(r₁→ r₂)=∑_n,n1 M_{(r,n), (r1,n1)} M⁺_{(r1,n1),(r,n)} .

The subscript '+' refers to Hermitian conjugation here.
4. Suppose one multiplies the state by element n of N. Since one has a tensor product structure, the trace over N separates out so that one obtains a factor Tr(n⁺n) giving a multiplicative constant just as is obtained by multiplying the states appearing in the ordinary S-matrix by a constant. Note that the measurement resolution could be different for initial and final states: in this case the smaller algebra would serve as measurement algebra.

5. The condition that N acts like complex numbers has not been used yet and the above picture makes sense even without this condition. For the ordinary S-matrix the action of complex number on final state affects S-matrix in the same manner as the action of its conjugate on initial state. Generalizing, in positive energy ontology S-matrix elements would be anti-linear with respect to (say) initial quantum states as inner product like and the action of element n on the final state would be equal to the action of n⁺ on the initial state. In zero energy ontology M-matrix is bilinear in positive and negative energy parts of the zero energy state. The condition that N acts like complex numbers means that the action of an element of N on the final final state in M-matrix induces the same effect as the action of n (-rather than that of n⁺) on the initial state. This condition is rather powerful and expected to determine the M-matrix highly uniquely.

6. This picture does not require gauge invariance with respect to N. One can however consider also a stronger condition stating that the action of unitary elements of N act as gauge symmetries. In zero energy ontology this would mean that symmetries represent elements of infinite-dimensional orthogonal group.

7. Connes tensor product makes sense also in construction of many-particle states and gives rise to irreducible entanglement not visible in measurement resolution.

2. How unique the Connes tensor product really is?

I have used to state in rather light hearted manner that Connes tensor product is highly unique but what the situation really is? Is my belief just a folk wisdom that I have gather somewhere? Let us try to think ourselves. How unique the M-matrix could be?

1. The simplest solution to the conditions could be written as a formal tensor product M=M₁⊗ M₂, where M₁ is M-matrix in M/N and M₂ projection operator P_N in to N. The appearance of P_NN would say that in N degrees of freedom entanglement coefficients are identical and basic condition would be satisfied since P_N would commute with elements of N. If one replaces P_N with a more general operator, it must also commute with all elements of N, which means that it must represent direct sum of HFFs of type II₁ since the definition property of factors is that only unit matrix commutes with all operators of the factor.

2. This representation would suggest that M₁ can be chosen completely freely as an analog of a complex square root of Hermitian density matrix. The skeptic would conclude that Connes tensor product does not say anything about the physically interesting part of M-matrix!

3. This formal representation very probably does not make sense as such since M₁ is tensor product of quantum spinor spaces and tensor product property might reveal itself only as a factorization of trace into a product of traces over M/N and N parts of the tensor product like structure.

This argument must be of course take as childish babbling of a poor physicist knowing nothing about the delicacies of this branch of mathematics. It is quite possible that the delicacies of the Connes tensor product bring in uniqueness.

3. Is the resulting M-matrix realistic?

Are there hopes that the resulting M-matrix is realistic?

1. The resulting M-matrix defined in quantum Hilbert space has fractal quantum dimension. Since the unitary representations of quantum groups are finite-dimensional, one can expect that finite-dimensionality results. Therefore skeptic would ask whether the use of mere HFFs of type II₁ is enough and whether the outcome is just a topological S-matrix of braid theories. This would be dissapointing. One can however easily add factor of type I as a tensor product factor to give HFF of type II giving also the structure of ordinary wave mechanics.

2. The physical picture is that the orbits of fermions and antifermions at light-like 3-surfaces defined braids and the topological M-matrix can be assigned with the legs of generalized Feynman diagrams. The remaining configuration space degrees of freedom could correspond to factors of type I.

3. In finite measurement resolution space-time surface decomposes to strings connecting the braid strands at different light-like 3-surfaces and this stringy interaction should guarantee that the theory does not reduce to a topological QFT. This interaction is between braids so that something new is needed. Does this something new allow a description in terms of finite measurement resolution too or is factor of type I enough? Or could the almost TQFT property mean that N does not act anymore like complex numbers so that Connes tensor product is deformed?
4. One can also wonder what is the role of the dynamics in the parameters labeling the copies of HFF. One interpretaton is in terms of zero modes for the Kähler metric of world of classical worlds and therefore as non-quantum fluctuating degrees of freedom. One can also consider interpretation of superposition over these parameters in terms of thermodynamical degrees of freedom.

4. Should one replace HFFs of type II₁ with HFFs of type III₁?

These skeptic arguments encourage a serious reconsideration of an alternative approach for achieving uniqueness by extending the HFF of type II₁ with HFF of type III₁ (I have considered this generalization already earlier). The dream would be that any M-matrix with a finite measurement resolution is obtained from a universal M-matrix with infinite measurement resolution existing in some sense by tracing over HFF N of type III₁ and multiplying by the projector to N. Tomita-Takesaki theorem raises the hope that this universal M-matrix indeed exists and is unique apart from the inner automorphisms of HFF of type III₁ and complex parameter t whose real and imaginary parts have interpretation as time and temperature.

1. In quantum field theories in Minkowski space hyper-finite factors of type III₁ appear and it would not be surprising that they would appear also in quantum TGD. These factors can be expressed as so called crossed products of hyper-finite factors of type hyper-finite II_∞ factor and real numbers, whereas the latter correspond to a tensor product of hyper-finite II₁ factor and I_∞ factor (for the basic wisdom about von Neumann algebras see this).

2. One can assign to any M-matrix in factor M an M-matrix associated with any quantum space M/N by tracing over N. This could make sense also for HFFs of type III₁. In this case the trace Tr(n^*n) over N would be always infinite so that multiplication by operator of n of N would be analogous to a multiplication of the moduli square of S-matrix elements with an infinite scale factor. Maybe this relates to infinite multiplicative renormalization factors appearing in quantum field theories.

3. Tomita-Takesaki theorem assumes a von Neuman algebra acting on complex Hilbert space and the existence of a cyclic state Ω of Hilbert space (vacuum state in physics). Furthermore, the existence of a faithful state ω(x) = (xΩ,Ω) satisfying by definition ω(x^*x) > 0 is assumed. Faithful state is analogous to a thermodynamical ensemble. For any anti-linear operator S one has the polar decomposition S=JΔ^1/2, where Δ = S^*S > 0 is positive self-adjoint operator and J is anti-unitary involution with maps the algebra acting in Hilbert space to the algebra commutating with it by x→ JxJ. ω→ σ^ω_t = AdΔ^it is canonical "evolution" associated with ω and maps von Neumann algebra to itself. This automorphism is unique apart of inner automorphism and thus characterizes the von Neumann algebra itself. So called KMS condition holds true: ω(yx) =ω(σ_t^ω(x)y), t→ i, in the sense of analytic continuation. This condition is easy to understand by checking it for matrix algebras.

4. The facts that the automorphism defined by Δ characterizes the algebra itself and the automorphism is non-trivial for factors of type III led Connes and Rovelli to propose that thermodynamics represents a deeper level of physics than Hamiltonian (unique apart from inner automorphism) defining it and one could define Lorentz invariance thermodynamical time using thermodynamics for von Neumann algebras of type III.

5. In zero energy ontology where M-matrix corresponds to a generalized thermodynamical state a variant of this idea suggests itself. The automorphism for hyper-finite factor of type III₁ would define M-matrix depending on complex parameter t apart from inner automorphism of the factor. The real part of the parameter t would correspond to the Lorentz invariant temporal distance between the tips of CD quantized in powers of 2. Imaginary part of t would correspond to the inverse temperature.

5. The notion of 2-vector space and entanglement with finite measurement resolution

John Baez and collaborators are playing with very formal looking formal structures by replacing vectors with vector spaces. Direct sum and tensor product serve as basic arithmetic operations for vector spaces and one can define category of n-tuples of vectors spaces with morphisms defined by linear maps between vectors spaces of the tuple. n-tuples allow also elementwise product and sum. The obtain results which make them happy. For instance, the category of linear representations of give group forms 2-vector spaces since direct sums and tensor products of representations as well as n-tuples make sense. The 2-vector space however looks more or less trivial from the point of physics.

The situation could become more interesting in quantum measurement theory with finite measurement resolution described in terms of inclusions of hyperfinite factors of type II₁. The reason is that Connes tensor product replaces ordinary tensor product and brings in interactions via irreducible entanglement as a representation of finite measurement resolution. The category in question could give Connes tensor products of quantum state spaces and describing interactions. For instance, one could multiply M-matrices via Connes tensor product to obtain category of M-matrices having also the structure of 2-operator algebra.

1. The included algebra represents measurement resolution and this means that the infinite-D sub-Hilbert spaces obtained by the action of this algebra replace the rays. Sub-factor takes the role of complex numbers in generalized QM so that one would obtain non-commutative quantum mechanics. For instance, quantum entanglement for two systems of this kind would not be between rays but between infinite-D subspaces corresponding to sub-factors. One could build a generalization of QM by replacing rays with sub-spaces. It seems that quantum group concept does more or less this: the states in representations of quantum groups could be seen as infinite-dimensional Hilbert spaces.

2. One could speak about both operator algebras and corresponding state spaces modulo finite measurement resolution as quantum operator algebras and quantum state spaces with fractal dimension defined as factor space like entities obtained from HFF by dividing with the action of included HFF. Possible values of the fractal dimension are fixed completely for Jones inclusions. Maybe these quantum state spaces could define the notions of quantum 2-Hilbert space and 2-operator algebra via direct sum and tensor production operations. Fractal dimensions would make the situation interesting both mathematically and physically.

Suppose one takes the fractal factor spaces as the basic structures and keeps the information about inclusion.

1. Direct sums for quantum vectors spaces would be just ordinary direct sums with HFF containing included algebras replaced with direct sum of included HFFs.

2. The tensor products for quantum state spaces and quantum operator algebras are not anymore trivial. The condition that measurement algebras act effectively like complex numbers would require Connes tensor product involving irreducible entanglement between elements belonging to the two HFFs. This would have direct physical relevance since this entanglement cannot be reduced in state function reduction.

3. The sequences of super-conformal symmetry breakings identifiable in terms of inclusions of super-conformal algebras and corresponding HFFs could have a natural description using the 2-Hilbert spaces and quantum 2-operator algebras.

For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-matrix".

My defence statement

Following von Neumann known from his factors of type I, II, and III, one could classify mathematicians to those of type I and type II. I hope that mathematicians do not feel insulted by this kind of classifications;-).

Mathematicians of type I believe in the uniqueness of those mathematical structures, which are really God given. Classical number fields and groups would be the basic example about God-given-ness. Also physics would be fixed uniquely by the condition of the mathematical existence of the theory and by its maximal richness.

Mathematicians of type II prefer genericity. Everything that can be imagined is possible. In physics this would mean that space-time dimension four just happens to be 4 in this particular corner of the multiverse and symmetries of physics - if real rather than figments of our imagination - just happen to be just those of standard model in this little pocket of multiverse.

String theorist began as mathematicians of type I and were fascinated by the idea that they had the theory of everything allowing to calculate proton mass to arbitrary number digits within decade or two. As they suffered a phase transition to M-theorists they woke up as mathematicians of type II and realized how wonderful it after all is that M-theory predicts nothing. Lubos is one of few exceptions. Personally I am also still of type I and I can only represent excuses for my conservatism. I hope that jury bothers to listen for humanitarian reasons if not for anything else.

1. The Kähler geometry of the worlds of classical worlds is unique already for loop spaces. In 3-D context there are excellent hopes that even the imbedding and therefore symmetries of physics can be fixed from the existence condition. The basic outcome is that infinite-dimensional symmetries are not figments of our imagination: they are absolutely essential for the infinite-dimensional geometric existence. Extended super-conformal invariance indeed fixes space-time dimension correctly and implies M⁴×S decomposition for imbedding space.

2. The isometries of this particular infinite-D space which has the special property that it exists and also of corresponding finite-dimensional imbedding space must have very special meaning. Number theoretic one is the natural guess. Classical number fields and their complexifixations are natural here.

3. The choice of S remains to be fixed from number theoretic considerations. The existence of M⁸-M⁴×CP₂ duality ("number theoretical compactification") with M⁸ is interpreted as hyper-octonionic plane of complexified octonions with Minkowskian metric, means the possibility to map space-time surfaces identified as hyper-quaternionic surfaces of M⁸ to M⁴xCP₂. This map preservers induced metric and Kahler form. Standard model symmetries are the outcome.

M⁸-M⁴× CP₂ duality implies also the core element of gauge invariance and stringy picture.

1. Only if the hyper-quaternionic plane assignable to a point of space-time surface contains preferred M² (hypercomplex plane) of M⁸ having interpretation as plane of non-physical polarizations, it can be labeled by a point of CP₂ so that the map of X⁴ in M⁸ to X⁴ in M⁴×CP₂ exists naturally and maps M⁴ point to M⁴ point and hyper-quaternionic plane to CP₂ point.This is the physical essence of gauge conditions. It seems to be possible to assume that the choices of the preferred polarization plane is local.

2. The duality also implies decomposition of space-time surface to string world sheets parameterized by partonic 2-surface and in finite measurement resolution implying the replacement of partonic 2-surface by a discrete set of points, the replacement of space-time surface with string world sheets.

Therefore the existence of the geometry of the world of classical worlds plus classical number theory would imply both gauge invariance and stringy description in finite measurement resolution besides M⁴×CP₂ and 4-dimensionality of space-time.

That was my defence. Jury can decide;-).

Condensed matter monopoles found

Kram sent me a New Scientist story telling about the finding of condensed matter monopoles. I decided to write something about this but Lubos was faster than me in his reactions and I can only recommend his blog for explanation what Dirac monopoles are and what the condensed matter monopoles are not. Lubos gives also link to a popular article in Nature and also other links. Unfortunately, the article Dirac Strings and Magnetic Monopoles in Spin Ice Dy2Ti2O7 itself is not freely accessible.

As Lubos tells, the monopoles are not Dirac monopoles with quantized magnetic charge, which are not allowed by the gauge invariance of Maxwell equations in topologically trivial space-time. They do not seem to be GUT monopoles either. Rather, they seem to correspond to magnetic flux tubes having opposite effective magnetic charges at their ends. This kind of objects are imaginable even in the standard Maxwellian theory. It is however important to notice that the flux must continue beyond the ends by a conservation of magnetic flux if Maxwell's equation in flat space are true so that the ends are fictive to some extent. What one can think in Minkowski space is the analog of a bar magnet with return flux dispersing at the ends of magnet.

These Dirac strings are very interesting from TGD point of view since CP₂ has a non-trivial topology. The fact that the second homology group is non-trivial (integers) means that there are non-contractible 2-surfaces in CP₂. These surfaces are characterized homologically by integer valued Kähler magnetic flux. These monopoles are not Dirac monopoles but homological monopoles for which the topological half of Maxwell's equations remains true (Faraday's induction law and vanishing of divergence of magnetic field).

Since the projection of Kähler form to space-time defines Maxwell's field (part of electromagnetic field and under some conditions all of it) these homological monopoles can appear at space-time level.

1. If space-time surfaces are orientable, these monopoles appear always as pairs with opposite Kähler magnetic charges.

2. For CP₂ type vacuum extremals representing elementary fermions the flux is confined inside CP₂ rather than flowing radially out. Same is true for a piece of CP₂ type vacuum extremal representing gauge boson as a wormhole contact.

3. If one allows non-orientable space-time surfaces so that homology becomes Z₂ valued also genuine monopoles carrying even value of magnetic charge are possible. I would guess that CP₂ projection of space-time surface must be non-orientable in this case. Klein bottle and Möbius strip imbedded in CP₂ could server as basic examples. By gluing handles one obtains more complex non-orientable 2-surfaces.

Let us restrict the consideration mostly to the orientable case.

1. Magnetic flux tubes at the ends of string like objects (cosmic strings with enormous string tension) would be the basic realization. Simplest extremal is just Cartesian product X²× Y², where X² is string orbit in M⁴ and Y² a complex sub-manifold of CP₂. Cosmic strings are basic objects in TGD inspired very early cosmic string dominated cosmology when space-time sheets had not yet emerged. During cosmic evolution these flux tubes would widen to magnetic flux tubes carrying arbitrary weak magnetic field. These flux tubes could (dare I say "would"?) be still present. I would guess that non-orientable 2-surfaces of CP₂ do not allow a representation as complex manifolds so that I would not expect them to give rise to cosmic string like objects.

2. Fermions correspond in TGD Universe to light-like 3-surfaces separating so called CP₂ type vacuum extremal with Euclidian signature of metric from space-time sheet with Minkowskian signature of metric. This light-like 3-surface can be interpreted as an orbit of a partonic 2-surface and besides fermionic quantum numbers could carry magnetic charge so that magnetically charged variants leptons and quarks could exist. A good guess is that these states are very massive. One would have a long ranged radial magnetic field and the second member of monopole pair could correspond to the outer boundary of the space-time sheet. Whether boundary conditions allow this is not clear.

3. Also bosonic variants of monopoles are possible. Gauge bosons are identified as wormhole contacts (piece of CP₂ type vacuum extremal) connecting space-time sheets with Minkowskian signature of metric and there would be pair of throats carrying opposite magnetic charges. Throats could carry fermionic and anti-fermionic quantum numbers so that all gauge bosons and possible Higgs particle could have magnetically charged variants. In two-sheeted space-time picture one would have magnetic dipoles with magnetic charges extremely near to each other . From the point of view of either space-time sheet one would have monopole carrying classical magnetic field giving rise to a genuine force.

4. I have proposed that magnetic flux in super-conductors could be transferred between space-time sheets through these kind of wormhole contacts carrying at its throats opposite magnetic charges possibly carrying also other elementary particle numbers and behaving as bosons. The magnetic flux tubes observed in superconductors could enter to the super-conductor from a larger space-time sheet and return to it through this kind of wormhole throats.

There is also a much more mundane possibility but still representing something new. Flux tubes could be just flux tubes with outer boundary (of form X²× S², D² disk, rather than of form X¹× Y². Y² closed manifold in CP₂) and beginning from the boundary of space-time sheet or from wormhole throat and carrying magnetic flux which need not satisfy Dirac condition. Condensed matter monopoles could be this kind of objects.

Notice that spin glasses in which the Dirac fluxes are observed are systems in which the direction of magnetization is frozen in a given patch but varies from patch to patch. In TGD framework the notion of 4-D spin glass landscape emerges naturally from the vacuum degeneracy of Kähler action meaning same for 4-D space-time region so that kind of dynamical spin glass would be in question.

For background see the chapters General View About Physics in Many-Sheeted Space-Time: Part I and General View About Physics in Many-Sheeted Space-Time: Part II of "Physics in Many-Sheeted Space-time".

Videos about TGD

I have at my home page videos summarizing both the basic ideas of TGD and the visions behind quantum TGD. My intention is to build a kind of video library about TGD gradually.

The motivations for seeing this trouble are two-fold. The amount of material at my home page is so enormous that many a visitor might leave it in despair. These short information bursts containing only the essentials (because of my laziness!;-)) might help to build an overall view.

There are also selfish motivations. Talking and explaining is very important for anyone making research. For reasons which I do not want to go here, this has not been possible for me. The preparing of the videos help in this respect. This talking to web camera also feeds the theory literally into the spine and allows easier identification of inconsistencies in texts and even some new idea can pop up now and then. I hope my broken English is not an insurmountable difficulty.

About life as a scientific dissident in Finland

Life as a scientific dissident in Finland keeps the stress hormones level high. I have been unemployed for most of my professional life. This has been the only possible manner for me to do science. In a norm ethics based on dead rules my strategy of survival could be regarded as a misuse of unemployment money. My ethics however relies on real values rather than blind rules, and my basic values imply that I would be criminal, if I would waste my life by doing something just for money and gaining easy social acceptance.

I must emphasize that I have had no other options than this. A person whose work can be regarded as Nobel risk becomes automatically Out-of-Law in Finnish academic circles. In my case this happened when John Wheeler- one of the greatest physicists of last century and teacher of Feynman- wrote a review about my first published paper and regarded the work as brilliant. They could not prevent the publication of the work as thesis but the Dear Brother Net has taken care that I have had not received not a single coin from a work that I have done these 28 years after the thesis. Of course, I did not receive any support even before this but for a short period after thesis I really thought of having the same rights as those who had needed an advisor for their thesis.

I received the latest dose of stress hormones yesterday. First some background. The right wing has been for some years in power in Finland and they have done a lot to make my life difficult. For instance, I must participate regularly into meetings in which plans are made in order to that I would get a job in future. This is of course just a theater: people in employment office know that I am fully occupied top scientist and would make a hunger strike if forced to stop my work. They are decent human beings and regret that they have no other option than to force me to participate this farce producing a lot of trash paper and wasting money of tax payers.

The newest bright idea of these right wing politicians is what they call "activation" of long term un-employed people. The victim receives a letter in which he is invited to a discussion in which "we together decide how to help you". Also I received this kind of letter yesterday. In practice "activation" means frightening of the victim to take a job of any kind irrespective of the background education. The letter makes clear that if I am not willing to participate this "activation" process, I will loose both the unemployment money and the support from the social office. This is against constitutional law guaranteeing a minimum income so that I will not die to hunger but these right wing politicians do not care too much about this kind of details.

I took a phone call to the employment office and told that I do not need any help. I am not an alcoholist, I do not use drugs, I am not socially handicapped, I have no problems of mental health. Simply: I do not want to be "activated" since I am extremely active already now. I am sitting at computer terminal 8-12 hours per day and thinking intensively also the rest of time. I have written during last month a new chapter to one of books, transformed all 15 books to net books with references realized as web links, just now I am busily preparing video lectures about TGD, I am doing my best to communicate my life work through my homepage and via discussions, etc., etc. This all is in vain. I am not paid for this activity so that I am not "active" in the sense they define activity. Therefore I need "help" and they are going to "help" me irrespective of whether I want or not.

What can I do? Should I try to tell these decision makers or perhaps even some colleagues in Finland that the best manner to help me would be an address to highest decisions makers containing sufficiently many names and telling about the horrible situation in the academic world of Finland: is it really impossible to anything for this mafia of old envious men? Of course not! This would be useless since all these people known that in Finland there are no mafias - and certainly not in the Finnish academic world. Besides this, Finland is the only country in the world where there is no corruption -as also media has told us all these years! Think about that! Instead, we have this wonderful Dear Brother Net. No written documents... if you understand what I mean;-)! It is not even necessarily to say anything aloud...;-)! Just a favor and the counter favor when it is needed;-)!