Could N=2 or N=4 SUSY have something to do with TGD?

N=4 SYM has been the theoretical laboratory of Nima and others. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form meant to be should be accessible to ordinary physicist.

N=4 SYM is definitely a completely exceptional theory and one cannot avoid the question whether it could in some sense be part of fundamental physics. In TGD framework right handed neutrinos have remained a mystery: whether one should assign space-time SUSY to them or not. Could they give rise to N=2 or N=4 SUSY with fermion number conservation?

Earlier results

My latest view is that fully covariantly constant right-handed neutrinos decouple from the dynamics completely. I will repeat first the earlier arguments which consider only fully covariantly constant right-handed neutrinos.

1. N=1 SUSY is certainly excluded since it would require Majorana property not possible in TGD framework since it would require superposition of left and right handed neutrinos and lead to a breaking of lepton number conservation. Could one imagine SUSY in which both MEs between which particle wormhole contacts reside have N=2 SUSY which combine to form an N=4 SUSY?
   
   
2. Right-handed neutrinos which are covariantly constant right-handed neutrinos in both M⁴ degrees of freedom cannot define a non-trivial theory as shown already earlier. They have no electroweak nor gravitational couplings and carry no momentum, only spin.
   

   The fully covariantly constant right-handed neutrinos with two possible helicities at given ME would define representation of SUSY at the limit of vanishing light-like momentum. At this limit the creation and annihilation operators creating the states would have vanishing anticommutator so that the oscillator operators would generate Grassmann algebra. Since creation and annihilation operators are hermitian conjugates, the states would have zero norm and the states generated by oscillator operators would be pure gauge and decouple from physics. This is the core of the earlier argument demonstrating that N=1 SUSY is not possible in TGD framework: LHC has given convincing experimental support for this belief.
   

Could massless right-handed neutrinos covariantly constant in CP₂ degrees of freedom define N=2 or N=4 SUSY?

Consider next right-handed neutrinos, which are covariantly constant in CP₂ degrees of freedom but have a light-like four-momentum. In this case fermion number is conserved but this is consistent with N=2 SUSY at both MEs with fermion number conservation. N=2 SUSYs could emerge from N=4 SUSY when one half of SUSY generators annihilate the states, which is a basic phenomenon in supersymmetric theories.

1. At space-time level right-handed neutrinos couple to the space-time geometry - gravitation - although weak and color interactions are absent. One can say that this coupling forces them to move with light-like momentum parallel to that of ME. At the level of space-time surface right-handed neutrinos have a spectrum of excitations of four-dimensional analogs of conformal spinors at string world sheet (Hamilton-Jacobi structure).
   

   For MEs one indeed obtains massless solutions depending on longitudinal M² coordinates only since the induced metric in M² differs from the light-like metric only by a contribution which is light-like and contracts to zero with light-like momentum in the same direction. These solutions are analogs of (say) left movers of string theory. The dependence on E² degrees of freedom is holomorphic. That left movers are only possible would suggest that one has only single helicity and conservation of fermion number at given space-time sheet rather than 2 helicities and non-conserved fermion number: two real Majorana spinors combine to single complex Weyl spinor.
   

   
   

2. At imbedding space level one obtains a tensor product of ordinary representations of N=2 SUSY consisting of Weyl spinors with opposite helicities assigned with the ME. The state content is same as for a reduced N=4 SUSY with four N=1 Majorana spinors replaced by two complex N=2 spinors with fermion number conservation. This gives 4 states at both space-time sheets constructed from ν_R and its antiparticle. Altogether the two MEs give 8 states, which is one half of the 16 states of N=4 SUSY so that a degeneration of this symmetry forced by non-Majorana property is in question.
   

Is the dynamics of N=2 or N=4 SYM possible in right-handed neutrino sector?

Could N=2 or N=4 SYM be a part of quantum TGD? Could TGD be seen a fusion of a degenerate N=4 SYM describing the right-handed neutrino sector and string theory like theory describing the contribution of string world sheets carrying other leptonic and quark spinors? Or could one imagine even something simpler?

What is interesting that the net momenta assigned to the right handed neutrinos associated with a pair of MEs would correspond to the momenta assignable to the particles and obtained by p-adic mass calculations. It would seem that right-handed neutrinos provide a representation of the momenta of the elementary particles represented by wormhole contact structures. Does this mimircry generalize to a full duality so that all quantum numbers and even microscopic dynamics of defined by generalized Feynman diagrams (Euclidian space-time regions) would be represented by right-handed neutrinos and MEs? Could a generalization of N=4 SYM with non-trivial gauge group with proper choices of the ground states helicities allow to represent the entire microscopic dynamics?

Irrespective of the answer to this question one can compare the TGD based view about supersymmetric dynamics with what I have understood about N=4 SYM.

1. In the scattering of MEs induced by the dynamics of Kähler action the right-handed neutrinos play a passive role. Modified Dirac equation forces them to adopt the same direction of four-momentum as the MEs so that the scattering reduces to the geometric scattering for MEs as one indeed expects on basic of quantum classical correspondence. In ν_R sector the basic scattering vertex involves four MEs and could be a re-sharing of the right-handed neutrino content of the incoming two MEs between outgoing two MEs respecting fermion number conservation. Therefore N=4 SYM with fermion number conservation would represent the scattering of MEs at quantum level.
   
   
2. N=4 SUSY would suggest that also in the degenerate case one obtains the full scattering amplitude as a sum of permutations of external particles followed by projections to the directions of light-like momenta and that BCFW bridge represents the analog of fundamental braiding operation. The decoration of permutations means that each external line is effectively doubled. Could the scattering of MEs can be interpreted in terms of these decorated permutations? Could the doubling of permutations by decoration relate to the occurrence of pairs of MEs?
   

   One can also revert these questions. Could one construct massive states in N=4 SYM using pairs of momenta associated with particle with label k and its decorated copy with label k+n? Massive external particles obtained in this manner as bound states of massless ones could solve the IR divergence problem of N=4 SYM.
   
   

3. The description of amplitudes in terms of leading singularities means picking up of the singular contribution by putting the fermionic propagators on mass shell. In the recent case it would give the inverse of massless Dirac propagator acting on the spinor at the end of the internal line annihilating it if it is a solution of Dirac equation.
   

   The only way out is a kind of cohomology theory in which solutions of Dirac equation represent exact forms. Dirac operator defines the exterior derivative d and virtual lines correspond to non-physical helicities with dΨ ≠ 0. Virtual fermions would be on mass-shell fermions with non-physical polarization satisfying d²Ψ=0. External particles would be those with physical polarization satisfying dΨ=0, and one can say that the Feynman diagrams containing physical helicities split into products of Feynman diagrams containing only non-physical helicities in internal lines.
   
   

4. The fermionic states at wormhole contacts should define the ground states of SUSY representation with helicity +1/2 and -1/2 rather than spin 1 or -1 as in standard realization of N=4 SYM used in the article. This would modify the theory but the twistorial and Grassmannian description would remain more or less as such since it depends on light-likeneness and momentum conservation only.
   

3-vertices for sparticles are replaced with 4-vertices for MEs

In N=4 SYM the basic vertex is on mass-shell 3-vertex which requires that for real light-like momenta all 3 states are parallel. One must allow complex momenta in order to satisfy energy conservation and light-likeness conditions. This is strange from the point of view of physics although number theoretically oriented person might argue that the extensions of rationals involving also imaginary unit are rather natural.

The complex momenta can be expressed in terms of two light-like momenta in 3-vertex with one real momentum. For instance, the three light-like momenta can be taken to be p, k, p-ka, k= ap_R. Here p (incoming momentum) and p_R are real light-like momenta satisfying p⋅ p_R=0 with opposite sign of energy, and a is complex number. What is remarkable that also the negative sign of energy is necessary also now.

Should one allow complex light-like momenta in TGD framework? One can imagine two options.

1. Option I: no complex momenta. In zero energy ontology the situation is different due to the presence of a pair of MEs meaning replaced of 3-vertices with 4-vertices or 6-vertices, the allowance of negative energies in internal lines, and the fact that scattering is of sparticles is induced by that of MEs. In the simplest vertex a massive external particle with non-parallel MEs carrying non-parallel light-like momenta can decay to a pair of MEs with light-like momenta. This can be interpreted as 4-ME-vertex rather than 3-vertex (say) BFF so that complex momenta are not needed. For an incoming boson identified as wormhole contact the vertex can be seen as BFF vertex.
   

   To obtain space-like momentum exchanges one must allow negative sign of energy and one has strong conditions coming from momentum conservation and light-likeness which allow non-trivial solutions (real momenta in the vertex are not parallel) since basically the vertices are 4-vertices. This reduces dramatically the number of graphs. Note that one can also consider vertices in which three pairs of MEs join along their ends so that 6 MEs (analog of 3-boson vertex) would be involved.
   
   

2. Option II: complex momenta are allowed. Proceeding just formally, the (g₄)^1/2 factor in Kähler action density is imaginary in Minkowskian and real in Euclidian regions. It is now clear that the formal approach is correct: Euclidian regions give rise to Kähler function and Minkowskian regions to the analog of Morse function. TGD as almost topological QFT inspires the conjecture about the reduction of Kähler action to boundary terms proportional to Chern-Simons term. This is guaranteed if the condition j_K^μA_μ=0 holds true: for the known extremals this is the case since Kähler current j_K is light-like or vanishing for them. This would seem that Minkowskian and Euclidian regions provide dual descriptions of physics. If so, it would not be surprising if the real and complex parts of the four-momentum were parallel and in constant proportion to each other.
   

   This argument suggests that also the conserved quantities implied by the Noether theorem have the same structure so that charges would receive an imaginary contribution from Minkowskian regions and a real contribution from Euclidian regions (or vice versa). Four-momentum would be complex number of form P= P_M+ iP_E. Generalized light-likeness condition would give P_M²=P_E² and P_M⋅P_E=0. Complexified momentum would have 6 free components. A stronger condition would be P_M²=0=P_E² so that one would have two light-like momenta "orthogonal" to each other. For both relative signs energy P_M and P_E would be actually parallel: parametrization would be in terms of light-like momentum and scaling factor. This would suggest that complex momenta do not bring in anything new and Option II reduces effectively to Option I. If one wants a complete analogy with the usual twistor approach then P_M²=P_E²≠ 0 must be allowed.
   

Is SUSY breaking possible or needed?

It is difficult to imagine the breaking of the proposed kind of SUSY in TGD framework, and the first guess is that all these 4 super-partners of particle have identical masses. p-Adic thermodynamics does not distinguish between these states and the only possibility is that the p-adic primes differ for the spartners. But is the breaking of SUSY really necessary? Can one really distinguish between the 8 different states of a given elementary particle using the recent day experimental methods?

1. In electroweak and color interactions the spartners behave in an identical manner classically. The coupling of right-handed neutrinos to space-time geometry however forces the right-handed neutrinos to adopt the same direction of four-momentum as MEs has. Could some gravitational effect allow to distinguish between spartners? This would be trivially the case if the p-adic mass scales of spartners would be different. Why this should be the case remains however an open question.
   
   
2. In the case of unbroken SUSY only spin distinguishes between spartners. Spin determines statistics and the first naive guess would be that bosonic spartners obey totally different atomic physics allowing condensation of selectrons to the ground state. Very probably this is not true: the right-handed neutrinos are delocalized to 4-D MEs and other fermions correspond to wormhole contact structures and 2-D string world sheets.
   

   The coupling of the spin to the space-time geometry seems to provide the only possible manner to distinguish between spartners. Could one imagine a gravimagnetic effect with energy splitting proportional to the product of gravimagnetic moment and external gravimagnetic field B? If gravimagnetic moment is proportional to spin projection in the direction of B, a non-trivial effect would be possible. Needless to say this kind of effect is extremely small so that the unbroken SUSY might remain undetected.
   
   

3. If the spin of sparticle be seen in the classical angular momentum of ME as quantum classical correspondence would suggest then the value of the angular momentum might allow to distinguish between spartners. Also now the effect is extremely small.
   

For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-dimensional Geometry" or the article Could N =2 or N =4 SUSY be a part of TGD after all?.

Scattering Amplitudes and the Positive Grassmannian

Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima Arkani-Hamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article.

All scattering amplitudes have on shell amplitudes for massless particles as building bricks

The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible.

In TGD framework quantum classical correspondence requires a space-time correlate for the on mass shell property and it indeed exists. The mathematically ill-defined path integral over all 4-surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3-D ends at the light-like boundaries of causal diamond (Euclidian/Minkowskian space-time regions give real/imaginary Chern-Simons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds.

Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4-D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition.

Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at space-time surfaces, which I call Hamilton-Jacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied.

Massless extremals and twistor approach

The decomposition M⁴=M²× E² is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M² fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel space-time sheets is crucial.

Can one find a concrete identification for M²× E² decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by p-adic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle?

Candidates for this kind of preferred extremals indeed exist. Local M²× E² decomposition and light-like longitudinal massless momentum assignable to M² characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single space-time sheet carrying a conserved light-like M² momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a time-like conserved four-momentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs.

The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of p-adic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework.

Scattering amplitudes as representations for braids whose threads can fuse at 3-vertices

Just a little comment about the content of the article. The main message of the article is that non-equivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines - or to be more precise - decorated permutations which replace permutation group S_n with n! elements with its decorated version containing 2ⁿn! elements. Besides 3-vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation.

This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.

1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).
   

   
   

2. One can assign to the lines of generalized Feynman diagrams lines in M² characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.
   

"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4-D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for sub-manifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word! Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima Arkani-Hamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article.

All scattering amplitudes have on shell amplitudes for massless particles as building bricks

The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible.

In TGD framework quantum classical correspondence requires a space-time correlate for the on mass shell property and it indeed exists. The mathematically ill-defined path integral over all 4-surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3-D ends at the light-like boundaries of causal diamond (Euclidian/Minkowskian space-time regions give real/imaginary Chern-Simons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds.

Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4-D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition.

Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at space-time surfaces, which I call Hamilton-Jacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied.

Massless extremals and twistor approach

The decomposition M⁴=M²× E² is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M² fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel space-time sheets is crucial.

Can one find a concrete identification for M²× E² decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by p-adic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle?

Candidates for this kind of preferred extremals indeed exist. Local M²× E² decomposition and light-like longitudinal massless momentum assignable to M² characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single space-time sheet carrying a conserved light-like M² momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a time-like conserved four-momentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs.

The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of p-adic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework.

Scattering amplitudes as representations for braids whose threads can fuse at 3-vertices

Just a little comment about the content of the article. The main message of the article is that non-equivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines - or to be more precise - decorated permutations which replace permutation group S_n with n! elements with its decorated version containing 2ⁿn! elements. Besides 3-vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation.

This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.

1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).
   

2. One can assign to the lines of generalized Feynman diagrams lines in M² characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.
   

"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4-D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for sub-manifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word!

For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-dimensional Geometry" or the article Could N =2 or N =4 SUSY be a part of TGD after all?.

Firewall mania

Multiverse mania has transformed to firewall mania. The posting of Peter Woit explains sociological background and gives a lot of links to articles about the newest fashion of the declining theoretical physics.

The 28 years of superstring models have led us to where we started from. The work with M-theory landscape is not very rewarding, and it is much sexier to produce wordy arguments about black-hole firewall. There are a lot heated debates and many conferences can be organized. Few years will pass until some new buzzword pops up and firewalls are forgotten. "Mountains are giving birth to mouse" as the text book of Latin from my school days expressed it.

I have said this many times earlier but repeat it: It would be really a high time to make questions about fundamentals. What happens in the interior of black holes? How the theory of general relativity could be modified to get free of its singularities and from its problems with the basic conservation laws? Could we generalize super string models to get 4-D space-time in natural manner out of the theory?

We refuse to do this because we have decided that M-theory is the final answer, and we refuse to consider the possibility that these nice formulas for black-hole entropy which we cannot test empirically might have nothing to do with reality. Somehow just those beliefs which are not testable are always the strongest ones. A related observation: the most noisy advocates of superstrings seem to do nothing to develop it! Perhaps this is natural, healthy laziness deriving from sub-conscious knowledge of the sad truth.

I have never been accepted to the "circles" so that I have not felt the social pressure of fashion, and have been free to develop an approach providing answers to the questions stated above and to many others too. I do not wish to retype what I have already written so that I give a link to the earlier blog article Do blackholes and blackhole evaporation have TGD counterparts?.

Particle physics Christmas rumor

We are not protected against particle physics rumors even during Christmas. This time the rumor was launched from the comment section of Peter Woit's blog and soon propagated to the blogs of Lubos and Phil Gibbs.

The rumor says that ATLAS has observed 5 sigma excess of like sign di-muon events. This would suggests a resonance with charge +/- 2 and muon number two. In the 3-triplet SUSY model there is a Higgs with charge 2 but the lower limit for its mass is already now around 300-400 GeV. Rumors are usually just rumors and at this time the most plausible interpretation is as a nasty joke intended to spoil the Christmas of phenomenologists. Lubos however represents a graph from a publication of ATLAS based on 2011 data giving a slight support for the rumor. The experiences during last years give strong reasons to believe that statistical fluctuation is in question. Despite this the temptation to find some explanation is irresistible.

TGD view about color allows charge 2 leptomesons

TGD color differs from that of other unified theories in the sense that colored states correspond to color partial waves in CP₂. Most of these states are extremely massive but I have proposed that leptons can appear also in color octet partial waves with light masses and there is indeed some evidence for pion like states with mass very near to 2m_L for all charged lepton generations decaying to lepton-antilepton pairs and gamma pairs also p-adically scaled up variant having masses coming as octaves of the lowest state is reported for the tau-pion (see this).

Since leptons move in triality zero color partial waves, color does not distinguish between lepton and anti-lepton so that also leptons with the same charge can in principle form a pion-like color singlet with charge +/-2. This is of course not possible for quarks. In the recent case the p-adic prime should be such that the mass for the color octet muon is 105/2 GeV which is about 2⁹m(μ), where m(μ) =105.6 MeV is the mass of muon. Therefore the color octet muons would correspond to p≈2^k, k=k(μ)-2× 9=113-18 =95, which not prime but is allowed by the p-adic length scale hypothesis.

But why just k=95? Is it an accident that the scaling factor is same as between the mass scales of the ordinary hadron physics characterized by M₁₀₇ and M₈₉ hadron physics? If one applies the same argument to tau leptons characterized by M₁₀₇, one finds that like sign tau pairs should result from pairs of M₈₉ tau leptons having mass m=512×1.776 GeV= 909 GeV. The mass of resonance would be twice this. For electron one has m= 512*.51 MeV= 261.6 MeV with resonance mass equal to 523.2 MeV. Skeptic would argue that this kind of states should have been observed for long time ago if they really exist.

Production of parallel gluon pairs from the decay of strings of M₈₉ hadron physics as source of the leptomesons?

The production mechanism would be via two-gluon intermediate states. Both gluons would decay to unbound colored lepton-antilepton pair such that the two colored leptons and two antileptons would fuse to form two like sign lepton pairs. This process favors gluons moving in parallel. The required presence of also other like sign lepton pair in the state might allow to kill the hypothesis easily.

The presence of parallel gluons could relate to the TGD explanation for the correlated charged particle pairs observed in proton proton collisions (QCD predicts quark gluon plasma and the absence of correlations) in terms of M₈₉ hadron physics (see the earlier posting). The decay of M₈₉ string like objects is expected to produce not only correlated charged pairs but also correlated gluon pairs with members moving in parallel or antiparallel manner. Parallel gluons could produce like sign di-muons and di-electrons and even pairs of like sign μ and e. In the case of ordinary hadron physics this mechanism would not be at work so that one could understand why resonances with electron number two and mass 523 MeV have not been observed earlier.

Even leptons belonging to different generations could in principle form this kind of states and Phil Gibbs has represented a graph which he interprets as providing indications for a state with mass around 105 GeV decaying to like sign μe pairs. In this case one would however expect that mass is roughly 105/2 GeV since electron is considerably lighter than muon in given p-adic length scale.

The decay of bound states of two colored leptons with same (or opposite) charge would require a trilinear coupling gLL₈ analogous to magnetic moment coupling. Color octet leptons L₈ would transform to ordinary leptons by gluon emission.

To sum up, if the rumor is true M₈₉ hadron physics is beginning to demonstrate its explanatory power. The new hadron physics would explain the correlated charged particle pairs not possible to understand in high energy QCD. The additional gamma pair background resulting from the decays of M₈₉ pions could explain the two-gamma anomaly of Higgs decays, and also the failure to get same mass for the Higgs from ZZ and gamma-gamma decays. One should not forget that M₈₉ pion explains the Fermi bump around 135 GeV. It would also explain the anomalous like sign lepton pairs if one accepts TGD view about color.

For background see the chapter New Particle Physics Predicted by TGD: Part I of "p-Adic Length Scale Hypothesis and Hierarchy of Planck constants".

The recent situation concerning Higgs

Most bloggers have said something about the latest ATLAS results concerning Higgs. I also mentioned this issue in previous posting: because of the importance of Higgs issue I glue below what I said earlier.

Two-gamma anomaly persists but bloggers still want to forget it. Additional anomaly manifesting itself as different estimates for Higgs mass from the observation of decays to gamma pairs and Z pairs has emerged. It is very difficult to believe that there could be two Higgses with very nearly the same mass. The neglect of the existence of some wide resonance (in TGD Universe M₈₉ pion decaying to gamma pairs) producing two-gamma background could lead two-gamma excess and also to problems in mass determination. Phil Gibbs mentions also the digamma excess which ranges up to 200 GeV. Sooner or later one must perhaps take it seriously.

Higgs has been a stone in the toe of TGD. The problem has been the lack of classical space-time correlate for it. No wonder that in the case of Higgs I have developed a large number of alternative scenarios with and without Higgs like particle.

At this moment it seems clear that Higgs like particle exists although it is far from clear whether it has standard model couplings. If TGD has QFT limit and if one believes that Higgs mechanism is the only manner to model the particle massivation in QFT context, then Higgs mechanism would provide a mimicry of p-adic massivation but not its fundamental description. p-Adic thermodynamics is required for a microscopic description. Higgs vacuum expectation could have space-time counterpart at microscopic level and correspond to CP₂ part for the trace of the second fundamental form assignable to string world sheet (if string world sheet is minimal surface in space-time as one might expect, it is not minimal surface in imbedding space (meaning vanishing Higgs expectation) except under very special conditions).

The too high decay rate of Higgs like state to gamma pairs is still reported and the mass of Higgs seems to depend slightly on whether it is determined from the production of gamma pairs or Z pairs. This suggests that also something else than Higgs is there. TGD candidate for this something else would be the pion of M₈₉ hadron physics to be discussed below. By a naive scaling estimate for its width as Γ∼ α_s M one would obtain width of order 20 GeV.

The identification as the 135 GeV particle for which Fermi telescope finds evidence as M₈₉ pion is rather suggestive. This suggests that the anomalously high rate for the production of gamma pairs could be due to the decays of M₈₉ pion providing an additional background. Due to this background also the determination of the mass of the Higgs like state could lead to different results for gamma pairs and Z pairs in ATLAS.

The rate for the production of gamma pairs is somewhat too high up to cm energy of gamma pair of order 200 GeV. May be this effect could be understood in terms of satellites of M₈₉ pion with mass difference of order 20 GeV. These satellites would be scaled up variants of satellites of ordinary pion(and also other hadrons) for which evidence has been found recently and explained in TGD framework in terms of infared Regge trajectories. Of course, not a single particle physicist in CERN takes this kind of idea seriously since ordinary low energy hadron physics is regarded as a closed chapter of particle physics in higher energy circles.

Both Fermi satellite and LHC have provided interesting data concerming the existence of M₈₉ hadron physics. The standard interpretation for the unexpected correlations for charged particle pairs meaning that they tend move either in parallel or antiparallel manner in heavy ion collisions detected already by RHIC for seven years ago and - even more surprisingly - in proton proton collisions detected by LHC for about two years ago are in terms of color spin glass. In quark gluon plasma one does not expect the correlations. Color spin glass has got support from AdS/CFT correspondence but the model is not fully consistent with the experimental data.

TGD suggests an interpretation in terms of decays of string like objects possible in low energy M₈₉ hadron physics but not in high energy QCD. The 135 GeV particle suggested by Fermi data could be pion of M₈₉ physics rather than dark matter particle.

We must however wait patiently until statistics possibly shows that these effects are real. Until this possibly happens colleagues continue to believe on standard model and direct their efforts to the elimination of new variants of SUSY.

Progress during last year in TGD III: consciousness and quantum biology

From the point of view of physicalism biology and neuroscience can be seen as gigantic collections of anomalies and a treasure trove for a theoretician with an open mind. By its inherent fractality TGD predicts new physics in all scales and the new view about quantum jump and predicts mechanisms of macroscopic quantum coherence make the attempts to explain these anomalies irresistible.

TGD inspired theory of consciousness

The key problem of TGD inspired theory of consciousness has been from the beginning the relationship between geometric time and subjective time assigned to the sequence of quantum jumps with quantum jump identified as moment of consciousness. Zero energy ontology poses powerful constraints on the picture and it is now possible to understand how the arrow of geometric time is induced both at the space-time level and the level of imbedding space.

The resulting vision is in conflict with the existing belief system identifying these two times and known to lead to paradoxes. Situation is same as in the case of revolution implied by special relativity: effects like time dilation were highly counter intuitive in the world view which assumed absolute time.

The identification of precise mechanism for what dualist would cal matter mind interaction are also important and the proposed crazy explanation of the reported psychokinetic effects on bits sequences stored in computer memory could apply also to living matter.

• The mystery of time again
  
  
• Quantum dynamics for the moduli associated with CDs and the arrow of geometric time
  
  
• Two attempts to understand PK
  

The notion of magnetic body

The concept of magnetic body derives from the identification of classical fields in terms of induced gauge potentials and topological field quantization. The hierarchy of Planck constants predicts hierarchy of macrocopically quantum coherent phases assigned with the magnetic bodies. The dynamics of the magnetic body includes phase transitions changing the value of Planck constant and thus inducing change of quantum lengths and reconnections of flux tubes changing the topology of the web formed by the flux tubes. These universal mechanisms allow completely new insights to bio-catalysis, to the ability of biomolecules to find themselves in the dense molecular soup, and on the synchrony of bio-chemical reactions. One obtains also a concrete realization for the idea about living matter as a hologram.

• DC currents of Becker: part I
  
  
• DC currents of Becker: part II
  
  
• DC currents of Becker: part III
  
  
• Negentropic entanglement, metabolism, and acupuncture
  
  
• Could magnetic flux tubes make possible effective holograms
  
  
• DNA, speech, music, and ordinary sound
  
  
• Magnetic body, DNA replication, and mitosis
  
  
• TGD inspired view about water memory
  
  
• Some considerations relating to the dynamics of quasicrystals
  
  

Negentropic entanglement and life in the intersection of matter and mind

The notion of negentropic entanglement encourages the identification of life as something residing in the intersection of realities and p-adicities ("matter and mind"). From TGD viewpoint biology and neurosciences become treasure troves of anomalies with precise data. EEG and its various variants generalize to fractal hierarchies in frequency domain and can be interpreted as a tool for communications between biological and magnetic body. Paranormal phenomena (or whatever word one wants to use) turn out to have an explanation in terms of the mechanisms used by the magnetic body to control biological body and receive sensory input from it.

Metabolism has an interpretation as a manner to transfer or generate negentropic entanglement serving as a quantum correlate of conscious information.

• Quantum view about metabolism
  
  
• How to build a quantum computer from magnetic flux tubes?
  
  
• The study of photosynthesis reveals new physics
  
  
• Metabolic energy, ATP, and negentropic entanglement
  

Models for DNA and genetic code

I have done considerable work to develop speculative models about DNA based on the notion of magnetic body. For instance, the model for DNA as a topological quantum computer relies on the assumption that DNA nucleotides and lipids of cell membrane are connected by magnetic flux tubes whose braiding defines the programs of resulting topological quantum computer. This is just a one particular example since flux tube connections would quite generally make living system to behave as a coherent whole and braiding would realized both tqc programs and memory.

• Constraints on the fermionic realization of genetic code from the model for color qualia
  
  
• A model for remote gene expression
  

Progress during last year in TGD II: theoretical aspects of TGD

In this posting I will summarize the progress made in TGD itself during this year. One category of problems relates to the interpretation of TGD and its relationship to existing theories, to the understanding of the preferred extremals of Kähler action and solutions of the modified Dirac action, to the construction of the generalized Feynman diagrams and to a more precise view about what particles are in this framework. Zero energy ontology is now a basic pillar of TGD and should be understood better. One should also develop the understanding about the fusion of real physics and various p-adic physics inspired by the success of p-adic mass calculations and required by number theoretical universality, about the effective hierarchy of Planck constants predicting dark matter hierarchy, and about hyperfinite factors allowing to realize the notion of finite measurement/cognitive resolution.

The interpretation of TGD

Last years have meant impressive progress in the interpretation of TGD both at classical and quantum level: here one could speak about refinement of the ontology of TGD. These levels are of course related by quantum classical correspondence and this principle has demonstrated its amazing power. The notion of many-sheeted space-time is central.

Basic argument against TGD

Perhaps the strong objection against TGD is that linear superposition for classical fields is lost. The linear superposition is however central starting point of field theories. Many-sheeted space-time allows to circumvent this argument about which I became conscious of just during this year - about 34 year after the discovery of TGD!

The replacement of linear superposition of fields with the superposition of their effecs meaning that sum is replaced with set theoretic union for space-time sheets. This simple observation has far reaching consequences: it becomes possible to replace the dynamics for a multitude of fields with the dynamics of space-time surfaces with only 4 imbedding space coordinates as primary dynamical variables.

• Basic objection against TGD
  
  
• Standing waves in TGD
  

Quantum classical correspondence and quantum ergodicity

Quantum classical correspondence has been one of the guiding principles of TGD. The newest conjecture generated by quantum classical correponds is quantum ergodicity. Quantum ergodicity states that quantal correlation functions for classical field like quantities in zero energy state are identical with the classical correlation functions for a any preferred extremal in their superposition. Single preferred extremal represents entire zero energy state so that all space-time surfaces in the quantum superposition of parallel classical worlds are equivalent observationally.

Although zero energy ontology (S-matrix is relaced with M-matrix definign "square root" of density matrix) and 4-D spin glass degeneracy suggest that this principle is satisfied only by the outcomes of state function reduction, it is extremely powerful if it really works.

The effective hierarchy of Planck constants and dark matter

In TGD framework dark matter could be understand as phases of matter with scaled up value of effective Planck constant. This explanation distinguishes sharply between TGD and competitors and leads to a completely new view about quantum biology based on the notion of magnetic body carrying dark matter. TGD leads also to a view about dark energy as Kähler magnetic energy.

The theoretical understanding of the effective hierarchy of Planck constants in terms of space-time topology has developed during this idea. The newest idea is that the effective n-sheeted cover of imbedding space to which effective value of Planck constant hbareff=nhbar is assigned corresponds to an n-furcation natural in the non-linear dynamics of Kähler action by its enormous vacuum degeneracy. The list about the applications to living matter has been steadily growing.

• Updated view about the hierarchy of Planck constants
  
  
• Dark matter hierarchy and fractional quantum Hall effect
  
  
• What about the relationship of gravitational Planck constant to ordinary Planck constant
  
  
• How the effective hierarchy of Planck constants could reveal itself in condensed matter physics?
  

Strong form of general coordinate invariance and holography

Strong form of general coordinate invariance (GCI) implies strong form of holography. The outcome is TGD counterpart of AdS/CFT correspondence. Bulk is replaced with space-time surface and strings with 2-D string world sheets carrying fermionic fields (right handed neutrino is exception and is delocalized into entire space-time surface). Partonic 2-surfaces at the boundaries of CDs and the 4-D tangent space data at them would code for the quantum dynamics. Interior dynamics would provide classical correlates for quantum states - say classical correlation functions identical to their quantum counterparts if quantum ergodicity holds true.

One outcome is a new view about black holes replacing the interior of blackhole with a space-time region of Euclidian signature of induced metric and identifiable as analogs of lines of generalized Feynman diagrams. In fact, black hole interiors are only special cases of Eucdlian regions which can be assigned to any physical system. This means that the description of condensed matter as AdS blackholes is replaced in TGD framework with description using Euclidian regions of space-time.

• Black holes and TGD
  
  
• Do blacholes and blackhole evaporation have TGD counterparts?
  

Zero energy ontology

Zero energy ontology (ZEO) has inspired detailed development of the views about the relationship between geometric and subjective time and led to highly non-trivial picture challenging the existing beliefs. No final conclusions are possible yet but I believe that the understanding continues to grow. ZEO has led also to a highly detailed view about generalized Feynman diagrammatics.

• Anatomy of quantum jump in zero energy ontology
  
  
• Riemann zeta and quantum theory as square root of thermodynamics
  
  
• The mystery of time again
  
  
• Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology?
  
  
• Does the square root of p-adic thermodynamics exist?
  
  
• Quantum dynamics for the moduli associated with CDs and the arrow of geometric time
  

p-Adic physics

p-Adic physics, p-adic mass calculations, p-adic length scale hypothesis and the integration of various p-adic physics and real physics to a bigger whole have continued to be sources of challenges and inspiration. In biological and neuroscience applications number theoretic entropy identifiable as negentropy has led to a vision about living matter as something residing in the intersection of real and p-adic physics.

• p-Adic homology and finite measurement resolution
  
  
• Constraints on super-conformal invariance from p-adic mass calculations and ZEO
  

Hyperfinite factors

The mathematics of hyperfinite factors is too difficult for an theoretical physicist like me but still considerable progress has taken place. All new visions generate a great burst of ideas and so did also hyperfinite factors as I realized their importance for about seven years ago. The subsequent progress has been mostly selection of the fittest ones with internal consistency defining the most powerful evolutionary pressure. During this year emerged a Fresh view about hyperfinite factors.

Twistors and TGD

Twistors are the hot topic of recent day theoretical physics. ZEO allows to make conjectures about connections with the twistor approach. I do not have the needed technical competence so that I can only try to see the situation at the level of principles.

The basic physical idea of generalized Feynman diagrammatics is that the fermions propagating along the lines of generalized Feynman diagrams (associated with braids at light-like 3-surfaces at which the signature of the induced metric changes) are massless. The condition that also virtual fermions are massless leads to a non-trivial diagrammatics if the sign of the energy can have both values for the virtual wormhole throats. The condition is extremely powerful and eliminates most diagrams and also both IR and UV divergences and allows to understand how the massivation of external particle can be consistent with conformal invariance.

Masslessness is the basic condition making possible to express kinematics in terms of twistors. What one expects is that after functional integration the resulting amplitudes satisfy the Yangian symmetry characterizing also twistor diagrams and fixing them to a high degree. Also Yangian symmetry has a natural generalization in TGD framework.

• Proposal for a twistorial description of generalized Feynman graphs
  
  
• Twistor revolution and TGD
  
  
• Still one attempt to understand generalized Feynman diagrams
  

Preferred extremals and solutions of the modified Dirac equation

The last years have meant rapid progress in the understanding of TGD at the fundamental level. The key concept of TGD is the notion of preferred extremal. It roughly means that one can assign to a collection of 3-surfaces at the ends of a causal diamond (CD) a unique space-time surface. A realization of holography would be in question.

Strong form of GCI leads to the strong form of holography stating that light-like 3-surfaces connecting the ends of CD and space-like 3-surfaces the ends of CD are equivalent choices. Therefore partonic 2-surface defined as their intersections plus 4-D tangent space data at them would fix the quantum physics in ZEO.

One can of course wonder whether the space-like 3-surfaces and light-like 3-surfaces are completely unique from the preferred extremal property or whether they are unique apart from conformal transformations assignable to the light-like coordinate of light-like 3-surface or to the light-like coordinate of δ CD× CP₂. This interpretation would conform with the interpretation of these conformal transformations as gauge transformations. These transformations would extend to ordinary conformal transformations at string world sheets (with Minkowskian signature and hypercomplex structure), which are carriers of spinor fields in the proposed general solution of the modified Dirac equation based on the requirement of the conservation of electric charge.

But what are these preferred extremals? This is the key question. The first guess was that they are absolute minima of Kähler action. This option did not resonate with the number theoretic visions for the simple reason that minimization for p-adic valued functions does not make sense. I have gradually gained rather detailed knowledge about propertie of preferred extremals. For instance, the reduction of the Kähler action to Chern-Simons terms coming from Minkowskian and Euclidian regions (and differing by imaginary unit) gives a partial realization of the holography and boils down to the vanishing of j_{K^μAμ, where jK denotes Kähler current.}

What I regard as a breakthrough came during this year and reduces the construction of preferred extremals to a generalization of the notion of complex structure. In Minkowskian regions of space-time surface I call this structure Hamilton-Jacobi structure identifiable as a composite of complex structure and hyper-complex structure. In Hamilton-Jacob coordinates the field equations reduce to conditions on 4-metric completely analogous to the condition g_zz=0 for Euclidian string world sheets.

The field equations are equivalent to minimal surface equations although action is of course something different from 4-volume which is definitely an unphysical choice. As a consequence, induced gamma matrices satisfy the consistency condition D_μΓ^μ=0 giving additional supercharges besides those associated with modified gamma matrices.

Also Einstein equations with cosmological term follow as a consistency condition requiring that the Maxwell energy momentum tensor has vanishing divergence. Newton's constant and cosmological constant are predictions rather than inputs as in the standard theory. Given space-time sheet has constant value of Ricci scalar which has far reaching consequences.

I already mentioned that also a beautiful solution ansatz for the modified Dirac equations emerges from the conservation of electric charged defined in spinorial sense. All modes of the induced spinor field except right-handed neutrino are localized on 2-D string world sheets. This implies automatically braid picture: the boundaries of the string world sheets at light-like and space-like 3-surfaces are 1-D curves identifiable as light-like and space-like braids strands. This also conforms with the vision that discretization at partonic 2-surfaces identifiable as a space-time correlate of finite measurement resolution follows from the dynamics automatically so that the dynamics is highly self-referential.

• What kind of preferred extremals Maxwell phase could correspond?
  
  
• About deformations of known extremals of Kähler action
  
  
• Under what conditions electric charge is conserved
  
  
• How quaternionicity of space-time could be realized?
  
  
• Recent vision about preferred extremals
  
  
• The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents
  
  
• Quantum criticality and electro-weak gauge symmetries
  
  
• The importance of being light-like
  
  
• About definition of Hamilton-Jacobi structure
  
  
• Gauge symmetries in TGD framework
  
  
• M⁸-H duality, preferred extremals, criticality, and Mandelbrot fractals
  
  
• Preferred extremals of Kähler action as constant curvature manifolds whose geometric invariants are topological invariants
  
  
• How coupling constant evolution could make itself visible as properties of preferred extremals?
  

The improved understanding of the modified Dirac action leads also to a detailed microscopic view about elementary particle as closed string like objects. Knotting is one topological phenomenon possible because of the dimension of space-time surface. This picture has concrete implications for the model of the observed elementary particles.

• The counterpart of Ads₅ duality in TGD
  
  
• Is there a connection between preferred extremals and AdS₄4/CFT correspondence?
  
  
• Electron as a trefoil or something more general?
  

Mathematical ideas inspired by TGD

I do not regard myself as a mathematician in the technical sense of the word. TGD has however forced to generalize existing mathematical concepts, and to even formulate new mathematical notions besides the use of existing mathematics still relatively new for theoretical physicists.

Classical TGD relies on existing mathematical concepts and the main challenges are the understanding of preferred extremals and solutions of the modified Dirac equation. There are several independent conjectures about preferred extremals which should be proven to be right or wrong.

Quantum TGD provides challenging examples of yet non-existing mathematics such as the generalization of loop space geometry to that of WCW and construction of WCW spinor structure. Infinite-dimensional isometry group fixes these structures more or less uniquely but there is huge amount of work to do. The theory hyper-finite factors seems to require an approach generalizing the existing thermodynamical approach to TGD framework which can be seen as a "square root" of thermodynamics. Number theoretical universality requires the fusion of p-adic and real number fields to a larger structure and there are several challenges involved (say precise definition of the notion of integral in p-adic context) attacked also by the leading mathematicians of our time. Infinite primes is a TGD inspired notion having an interpretation as a repeated second quantization of a super-symmetric arithmetic quantum field theory with states labelled by primes and their generalization to infinite primes. There are strong indications that this hierarchy relates directly to various hierarchies of quantum TGD.

During this year a burst of really crazy ideas was inspired by TGD, and I still feel myself rather uneasy with this stuff. Certainly it takes years to find the surviving ideas if any. The key idea is roughly that the standard arithmetics with its sum and product generalize to an arithmetics of Hilbert spaces with sum replaced with direct sum and product with tensor product. This suggests a calculus of Hilbert spaces: one can define Hilbert spaces with dimension which can be also negative integer, rational, algebraic, or even transcendental, one can define Taylor series of Hilbert spaces. By mapping Hilbert space to a single number defined by its dimension one would obtain the ordinary calculus. Everything that can be done with ordinary numbers could be done with Hilbert spaces. This construction can be repeated just like the construction of infinite primes. One can replace the points of Hilbert space with Hilbert spaces and so on... This of course goes completely over the human head.

The question is whether this crazy construction might have some sensible physical interpretation. What made me to take this crazy idea half-seriously was that for generalized Feynman diagrams there are two basic vertices: the generalization of the 3-vertex of ordinary Feynman diagrams having interpretation as tensor product and the generalization of stringy 3-vertex having very natural interpretation in terms of direct sum (in string models the interpretation is in terms of tensor product).

• Number theoretical universality and quantum arithmetics, renormalization, and relation of TGD to N=4 SYM
  
  
• Does 2-adic quantum arithmetics explain p-adic length scale hypothesis?
  
  
• Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?
  
  
• http://matpitka.blogspot.fi/2012/03/hilbert-p-adics-hierarchy-of-planck.html
  
  
• Quantum mechanics as Quantum mathematics
  

Progress during last year in TGD I: Particle physics

In this and subsequent postings I try to present an overview about the basic themes that have motivated blog articles during this year with links to the appropriate postings. This also helps me to get a bird's eye of view to what I have been doing during the year;-).

In these postings I will consider first the experimental side, then theoretical aspects of TGD, and finally quantum biology and TGD inspired theory of consciousness.

One can say that on the experimental side the year has been dominated by Higgs, SUSY, and dark matter. On the theoretical side the developments related to the understanding of the preferred extremals of Kähler action and of solutions of the modified Dirac action have dominated the scene but also other important ideas and insights have emerged. In quantum biology new applications for the notions of magnetic body and negentropic entanglement have emerged. In TGD inspired theory of consciousness zero energy ontology (ZEO) has led to a more detailed view about the relationship between geometric time and experienced time leading to highly non-trivial modification of existing manners of thinking.

The media-hot issues have been mostly in particle physics sector. The buzz words have been Higgs, SUSY, and dark matter. Scaled up variante of hadron physics is one of the most important TGD predictions but represents something totally new for the mainstream and blogger community. The results from LHC and Fermi satellite have been especially interesting in this respect.

1. Higgs issue

Higgs has been a stone in the toe of TGD. The problem has been the lack of classical space-time correlate for it.
No wonder that in the case of Higgs I have developed a large number of alternative scenarios with and without Higgs like particle.

At this moment it seems clear that Higgs like particle exists although it is far from clear whether it has standard model couplings. If TGD has QFT limit and if one believes that Higgs mechanism is the only manner to model the particle massivation in QFT context, then Higgs mechanism would provide a mimicry of p-adic massivation but not its fundamental description. p-Adic thermodynamics is required for a microscopic description. Higgs vacuum expectation could have space-time counterpart at microscopic level and correspond to CP₂ part for the trace of the second fundamental form assignable to string world sheet (if string world sheet is minimal surface in space-time as one might expect, it is not minimal surface in imbedding space (meaning vanishing Higgs expectation) except under very special conditions).

The too high decay rate of Higgs like state to gamma pairs is still reported and the mass of Higgs seems to depend slightly on whether it is determined from the production of gamma pairs or Z pairs. This suggests that also something else than Higgs is there. TGD candidate for this something else would be the pion of M₈₉ hadron physics to be discussed below. By a naive scaling estimate for its width as Γ∼ α_s M one would obtain width of order 20 GeV.

The identification as the 135 GeV particle for which Fermi telescope finds evidence as M₈₉ pion is rather suggestive. This suggests that the anomalously high rate for the production of gamma pairs could be due to the decays of M₈₉ pion providing an additional background. Due to this background also the determination of the mass of the Higgs like state could lead to different results for gamma pairs and Z pairs in ATLAS.

The rate for the production of gamma pairs is somewhat too high up to cm energy of gamma pair of order 200 GeV. May be this effect could be understood in terms of satellites of M₈₉ pion with mass difference of order 20 GeV. These satellites would be scaled up variants of satellites of ordinary pion(and also other hadrons) for which evidence has been found recently and explained in TGD framework in terms of infared Regge trajectories. Of course, not a single particle physicist in CERN takes this kind of idea seriously since ordinary low energy hadron physics is regarded as a closed chapter of particle physics in higher energy circles.

• Is it really Higgs?
  
  
• Indeed, is it really Higgs?
  
  
• Problems with Higgs interpretation of 125 GeV signal
  
  
• At the eve of discovery
  
  
• Is it really Higgs?
  
  
• Higgs as a belief
  
  
• Why Higgs is not favored in TGD framework
  
  
• Is it really Higgs?
  
  
• Is it indeed M₈₉ pion
  
  
• Pseudo-scalar JHiggs as Euclidian pion
  
  
• Cautious conclusions concerning gauge boson massivation
  
  
• The decays of Higgs like particle to bbar pairs observed?
  
  
• Could Higgs mechanism provide a description of p-adic particle massivation at QFT limit?
  
  
• Two possible views about Higgs like states in TGD
  
  
• To deeper waters
  
  
• Higgs like state according to TGD after HCP₂₀₁₂
  
  
• About the basic assumptions behind p-adic mass calculations
  

2. M₈₉ hadron physics

M₈₉ hadron physics is one of the key "almost"-predictions of TGD at LHC. Both Fermi satellite and LHC have provided interesting data in this respect. The standard interpretation for the unexpected correlations for charged particle pairs meaning that they tend move either in parallel or antiparallel manner in heavy ion collisions detected already by RHIC for seven years ago and - even more surprisingly - in proton proton collisions detected by LHC for about two years ago are in terms of color spin glass. In quark gluon plasma one does not expect the correlations. Color spin glass has got support from AdS/CFT correspondence but the model is not fully consistent with the experimental data.

TGD suggests an interpretation in terms of decays of string like objects possible in low energy M₈₉ hadron physics but not in high energy QCD. The 135 GeV particle suggested by Fermi data could be pion of M₈₉ physics rather than dark matter particle.

• Indications for M₈₉ hadron physics from cosmic rays
  
  
• Confirmation for Fermi excess
  
  
• Is it indeed M₈₉ pion?
  
  
• LHC produced might have produced new matter: are M₈₉ hadrons in question?
  

3. New hadron physics suggested by TGD

TGD view about strong interactions differs in many respects from that provided by QCD. In particular, the interpretation of color quantum numbers is not as spin like quantum numbers but in terms of partial waves in CP₂ degrees of freedom. The many-sheeted space-time also leads to a view about both partons and hadrons as 3-D surfaces and the notion of color magnetic body is expected to be central in the description of hadrons at low energies.

There exists recent evidence for satellites of ordinary hadrons with mass differences having the scale of 20-40 MeV. TGD suggest an explanation in terms of new physics assignable to IR color magnetic flux tubes. This physics should make itself visible also in M₈₉ physics via satellites of M₈₉ hadrons, in particular pion whose decays would provide additional gamma pair background perhaps relating to the too high decay rate of Higgs like state to gamma pairs.

• About the construction of mesons and elementary bosons in TGD Universe
  
  
• Low mass exotic mesonic structures as evidence for dark scaled down variants of weak bosons?
  

4. N=1 SUSY

LHC has reported very strong bounds on the parameters of the models based on N=1 SUSY and the models are getting increasinly complicated. This is also a bad news for super string models. In fact, the Russian discoverer of the supersymmetry believes that something is badly wrong with the standard SUSY, and one should try something more imaginative rather than tinkering with models which do not work. He even talks about a lost generation of theoretical physicists. I can only agree.

N =1 SUSY and thus standard SUSY is excluded in TGD framework from the beginning by the dimension 8 of the imbedding space. For long time I however thought that covariantly constant right-handed neutrino could produce it approximately. It seems now that this is not the case although one has different kind of badly broken large N SUSY. The core of argument is that since covariantly constant right handed neutrino decouples from all interactions (even gravitational!), its behavior cannot combine with particle to form sparticle as strongly spin-correlated pair so that right-handed neutrinos behave as their own phase.

• he particle spectrum predicted by TGD and TGD based SUSY
  
  
• The role of the right-handed neutrino in TGD based view about SUSY
  
  
• Realization of large N SUSY in TGD
  
  
• Why standard space-time SUSY is not possible in TGD framework?
  

5. Dark matter

Dark matter is one of the hot topics of the recent day physics. TGD view about various forms of dark matter differs dramatically from the standard views and means different interpretation for the observations interpreted as indications for the existence of dark matter. The hierarchy of phases of matter characterized by an effective value of Planck constant coming as a multiple of Planck constant would behave like dark matter as far as vertices of Feynman diagrams are considered.

Galactic dark matter could be identified as Kähler magnetic energy of magnetic flux tubes originated from primordial cosmic strings. One can assign to these objects a gigantic value of an effective Planck constant as "gravitational Planck constant". The magnetic energy has an identification as dark energy in TGD framework. Distant stars in the galactic plane are predicted to have contant velocity spectrum without any further assumptions. The motion of astrophysical objects would be however free along the cosmic string containing galaxies around it like pearls in necklace.

During the last year Fermi satellite has produced valuable data consistent with TGD view.

• Has Fermi detected dark matter?
  
  
• Standard views about cosmic rays and galactic dark matter in difficulties
  
  
• New blow against standard view about galactic dark matter
  
  
• Dark haze
  
  
• Confirmation for Fermi excess
  
  
• Dark magnetism
  
  
• Giant dark matter bridge between galaxy clusters discovered
  

6. Miscellaneous

There are also other new physics topics related to physics that I have written about.

• New evidence for anomalies of radio-active decay rates
  
  
• Gamma ray diffraction from silicon prism?
  
  
• Do galaxies have preferred handedness?
  

Is there a connection between preferred extremals and AdS4/CFT correspondence?

The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4-D space-times with hyperbolic metric provide canonical representation for a large class of four-manifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals.

4-D hyperbolic space with Minkowski signature is locally isometric with AdS₄. This suggests a connection with AdS₄/CFT correspondence of M-theory. The boundary of AdS would be now replaced with 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of complex coordinate at light-like boundary. AdS would presumably represent a special case of a more general family of space-time surfaces with constant Ricci scalar satisfying Einstein-Maxwell equations and generalizing the AdS₄/CFT correspondence.

For the ordinary AdS₅ correspondence empty M⁴ is identified as boundary. In the recent case the boundary of AdS₄ is replaced with a 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. String world sheets have boundaries along light-like 3-surfaces and space-like 3-surfaces at the light-like boundaries of CD. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of hyper- complex coordinate at light-like 3-surface. AdS₅× S⁵ of M-theory context is replaced by a 4-surface of constant Ricci scalar in 8-D imbedding space M⁴× CP₂ satisfying Einstein-Maxwell equations. A generalization of AdS₄/CFT correspondence would be in question. Note however that the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS₄ instead of AdS₄.

These observations give motivations for finding whether AdS₄ or dS₄ or both allow an imbedding as vacuum extremal to M⁴× S²⊂ M⁴× CP₂, where S² is a homologically trivial geodesic sphere of CP₂. It is easy to guess the general form of the imbedding by writing the line elements of, M⁴, S², and AdS₄.

1. The line element of M⁴ in spherical Minkowski coordinates (m,r_M,θ,φ) reads as
   

   ds²= dm²-dr_M²-r_M²dΩ² .
   
   

2. Also the line element of S² is familiar:
   
   

   ds²=- R²(dΘ²+sin²(θ)dΦ²) .
   
   

3. By visiting in Wikipedia one learns that in spherical coordinate the line element of AdS₄ is given by
   

   ds²= A(r)dt²-(1/A(r))dr²-r²dΩ² ,
   

   A(r)= 1+y² , y = r/r₀ .
   
   

4. From these formulas it is easy to see that the ansatz is of the same general form as for the imbedding of Schwartschild-Nordstöm metric:
   

   m= Λ t+ h(y) , r_M= r ,
   Θ = s(y) , Φ= ω× (t+f(y)) .
   

   The non-trivial conditions on the components of the induced metric are given by
   

   g_tt= Λ²-x²sin²(Θ) = A(r) ,
   

   g_tr= 1/r₀[Λ dh/dy -x²sin²(θ) df/dr]=0 ,
   

   g_rr= 1/r₀²[(dh/dy)² -1- x²sin²(θ)(df/dy)²- R²(dΘ/dy)²]= -1/A(r) ,
   

   x=Rω .
   
   

By some simple algebraic manipulations one can derive expressions for sin(Θ), df/dr and dh/dr.

1. For Θ(r) the equation for g_tt gives the expression
   

   sin²(Θ)= P/x² ,
   

   P= Λ² -A =Λ²-1-y² .
   

   The condition 0≤ sin²(Θ)≤ 1 gives the conditions
   

   (Λ²-x²-1)^1/2 ≤ y≤ (Λ²-1)^1/2 .
   

   Clearly only a spherical shell is possible.
   
   

2. From the vanishing of g_tr one obtains
   

   dh/dy = ( P/Λ)× df/dy ,
   
   
   

3. The condition for g_rr gives
   

   (df/dy)² =[r₀²/AP]× [A^-1-R²(dΘ/dy)²] .
   

   Clearly, the right-hand side is positive if P≥ 0 holds true and RdΘ/dy is small.
   From this condition one can solved by expressing dΘ/dy using chain rule as
   

   (dΘ/dy)²=x²y²/[P (P-x²)] .
   

   One obtains
   

   (df/dy)² = [Λ r₀²y²/AP]× [(1+y²)^-1 -x²(R/r₀)² [P(P-x²)]^-1)] .
   

   The right hand side of this equation is non-negative for certain range of parameters and variable y.
   Note that for r₀>> R the second term on the right hand side can be neglected. In this case it is easy to integrate f(y).
   
   

The conclusion is that AdS₄ allows a local imbedding as a vacuum extremal. Whether also an imbedding as a non-vacuum preferred extremal to homologically non-trivial geodesic sphere is possible, is an interesting question. The only modification in the case of De Sitter space dS₄ is the replacement of the function A= 1+y² appearing in the metric of AdS₄ with A=1-y². Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. p-Adic length scale hypothesis motivates the conjecture that stars indeed have onion-like layered structure consisting of shells, whose radii are consistent with p-adic length scale hypothesis. This brings in mind also Titius-Bode law.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Physics as Infinite-dimensional Geometry", or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".

How coupling constant evolution could make itself visible as properties of preferred extremals?

Quantum classical correspondence states that all aspects of quantum states should have correlates in the geometry of preferred extremals. In particular, various elementary particle propagators should have a representation as properties of preferred extremals. This would allow to realize the old dream about being able to say something interesting about coupling constant evolution although it is not yet possible to calculate the M-matrices and U-matrix. Hitherto everything that has been said about coupling constant evolution has been rather speculative arguments except for the general vision that it reduces to a discrete evolution defined by p-adic length scales. General first principle definitions are much more valuable than ad hoc guesses even if the latter give rise to explicit formulas.

In quantum TGD and also at its QFT limit various correlation functions in given quantum state code for its properties. These correlation functions should have counterparts in the geometry of preferred extremals. Even more: these classical counterparts for a given preferred extremal ought to be identical with the quantum correlation functions for the superposition of preferred extremals.

1. The marvelous implication of quantum ergodicity would be that one could calculate everything solely classically using the classical intuition - the only intuition that we have. Quantum ergodicity would also solve the paradox raised by the quantum classical correspondence for momentum eigenstates. Any preferred extremal in their superposition defining momentum eigenstate should code for the momentum characterizing the superposition itself. This is indeed possible if every extremal in the superposition codes the momentum to the properties of classical correlation functions which are identical for all of them.
   
   
2. The only manner to possibly achieve quantum ergodicity is in terms of the statistical properties of the preferred extremals. It should be possible to generalize the ergodic theorem stating that the properties of statistical ensemble are represented by single space-time evolution in the ensemble of time evolutions. Quantum superposition of classical worlds would effectively reduce to single classical world as far as classical correlation functions are considered. The notion of finite measurement resolution suggests that one must state this more precisely by adding that classical correlation functions are calculated in a given UV and IR resolutions meaning UV cutoff defined by the smallest CD and IR cutoff defined by the largest CD present.
   
   
3. The skeptic inside me immediately argues that TGD Universe is 4-D spin glass so that this quantum ergodic theorem must be broken. In the case of the ordinary spin classes one has not only statistical average for a fixed Hamiltonian but a statistical average over Hamiltonians. There is a probability distribution over the coupling parameters appearing in the Hamiltonian. Maybe the quantum counterpart of this is needed to predict the physically measurable correlation functions.
   

   Could this average be an ordinary classical statistical average over quantum states with different classical correlation functions? This kind of average is indeed taken in density matrix formalism. Or could it be that the square root of thermodynamics defined by ZEO actually gives automatically rise to this average? The eigenvalues of the "hermitian square root " of the density matrix would code for components of the state characterized by different classical correlation functions. One could assign these contributions to different "phases".
   
   

4. Quantum classical correspondence in statistical sense would be very much like holography (now individual classical state represents the entire quantum state). Quantum ergodicity would pose a rather strong constraint on quantum states. This symmetry principle could actually fix the spectrum of zero energy states to a high degree and fix therefore the M-matrices given by the product of hermitian square root of density matrix and unitary S-matrix and unitary U-matrix having M-matrices as its orthonormal rows.
   
   
5. In TGD inspired theory of consciousness the counterpart of quantum ergodicity is the postulate that the space-time geometry provides a symbolic representation for the quantum states and also for the contents of consciousness assignable to quantum jumps between quantum states. Quantum ergodicity would realize this strongly self-referential looking condition. The positive and negative energy parts of zero energy state would be analogous to the initial and final states of quantum jump and the classical correlation functions would code for the contents of consciousness like written formulas code for the thoughts of mathematician and provide a sensory feedback.
   
   

How classical correlation functions should be defined?

1. General Coordinate Invariance and Lorentz invariance are the basic constraints on the definition. These are achieved for the space-time regions with Minkowskian signature and 4-D M⁴ projection if linear Minkowski coordinates are used. This is equivalent with the contraction of the indices of tensor fields with the space-time projections of M⁴ Killing vector fields representing translations. Accepting ths generalization, there is no need to restrict oneself to 4-D M⁴ projection and one can also consider also Euclidian regions identifiable as lines of generalized Feynman diagrams.
   

   Quantum ergodicity very probably however forces to restrict the consideration to Minkowskian and Euclidian space-time regions and various phases associated with them. Also CP₂ Killing vector fields can be projected to space-time surface and give a representation for classical gluon fields. These in turn can be contracted with M⁴ Killing vectors giving rise to gluon fields as analogs of graviton fields but with second polarization index replaced with color index.
   
   

2. The standard definition for the correlation functions associated with classical time evolution is the appropriate starting point. The correlation function G_XY(τ) for two dynamical variables X(t) and Y(t) is defined as the average G_XY(τ)=∫_T X(t)Y(t+τ)dt/T over an interval of length T, and one can also consider the limit T→ ∞. In the recent case one would replace kenotau with the difference m₁-m₂=m of M⁴ coordinates of two points at the preferred extremal and integrate over the points of the extremal to get the average. The finite time interval T is replaced with the volume of causal diamond in a given length scale. Zero energy state with given quantum numbers for positive and negative energy parts of the state defines the initial and final states between which the fields appearing in the correlation functions are defined.
   
   
3. What correlation functions should be considered? Certainly one could calculate correlation functions for the induced spinor connection given electro-weak propagators and correlation functions for CP₂ Killing vector fields giving correlation functions for gluon fields using the description in terms of Killing vector fields. If one can uniquely separate from the Fourier transform uniquely a term of form Z/(p²-m²) by its momentum dependence, the coefficient Z can be identified as coupling constant squared for the corresponding gauge potential component and one can in principle deduce coupling constant evolution purely classically. One can imagine of calculating spinorial propagators for string world sheets in the same manner. Note that also the dependence on color quantum numbers would be present so that in principle all that is needed could be calculated for a single preferred extremal without the need to construct QFT limit and to introduce color quantum numbers of fermions as spin like quantum numbers (color quantum numbers corresponds to CP₂ partial wave for the tip of the CD assigned with the particle).
   
   
4. What about Higgs like field? TGD in principle allows scalar and pseudo-scalars which could be called Higgs like states. These states are however not necessary for particle massivation although they can represent particle massivation and must do so if one assumes that QFT limit exist. p-Adic thermodynamics however describes particle massivation microscopically.
   

   The problem is that Higgs like field does not seem to have any obvious space-time correlate. The trace of the second fundamental form is the obvious candidate but vanishes for preferred extremals which are both minimal surfaces and solutions of Einstein Maxwell equations with cosmological constant. If the string world sheets at which all spinor components except right handed neutrino are localized for the general solution ansatz of the modified Dirac equation, the corresponding second fundamental form at the level of imbedding space defines a candidate for classical Higgs field. A natural expectation is that string world sheets are minimal surfaces of space-time surface. In general they are however not minimal surfaces of the imbedding space so that one might achieve a microscopic definition of classical Higgs field and its vacuum expectation value as an average of one point correlation function over the string world sheet.
   
   

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".