## Saturday, March 26, 2011

### Does high temperature super-conductivity involve a new phase of matter?

Kram sent a link to a Science Daily popular article titled "High-Temperature Superconductor Spills Secret: A New Phase of Matter?" (see also this). For more details see the article in Science.

Zhi-Xun Shen of the Stanford Institute for Materials and Energy Science (SIMES), a joint institute of the Department of Energy's SLAC National Accelerator Laboratory and Stanford University led the team of researchers, which discovered that in the temperature region between the pseudo gap temperature and genuine temperature for the transition to super-conducting phase there exists a new phase of matter. The new phase would not be super-conducting but would be characterized by an order of its own which remains to be understood. This phase would be present also in the super-conducting phase.

The announcement does not come as a complete surprise for me. For few years ago I developed the model high Tc superconductivity (see this and this) and a new phase of matter is what TGD indeed predicts. This phase would consist of Cooper pairs of electrons with a large value of Planck constant but associated with magnetic flux tubes with short length so that no macroscopic supra currents would be possible.

The transition to super-conducting phase involves long range fluctuations at quantum criticality and the analog of a phenomenon known as percolation. For instance, the phenomenon occurs for the filtering of fluids through porous materials. At critical threshold the entire filter suddenly wets as fluid gets through the filter. Now this phenomenon occurs for magnetic flux tubes carrying the Cooper pairs. At criticality the short magnetic flux tubes fuse by reconnection to form long ones so that supra currents in macroscopic scales become possible.

It is not clear whether this prediction is consistent with the finding of Shen and others. The simultaneous presence of short and long flux tubes in macroscopically super-conducting phase is certainly consistent with TGD prediction. The situation depends on what one means with super-conductivity. Is super-conductivity super-conductivity in macroscopic scales only or should one call also short scale super-conductivity not giving rise to macroscopic super currents as super-conductivity. In other words: do the findings of Shen's team prove that the electrons above gap temperature do not form Cooper pairs or only that there are no macroscopic supra currents?

Whether the model works as such or not is not a life and death question for the TGD based model. One can quite well imagine that the first phase transition increasing hbar does not yet produce electron Compton lengths long enough to guarantee that the overlap criterion for the formation of Cooper pairs is satisfied. The second phase transition increasing hbar would do this and also scale up the lengths of magnetic flux tubes making possible the flow of supra currents as such even without reconnections. Also reconnections making possible the formation of very long flux tubes could be involved and would be made possible by the increase in the length of flux tubes.

For background see the chapter Super-Conductivity in Many-Sheeted Space-Time of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" or the chapter Bio-Systems as Super-Conductors: Part I of "Quantum Hardware of Living Matter".

## Wednesday, March 23, 2011

### Summary about breakthrough in the construction of U-matrix and understanding of twistorialization

During last few days a dramatic development in the understanding of the notions of U-matrix, M-matrix and S-matrix- a trinity of matrices replacing in zero energy ontology the notion of S-matrix of positive energy ontology and allowing fusion of thermodynamics with quantum theory. Also twistorialization reduces to pure group theory-albeit infinite-dimensional: zero energy states define a Yangian algebra generalizing that of Grassmannian twistor approach. For this reason I feel it as appropriate to add a further posting trying to give bird's eye of view about the ideas since previous posting is rather long and detailed and reader easily loses the thread.

1. The realization that the hermitian square roots of density matrices form infinite-D unitary algebra and that their commutativity with universal S-matrix implies that zero energy states define the generalization of Kac-Moody algebra became only after I had realized the possibility to construct U-matrix. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1). It is this observation which reduces the construction of U-matrix (or matrices if they form algebra) to that for S is expected to correspond directly to the ordinary S-matrix. A possible interpretation of the algebra of U-matrices is in terms of scales of CDs coming as positive integer powers of two. Another possibility more in line with the usual interpretation of S-matrix as time evolution operator is that scales of CDs come as integers and these integers correspond to powers of S.

What is so fascinating is that zero energy states themselves define the symmetry algebra of the theory and that this algebra can be interpreted as a generalization of Yangian responsible for the successes of Grassmannian twistor approach by replacing finite-dimensional conformal group of Minkowski space with infinite-dimensional super-conformal algebras associated with partonic 2-surfaces in accordance with the replacement of point-like particles with surfaces. The basic characteristic of Yangian algebra is the multilocality of its generators and zero energy states are indeed multilocal since they involve partonic surfaces at both light-like boundaries of CD. Quantum TGD reduces to pure group theory! Note only states but also dynamics is coded completely by symmetries since M-matrices code for quantum dynamics! This powerful aspect of zero energy ontology I have not realized before.

2. In ordinary QFT Feynman diagrams are purely algebraic objects. In TGD framework they reduce to space-time topology and geometry with Euclidian regions of space-time surfaces having interpretation as generalized Feynman diagrams. At the vertices of generalized Feynman diagrams in coming partonic 2-surfaces meet just like in ordinary Feynman diagrams which means deep difference from string theory. A more general assumption is that entire 4-D lines of generalized Feynman diagram meet at vertices. This could apply to the Euclidian regions only.

There is also a second kind of branching involved with the hierarchy of Planck constants. In Minkowskian regions similar meeting would take place for the branches of space-time sheets with same values of canonical momentum densities of Kähler action at the ends of CDs and have interpretation in terms of fractionization and hierarchy of Planck constants. The value of Planck constant for single branch would be effectively and integer multiple of the ordinary one. For the entire multi-sheeted structure describable naturally in terms of singular covering space of M4× CP2 it would be just the ordinary value.

3. Zero energy ontology with massless external wormhole wormholes implies as such twistorialization of the theory although external wormhole momenta must be assumed to be massive bounds states of massless throats (see this). This also guarantees exact Yangian symmetry and the absence of IR divergences. If also virtual wormhole throats are massless, twistorialization takes place in strong sense. This is possible only in zero energy ontology and accepting the identification of wormhole throats as basic building blocks of particles.

4. The notion of a bosonic emergence means that bosonic propagators emerge as radiative loops for wormhole contacts. The emergence generalizes to all states associated with wormhole contacts and also to flux tubes having wormhole contacts at their ends. What is nice that coupling constants emerge as normalization factors of propagators. Note that for single wormhole throat as opposed to wormhole contact having two throats bosonic propagator would result as a product of two collinear fermionic propagators and have the standard form. For states with higher total number of fermions and anti-fermions the propagator of wormhole throat behaves as pn, n>2. Here however p is replaced with what I call pseudo-momentum.

5. Number theoretical universality suggest that at given level (CD) only finite sum of diagrams appears: otherwise there is a danger that one obtains sum of rational functions which is not rational anymore. This gives strong constraints on generalized Feynman diagrams at the lowest level of the hierarchy. This follows naturally if twistor diagrams are identified as sums of Feynman diagrams which are irreducible in the sense that they do not represent two subsequence scatterings. Only these diagrams contribute to twistor diagram and the number of these diagrams is finite if all particles have small mass (even photon which would eat the remaining Higgs component).

6. Category theoretical approach to TGD based on planar operad proposed for few years ago fits nicely with the twistorial construction of amplitudes interpreting radiative corrections in terms of CDs within CDs picture. The generalized Feynman diagrams with radiative corrections define the analog of planar operad with disk containing within itself disks containing.... replaced with causal diamond containing causal diamonds containing....

To make this more concrete it deserves to describe TGD counterparts for the recursion formula and duality between descriptions using twistors and momentum twistors.

1. The great victory of twistor approach is the recursion formula for the amplitudes (see also the representation in TGD framework) applying to all planar diagrams of N=4 SYM becoming an exact formula at the large N limit for gauge group SU(N). In the recent case the infinite-dimensional character of the Yangian symmetry algebra of S-matrix could be correlate for large N limit so that the planar limit should make sense. Also the fact that string worlds sheets are an essential aspect of TGD approach suggests that stringy picture deduced by t'Hooft for gauge theories at this limit implies planarity. What is relevant in the recent case is the general structure of the reduction formula, not the details which as such are of course extremely interesting also in TGD framework since Grassmannian amplitudes are claimed to provide a universal representation of Yangian invariants.

The recursive formula expresses scattering amplitude with n external particles with k negative helicities up to l loops is expressible as a sum of two terms. The first term-referred to as classical contribution- involves a fusion of twistor amplitudes with smaller number of particles and with the number of loops not larger than l by a procedure used already for tree diagrams. Second term - called quantum contribution- involves l loops and is irreducible in the sense that it is not expressible as a fusion of lower amplitudes and is obtained from n+2 particle by a process eliminating two particles. The identification of the TGD counterparts of these terms is obvious. The "classical" term corresponds to the proposed fusion of the lower level amplitudes associated with polygons for sub-CDs. The "quantum" term corresponds to the contribution appearing at the level of CD itself and involves genuine loops in Feynman sense but only a finite number of them.

Since zero energy states correspond to generators of Yangian algebra or rather- its Kac-Moody variant with integer power of phase factor identified as integer power of S, the recursion formula might allow an interpretation as a direct counterpart for the recursive definition of Yangian algebra in terms of relations allowing the construction of generators labeled by non-negative integers.

2. One of the fascinating findings of twistor Grassmannian approach is that conformal invariance and its dual correspond in twistor approach to descriptions in terms of twistors in ordinary Minkowski space by starting from Feynman diagrams and in terms of momentum twistors in its dual by starting from Wilson loops. Also this duality has counterpart in TGD.

String world sheets are an essential part of quantum TGD and the translation of Witten's work with knots to TGD context led to a precise identification of string world sheets and a deep connection between TGD and the theory of knots, braids, braid cobordisms, and 2-knots emerges (see this).

Amusingly, the basic idea of this connection emerged from the model of DNA as topological quantum computer developed for few years ago. The braiding defining the quantum computation is time-like and can be illustrated using dance metaphor: the world lines of dancers define the running topological computation program. If you connect the feet of dancers to a wall with threads (dancers are lipids at cell membrane forming 2-D liquid, wall is represented by DNA nucleotide sequence, and threads are magnetic flux tubes), the threads entangle during dance and give rise to a space-like braiding and code the computer program to memory: a fundamental mechanism of memory. These braidings are clearly dual and this duality relates closely to the duality just to the duality between Feynman graphs and Wilson loops! The time evolution of this space-like braiding defines braid cobordism and also a 2-knot.

The natural implication of strong form of holography made possible by preferred extremal (Bohr orbit in generalized sense) property of space-time surfaces is that the descriptions in terms of string world sheets and partonic 2-surfaces are dual. The twistorial representation of this duality is as the duality of descriptions in terms of Feynman diagrammatics in ordinary space-time and Wilson sheets-rather than loops- in the dual space-time assigned with region momenta.

To conclude, it seems that all the basic visions about quantum TGD combine to single coherent whole! Maybe the long march is in some sense over!

For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-Matrix". The recent results are summarized at the end of the chapter. I added to my homepage also an article summarizing the recent progress.

By the way, there is excellent video of a lecture by Nima Arkani-Hamed to mathematicians with title Grassmannians, Polytopes and Quantum Field Theory. If you identify yourself as a theoretician do not miss it! Just the opportunity to experience Nima's enthusiasm, joy, and energy convinces that theoretical physics can also be an exploration and adventure.

### Universal formula for the hermitian square roots of density matrix

The construction of U-matrices discussed in previous posting led as a side product to a general formula for the M-matrices. Although the result is added to the previous posting, it deserves a separate discussion since it can be seen as a victory of the zero energy ontology providing zero energy states with Lie algebra structure and allowing to interpret hermitian square roots of density matrices as observables commuting with universal S-matrix. A generalization of Kac-Moody algebra emerges in which powers of S correspond to powers exp(i n φ).

Zero energy ontology replaces S-matrix with M-matrix and groups M-matrices to rows of U-matrix. S-matrix appears as factor in the decomposition of M-matrix to a product of hermitian square root of density matrix and unitary S-matrix interpreted in standard sense.

Mi1/2i .

Note that one cannot drop the S-matrix factor from M-matrix since M-matrix is neither unitary nor hermitian and the dropping of S would make it hermitian. The analog of the decomposition of M-matrix to the decomposition of Schrödinger amplitude to a product of its modulus and of phase is obvious.

The interpretation is in terms of square root of thermodynamics. This interpretation should give the analogs of the Feynman rules ordinary quantum theory producing unitary matrix when one has pure quantum states so that density matrix is projector in 1-D sub-space of state space (for hyper-finite factors of type II1 something more complex is required).

This is the case. M-matrices are in this case just the projections of S-matrix to 1-D subspaces defined by the rows of S-matrix. The state basis is naturally such that the positive energy states at the lower boundary of CD have well-defined quantum numbers and superposition of zero energy states does not contain different quantum numbers for the positive energy states. The state at the upper boundary of CD is the state resulting in the interaction of the particles of the initial state. Unitary of the resulting U-matrix reduces to that for S-matrix.

A more general situation allows square roots of density matrices which are diagonalizable hermitian matrices satisfying the orthogonality condition that the traces

Tr(ρ1/2iρ1/2j)=δij .

The matrices span the Lie algebra of infinite-dimensional unitary group. The hermitian square roots of M-matrices would reduce to the Lie algebra of infinite-D unitary group. This does not hold true for zero energy states.

If one however assumes that S commutes with the algebra spanned by the square roots of density matrices and allows powers of S one obtains a larger algebra complely analogous to Kac-Moody algebra in the sense that powers of S takes the role of powers of exp(i nφ) in Kac-Moody algebra generators. The commutativity of S and density matrices means that the square roots of density matrices span symmetry algebra of S. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1) so that the symmetry algebra is huge in infinite-D case.

A possible interpretation for the sub-space spanned by M-matrices proportional to Sn is in terms of the hierarchy of CDs. If one assumes that the size scales of CDs come as octaves 2m of a fundamental scale then one would have m=n. Second possibility is that scales of CDs come as integer multiples of the CP2 scale: in this case the interpretation of n would be as this integer: this interpretation conforms with the intuitive picture about S as TGD counterpart of time evolution operator. This interpretation could also make sense for the M-matrice associated with the hierarchy of dark matter for which the scales of CDs indeed come as integers multiples of the basic scale.

If the square roots of density matrices are required to have only non-negative eigenvalues -as I have carelessly proposed in some contexts,- only projection operators are possible for Cartan algebra so that only pure states are possible. If one allows both signs one can have more interesting density matrices and this is the only manner to obtain square root of thermodynamics. Note that the standard representation for the Cartan algebra of finite-dimensional Lie group corresponds to non-pure state. For ρ=Id one obtains M=S defining the ordinary S-matrix. The orthogonality of this zero energy state with respect to other ones requires

Tr(ρi1/2)=0

stating that SU(N=∞) Lie algebra element is in question.

The reduction of the construction of U to that of S is an enormous simplification and reduces to the problem of finding the TGD counterpart of S-matrix. Note that the finiteness of the norm of SS=Id requires that hyper-finite factor of type II1 is in question with the definining property that the infinite-dimensional unit matrix has unit norm. This means that state function reduction is always possible only into an infinite-dimensional subspace only.

The natural guess is that the Lie algebra generated by zero energy states is just the generalization of the Yangian symmetry algebra of N=4 SUSY postulated to be a symmetry algebra of TGD (see this). The characteristic feature of the Yangian algebra is the multi-locality of its generators. The generators of the zero energy algebra are zero energy states and indeed form a hierarchy of multi-local objects defined by partonic 2-surfaces at upper and lower light-like boundaries of causal diamonds. Zero energy states themselves would define the symmetry algebra of the theory and the construction of quantum TGD also at the level of dynamics -not only quantum states in sense of positive energy ontology- would reduce to the construction of infinite-dimensional Lie-algebra! It is hard to imagine anything simpler!

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

## Sunday, March 20, 2011

For some reason the comments to the most recent postings are not visible in blog although I have received the emails. My highest hope is that I am not a victim of virus attack again. The last really massive virus attack spoiled my Christmas and it took two months to recover from it. It also forced to change Windows to Mac. My sincere wish is that the problem could be cured. If someone has encountered similar problem and got rid of it somehow, I would be happy to hear the recipe. In any case, sorry for inconvenience.

## Friday, March 18, 2011

### The master formula for the U-matrix finally found?

In zero energy ontology U-matrix replaces S-matrix as the fundamental object characterizing the predictions of the theory. U-matrix is defined between zero energy states and its orthogonal rows define what I call M-matrices, which are analogous to thermal S-matrices of thermal QFTs. M-matrix defines the time-like entanglement coefficients between positive and negative energy parts of the zero energy state.

A dramatic development of ideas related to the construction of U-matrix has taken place during lthe ast year. In particular, twistorialization becomes possible in zero energy ontology and leads to the generalization of the Yangian symmetry of N=4 SUSY to TGD framework with the replacement of finite-dimensional super-conformal group of M4 with infinite-D super-conformal group assignable to partonic 2-surfaces. What is so beautiful is that the physical IR cutoff due to the formation of bound states of massless wormhole throats resolves the infrared divergence problem whereas UV divergences are solved by on mass shell propagation of wormhole throats for virtual particles. This also guarantees that Yangian invariance is not lost. There are excellent reasons to expect that the twistorial constructions generalize.

What is the master formula for the U-matrix?

The basic challenge is however still there and boils down to a simple question represented in the title. This master formula should be something extremely simple and should generalize the formula for S-matrix defined between positive energy states and identified formally as the exponential of Hamiltonian operator. In TGD framework the notion of unitary time development is given up so that something else is required and this something else should be manifestly Lorentz invariant and characterize the interactions.

This formula should be something extremely simple and should generalize the formula for S-matrix defined between positive energy states and identified formally as the exponential of Hamiltonian operator. In TGD framework the notion of unitary time development is given up so that something else is required and this something else should be manifestly Lorentz invariant and characterize the interactions.

Thinking the problem from this point of view allows only one answer: replace the time evolution operator defined by the Hamiltonian with the exponent for the action containing both bosonic and fermionic term. Bosonic term is the action for preferred extremal of Kähler action, which is indeed the unique Lorentz invariant defining interactions! Fermionic term would given by Chern-Simons Dirac action associated with light-like three surfaces and space-like 3-surfaces at the ends of CDs. The formula is as simple as it is obvious and still I had to use 32 years to discover it!

It took however one day to realize that the situation is not so simple as one might think first. The question is whether this action should be interpreted as the counterpart of action or effective action obtained by performing path integral in presence of external sources in QFT framework. Since one restricts space-time surfaces to preferred extremals so that there is no path integral, the only possible interpretation as the effective action. Also the condition that one obtains fermionic propagators correctly allows only this interpretation. For the Chern-Simons Dirac action the propagator would be the inverse of the correct propagator which obviously makes no sense. For the corresponding effective action the kinetic term is replaced with propagator and correct fermionic Feynman rules result when spinor basis selected to represent generalized eigenstates of the Chern-Simons Dirac operator.

The action interpreted as a counterpart of QFT effective action reduces to the sum of fermionic and bosonic terms. To make the representation more fluent I will mean with 3-surfaces in the following either the light-like orbits of wormhole throats at which the signature of the induced metric changes or the ends of space-time sheets at the boundaries of CDs. Note that it is possible to have CDs within CDs and these give rise to loop corrections having interpretation as zero energy states in shorter length scale. Finite measurement resolution means that one integrates over these degrees of freedom below the resolution scale. This gives rise to discrete variant of gauge coupling evolution based on scalings by factor two for CDs.

The next unpleasant question was whether this U-matrix is actually only the S-matrix appearing in the expression of a given M-matrix as a product of a hermitian square root of density matrix and unitary S-matrix having interpretation as the TGD counterpart of the ordinary S-matrix. The physical picture suggests this strongly. This observation led to a realization that the square roots of density matrices can be identified as generators of infinite-dimensional Lie-algebra of unitary matrices. Unit norm requires that hyper-finite factor of type II1 is in question. The basic implication is that the whole construction reduces to that for unitary S-matrix. This picture looks rather realistic but many details remain to be clarified and I have added lacking details to the posting gradually.

Universal formula for the hermitian square roots of density matrix

Zero energy ontology replaces S-matrix with M-matrix and groups M-matrices to rows of U-matrix. S-matrix appears as factor in the decomposition of M-matrix to a product of hermitian square root of density matrix and unitary S-matrix interpreted in standard sense.

Mi1/2i .

Note that one cannot drop the S-matrix factor from M-matrix since M-matrix is neither unitary nor hermitian and the dropping of S would make it hermitian. The analog of the decomposition of M-matrix to the decomposition of Schrödinger amplitude to a product of its modulus and of phase is obvious.

The interpretation is in terms of square root of thermodynamics. This interpretation should give the analogs of the Feynman rules ordinary quantum theory producing unitary matrix when one has pure quantum states so that density matrix is projector in 1-D sub-space of state space (for hyper-finite factors of type II1 something more complex is required).

This is the case. M-matrices are in this case just the projections of S-matrix to 1-D subspaces defined by the rows of S-matrix. The state basis is naturally such that the positive energy states at the lower boundary of CD have well-defined quantum numbers and superposition of zero energy states does not contain different quantum numbers for the positive energy states. The state at the upper boundary of CD is the state resulting in the interaction of the particles of the initial state. Unitary of the resulting U-matrix reduces to that for S-matrix.

A more general situation allows square roots of density matrices which are diagonalizable hermitian matrices satisfying the orthogonality condition that the traces

Tr(ρ1/2iρ1/2j)=δij .

The matrices span the Lie algebra of infinite-dimensional unitary group. The hermitian square roots of M-matrices would reduce to the Lie algebra of infinite-D unitary group. This does not hold true for zero energy states.

If one however assumes that S commutes with the algebra spanned by the square roots of density matrices and allows powers of S one obtains a larger algebra complely analogous to Kac-Moody algebra in the sense that powers of S takes the role of powers of exp(i nφ) in Kac-Moody algebra generators. The commutativity of S and density matrices means that the square roots of density matrices span symmetry algebra of S.

A possible interpretation for the sub-space spanned by M-matrices proportional to Sn is in terms of the hierarchy of CDs. If one assumes that the size scales of CDs come as octaves 2m of a fundamental scale then one would have m=n. Second possibility is that scales of CDs come as integer multiples of the CP2 scale: in this case the interpretation of n would be as this integer: this interpretation conforms with the intuitive picture about S as TGD counterpart of time evolution operator. This interpretation could also make sense for the M-matrice associated with the hierarchy of dark matter for which the scales of CDs indeed come as integers multiples of the basic scale.

If the square roots of density matrices are required to have only non-negative eigenvalues -as I have carelessly proposed in some contexts,- only projection operators are possible for Cartan algebra so that only pure states are possible. If one allows both signs one can have more interesting density matrices and this is the only manner to obtain square root of thermodynamics. Note that the standard representation for the Cartan algebra of finite-dimensional Lie group corresponds to non-pure state. For ρ=Id one obtains M=S defining the ordinary S-matrix. The orthogonality of this zero energy state with respect to other ones requires

Tr(ρi1/2)=0

stating that SU(N=∞) Lie algebra element is in question.

The reduction of the construction of U to that of S is an enormous simplification and reduces to the problem of finding the TGD counterpart of S-matrix. Note that the finiteness of the norm of SS=Id requires that hyper-finite factor of type II1 is in question with the definining property that the infinite-dimensional unit matrix has unit norm. This means that state function reduction is always possible only into an infinite-dimensional subspace only.

The natural guess is that the Lie algebra generated by zero energy states is just the generalization of the Yangian symmetry algebra of N=4 SUSY postulated to be a symmetry algebra of TGD (see this). The characteristic feature of the Yangian algebra is the multi-locality of its generators. The generators of the zero energy algebra are zero energy states and indeed form a hierarchy of multi-local objects defined by partonic 2-surfaces at upper and lower light-like boundaries of causal diamonds. Zero energy states themselves would define the symmetry algebra of the theory and the construction of quantum TGD also at the level of dynamics -not only quantum states in sense of positive energy ontology- would reduce to the construction of infinite-dimensional Lie-algebra! It is hard to imagine anything simpler!

Bosonic part of the action

Consider now the bosonic part of the action in detail.

1. The first term is the exponent of Kähler action which is purely classical quantity defining vacuum functional as the exponent of the modified Dirac action for the interior. Since there is no path integral over 4-surfaces, the only possible interpretation for Kähler action is as the counterpart of the effective action of quantum field theories to which one can indeed assign unique field pattern one the boundary values are fixed. For the preferred extremals with boundary conditions satisfying the weak form of electric-magnetic duality Kähler action reduces to Chern-Simons term with a constraint guaranteing the weak form of electric-magnetic duality. This constraint implies that the theory does not reduce to topological QFT. One must perform functional integral over 3-surfaces.

2. What is interesting that the Kähler action reduces to Chern-Simons action with constraint term. Could one replace exponent of real Kähler action with the imaginary one so that the situation would resemble very strongly ordinary QFT? Note however that one can also consider the replacement of imaginary unit with real unit in Chern-Simons action exponential and that in Abelian case the quantization argument for the coefficient of Chern-Simons action does not apply: the coefficient is however fixed by the weak form of electric-magnetic duality. In fact unitarity does not allow imaginary exponent: a simpler example is function space endowed with inner product defined by integration with weighting by exponent of some function. Unitarity requires real exponent.

3. Bosonic term involves also measurement interaction term which formally reduces to an addition of gauge part to Kähler gauge potential linear in momentum, color isospin and hyper charge, and possible other measured quantum numbers. This term couples space-time geometry to conserved quantum numbers and in this manner guarantees quantum classical correspondence. This term is added either to interior or with opposite sign to 3-surfaces but not both and therefore does not reduce to gauge transform. This term induces to Chern-Simons term at boundary an effective gauge term as addition to the induced Kähler gauge potential appearing in the Chern-Simons Dirac action. There it is not necessary add this term separately as done earlier.

Fermionic part of the action

It took some time to understand the identification of the fermionic term of the action.

1. By holography the fermionic term should reduce to modified Chern-Simons Dirac action with kinetic term replaced with its inverse. Otherwise kinetic term would replace propagator in the perturbative expansion. This replacement is new as compared to the earlier work.

2. The assumption familiar already from earlier work \cite{Dirac} is that spinors are generalized eigen modes of Chern-Simons Dirac operator with eigenvalues given by λ-1γk, where λk having only M4 components is what I have called pseudo-momentum having region momentum as in Grassmannian approach to twistorialization. This gives the analog of massless propagator. The natural assumption is that pseudo-momenta relate to the massless incoming and outgoing momenta propagating along wormhole lines via twistorial formula: in other words, the difference of pseudomomenta in the vertex of polygon to which external particle line is attached equals to the incoming real massless momentum. This allows to identify virtual particles as composites of massless wormhole throats. Incoming particles consists also of massless wormhole throats but are bound states so that their mass is quantized. The precise relationship between pseudo-momenta and real massless momenta in loops remains to be understood.

3. One could postulate the form of the fermionic effective action directly. It is also possible to obtain it by interpreting Chern-Simons Dirac action as being associated with primary spinor field and the spinor fields associated with the interior as the analog of external spinor source. These fields can be coupled to each other in standard manner by the term ΨbarΦ + ΦbarΨ, which couples quark and lepton chiralities but does not lead to the breaking of baryon and lepton number conservation in perturbation theory as terms of form ΨbarΨ and ΦbarΦ would lead. The Grassmannian path integral over Φ gives the fermionic effective action as the integral of Ψbar D-1CSDΨ over 3-surface with D-1 identified as the propagator for Chern-Simons Dirac action. The assumption that spinors are generalized eigenmodes of D at the 3-surface implies the reduction of propagator to 1/&lambdakγk in the basis of generalized eigen modes.
4. In the spirit of holography the resulting fermionic effective action reduces to the terms assignable to 3-surfaces (as defined above) since in the interior Kähler Dirac equation is satisfied. Although Kähler Dirac action vanishes, its function of Kähler Dirac equation is highly non-trivial in holography since it correlates the modes of the induced spinor fields at different wormhole throats. One ask whether one should add to the fermionic effective action also measurement interaction term. Since this term correspond formally to a gauge term in Kähler gauge potential and is already induced by the corresponding bosonic term, the addition of this term seems un-necessary.

Definition of U-matrix

The definition of U-matrix would be shockingly simple. Just the exponential of the measurement interaction term Chern-Simons Dirac action besides Kähler action reducing to Chern-Simons term and defining the weight for the functional integral over 3-surface. What is encouraging that the resulting U-matrix would be more or less the same as the one expected on basis of heuristic considerations.

1. The basis for bare zero energy states is obtained by using pairs of positive and negative energy states assigned to the boundaries of CD and having opposite quantum numbers. The action of the exponent of Kähler action generates from these states "dressed" states and U-matrix is defined between these stressed states and bare states. M-matrix in turn is defined by the action of L on given bare zero energy states as entanglement coefficients.

2. U-matrix is automatically unitary in the fermionic degrees of freedom since the measurement interaction term is Hermitian operator. In bosonic degrees of freedom one expects unitarity by the analogy with finite dimensional function space endowed with inner product with vacuum functional defining the weighting. This would mean a beautiful solution to the long standing problem of how to achieve unitarity.

3. There are strong reasons to believe that a duality prevails in the sense that one can restrict the interior part of action to either the Euclidian regions of space-time surfaces defining 4-D Feynman diagram or to their Minkowskian exterior. Number theoretic vision suggests this duality and the recent considerations (see also the recent posting) support the same conclusion. Obviously this duality brings in mind Wick rotation of quantum field theories.

4. The fermionic action corresponds formally to free action so that there are no explicit interaction vertices: the situation in the geometric formulation of string theory is same. There is however no need for non-linear interaction terms which are also responsible for the divergences of quantum field theories. The interaction terms are replaced with topological interaction vertex at which the light-like 3-surfaces associated defining the orbits of partonic 2-surfaces (wormhole throats) meet like lines of the ordinary Feynman diagram. Note that this vertex distinguishes between TGD and string models where trouser vertex is a typical vertex: in TGD framework this kind of geometric decay does not correspond to particle decay but to the propagation of particle along different paths. The conservation of quantum numbers is required at the vertices. Also massless-ness property for the particles propagating along the lines is natural in zero energy ontology and makes possible twistorialization with the constraint that physical particles are massive bound states of massless wormhole throats.

5. It might be also possible to treat the fermions using fermionic path integral. This treatment would assign to each wormhole throat orbit and space-like 3-surface at either end of CD a propagato analogous to the inverse of the propagator of massless propagator. It does not however diverge for on mass shell states since projections of momenta and Cartan algebra color quantum numbers (and more general quantum numbers) with modified gamma matrices are in question. This allows to avoid the problem caused by on mass shell property of virtual particles (off mass shell property means only that the bound state constraint for external particles is given up and that wormhole throats can have also negative sign of energy).

6. The non-trivial propagation of state with total number n of fermions and antifermions is possible only if n contractions of the propagator appears along the line (otherwise one would obtain only quark lepton contractions forbidden by conservation laws). This implies the proportionality 1/pn of the propagator so that only total fermion number n=1, 2 is possible for non-vacuum wormhole throat. This proportionality was earlier deduced from the SUSY limit of TGD based on a generalization of SUSY algebra. As a consequence, wormhole contact having two throats can carry at most spin 2 and the large SUSY defined by the fermionic oscillator operators is badly broken and effectively reduced to that generated by the right-handed neutrino which is also broken.

7. The assumption that all particles have non-vanishing mass means that given state can decay only to a virtual state with finite number of particles. This together with massless propagation along virtual lines simplifies enormously the perturbation series and is expected to imply finiteness.

8. The integration over WCW could spoil the unitarity. Although the exponent of Kähler action is positive it could give rise to divergent integral if the Kähler action has definite sign. The reduction to Chern-Simons term does not make obvious the positivity. If one believes on Minkowskian-Euclidian duality in the sense that one can define vacuum functional either as the exponent of Kähler action for the Minkowskian or Euclidian regions, one obtains definite sign for the Kähler function since for the Euclidian signature Kähler action indeed has definite sign.

What is remarkable that in Chern-Simons term the non-analytic 1/gK2 dependence on Kähler coupling strength disappears by the the weak form of electric-magnetic duality so that perturbation series with respect to the small parameter gK2 should make sense. One expects that this expansion gives small contributions to coupling constants determined in lowest order by bosonic emergence and involving fermionic loops.

9. The resulting Feynman diagrammatics differs from the standard one in many respects. The lines of Feynman diagrams are replaced with 3-surfaces in the sense specified above. Only a very restricted subset of loops are allowed classically by preferred extremals. The massless on mass shell property for wormhole throat momenta indeed allows very restricted phase space for loops. If all particles are massive bound states of massless wormhole throats intermediate virtual particles states with positive energies can contain only a finite number of particles so that the situation simplifies dramatically. The already mentioned collinear many-fermion states with propagator behaving like 1/pn, n>2 are also present. Hence on can ask whether a more appropriate identification of generalized Feynman diagrams might be as counterparts of twistor diagrams.

What is the relationship of generalized Feynman diagrams to twistor diagrams?

The general idea about the construction of U-matrix gives strong support for the existing heuristics and provides a connection with category theoretical ideas (planar operads and generalized Feynman diagramatics (see this and also suggests a generalization of twistor diagrammatics. Many questions of course remain unanswered. The basic question is the relationship of generalized Feynman diagrams with twistor diagrams. There are arguments favoring also the interpretation as direct counterparts of twistor diagrams. The following series of arguments however favors Feynman diagram interpretation and leads to a precise connection between the two diagrammatics. The arguments rely on following general ideas which deserve to be restated.

1. The realization that the hermitian square roots of density matrices form infinite-D unitary algebra and that their commutativity with universal S-matrix implies that zero energy states define the generalization of Kac-Moody algebra became only after I had realized the possibility to construct U-matrix. It is this observation which reduces the construction of U-matrix (or matrices if they form algebra) to that for S is expected to correspond directly to the ordinary S-matrix. A possible interpretation of the algebra of U-matrices is in terms of scales of CDs coming as positive integer powers of two. Another possibility more in line with the usual interpretation of S-matrix as time evolution operator is that scales of CDs come as integers and these integers correspond to powers of S.

2. In ordinary QFT Feynman diagrams are purely algebraic objects. In TGD framework they reduce to space-time topology and geometry with Euclidian regions of space-time surfaces having interpretation as generalized Feynman diagrams. At the vertices of generalized Feynman diagrams in coming partonic 2-surfaces meet just like in ordinary Feynman diagrams which means deep difference from string theory. A more general assumption is that entire 4-D lines of generalized Feynman diagram meet at vertices. This could apply to the Euclidian regions only.

There is also a second kind of branching involved with the hierarchy of Planck constants. In Minkowskian regions similar meeting would take place for the branches of space-time sheets with same values of canonical momentum densities of Kähler action at the ends of CDs and have interpretation in terms of fractionization and hierarchy of Planck constants. The value of Planck constant for single branch would be effectively and integer multiple of the ordinary one. For the entire multi-sheeted structure describable naturally in terms of singular covering space of M4× CP2 it would be just the ordinary value.

3. Zero energy ontology with massless external wormhole wormholes implies as such twistorialization of the theory although external wormhole momenta must be assumed to be massive bounds states of massless throats. This also guarantees exact Yangian symmetry and the absence of IR divergences. If also virtual wormhole throats are massless, twistorialization takes place in strong sense. This is possible only in zero energy ontology and accepting the identification of wormhole throats as basic building blocks of particles. Zero energy ontology leads also to an unexpected connection between infinite-dimensional Lie algebras and the space of allowed Hermitian square roots of density matrices multiplying unitary S-matrix in M-matrix.

4. The notion of bosonic emergence means that bosonic propagators emerge as radiative loops for wormhole contacts. The emergence generalizes to all states associated with wormhole contacts and also to flux tubes having wormhole contacts at their ends. What is nice that coupling constants emerge as normalization factors of propagators. Note that for single wormhole throat as opposed to wormhole contact having two throats bosonic propagator would result as a product of two collinear fermionic propagators and have the standard form. For states with higher total number of fermions and ant-ifermions the propagator of wormhole throat behaves as pn, n>2. Here however p is replaced with what I call pseudo-momentum.

5. Number theoretical universality suggest that at given level (CD) only finite sum of diagrams appears: otherwise there is danger that one obtains sum of rational functions which is not rational anymore. This gives strong constraints on generalized Feynman diagrams at the lowest level of the hierarchy.

6. Category theoretical approach based on planar operad proposed for few years ago kenocite{categorynew fits nicely with the twistorial construction of amplitudes interpreting radiative corrections in terms of CDs within CDs picture.

1. What is the correct identification of pseudo-momenta

The modified Dirac equation gives as generalized eigenvalues the quantities λkγk. I have christen λ as f pseudo-momentum and proposed number theoretic quantization rules for the values of pseudo-momenta kenocite{Dirac The physical interpretation of pseudo-momenta is still open as is also their relationship to massless on mass shell momenta propagating in wormhole throats associated with virtual particles. It is convenient to consider wormhole contact with two wormhole throats as a representation of incoming or virtual particle. The questions are following.

1. Is there a summation over pseudo-momenta for wormhole throats such that the sum of pseudo-momenta equals to the total exchanged real momentum associated with the wormhole contact. The real momenta on virtual line would be massless and give strong kinematic conditions on phase space allowed in loops.

Physical propagators from wormhole contacts would result as self energy loops for pseudo-momenta and there is the danger of getting divergences unless one uses the number theoretic conditions to reduce the summation as proposed. This picture would realize the idea about the emergence of bosonic propagators as fermionic radiative corrections and also more general propagators. Coupling constants would be predicted and appear in the normalization of bosonic propagators. Note that also the integration over WCW degrees of freedom affects the values of coupling constants.

The question is how strong additional conditions the number theoretic quantization of pseudo-momenta poses on the exchanged massless real momenta depends on the strength of number theoretical conditions. Are these conditions sensible?

2. Can one really identify pseudo-momenta really identifiable as region momenta of the twistor approach as I have cautiously suggested? The above line of arguments does not encourage this interpretation. Whether the identification makes sense can be tested immediately by looking for the dependence of Grassmannian twistor amplitudes on pseudo-momenta. If it is of standard propagator form one can consider this interpretation.

2. Connection between generalized Feynman diagrams and generalized twistor diagrams

The connection between generalized Feynman diagrams and generalized twistor diagrams should be understood.

1. The natural manner to identify twistor diagrams for a given CD without radiative corrections given by the addition of sub-CDs would be as the diagrams obtained by connecting the points or upper and lower boundaries of CD to form a polygon. There are several manners to do this. The differences of region-momenta would give the massless momenta for each external wormhole throat. Region momenta would have nothing to do with pseudo-momenta.

2. Twistor diagrams would represent sum for a subset of allowed generalized Feynman diagrams with massless particles in internal lines. On mass shell condition for massless wormhole throats restricts dramatically the number of contributing diagrams and the assumption that all particles have at least small mass means that particle numbers in intermediate states are finite. One however obtains infinite number of diagrams obtained as series of allowed diagrams. The problem is that although individual diagrams give rational functions, an infinite sum of them leads out from the algebraic extensions of p-adic numbers and rationals. This does not conform with number theoretic universality.

Therefore only irreducible diagrams not decomposing to series of allowed scatterings are allowed. As a consequence only finite number of diagrams are possible. The sum of these diagrams would correspond to a given basic twistor diagram. One could consider also the condition that at given length scaled determined by CD only tree diagrams are allowed. but this option looks ad hoc.

The addition of sub-CD:s would give radiative corrections from shorter length scales and the depth of the hierarchy of CDs within CDs hierarchy defines the IR and UV cutoffs and measurement resolution. If one accepts the assumption that the sizes of CD come as octaves of CP2 time scale, there would be natural IR and UV cutoffs on the values of pseudo-momenta from p-adic lentth scale hypothesis so that the amplitudes should remain finite and there would no fear about the loss of number theoretic universality. Note that in zero energy ontology cutoffs would characterize physical states themselves rather than restrictions of physicist only.

3. Diagrammatics based on gluing of twistor amplitudes

Radiative corrections n shorter scales than that of CD would result from the gluing of basic amplitudes for CDs within CDs.

1. Radiative corrections could be organized in terms of twistor diagrams. The rule transforming twistor polygons to simplest Feynman diagrams is standard duality replacing polygon with external lines at vertices with a bundle of lines obtained by connecting external lines to same point in the interior of the polygon. For triangle this gives three vertex. For n-polygon this would give n-vertex which corresponds to tree diagram as a Feynman diagram.

For instance, one can understand self energy corrections in this framework in terms of two twistorial triangles with two edges of both connected by two lines. Again on mass shell massless holds true for the throats. Vertex correction corresponds to triangle triangle within triangle with vertices of the inner triangle connected to the vertices of the outer triangle.. One obtains radiative corrections from this picture.

2. Also now one can have loops obtained as a closed ring of polygons connected to each other. There are also much more complex configurations of polygons. Unless one allow splitting of wormhole contacts the wormhole lines associated with a given wormhole throat end up to single CD.

3. For an outgoing pair of wormhole lines from given CD the wormhole throats should have same sign of energy: this would mean that only time-like momenta can propagate between CDs so that space-like loop momenta would be possible only for the fundamental radiative corrections. This would a further strong restriction on the amplitudes and space-like momentum exchanges would come from the fundamental level involving only a finite number of diagrams.

Is this good or bad? If bad, should one be ready to assign independent CDs with the two wormhole throats? Or should the interpretation be that the wormhole contact is split and wormhole throats propagate in two different time directions? But is it possible to speak about single space-like momentum exchange if the wormhole contact is split. Note that pseudo-momentum propagator for wormhole throat would still make sense. This line of thought does not look attractive.

4. Massless particles assigned with wormhole lines connecting the polygons and net pseudo-momenta are not on mass shell. Apart from time-likeness of net momenta, the rules for the propagators seem exactly the same as for polygons. These rules would summarize how radiative corrections from shorter scales are obtained.

4. The generalization of the recursion formula to TGD framework

The great victory of twistor approach is the recursion formula for the amplitudes (see also the representation in TGD framework) applying to all planar diagrams of N=4 SYM becoming an exact formula at the large N limit for gauge group SU(N). In the recent case the infinite-dimensional character of the Yangian symmetry algebra of S-matrix could be correlate for large N limit so that the planar limit should make sense. Also the fact that string worlds sheets are an essential aspect of TGD approach suggests that stringy picture deduced by t'Hooft for gauge theories at this limit implies planarity. What is relevant in the recent case is the general structure of the reduction formula, not the details which as such are of course extremely interesting also in TGD framework since Grassmannian amplitudes are claimed to provide a universal representation of Yangian invariants.

The recursive formula expresses scattering amplitude with n external particles with k negative helicities up to l loops is expressible as a sum of two terms. The first term-referred to as classical contribution- involves a fusion of twistor amplitudes with smaller number of particles and with the number of loops not larger than l by a procedure used already for tree diagrams. Second term - called quantum contribution- involves l loops and is irreducible in the sense that it is not expressible as a fusion of lower amplitudes and is obtained from n+2 particle by a process eliminating two particles. The identification of the TGD counterparts of these terms is obvious. The "classical" term corresponds to the proposed fusion of the lower level amplitudes associated with polygons for sub-CDs. The "quantum" term corresponds to the contribution appearing at the level of CD itself and involves genuine loops in Feynman sense but only a finite number of them.

Generalized twistor diagrams and planar operads

The resulting diagrams would have very close resemblance to structures known as planar operads (see this and this) associated with both topological quantum field theories and subfactors of von Neumann algebras. Planar operads provide a graphic representation of these structures. Since TGD corresponds to almost topological QFT and since WCW ("world of classical worlds") Clifford algebras correspond to von Neumann algebras known as hyper-finite factors of type II_1 (see this), the natural expectation is that generalized Feynman diagrams correspond to planar operads. This is indeed what I proposed for three years ago here but with disks replaced with CDs so that a the recent view unifies several earlier visions.

An additional motivation for the operad picture came from the notion of super-symplectic analog of super-conformal field theory motivated by the assumption that the symplectic transformations of δ M4+/-× CP2 act as isometries of WCW. The fusion rules of super-symplectic QFT lead to an infinite hierarchy of algebras forming an operad.

The basic structure of planar operad is very much reminiscent of generalized twistor diagrams.

1. One has essentially disks within disks connected by lines. The modification is obvious. Replace disks within disks disks with CDs within CDs and assign to the upper resp. lower boundaries of CDs positive resp. negative energy states. Many-sheeted space-time allows locally two CDs above each other corresponding to the identification of particles as wormhole contacts.

2. The planarity of the operad would be an obvious correlate for the absence of non-planar loops in twistor approach to N=4 SUSY making it problematic. Stringy picture actually suggests the absence of non-planar diagrams. The proposed generalization of twistor diagrammatics allowing arbitrary polygons within polygons structure might be enough to compensate for the absence of non-planar diagrams.

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

## Thursday, March 17, 2011

### Are the masses of top and antitop quark different?

Measurement of the mass difference between t and tbar quarks is the title of the e-print by a group working in Fermilab. The first name in the long list of authors is T. Aaltonen, which is very finnnish sounding. The finding of the group is that top-tbar mass difference is

Δ M= Mt-Mtbar= -3.3+/- 1.7 GeV.

The best fit is obtained with

Δ M=-4 GeV.

For top quark mass Mt= 170 GeV this means Δ M/M≈ 2.3 per cent. The result deviates from CPT-symmetric expectation Δ M=0 at 2σ level. Also D0 collaboration has got similar results two years earlier (PRL 103, 132001 (2009)) but at time the errors bars were so large that the finding was consistent with CPT symmetry.

The finding encourages to consider the possibility of CPT breaking seriously. In TGD framework a very strong form of apparent CPT breaking results if fermion and anti-fermion correspond to different values of p-adic prime so that mass scales differ by a multiple of half octave. The different choices of the p-adic mass scale would be induced by the interaction with environment. This option might explain the observations suggesting that neutrino and antineutrino masses and mixing matrices are different without introducing sterile neutrino: sterile neutrino would correspond to neutrino but in different p-adic length scale. In the recent case this option is excluded by the smallness of the mass difference. In zero energy ontology, which assigns to elementary particles size scale which which is macroscopic, one can however consider a more delicate breaking of CPT induced by the interactions with environment.

What CPT and CPT breaking do mean?

To begin, recall that CPT breaking would mean that the invariance condition

P(Ψif)= P(θΨf,θΨi)

for probabilities fails to be satisfied. Here θ is shorthand for CPT. The permutation of initial and final states is what distinguishes T and thus CPT from ordinary symmetries and means that T must be realized anti-linearly. In standard QFT P and T have geometric meaning whereas C does not. In TGD framework also C is geometric and this means that one must reconsider CPT and its tests based on phenomenological models.

CPT symmetry is one of the basic tenets of quantum field theory. In particular, the breaking of CPT requires the breaking of Lorentz invariance in standard QFT framework. In TGD framework the situation is actually different as I realized only now! The reason is that also charge conjugation is induced by a geometric transformation just like P and T. C indeed involves complex conjugation of CP2 coordinates, and one can quite well consider a situation in with T and P are unbroken and only C is broken so that CPT is broken. What actually happens depends on the detailed action of the symmetries on the modified Dirac action.

Some fact about zero energy ontology

Before one one can proceed, one must consider in more detail the notion of CD. CD is a product of CD proper defined as intersection of future and past directed light-cones of M4 and of CP2. The scales of CDs are assumed to come in powers of two of CP2 scale to explain p-adic length scale hypothesis (one can consider also prime and even integer multiples). What is of utmost significance is that these scales are macroscopic. Poincare transformations affect CDs and give rise to a moduli space for CDs. In the case of CP2 this is not the case unless one introduces additional physically well motivated structure.

Quite generally, this additional structure corresponds to the choice choice of quantization axes for various isometry currents realized at the level of the geometry of world of classical worlds which decomposes to a union of the geometries assigned with difference CDs labelled by moduli specifying the choice of quantization axes. In the case of M4 the line joining the tips of CD defines a unique rest system with origin at the middle point of the line and selects quantization axes of energy. The direction of spin quantization axes is fixed if one introduces preferred plane M2 physically analogous to the preferred plane of unphysical polarizations. This plane is fixed also by number theoretical vision and correspond to hyper-octonionic plane of complexified octonions highly relevant for the number theoretic formulation of TGD.

One must also introduce CP2 coordinates transforming linearly with respect to U(2) sub-group. The choice of preferred point of CP2 at either boundary of CD allows to fix complex coordinates of CP2 only apart from U(2) rotation. Hyper-charge quantization axes is fixed but color isospin direction remains free. In fact, there is a preferred color isospin generator leaving the points of the geodesic sphere invariant whereas hypercharge generator induces phase multiplication. By choosing two preferred points of CP2 assigned to the opposite boundaries of CD one can identify the geodesic line connecting the points as a flow line of color isospin rotations so that the quantization axes are fixed.

The choices of preferred plane M2 and preferred geodesic sphere make sense also at the level of the preferred extremals of Kähler action and this leads to a concrete realization of the conjectured slicing of the space-time surface by string world sheets having braid strands at their ends at light-like wormhole throats carrying particle quantum numbers.

The vision about how quantum TGD gives rise to symplectic theory of knots, braids, braid cobordisms, and of two-knots (see the previous posting) led to the realization that preferred extremals should involve preferred geodesic sphere of CP2, whose inverse image under imbedding map assigns to the space-time surface unique complex of stringy two surfaces. These stringy two-surfaces define braid cobordisms and 2-knots and provide also the reduction of quantum TGD to string theory like structure in finite measurement resolution meaning the replacement of the orbits of partonic 2-surfaces with braids.

Charge conjugation is geometric in TGD framework

Charge conjugation in TGD Universe involves complex conjugation of CP2 coordinates. Complex conjugation commutes with color rotations only if they belong to a subgroup U(2) ⊂ SU(3) leaving a preferred point of CP2 invariant remaining invariant also under C just like the origin of M4 remains invariant under P and T. The situation differs from that for P and T decisively since the scale of CP2 is about 104 Planck lengths. More general color rotations acting non-linearly and affecting non-trivially on the preferred point do not commute with C.

A simple example is provided by sphere. In this case C would act in complex coordinates as φ → -φ, where φ is the phase angle of the complex coordinate with origin at the preferred point of the sphere. The action obviously depends on the choice of the preferred point.

The situation is therefore same as for P and T which also fail to commute with Poincare group and commute only with Lorentz transformations leaving the selected space-time point fixed. In TGD framework this point would correspond naturally to the center of the line connecting the tips of the causal diamond proper.

The action of C on physical states involves a linear transformation of spinors transformation besides the geometric action. The details of this action were discussed already in my thesis for almost three decades ago and the reader can consult the appendix of some of the books about TGD or the little article titled The Geometry of CP2 and its Relationship to Standard Model as the appendix of an article series summarizing Quantum TGD published in Prespacetime Journal. What is essential is that the action of C does does not commute with color rotations acting on the moduli of CD unless they belong to the U(2) subgroup leaving the geodesic sphere invariant. One can define C for the two boundaries of CD by requiring that the corresponding geodesic spheres remain invariant under C.

The action of CPT in zero energy ontology

The action of CPT is following.

1. First one applies P and T. If one assumes that the preferred point of M4 corresponds to the middle point of the line connecting the tips of CD proper, these transformations permute upper and lower boundaries of CD proper. This is indeed a very natural requirement and means that positive and negative energy parts of the quantum state serving as counterparts of initial and finals states in positive energy ontology are permuted just as they are permuted in CPT. That T is realized anti-linearly conforms with the fact that T does leave invariant the boundary of CD proper.

2. Next one applies C involving complex conjugation which in general affects the moduli of CD. If C is chosen differently at the opposite boundaries it leaves the corresponding moduli invariant but since CPT involves the permutation of positive and negative energy states the moduli of CD are changed since the preferred point of upper boundary becomes the preferred point of the lower boundary and vice versa.

Only in the case that the preferred points assigned to the upper and lower boundaries are same, this does not happen but in this case the quantization axes are not completely fixed which could make sense only if color isospin of all particles or at least of the positive (and negative) energy part of the zero energy state vanishes. Unless the CD has a wave function in the space of moduli which is constant, a spontaneous and a purely geometric breaking of C symmetry is induced. The breaking would be highly analogous to the breaking of rotational symmetry in spontaneous magnetization taking place in many particle systems.

3. The size scale of the CD proper is macroscopic even for elementary particles and corresponds to the secondary p-adic length scale associated with the particle. For electron with p= M127=2127-1 this time scale is T(2,127)= .1 seconds, defining the fundamental biological rhythm. For u and d quarks it is of order millisecond and for t quark characterized by p≈ 293 it is given by

T(2,93) 2-127+93× T(2,127)≈5.8×10-12 seconds.

The corresponding length scale is 1.74 mm and is macroscopic. There are very many particles in CD of this size scale which suggests the possibility of spontaneous C breaking inducing by a localization in the moduli space of CDs implying the breaking of the CPT invariance condition. The many-particle system would be present since the CDs assignable to individual quarks intersect which suggests that they correspond to common CD. The non-invariance of the many-particle system under CPT could also result from that under PT operation in macroscopic situation.

1. The mass difference should depend on the moduli of CDs characterizing color quantization axes and characterize by preferred points of CP2 assigned with future and past boundaries of CD. A natural measure for the symmetry breaking is defined by the geodesic distance -call it s- between the preferred points so that one expects that the mass of top quark assigned with a particular CD involves a small contribution depending on s. This distance is however not changed in C.

The additional contribution to the mass should contain a term which is odd under C (most naturally), CP, or CPT. Could the oddness come from the spontaneous symmetry breaking giving rise to an interaction term with environment affecting the mass of particle and antiparticle in different manner? This oddness would be analogous to the oddness of the interaction energy of magnetic dipole with an external magnetic field.

2. The relative mass difference is of order 2 per cent. What determines this scale? Could the mass difference be proportional to fine structure constant? This could be the case if the electromagnetic interaction of the top quark with environment defined by CD induces the CPT odd term to the mass of the particle? What is the role of the magnetic flux tube containing top at the second end and neutrino pair neutralizing its weak isospin at the second end?

Questions

This picture inspires several questions.

1. Can one consider C breaking without the presence of P and T breakings? If the CP breaking assigned with kaon-antikaon system and other neutral meson systems is CP breaking in TGD sense, does it involve the breaking of T at all? The answers to these questions are not obvious since the tests of discrete symmetries rely on the standard view about charge conjugation lacking totally the geometric aspect of C in TGD Universe.

2. Could it be that the different topological mixings of U and D quarks inducing in turn CKM mixing are induced by C breaking basically so that the mass differences would correlate directly with CKM mixing parameters?

3. Is the geometric view about about breaking of C relevant for the understanding of matter antimatter asymmetry? I have considered several models of the generation of matter antimatter asymmetry, one of them assuming that antimatter is eaten by long cosmic strings with breaking induced by the Kähler electric fields inducing small difference in the densities of fermions and anti-fermions outside cosmic strings. Could matter antimatter asymmetry be mathematically analogous to chiral selection in living matter so that P would be only replaced with C? Whether the geometric view about C is relevant for the understanding of the matter antimatter asymmetry must be however left open question. Different masses for fermions anti-fermions could however help to understand why this kind of separation takes place.

4. C acts in CP2 and in color degrees of freedom. Does this mean that for non-colored states C is not broken and that s CP breaking is present only for quarks but not for leptons? The answers to these questions are not obvious since in TGD framework M4×CP2 spinor harmonics correspond to color partial waves which have wrong correlation with electro-weak quantum numbers. Only covariantly constant right-handed neutrino spinor generating supersymmetry can move in color single partial wave. The physical color assignments are the result of a state construction involving super-conformal algebra with algebra elements carrying color.

For background see the chapter p-Adic Particle Massivation: New Physics of "p-Adic Length Scale Physics and Hierarchy of Planck Constants".

## Monday, March 14, 2011

### Some TGD inspired comments about Higgs, ew symmetry breaking, and SUSY

In the following still some comments about TGD based view about symmetry breaking, Higgs, electroweak symmetry breaking, and SUSY. There are several unclear issues at the level of detais. This is thanks to my unforgivable laziness in writing down the details. The results from LHC are however so fascinating that they force me to win my laziness. In the following I try to clarify my thoughs.

1. Is the earlier conjectured pseudoscalar Higgs there at all?

Spin 1 gauge bosons and Higgs differ only by different spin direction of fermions at opposite wormhole throats. For spin 1 gauge bosons the 3-momenta at two wormhole throats cannot be parallel if if one wants non-vanishing spin component in the direction of moment. 3-momenta are most naturally opposite for the massless states at throats. This forces massivation for all gauge bosons and even graviton and this in turn requires Higgs even in the case of gluons.

Question: Could the parity properties of the couplings of gauge boson and corresponding Higgs transforming like 3+1 under SU(2) (this is due to the special character of imbedding space spinors) be exactly the same? Higgs would couple like a mixture of scalar and pseudoscalar to fermions just as weak gauge bosons couple and the mixture would be just the same. If there are no axial variants of vector gauge bosons there should exist no pseudoscalar Higgs. The nonexistence of axial variants of vector gauge bosons is suggested by quantum classical correspondence: only gauge bosons having classical space-time correlates as induced gauge potentials should be allowed, nothing else. Note that color variant of Higgs would exist and would be eaten by gluons to get mass.

2. Could Higgs mechanism lead to the disappearance of also Higgsinos?

The similarity of the construction of gauge bosons and Higgsinos as pairs of wormhole throats containing fermion and antifermion encourages to think that Higgs mechanism is invariant under supersymmetries. If so, also Higgsinos would be eaten and one would have massive super-symmetric gauge theory with fermions with photon and other massless particle possessing a tiny mass. This looks very simple. The testable implication would be that only weak gauginos should contribute to muon g-2 anomaly.

3. Electroweak symmetry breaking

The recent view about electroweak symmetry breaking is less than year old. The basic realization was that wormhole throats carrying elementary particle quantum numbers possess Kähler magnetic charge (in homological sense-CP2 has non-trivial second homology). This magnetic charge must be compensated and this is achieved if the particle wormhole throat is connected to a second wormhole throat by a magnetic flux tube. The second wormhole would carry a weak charge of neutrino pair compensating the weak isospin of the particle so that weak interactions would be screened above the weak length scale. For colored states the compensation could also occur in longer length scale and corresponds to color confinement.

This does not actually require the length scale of flux tubes associated with all elementary particles to be the weak length scale as I have thought. Rather, the flux tube length for a particle at rest could correspond to the Compton length of the particle. For instance, for electron the maximal flux tube length would be about 10-13 meters. For particles not at rest the length would get shorter by length contraction. For very light but massive particles such as photon and graviton the maximum length of flux tube would be very long. The interaction of very low energy photons and gravitons would be essentially classical and induced by the classical oscillations of induced gauge fields induced by a long flux tube connecting the interacting systems. For high energy quanta this interaction would be essentially quantal and realized as absorption of quanta with flux tube length -essentially wave length of quantum- much shorter than the distance between the interacting systems. Gravitational waves would interact essentially classically even when absorbed since absorption would mean that the flux tube would connects two parts of the measurement apparatus. For large hbar gravitons the length of flux tube could correspond to the distance between interaction systems.

A fascinating possibility is that electronic Cooper pairs of superconductors with large value of hbar, could correspond to long flux tubes with electron's quantum numbers at both ends. Maybe this takes place in high Tc super conductors.

4. Some details of the SUSY predictions

TGD SUSY differs from the standard SUSY in many respects.

1. All fermionic oscillator operators assignable to the wormhole throats generate supersymmetries. These oscillator operators differ from ordinary ones in that they do not have momentum label and momentum can be only assigned to the entire state. Therefore the interpretation of all states assignable to wormhole throats as large SUSY multiplet is possible. This SUSY is badly broken and there is hierarchy of breakings defined by the interactions inducing the breaking in turn define by the quantum numbers of SUSY generators. For quark generators the breaking is largest and the smallest breaking is associated with the oscillator operators assignable to right-handed neutrinos since they have only gravitational interactions.

2. The symmetry generators are not Majorana spinors and this does not lead to any difficulties as has been found. Only if one would try stringy quantization trying to define stringy diagrams in terms of stringy propagators defined by stringy form of super-conformal algebra, one would end up with difficulties. Majorana property is also excluded by the separate conservation of baryon and lepton number.

For single wormhole throat one can see the situation in terms of N=2 SUSY with right handed neutrino and its antiparticle appearing as SUSY generators carrying conserved fermion number. One can classify the superpartners by their right-handed neutrino number which is +/-1. For instance, for single wormhole throat one obtains fermion and its partner containing νR pair, and fermion number 0 and fermion number 2 sfermions. In the case of gauge bosons and Higgs similar degeneracy is obtained for both wormhole throats.

3. Since induced gamma matrices and modified gamma matrices are mixtures of M4 and CP2 gamma matrices right handed neutrino is mixed with the left handed neutrino meaning breaking of R-parity. The simplest decays of sparticles are of form P→P+ν and can be said to be gravitationally induced since the mixing of gamma matrices is indeed a characteristic phenomenon of induced spinor structure. Also more complex decays with neutrino replaced with charge lepton are possible. The basic signature is lonely lepton not possible in decays of weak bosons.

4. The basic outcome of SUSY QFT limit of TGD (see this) is that wormhole throat can carry only spin 0,1/2,1 corresponding to fermion and fermion pair if one wants to obtain standard propagator: otherwise one obtains 1/pn, n>2 and this is not an ordinary particle pole. The reason is that one cannot assign to fermionic oscillator operators independent momenta but only common momentum so they propagate effectively collinearly.

One can criticize this argument as being inconsistent with the twistorial approach combined with zero energy ontology implying that wormhole throats are massless even for on mass shell states. In this approach one in principle avoids completely the use of propagators which would of course diverge for on shell wormhole throats. Also for twistor diagrams the counterparts of virtual particles are massless and off shell. The so called region momentum replaces momentum in Grassmannian twistor approach and has a direct counterpart as eigenvalue of the modified Dirac operator so that the analog of propagator exists in TGD framework. Since QFT limit must be a reasonable approximation to the full theory, one might hope that the QFT based argument makes sense when one replaces momentum with region momentum (or pseudo momentum as I have called it in TGD framework).

5. Should one allow both nuR and its antiparticle as SUSY generators? This would mean more states as in standard SUSY for which only anti-nuR would be allowed for fermion. This would assign to a given wormhole throat with fermion number 1 spin 1 and spin 0 super partner and companion of fermion containing nuR-anti-nuR pair. For this state however propagator would behave like 1/p3 should that again strong SUSY breaking would occur for this extended SUSY. Only one half of SUSY would be broken weakly by the mixing of M4 and CP2 gamma matrices appearing in modified gamma matrices: the mixing would not involve weak or color interactions but could be said to be gravitational but not in the sense of abstract for geometry but induced geometry. The breaking of symmetries by this mechanicsm would be a beautiful demonstration that it is sub-manifold geometry rather than abstract manifold geometry that matters. Again string theorists managed to miss the point by effectively elimating induced geometry from the original string model by inducing the metric of space-time sheet as an independent variable. The motivation was that it became easy to calculate! The price paid was symmetry breaking mechanisms involving hundreds of three parameters.

6. Single wormhole contact could carry spin J=2 and give rise to graviton like state. If one constructs from this gravitino by adding right-handed neutrinos, and if SUSY QFT limit makes sense, one obtains particle with propagator decreasing faster at either throat so that gravitino in standard sense would not exist. This would represent strong SUSY breaking in gravitational sector. These results are of utmost importance since the basic argument in favor dimension D=10 or D=11 for the target space of superstring models is that higher dimensions would give fundamental massless particles with higher spin.

Note that the replament of wormhole throats by flux tubes having neutrino pair at the second end of the flux tube complicates the situation since one can add right handed neutrino also to the neutrino end. The SUSY QFT criterion would however suggest that these states are not particle like.

## Friday, March 04, 2011

### More about the strange asymmetry in t-tbar production

For top pairs with invariant mass above 450 GeV the asymmetry is claimed by CDF to be stunningly large 48+/-11 per cent. 3 times more often top quarks produced in qqbar annihilation prefer to move in the direction of q. If true this would favor color octet excitations of Z0as the most natural explanation since the asymmetry would be not only due to the interference of vector and axial vector exchanges but also due to the inherent parity breaking of colored Z0 couplings. The effect would provide further support for the identification of color quantum numbers in terms of color partial waves rather than as spin like quantum numbers. The earlier support comes from the evidence for colored excitations of leptons.

Addition: After a badly slept night I have come to new thoughts about the possible explanation of the effect. What is so weird (really weird when one begins to think the numbers!) that the outgoing top quark (t) remembers the direction of motion of quark q before annihilation to intermediate gluon which it should by the basic definition of annihilation diagram. For any exchange diagram the situation would be totally different: consider only Coulomb scattering! The quark q of the first proton would scatter from the quark of the second proton and transform to top quark in the scattering and keep its direction of motion in good approximation since small angle exchanges dominate due to the propagator factor. Flavor changing exchange diagrams are however not possible in the standard model world since the only flavor changing are charged weak currents and their contribution is negligible.

In the new physics inspired by TGD situation is however different! The identification of family replication phenomenon in terms of genus of the wormhole throats (see this) predicts that family replication corresponds to a dynamical SU(3) symmetry (having nothing to do with color SU(3)or Gell-Mann's SU(3)) with gauge bosons belonging to the octet and singlet representations. Ordinary gauge bosons would correspond besides the familar singlet representation also to exotic octet representation for which the exchanges induce neutral flavor changing currents in the case of gluons and neutral weak bosons and charge changing ones in the case of charged gauge bosons. The exchanges of the octet representation for gluons would explain the anomaly! Also electroweak octet could of of course contribute.

What is fantastic is that LHC will soon allow to decide whether this explanation is correct!

Addition: I noticed that this argument requires a more detailed explanation for what happens in the exchange of gauge boson changing the genus. Particles correspond to wormhole contacts. For topologically condensed fermions the genus of the second throat is that of sphere created when the fermionic CP2 vacuum extremal touches background space sheet. For bosons both wormhole throats are dynamical and the topologiies of both throats matter. The exchange diagram corresponds to a situation in which g=gi fermionic wormhole throat from past turns back in time spending some time as second throat of virtual boson wormhole contact and g=gf from future turns back in time and defines the second throat of virtual boson wormhole contact. The turning corresponds to gauge boson exchange represented by a wormhole contact with g=gi and g=gf wormhole throats. Ordinary gauge bosons are quantum superpositions of (g,g) pairs transforming as SU(3) singlets and SU(3) charged octet bosons are of pairs (g1,g2) g1≠ g2. In the absence of topological mixing of fermions inducing CKM mixing the exchange is possible only between fermions of same generation. The mixing is however large and changes the situation.

Addition: The following sayings by some celebrities of science deserve also to be added because the last three decades in theoretical particle physics have convincingly demonstrated their truth.

"The difference between stupidity and genius is that genius has its limit."

"Only two things are infinite: the universe and human stupidity, and I`m not sure about the universe."

See this and also the earlier posting which I have modified to take into account the newest twist.