Monday, July 30, 2012

The counterpart of AdS5 duality in TGD framework

The generalization of AdS5 duality of N=4 SYMs to TGD framework is highly suggestive and states that string world sheets and partonic 2-surfaces play a dual role in the construction of M-matrices. In the following I give an argument providing a "proof" of this duality and also demonstrating that for singular string world sheets and partonic 2-surfaces perturbative description of generalized Feynman diagrams is especially simple since string effectively reduces to point like particles.

Some terminology first.

  1. Let us agree that string world sheets and partonic 2-surfaces refer to 2-surfaces in the slicing of space-time region defined by Hermitian structure or Hamilton-Jacobi structure.
  2. Let us also agree that singular string world sheets and partonic 2-surfaces are surfaces at which the effective metric defined by the anticommutators of the modified gamma matrices degenerates to effectively 2-D one.
  3. Braid strands at wormhole throats in turn would be loci at which the induced metric of the string world sheet transforms from Euclidian to Minkowskian as the signature of induced metric changes from Euclidian to Minkowskian.
AdS5 duality suggest that string world sheets are in the same role as string world sheets of 10-D space connecting branes in AdS5 duality for N=4 SYM. What is important is that there should exist a duality meaning two manners to calculate the amplitudes. What the duality could mean now?
  1. Also in TGD framework the first manner would be string model like description using string world sheets. The second one would be a generalization of conformal QFT at light-like 3-surfaces (allowing generalized conformal symmetry) defining the lines of generalized Feynman diagram. The correlation functions to be calculated would have points at the intersections of partonic 2-surfaces and string world sheets and would represent braid ends.

  2. General Coordinate Invariance (GCI) implies that physics should be codable by 3-surfaces. Light-like 3-surfaces define 3-surfaces of this kind and same applies to space-like 3-surfaces. There are also preferred 3-surfaces of this kind. The orbits of 2-D wormhole throats at which 4-metric degenerates to 3-dimensional one define preferred light-like 3-surfaces. Also the space-like 3-surfaces at the ends of space-time surface at light-like boundaries of causal diamonds (CDs) define preferred space-like 3-surfaces. Both light-like and space-like 3-surfaces should code for the same physics and therefore their intersections defining partonic 2-surfaces plus the 4-D tangent space data at them should be enough to code for physics. This is strong form of GCI implying effective 2-dimensionality. As a special case one obtains singular string world sheets at which the effective metric reduces to 2-dimensional and singular partonic 2-surfaces defining the wormhole throats. For these 2-surfaces situation could be especially simple mathematically.

  3. The guess inspired by strong GCI is that string world sheet -partonic 2-surface duality holds true. The functional integrals over the deformations of 2 kinds of 2-surfaces should give the same result so tthat functional integration over either kinds of 2-surfaces should be enough. Note that the members of a given pair in the slicing intersect at discrete set of points and these points define braid ends carrying fermion number. Discretization and braid picture follow automatically.

  4. Scattering amplitudes in the twistorial approach could be thus calculated by using any pair in the slicing - or only either member of the pair if the analog of AdS5 duality holds true as argued. The possibility to choose any pair in the slicing means general coordinate invariance as a symmetry of the Kähler metric of WCW and of the entire theory suggested already early: Kähler functions for difference choices in the slicing would differ by a real part of holomorphic function and give rise to same Kähler metric of "world of classical worlds" (WCW). For a general pair one obtains functional integral over deformations of space-time surface inducing deformations of 2-surfaces with only other kind 2-surface contributing to amplitude. This means the analog of stringy QFT: Minkowskian or Euclidian string theory depending on choice.

  5. For singular string world sheets and partonic 2-surfaces an enormous simplification results. The propagators for fermions and correlation functions for deformations reduce to 1-D instead of being 2-D: the propagation takes place only along the light-like lines at which the string world sheets with Euclidian signature (inside CP2 like regions) change to those with Minkowskian signature of induced metric. The local reduction of space-time dimension would be very real for particles moving along sub-manifolds at which higher dimensional space-time has reduced metric dimenson: they cannot get out from lower-D sub-manifold. This is like ending down to 1-D black hole interior and one would obtain the analog of ordinary Feynman diagrammatics. This kind of Feynman diagrammatics involving only braid strands is what I have indeed ended up earlier so that it seems that I can trust good intuition combined with a sloppy mathematics sometimes works;-).

    These singular lines represent orbits of point like particles carrying fermion number at the orbits of wormhole throats. Furthermore, in this representation the expansions coming from string world sheets and partonic 2-surfaces are identical automatically. This follows from the fact that only the light-like lines connecting points common to singular string world sheets and singular partonic 2-surfaces appear as propagator lines!

  6. The TGD analog of AdS5 duality of N=4 SUSYs would be trivially true as an identity in this special case, and the good guess is that it is true also generally. One could indeed use integral over either string world sheets or partonic 2-sheets to deduce the amplitudes.
What is important to notice that singularities of Feynman diagrams crucial for the Grassmannian approach of Nima and others would correspond at space-time level 2-D singularities of the effective metric defined by the modified gamma matrices defined as contractions of canonical momentum currents for Kähler action with ordinary gamma matrices of the imbedding space and therefore directly reflecting classical dynamics.

For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Twistor revolution and TGD

Lubos wrote a nice summary about the talk of Nima Arkani Hamed about twistor revolution in Strings 2012 and gave also a link to the talk. It seems that Nima and collaborators are ending to a picture about scattering amplitudes which strongly resembles that provided bt generalized Feynman diagrammatics in TGD framework

TGD framework is much more general than N=4 SYM and is to it same as general relativity for special relativity whereas the latter is completely explicit. Of course, I cannot hope that TGD view could be taken seriously - at least publicly. One might hope that these approaches could be combined some day: both have a lot to give for each other. Below I compare these approaches.

The origin of twistor diagrammatics

In TGD framework zero energy ontology forces to replace the idea about continuous unitary evolution in Minkowski space with something more general assignable to causal diamonds (CDs), and S-matrix is replaced with a square root of density matrix equal to a hermitian l square root of density matrix multiplied by unitary S-matrix. Also in twistor approach unitarity has ceased to be a star actor. In p-Adic context continuous unitary time evolution fails to make sense also mathematically.

Twistor diagrammatics involves only massless on mass shell particles on both external and internal lines. Zero energy ontology (ZEO) requires same in TGD: wormhole lines carry parallely moving massless fermions and antifermions. The mass shell conditions at vertices are enormously powerful and imply UV finiteness. Also IR finiteness follows if external particles are massive.

What one means with mass is however a delicate matter. What does one mean with mass? I have pondered 35 years this question and the recent view is inspired by p-adic mass calculations and ZEO, and states that observed mass is in a well-defined sense expectation value of longitudinal mass squared for all possible choices of M2 ⊂ M4 characterizing the choices of quantization axis for energy and spin at the level of "world of classical worlds" (WCW) assignable with given causal diamond CD.

The choice of quantization axis thus becomes part of the geometry of WCW. All wormhole throats are massless but develop non-vanishing longitudinal mass squared. Gauge bosons correspond to wormhole contacts and thus consist of pairs of massless wormhole throats. Gauge bosons could develop 4-D mass squared but also remain massless in 4-D sense if the throats have parallel massless momenta. Longitudinal mass squared is however non-vanishing andp-adic thermodynamics predicts it.

The emergence of 2-D sub-dynamics at space-time level

Nima et al introduce ordering of the vertices in 4-D case. Ordering and related braiding are however essentially 2-D notions. Somehow 2-D theory must be a part of the 4-D theory also at space-time level, and I understood that understanding this is the challenge of the twistor approach at this moment.

The twistor amplitude can be represented as sum over the permutations of n external gluons and all diagrams corresponding to the same permutation are equivalent. Permutations are more like braidings since they carry information about how the permutation proceeded as a homotopy. Yang-Baxter equation emerge. The allowed braidings are minimal braidings in the sense that the repetitions of permutations of two adjacent vertices are not considered to be separate. Minimal braidings reduce to ordinary permutations. Nima also talks about affine braidings which I interpret as analogs of Kac-Moody algebras meaning that one uses projective representations which for Kac-Moody algebra mean non-trivial central extension. Perhaps the condition is that the square of a permutation permuting only two vertices which each other gives only a non-trivial phase factor. Lubos suggests an alternative interpretation for "affine" which would select only special permutations and cannot be therefore correct.

There are rules of identifying the permutation associated with a given diagram involving only basic 3-gluon vertex with white circle and its conjugate. Lubos explains this "Mickey Mouse in maze" rule in his posting in detail: to determine the image p(n) of vertex n in the permutation put a mouse in the maze defined by the diagram and let it run around obeying single rule: if the vertex is black turn right and if the vertex is white turn left. Eventually the mouse ends up to external vertex. The mouse cannot end up with loop: if it would do it, the rule would force it to run back to n after the full loop and one would have fixed point: p(n)=n. The reduction in the number of diagrams is enormous: the infinity of different diagrams reduces to n! diagrams!

  1. In TGD framework string world sheets and partonic 2-surfaces (or either or these if they are dual notions as conjectured) at space-time surface would define the sought for 2-D theory, and one obtains indeed perturbative expansion with fermionic propagator defined by the inverse of the modified Dirac operator and bosonic propagator defined by the correlation function for small deformations of the string world sheet. The vertices of twistor diagrams emerge as braid ends defining the intersections of string world sheets and partonic 2-surfaces.

    String model like description becomes part of TGD and the role of string world sheets in X4 is highly analogous to that of string world sheets connecting branes in AdS5× S5 of N=4 SYM. In TGD framework 10-D AdS5× S5 is replaced with 4-D space-time surface in M4× CP2. The meaning of the analog of AdS5 duality in TGD framework should be understood. In particular, could it be that the descriptions involving string world sheets on one hand and partonic 2-surfaces - or 3-D orbits of wormhole throats defining the generalized Feynman diagram- on the other hand are dual to each other. I have conjectured something like this earlier but it takes some time for this kind of issues to find their natural answer.

  2. As described in the article, string world sheets and partonic 2-surfaces emerge directly from the construction of the solutions of the modified Dirac equation by requiring conservation of em charge. This result has been conjectured already earlier but using other less direct arguments. 2-D "string world sheets" as sub-manifolds of the space-time surface make the ordering possible, and guarantee the finiteness of the perturbation theory involving n-point functions of a conformal QFT for fermions at wormhole throats and n-point functions for the deformations of the space-time surface. Conformal invariance should dictate these n-point functions to a high degree. In TGD framework the fundamental 3-vertex corresponds to joining of light-like orbits of three wormhole contacts along their 2-D ends (partonic 2-surfaces).

The emergence of Yangian symmetry

Yangian symmetry associated with the conformal transformations of M4 is a key symmetry of Grassmannian approach. Is it possible to derive it in TGD framework?

  1. TGD indeed leads to a concrete representation of Yangian algebra as generalization of color and electroweak gauge Kac-Moody algebra using general formula discussed in Witten's article about Yangian algebras (see the article).

  2. Article discusses also a conjecture about 2-D Hodge duality of quantized YM gauge potentials assignable to string world sheets with Kac-Moody currents. Quantum gauge potentials are defined only where they are needed - at string world sheets rather than entire 4-D space-time.

  3. Conformal scalings of the effective metric defined by the anticommutators of the modified gamma matrices emerges as realization of quantum criticality. They are induced by critical deformations (second variations not changing Kähler action) of the space-time surface. This algebra can be generalized to Yangian using the formulas in Witten's article (see the article).

  4. Critical deformations induce also electroweak gauge transformations and even more general symmetries for which infinitesimal generators are products of U(n) generators permuting n modes of the modified Dirac operator and infinitesimal generators of local electro-weak gauge transformations. These symmetries would relate in a natural manner to finite measurement resolution realized in terms of inclusions of hyperfinite factors with included algebra taking the role of gauge group transforming to each other states not distinguishable from each other.

  5. How to end up with Grassmannian picture in TGD framework? This has inspired some speculations in the past. From Nima's lecture one however learns that Grassmannian picture emerges as a convenient parametrization. One starts from the basic 3-gluon vertex or its conjugate expressed in terms of twistors. Momentum conservation implies that with the three twistors λi or their conjugates are proportional to each other (depending on which is the case one assigns white or black dot with the vertex). This constraint can be expressed as a delta function constraint by introducing additional integration variables and these integration variables lead to the emergence of the Grassmannian Gn,k where n is the number of gluons, and k the number of positive helicity gluons.

    Since only momentum conservation is involved, and since twistorial description works because only massless on mass shell virtual particles are involved, one is bound to end up with the Grassmannian description also in TGD.

Problems of the twistor approach from TGD point of view

Twistor approach has also its problems and here TGD suggests how to proceed. Signature problem is the first problem.

  1. Twistor diagrammatics works in a strict mathematical sense only for M2,2 with metric signature (1,1,-1,-1) rather than M4 with metric signature (1,-1,-1,-1). Metric signature is wrong in the physical case. This is a real problem which must be solved eventually.

  2. Effective metric defined by anticommutators of the modified gamma matrices (to be distinguished from the induced gamma matrices) could solve that problem since it would have the correct signature in TGD framework (see the article). String world sheets and partonic 2-surfaces would correspond to the 2-D singularities of this effective metric at which the even-even signature (1,1,1,1) changes to even-even signature (1,1,-1,-1). Space-time at string world sheet would become locally 2-D with respect to effective metric just as space-time becomes locally 3-D with respect to the induced metric at the light-like orbits of wormhole throats. String world sheets become also locally 1-D at light-like curves at which Euclidian signature of world sheet in induced metric transforms to Minkowskian.

  3. Twistor amplitudes are indeed singularities and string world sheets implied in TGD framework by conservation of em charge would represent these singularities at space-time level. At the end of the talk Nima conjectured about lower-dimensional manifolds of space-time as representation of space-time singularities. Note that string world sheets and partonic 2-surfaces have been part of TGD for years. TGD is of course to N=4 SYM what general relativity is for the special relativity. Space-time surface is dynamical and possesses induced and effective metrics rather than being flat.
Second limitation is that twistor diagrammatics works only for planar diagrams. This is a problem which must be also fixed sooner or later.
  1. This perhaps dangerous and blasphemous statement that I will regret it some day but I will make it;-). Nima and others have not yet discovered that M2 ⊂ M4 must be there but will discover it when they begin to generalize the results to non-planar diagrams and realize that Feynman diagrams are analogous to knot diagrams in 2-D plane (with crossings allowed) and that this 2-D plane must correspond to M2⊂ M4. The different choices of causal diamond CD correspond to different choices of M2 representing choice of quantization axes 4-momentum and spin. The integral over these choices guarantees Lorentz invariance. Gauge conditions are modified: longitudinal M2 projection of massless four-momentum is orthogonal to polarization so that three polarizations are possible: states are massive in longitudinal sense.

  2. In TGD framework one replaces the lines of Feynman diagrams with the light-like 3-surfaces defining orbits of wormhole throats. These lines carry many fermion states defining braid strands at light-like 3-surfaces. There is internal braiding associated with these braid strands. String world sheets connect fermions at different wormhole throats with space-like braid strands. The M2 projections of generalized Feynman diagrams with 4-D "lines" replaced with genuine lines define the ordinary Feynman diagram as the analog of braid diagram. The conjecture is that one can reduce non-planar diagrams to planar diagrams using a procedure analogous to the construction of knot invariants by un-knotting the knot in Alexandrian manner by allowing it to be cut temporarily.

  3. The permutations of string vertices emerge naturally as one constructs diagrams by adding to the interior of polygon sub-polygons connected to the external vertices. This corresponds to the addition of internal partonic two-surfaces. There are very many equivalent diagrams of this kind. Only permutations matter and the permutation associated with a given diagram of this kind can be deduced by the Mickey-Mouse rule described explicitly by Lubos. A connection with planar operads is highly suggestive and also conjecture already earlier in TGD framework.

For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Saturday, July 28, 2012

About the definition of Hamilton-Jacobi structure

I have talked in previous postings a lot about Hamilton-Jacobi structure without bothering to write detailed definitions. In the following I discuss the notion in more detail. Thanks for Hamed who asked for more detailed explanation.

Hermitian and hyper-Hermitian structures

The starting point is the observation that besides the complex numbers forming a number field there are hyper-complex numbers. Imaginary unit i is replaced with e satisfying e2=1. One obtains an algebra but not a number field since the norm is Minkowskian norm x2-y2, which vanishes at light-cone x=y so that light-like hypercomplex numbers x+/- e) do not have inverse. One has "almost" number field.

Hyper-complex numbers appear naturally in 2-D Minkowski space since the solutions of a massless field equation can be written as f=g(u=t-ex)+h(v=t+ex) whith e2=1 realized by putting e=1. Therefore Wick rotation relates sums of holomorphic and antiholomorphic functions to sums of hyper-holomorphic and anti-hyper-holomorphic functions. Note that u and v are hyper-complex conjugates of each other.

Complex n-dimensional spaces allow Hermitian structure. This means that the metric has in complex coordinates (z1,....,zn) the form in which the matrix elements of metric are nonvanishing only between zi and complex conjugate of zj. In 2-D case one obtains just ds2=gzz*dzdz*. Note that in this case metric is conformally flat since line element is proportional to the line element ds2=dzdz* of plane. This form is always possible locally. For complex n-D case one obtains ds2=gij*dzidzj*. gij*=(gji*)* guaranteing the reality of ds2. In 2-D case this condition gives gzz*= (gz*z)*.

How could one generalize this line element to hyper-complex n-dimensional case? In 2-D case Minkowski space M2 one has ds2= guvdudv, guv=1. The obvious generalization would be the replacement ds2=guivjduidvj. Also now the analogs of reality conditions must hold with respect to ui↔ vi.

Hamilton-Jacobi structure

Consider next the path leading to Hamilton-Jacobi structure.

4-D Minkowski space M4=M2× E2 is Cartesian product of hyper-complex M2 with complex plane E2, and one has ds2= dudv+ dzdz* in standard Minkowski coordinates. One can also consider more general integrable decompositions of M4 for which the tangent space TM4=M4 at each point is decomposed to M2(x)× E2(x). The physical analogy would be a position dependent decomposition of the degrees of freedom of massless particle to longitudinal ones (M2(x): light-like momentum is in this plane) and transversal ones (E2(x): polarization vector is in this plane). Cylindrical and spherical variants of Minkowski coordinates define two examples of this kind of coordinates (it is perhaps a good exercize to think what kind of decomposition of tangent space is in question in these examples). An interesting mathematical problem highly relevant for TGD is to identify all possible decompositions of this kind for empty Minkowski space.

The integrability of the decomposition means that the planes M2(x) are tangent planes for 2-D surfaces of M4 analogous to Euclidian string world sheet. This gives slicing of M4 to Minkowskian string world sheets parametrized by euclidian string world sheets. The question is whether the sheets are stringy in a strong sense: that is minimal surfaces. This is not the case: for spherical coordinates the Euclidian string world sheets would be spheres which are not minimal surfaces. For cylindrical and spherical coordinates hower M2(x) integrate to plane M2 which is minimal surface.

Integrability means in the case of M2(x) the existence of light-like vector field J whose flow lines define a global coordinate. Its existence implies also the existence of its conjugate and together these vector fields give rise to M2(x) at each point. This means that one has J= Ψ∇ Φ: Φ indeed defines the global coordinate along flow lines. In the case of M2 either the coordinate u or v would be the coordinate in question. This kind of flows are called Beltrami flows. Obviously the same holds for the transversal planes E2.

One can generalize this metric to the case of general 4-D space with Minkowski signature of metric. At least the elements guv and gzz* are non-vanishing and can depend on both u,v and z,z*. They must satisfy the reality conditions gzz*= (gzz*)* and guv= (gvu)* where complex conjugation in the argument involves also u↔ v besides z↔ z*.

The question is whether the components guz, gvz, and their complex conjugates are non-vanishing if they satisfy some conditions. They can. The direct generalization from complex 2-D space would be that one treats u and v as complex conjugates and therefore requires a direct generalization of the hermiticity condition

guz= (gvz*)*, gvz= (guz*)* .

This would give complete symmetry with the complex 2-D (4-D in real sense) spaces. This would allow the algebraic continuation of hermitian structures to Hamilton-Jacobi structures by just replacing i with e for some complex coordinates.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Friday, July 27, 2012

The importance of being light-like

The singular geometric objects associated with the space-time surface have become increasingly important in TGD framework. In particular, the recent progress has made clear that these objects might be crucial for the understanding of quantum TGD. The singular objects are associated not only with the induced metric but also with the effective metric defined by the anti-commutators of the modified gamma matrices appearing in the modified Dirac equation and determined by the Kähler action.

The singular objects associated with the induced metric

Consider first the singular objects associated with the induced metric.

  1. At light-like 3-surfaces defined by wormhole throats the signature of the induced metric changes from Euclidian to Minkowskian so that 4-metric is degenerate. These surfaces are carriers of elementary particle quantum numbers and the 4-D induced metric degenerates locally to 3-D one at these surfaces.

  2. Braid strands at light-like 3-surfaces are most naturally light-like curves: this correspond to the boundary condition for open strings. One can assign fermion number to the braid strands. Braid strands allow an identification as curves along which the Euclidian signature of the string world sheet in Euclidian region transforms to Minkowskian one. Number theoretic interpretation would be as a transformation of complex regions to hyper-complex regions meaning that imaginary unit i satisfying i2=-1 becomes hyper-complex unit e satisfying e2=1. The complex coordinates (z,z*) become hyper-complex coordinates (u=t+ex, v=t-ex) giving the standard light-like coordinates when one puts e=1.

The singular objects associated with the effective metric

There are also singular objects assignable to the effective metric. According to the simple arguments already developed, string world sheets and possibly also partonic 2-surfaces are singular objects with respect to the effective metric defined by the anti-commutators of the modified gamma matrices rather than induced gamma matrices. Therefore the effective metric seems to be much more than a mere formal structure.

  1. For instance, quaternionicity of the space-time surface could allow an elegant formulation in terms of the effective metric avoiding the problems due to the Minkowski signature. This is achieved if the effective metric has Euclidian signature ε × (1,1,1,1), ε=+/- 1 or a complex counterpart of the Minkowskian signature ε (1,1,-1,-1).

  2. String world sheets and perhaps also partonic 2-surfaces could be understood as singularities of the effective metric. What happens that the effective metric with Euclidian signature ε ×(1,1,1,1) transforms to the signature ε (1,1,-1,-1) (say) at string world sheet so that one would have the degenerate signature ε×(1,1,0,0) at the string world sheet.

    What is amazing is that this works also number theoretically. It came as a total surprise to me that the notion of hyper-quaternions as a closed algebraic structure indeed exists. The hyper-quaternionic units would be given by (1,I, iJ,iK), where i is a commuting imaginary unit satisfying i2=-1. Hyper-quaternionic numbers defined as combinations of these units with real coefficients do form a closed algebraic structure which however fails to be a number field just like hyper-complex numbers do. Note that the hyper-quaternions obtained with real coefficients from the basis (1,iI,iJ,iK) fail to form an algebra since the product is not hyper-quaternion in this sense but belongs to the algebra of complexified quaternions. The same problem is encountered in the case of hyper-octonions defined in this manner. This has been a stone in my shoe since I feel strong disrelish towards Wick rotation as a trick for moving between different signatures.

  3. Could also partonic 2-surfaces correspond to this kind of singular 2-surfaces? In principle, 2-D surfaces of 4-D space intersect at discrete points just as string world sheets and partonic 2-surfaces do so that this might make sense. By complex structure the situation is algebraically equivalent to the analog of plane with non-flat metric allowing all possible signatures (ε12) in various regions. At light-like curve either ε1 or ε2 changes sign and light-like curves for these two kinds of changes can intersect as one can easily verify by drawing what happens. At the intersection point the metric is completely degenerate and simply vanishes.

  4. Replacing real 2-dimensionality with complex 2-dimensionality, one obtains by the universality of algebraic dimension the same result for partonic 2-surfaces and string world sheets. The braid ends at partonic 2-surfaces representing the intersection points of 2-surfaces of this kind would have completely degenerate effective metric so that the modified gamma matrices would vanish implying that energy momentum tensor vanishes as also the induced Kähler field.

  5. The effective metric suffers a local conformal scaling in the critical deformations identified in the proposed manner. Since ordinary conformal group acts on Minkowski space and leaves the boundary of light-cone invariant, one has two conformal groups. It is not however clear whether the M4 conformal transformations can act as symmetries in TGD, where the presence of the induced metric in Kähler action breaks M4 conformal symmetry. As found, also in TGD framework the Kac-Moody currents assigned to the braid strands generate Yangian: this is expected to be true also for the Kac-Moody counterparts of the conformal algebra associated with quantum criticality. On the other hand, in twistor program one encounters also two conformal groups and the space in which the second conformal group acts remains somewhat mysterious object. The Lie algebras for the two conformal groups generate the conformal Yangian and the integrands of the scattering amplitudes are Yangian invariants. Twistor approach should apply in TGD if zero energy ontology is right. Does this mean a deep connection?

    What is also intriguing that twistor approach in principle works in strict mathematical sense only at signatures ε × (1,1,-1-1) and the scattering amplitudes in Minkowski signature are obtained by analytic continuation. Could the effective metric give rise to the desired signature? Note that the notion of massless particle does not make sense in the signature ε × (1,1,1,1).

These arguments provide genuine a support for the notion of quaternionicity and suggest a connection with the twistor approach.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Wednesday, July 25, 2012

Quantum criticality and electro-weak gauge symmetries

Quantum criticality is one of the basic guiding principles of Quantum TGD. What it means mathematically is however far from clear.

  1. What is obvious is that quantum criticality implies quantization of Kähler coupling strength as a mathematical analog of critical temperature so that the theory becomes mathematically unique if only single critical temperature is possible. Physically this means the presence of long range fluctuations characteristic for criticality and perhaps assignable to the effective hierarchy of Planck constants having explanation in terms of effective covering spaces of the imbedding space. This hierarchy follows from the vacuum degeneracy of Kähler action, which in turn implies 4-D spin-glass degeneracy. It is easy to interpret the degeneracy in terms of criticality.

  2. At more technical level one would expect criticality to corresponds deformations of a given preferred extremal defining a vanishing second variation of Kähler action. This is analogous to the vanishing of also second derivatives of potential function at extremum in certain directions so that the matrix defined by second derivatives does not have maximum rank. Entire hierarchy of criticalities is expected and a good finite-dimensional model is provided by the catastrophe theory of Thom. Cusp catastrophe is the simplest catastrophe one can think of, and here the folds of cusp where discontinuous jump occurs correspond to criticality with respect to one control variable and the tip to criticality with respect to both control variables.

  3. I have discussed what criticality could mean for modified Dirac action (see this) and claimed that it leads to the existence of additional conserved currents defined by the variations which do not affect the value of Kähler action. These arguments are far from being mathematically rigorous and the recent view about the solutions of the modified Dirac equation predicting that the spinor modes are restricted to 2-D string world sheets requires a modification of these arguments.
In the following these arguments are updated. The unexpected result is that critical deformations induce conformal scalings of the modified metric and electro-weak gauge transformations of the induced spinor connection at X2. Therefore holomorphy brings in the Kac-Moody symmetries associated with isometries of H (gravitation and color gauge group) and quantum criticality those associated with the holonomies of H (electro-weak-gauge group) as additional symmetries.

The variation of modes of the induced spinor field in a variation of space-time surface respecting the preferred extremal property

Consider first the variation of the induced spinor field in a variation of space-time surface respecting the preferred extremal property. The deformation must be such that the deformed modified Dirac operator D annihilates the modified mode. By writing explicitly the variation of the modified Dirac action (the action vanishes by modified Dirac equation) one obtains deformations and requiring its vanishing one obtains

δ Ψ=D-1(δ D)Ψ .

D-1 is the inverse of the modified Dirac operator defining the analog of Dirac propagator and δ D defines vertex completely analogous to γkδ Ak in gauge theory context. The functional integral over preferred extremals can be carried out perturbatively by expressing Δ D in terms of δ hk and one obtains stringy perturbation theory around X2 associated with the preferred extremal defining maximum of Kähler function in Euclidian region and extremum of Kähler action in Minkowskian region (stationary phase approximation).

What one obtains is stringy perturbation theory for calculating n-points functions for fermions at the ends of braid strands located at partonic 2-surfaces and representing intersections of string world sheets and partonic 2-surfaces at the light-like boundaries of CDs. δ D- or more precisely, its partial derivatives with respect to functional integration variables - appear atthe vertices located anywhere in the interior of X2 with outcoming fermions at braid ends. Bosonic propagators are replaced with correlation functions for δ hk. Fermionic propagator is defined by D-1.

After 35 years or hard work this provides for the first time a reasonably explicit formula for the N-point functions of fermions. This is enough since by bosonic emergence(se this) these N-point functions define the basic building blocks of the scattering amplitudes. Note that bosonic emergence states that bosons corresponds to wormhole contacts with fermion and antifermion at the opposite wormhole throats.

What critical modes could mean for the induced spinor fields?

What critical modes could mean for the induced spinor fields at string world sheets and partonic 2-surfaces. The problematic part seems to be the variation of the modified Dirac operator since it involves gradient. One cannot require that covariant derivative remains invariant since this would require that the components of the induced spinor connection remain invariant and this is quite too restrictive condition. Right handed neutrino solutions delocalized into entire X2 are however an exception since they have no electro-weak gauge couplings and in this case the condition is obvious: modified gamma matrices suffer a local scaling for critical deformations:

δ Γμ = Λ(x)Γμ .

This guarantees that the modified Dirac operator D is mapped to Λ D and still annihilates the modes of νR labelled by conformal weight, which thus remain unchanged.

What is the situation for the 2-D modes located at string world sheets? The condition is obvious. Ψ suffers an electro-weak gauge transformation as does also the induced spinor connection so that Dμ is not affected at all. Criticality condition states that the deformation of the space-time surfaces induces a conformal scaling of Γμ at X2, It might be possible to continue this conformal scaling of the entire space-time sheet but this might be not necessary and this would mean that all critical deformations induced conformal transformations of the effective metric of the space-time surface defined by {Γμ, Γν}=2 Gμν. Thus it seems that effective metric is indeed central concept (recall that if the conjectured quaternionic structure is associated with the effective metric, it might be possible to avoid problem related to the Minkowskian signature in an elegant manner).

Note that one can consider even more general action of critical deformation: the modes of the induced spinor field would be mixed together in the infinitesimal deformation besides infinitesimal electroweak gauge transformation, which is same for all modes. This would extend electroweak gauge symmetry. Modified Dirac equation holds true also for these deformations. One might wonder whether the conjecture dynamically generated gauge symmetries assignable to finite measurement resolution could be generated in this manner.

Thus the critical deformations would induce conformal scalings of the effective metric and dynamical electro-weak gauge transformations. Electro-weak gauge symmetry would be a dynamical symmetry restricted to string world sheets and partonic 2-surfaces rather than acting at the entire space-time surface. For 4-D delocalized right-handed neutrino modes the conformal scalings of the effective metric are analogous to the conformal transformations of M4 for N=4 SYMs. Also ordinary conformal symmetries of M4 could be present for string world sheets and could act as symmetries of generalized Feynman graphs since even virtual wormhole throats are massless. An interesting question is whether the conformal invariance associated with the effective metric is the analog of dual conformal invariance in N=4 theories.

Critical deformations of space-time surface are accompanied by conserved fermionic currents. By using standard Noetherian formulas one can write

Jμi= Ψbar Γμδi Ψ + δi ΨbarΓμΨ .

Here δ Ψi denotes derivative of the variation with respect to a group parameter labeled by i. Since δ Ψi reduces to an infinitesimal gauge transformation of Ψ induced by deformation, these currents are the analogs of gauge currents. The integrals of these currents along the braid strands at the ends of string world sheets define the analogs of gauge charges. The interpretation as Kac-Moody charges is also very attractive and I have proposed that the 2-D Hodge duals of gauge potentials could be identified as Kac-Moody currents. If so, the 2-D Hodge duals of J would define the quantum analogs of dynamical electro-weak gauge fields and Kac-Moody charge could be also seen as non-integral phase factor associated with the braid strand in Abelian approximation (the interpretation in terms of finite measurement resolution is discussed earlier).

One can also define super currents by replacing Ψbar or Ψ by a particular mode of the induced spinor field as well as c-number valued currents by performing the replacement for both Ψbar and Ψ. As expected, one obtains a super-conformal algebra with all modes of induced spinor fields acting as generators of super-symmetries restricted to 2-D surfaces. The number of the charges which do not annihilate physical states as also the effective number of fermionic modes could be finite and this would suggest that the integer N for the supersymmetry in question is finite. This would conform with the earlier proposal inspired by the notion of finite measurement resolution implying the replacement of the partonic 2-surfaces with collections of braid ends.

Note that Kac-Moody charges might be associated with "long" braid strands connecting different wormhole throats as well as short braid strands connecting opposite throats of wormhole contacts. Both kinds of charges would appear in the theory.

What is the interpretation of the critical deformations?

Critical deformations bring in an additional gauge symmetry. Certainly not all possible gauge transformations are induced by the deformations of preferred extremals and a good guess is that they correspond to holomorphic gauge group elements as in theories with Kac-Moody symmetry. What is the physical character of this dynamical gauge symmetry?

  1. Do the gauge charges vanish? Do they annihilate the physical states? Do only their positive energy parts annihilate the states so that one has a situation characteristic for the representation of Kac-Moody algebras. Or could some of these charges be analogous to the gauge charges associated with the constant gauge transformations in gauge theories and be therefore non-vanishing in the absence of confinement. Now one has electro-weak gauge charges and these should be non-vanishing. Can one assign them to deformations with a vanishing conformal weight and the remaining deformations to those with non-vanishing conformal weight and acting like Kac-Moody generators on the physical states?

  2. The simplest option is that the critical Kac-Moody charges/gauge charges with non-vanishing positive conformal weight annihilate the physical states. Critical degrees of freedom would not disappear but make their presence known via the states labelled by different gauge charges assignable to critical deformations with vanishing conformal weight. Note that constant gauge transformations can be said to break the gauge symmetry also in the ordinary gauge theories unless one has confinement.

  3. The hierarchy of quantum criticalities suggests however entire hierarchy of electro-weak Kac-Moody algebras. Does this mean a hierarchy of electro-weak symmetries breakings in which the number of Kac-Moody generators not annihilating the physical states gradually increases as also modes with a higher value of positive conformal weight fail to annihilate the physical state?

    The only manner to have a hierarchy of algebras is by assuming that only the generators satisfying n mod N=0 define the sub-Kac-Moody algebra annihilating the physical states so that the generators with n mod N≠ 0 would define the analogs of gauge charges. I have suggested for long time ago the relevance of kind of fractal hierarchy of Kac-Moody and Super-Virasoro algebras for TGD but failed to imagine any concrete realization.

    A stronger condition would be that the algebra reduces to a finite dimensional algebra in the sense that the actions of generators Qn and Qn+kN are identical. This would correspond to periodic boundary conditions in the space of conformal weights. The notion of finite measurement resolution suggests that the number of independent fermionic oscillator operators is proportional to the number of braid ends so that an effective reduction to a finite algebra is expected.

    Whatever the correct interpretation is, this would obviously refine the usual view about electro-weak symmetry breaking.

These arguments suggests the following overall view. The holomorphy of spinor modes gives rise to Kac-Moody algebra defined by isometries and includes besides Minkowskian generators associated with gravitation also SU(3) generators associated with color symmetries. Vanishing second variations in turn define electro-weak Kac-Moody type algebra.

Note that criticality suggests that one must perform functional integral over WCW by decomposing it to an integral over zero modes for which deformations of X4 induce only an electro-weak gauge transformation of the induced spinor field and to an integral over moduli corresponding to the remaining degrees of freedom.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Saturday, July 21, 2012

Why Higgs is not favored in TGD?

The discovery of a new spinless particle at LHC has dominated the discussions in physics blogs during last weeks. Quite many bloggers identify without hesitation the new particle as the long sought for Higgs although some aspects of data do not encourage the interpretation as standard model Higgs or possibly its SUSY variant. Maybe the reason is that it is rather imagine any other interpretation. In this article the TGD based interpretation as a pion-like state of scaled up variant of hadron physics is discussed explaining also why Higgs is not needed and why it cannot even perform the tasks posed for it in TGD framework.

Essentially single assumption, the separate conservation of quark and lepton numbers realized in terms of 8-D chiral invariance, excludes Higgs like states as also standard N=1 SUSY. This identification could explain the failure to find the decays to τ pairs and also the excess of two-gamma decays. The decays gauge boson pairs would be caused by the coupling of pion-like state to instanton density for electro-weak gauge fields. Also a connection with the dark matter researches reporting signal at 130 GeV and possibly also at 110 GeV suggests itself: maybe also these signals also correspond to pion-like states.

The detailed arguments can be found in the article Is it really Higgs? at my homepage.

Thursday, July 19, 2012

Higgs as a belief

I have been following the postings of Lubos Motl about Higgs (here is one example). It seems that Lubos has developed a real complex from Higgs. He has written posting after posting trying to convince the reader that the new spinless particle cannot be but Higgs. "Why Higgs had to be discovered" was the title of one posting. "Idiot" belongs to the vocabulary used about those who think differently.

I wonder why Lubos does he have this fix idee, one of many other extremist beliefs, many of them irrational. Is this related to assimilation with existing theory or with establishment? Lubos lived his childhood in a totalitarian society. Did he somehow inherit the strong tendency to totalitarian thinking in which arguments are justified by names? To me Lubos in many respects looks like a mirror image of a fanatic communist from Soviet Union.

As a theoretical physicist Lubos should be able to see that Higgs is just one explanation, and that one never can prove that Higgs exist, only its non-existence. At best its presence allows to explain the data but there are alternative explanations. Standard model Higgs seems to be even excluded already now if data are taken at face value. Here most of the bloggers follow text book thinking and neglect the data: Phil Gibbs is a refreshing exception in this respect.

How dangerous the Lubosian thinking is for theoretical physicist himself is, can be demonstrated by simple example.

  1. Suppose - in accordance with string models and TGD - that higher-dimensional space-time exists in some sense as the arena of dynamics.

  2. Suppose that baryon and lepton numbers are conserved separately and correspond to different chiralities of the spinors of higher-D space: there is indeed not a single proton decay to prove that this is not the case. One can also consider giving up the separate conservation but keeping the identification of quarks and leptons.
The argument is very simple if one accepts these assumptions.
  1. Suppose first that Higgs is scalar in this higher-dimensional space. With these assumptions the only conclusion is that Higgs decays to lepton-quark pairs and that the theory does not conserve baryon and lepton numbers separately. This is a contradiction with basic assumptions and also conflicts the existing data for the spin 0 boson.

  2. One could alternatively argue that Higgs is actually a vector in higher-D space having components only in "internal degrees of freedom". This would fix the dimension of internal space to D=4 and imbedding space would be 8-dimensional. In this case it becomes however very difficult to imagine how gauge bosons could "eat" the components of Higgs.
The entire Higgs paradigm falls down with single quite reasonable assumption: B and L are separately conserved and correspond to different chiralities of imbedding space spinors.

In TGD framework problem disappears in zero energy ontology.

  1. Higgs is not needed and p-adic thermodynamics describes particle massivation. A very important and testable outcome is that it is longitudinal mass squared which obeys p-adic thermodynamics: total mass squared vanishes and al particles are massless at basic level. This implies twistorial picture and Yangian symmetry and absence of IR and UV divergences since virtual particles are on mass shell massless particles with propagator defined by longitudinal momentum.

  2. But what about Lorentz invariance? Since causal diamonds characterized by longitudinal sub-space M^2 have Lorentz boosts as moduli, Lorentz invariance is not broken and one obtains a nice connection with the basic picture of QCD involving parton distributions depending on longitudinal momenta.

Lubos should be also able to see that Higgs is just a phenomenological description and that there must be a real microscopic theory behind Higgs in the case that it is needed. p-Adic thermodynamics is my proposal.

Tuesday, July 17, 2012

Constraints on super-conformal invariance from p-adic mass calculations and ZEO

The generalization of super-conformal symmetry to 4-D context is basic element of quantum TGD. Several variants for the realization of supersymmetry has been proposed. Especially problematic has been the question about whether the counterpart of standard SUSY is realized in TGD framework or not. Thanks to the progress made in the understanding of preferred extremals of Kähler action and of solutions of modified Dirac equation, it has become to develop the vision in considerable detail. As a consequence some existing alternative visions have been eliminated from consideration. One of them the proposal that Equivalence Principle might have realization in terms of coset representations. Second one is the idea that right-handed neutrino generating space-time supersymmetry might be in color partial wave so that sparticles would be colored: this would explain why sparticles are not observed at LHC. The new view providing a possible alternative explanation for the absence of sparticles has been discussed in previous posting.

Concerning the general understanding of super-conformal invariance in TGD framework, an important physical constraint comes from the success p-adic thermodynamics: superconformal invariance indeed forms a core element of p-adic mass calculations (see this, especially this).

  1. The first thing that one can get worried about relates to the extension of conformal symmetries. If the conformal symmetries generalize to D=4, how can one take seriously the results of p-adic mass calculations based on 2-D conformal invariance? There is no reason to worry. The reduction of the conformal invariance to 2-D one for the preferred extremals takes care of this problem. This however requires that the fermionic contributions assignable to string world sheets and/or partonic 2-surfaces - Super- Kac-Moody contributions - should dictate the elementary particle masses. For hadrons also symplectic contributions should be present (see this). This is a valuable hint in attempts to identify the mathematical structure in more detail.

  2. Zero Energy Ontology (ZEO) suggests that all particles, even virtual ones correspond to massless wormhole throats carrying fermions. As a consequence, twistor approach would work and the kinematical constraints to vertices would allow th cancellation of both UV and IR divergences. This would suggest that the p-adic thermal expectation value is for the longitudinal M2 momentum squared (the definition of CD selects M1⊂ M2⊂ M4 as also does number theoretic vision). Also propagator would be determined by M2 momentum. Lorentz invariance would be obtained by integration of the moduli for CD including also Lorentz boosts of CD. This is definitely something new from standard physics point of view, but suggested already by the first p-adic mass calculations. The fact that parton distributions in hadrons are functions of longitudinal momentum fraction, and the division of tangent space of M4 to longitudinal and transversal parts for gauge bosons suggests the same. Also number theoretical vision and the need to assign to realize the choice of spin quantization axis and time like direction defining the rest system in quantum measurement theory at the level of WCW geometry also this.

  3. In the original approach one allows states with arbitrary large values of L0 as physical states. Usually one would require that L0 annihilates the states. In the calculations however mass squared was assumed to be proportional L0 apart from vacuum contribution. This is a questionable assumption. ZEO suggests that total mass squared vanishes and that one can decompose mass squared to a sum of longitudinal and transversal parts. If one can do the same decomposition to longitudinal and transverse parts also for the Super Virasoro algebra then one can calculate longitudinal mass squared as a p-adic thermal expectation in the transversal super-Virasoro algebra and only states with L0=0 would contribute and one would have conformal invariance in the standard sense.

  4. The assumption motivated by Lorentz invariance has been that mass squared is replaced with conformal weight in thermodynamics, and that one first calculates the thermal average of the conformal weight and then equates it with mass squared. This assumption is somewhat ad hoc. ZEO however suggests an alternative interpretation in which one has zero energy states for which longitudinal mass squared of positive energy state derive from p-adic thermodynamics. Thermodynamics - or rather, its square root - would become part of quantum theory in ZEO. M-matrix is indeed product of hermitian square root of density matrix multiplied by unitary S-matrix and defines the entanglement coefficients between positive and negative energy parts of zero energy state.

  5. The crucial constraint is that the number of super-conformal tensor factors is N=5: this suggests that thermodynamics applied in Super-Kac-Moody degrees of freedom assignable to string world sheets is enough, when one is interested in the masses of fermions and gauge bosons. Super-symplectic degrees of freedom can also contribute and determine the dominant contribution to baryon masses. Should also this contribution obey p-adic thermodynamics in the case when it is present? Or does the very fact that this contribution need not be present mean that it is not thermal? The symplectic contribution should correspond to hadronic p-adic length prime rather the one assignable to (say) u quark. Hadronic p-adic mass squared and partonic p-adic mass squared cannot be summed since primes are different. If one accepts the basic rules (see this), longitudinal energy and momentum are additive as indeed assumed in perturbative QCD.

  6. Calculations work if the vacuum expectation value of the mass squared must be assumed to be tachyonic. There are two options depending on whether one whether p-adic thermodynamics gives total mass squared or longitudinal mass squared.

    1. One could argue that the total mass squared has naturally tachyonic ground state expectation since for massless extremals longitudinal momentum is light-like and transversal momentum squared is necessary present and non-vanishing by the localization to topological light ray of finite thickness of order p-adic length scale. Transversal degrees of freedom would be modeled with a particle in a box.

    2. If longitudinal mass squared is what is calculated, the condition would require that transversal momentum squared is negative so that instead of plane wave like behavior exponential damping would be required. This would conform with the localization in transversal degrees of freedom.

  7. For preferred extremals Einstein's equations with cosmological term are satisfied as a consistency condition guaranteing algebraization of the equations. Hence Equivalence Principle holds true. But what about possible quantum realization of Equivalence Principle in this framework? A possible quantum counterpart of Equivalence Principle could be that the longitudinal parts of the imbedding space mass squared operator for a given massless state equals to that for d'Alembert operator assignable to the modified Dirac action. The attempts to formulate this in more precise manner however seem to produce only troubles.

To sum up, the basic new element inspired by ZEO is that p-adic mass calculations are for the longitudinal momentum squared and that elementary particles and even hadrons are basically massless. This assumption looks certainly strange at first. Thank so the integration over boosts of CDs it is however consistent with Lorentz invariance and also allows to understand p-adic thermodynamics as thermodynamics for the transversal part of scaling generator L0. ZEO allows indeed superposition of pairs of positive and negative energies with different momenta for positive energy state without a loss of Lorentz invariance. Also p-adic thermodynamics finds a natural interpretation in terms of M-matrix defining square root of hermitian density matrix.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Monday, July 16, 2012

The role of the right-handed neutrino in TGD based view about SUSY

The general ansatz for the preferred extremals of Kähler action and application of the conservation of em charge to the modified Dirac equation have led to a rather detailed view about classical and TGD and allowed to build a bridge between general vision about super-conformal symmetries in TGD Universe and field equations.

  1. Equivalence Principle realized as Einstein's equations in all scales follows directly from the general assatz for preferred extremals implying that space-time surface has either hermitian or Hamilton-Jacobi structure (which of them depends on the signature of the induced metric).

  2. The general structure of Super Virasoro representations can be understood: super-symplectic algebra is responsible for the non-perturbative aspects of QCD and determines also the ground states of elementary particles determining their quantum numbers.

  3. Super-Kac-Moody algebras associated with isometries and holonomies dictate standard model quantum numbers and lead to a massivation by p-adic thermodynamics: the crucial condition that the number of tensor factors in Super-Virasoro represention is 5 is satisfied.

  4. One can understand how the Super-Kac-Moody currents assignable to stringy world sheets emerging naturally from the conservation of em charge defined as their string world sheet Hodge duals gauge potentials for standard model gauge group and also their analogs for gravitons. Also the conjecture Yangian algebra generated by Super-Kac-Moody charges emerges naturally.

  5. One also finds that right handed neutrino is in a very special role because of its lacking couplings in electroweak sector and its role as a generator of the least broken SUSY. All other modes of induced spinor field are restricted to 2-D string world sheets and partonic 2-surfaces. Right-handed neutrino allows also the mode delocalized to entire space-time surface or perhaps only to the Euclidian regions defined by the 4-D line of the generalized Feynman diagrams.

    In fact, in the following the possibility that the resulting sparticles cannot be distinguished from particles since the presence of right handed neutrino is not seen in the interactions and does not manifest itself in different spin structures for the couplings of particle and sparticle. This could explain the failure to detect spartners at LHC. Intermediate gauge boson decay widths however require that sparticles are dark in the sense of having non-standard value of Planck constant. Another variant of the argument assumes that 4-D right handed neutrinos are associated with space-time regions of Minkowskian signature and SUSY is defined for many-particle states rather than single particle states. It should be emphasized that TGD predicts that all fermions act as generators of badly broken supersymmetries at partonic 2-surfaces but these super-symmetries could correspond to much higher mass scale as that associated with the delocalized right-handed neutrino. The following piece of text summarizes the argument.

A highly interesting aspect of Super-Kac-Moody symmetry is the special role of right handed neutrino.

  1. Only right handed neutrino allows besides the modes restricted to 2-D surfaces also the 4D modes delocalized to the entire space-time surface. The first ones are holomorphic functions of single coordinate and the latter ones holomorphic functions of two complex/Hamilton-Jacobi coordinates. OnlyνR has the full D=4 counterpart of the conformal symmetry and the localization to 2-surfaces has interpretation as super-conformal symmetry breaking halving the number of super-conformal generators.

  2. This forces to ask for the meaning of super-partners. Are super-partners obtained by adding νR neutrino localized at partonic 2-surface or delocalized to entire space-time surface or its Euclidian or Minkowskian region accompanying particle identified as wormhole throat? Only the Euclidian option allows to assign right handed neutrino to a unique partonic 2-surface. For the Minkowskian regions the assignment is to many particle state defined by the partonic 2-surfaces associated with the 3-surface. Hence for spartners the 4-D right-handed neutrino must be associated with the 4-D Euclidian line of the generalized Feynman diagram.

  3. The orthogonality of the localized and de-localized right handed neutrino modes requires that 2-D modes correspond to higher color partial waves at the level of imbedding space. If color octet is in question, the 2-D right handed neutrino as the candidate for the generator of standard SUSY would combine with the left handed neutrino to form a massive neutrino. If 2-D massive neutrino acts as a generator of super-symmetries, it is is in the same role as badly broken supers-ymmeries generated by other 2-D modes of the induced spinor field (SUSY with rather large value of N) and one can argue that the counterpart of standard SUSY cannot correspond to this kind of super-symmetries. The right-handed neutrinos delocalized inside the lines of generalized Feynman diagrams, could generate N=2 variant of the standard SUSY.

1. How particle and right handed neutrino are bound together?

Ordinary SUSY means that apart from kinematical spin factors sparticles and particles behave identically with respect to standard model interactions. These spin factors would allow to distinguish between particles and sparticles. But is this the case now?

  1. One can argue that 2-D particle and 4-D right-handed neutrino behave like independent entities, and because νR has no standard model couplings this entire structure behaves like a particle rather than sparticle with respect to standard model interactions: the kinematical spin dependent factors would be absent.

  2. The question is also about the internal structure of the sparticle. How the four-momentum is divided between the νR and and 2-D fermion. If νR carries a negligible portion of four-momentum, the four-momentum carried by the particle part of sparticle is same as that carried by particle for given four-momentum so that the distinctions are only kinematical for the ordinary view about sparticle and trivial for the view suggested by the 4-D character of νR.
Could sparticle character become manifest in the ordinary scattering of sparticle?
  1. If νR behaves as an independent unit not bound to the particle it would continue un-scattered as particle scatters: sparticle would decay to particle and right-handed neutrino. If νR carries a non-negligible energy the scattering could be detected via a missing energy. If not, then the decay could be detected by the interactions revealing the presence of νR. νR can have only gravitational interactions. What these gravitational interactions are is not however quite clear since the proposed identification of gravitational gauge potentials is as duals of Kac-Moody currents analogous to gauge potentials located at the boundaries of string world sheets. Does this mean that 4-D right-handed neutrino has no quantal gravitational interactions? Does internal consistency require νR to have a vanishing gravitational and inertial masses and does this mean that this particle carries only spin?

  2. The cautious conclusion would be following: if delocalized νR and parton are un-correlated particle and sparticle cannot be distinguished experimentally and one might perhaps understand the failure to detect standard SUSY at LHC. Note however that the 2-D fermionic oscillator algebra defines badly broken large N SUSY containing also massive (longitudinal momentum square is non-vanishing) neutrino modes as generators.

2. Taking a closer look on sparticles

It is good to take a closer look at the delocalized right handed neutrino modes.

  1. At imbedding space level that is in cm mass degrees of freedom they correspond to covariantly constant CP2 spinors carrying light-like momentum which for causal diamond could be discretized. For non-vanishing momentum one can speak about helicity having opposite sign for νR and νRbar. For vanishing four-momentum the situation is delicate since only spin remains and Majorana like behavior is suggestive. Unless one has momentum continuum, this mode might be important and generate additional SUSY resembling standard N=1 SUSY.

  2. At space-time level the solutions of modified Dirac equation are holomorphic or anti-holomorphic.

    1. For non-constant holomorphic modes these characteristics correlate naturally with fermion number and helicity of νR . One can assign creation/annihilation operator to these two kinds of modes and the sign of fermion number correlates with the sign of helicity.

    2. The covariantly constant mode is naturally assignable to the covariantly constant neutrino spinor of imbedding space. To the two helicities one can assign also oscillator operators {a+/-,a+/-}. The effective Majorana property is expressed in terms of non-orthogonality of νR and and νRbar translated to the the non-vanishing of the anti-commutator {a+,a-}= {a-,a+}=1. The reduction of the rank of the 4× 4 matrix defined by anti-commutators to two expresses the fact that the number of degrees of freedom has halved. a+=a- realizes the conditions and implies that one has only N=1 SUSY multiplet since the state containing both νR and νRbar is same as that containing no right handed neutrinos.

    3. One can wonder whether this SUSY is masked totally by the fact that sparticles with all possible conformal weights n for induced spinor field are possible and the branching ratio to n=0 channel is small. If momentum continuum is present, the zero momentum mode might be equivalent to nothing.

What can happen in spin degrees of freedom in super-symmetric interaction vertices if one accepts this interpretation? As already noticed, this depends solely on what one assumes about the correlation of the four-momenta of particle and νR.

  1. For SUSY generated by covariantly constant νR and νRbar there is no neutrino four-momentum involved so that only spin matters. One cannot speak about the change of direction for νR. In the scattering of sparticle the direction of particle changes and introduces different spin quantization axes. νR retains its spin and in new system it is superposition of two spin projections. The presence of both helicities requires that the transformation νR→ νRbar happens with an amplitude determined purely kinematically by spin rotation matrices. This is consistent with fermion number conservation modulo 2. N=1 SUSY based on Majorana spinors is highly suggestive.

  2. For SUSY generated by non-constant holomorphic and anti-holomorphic modes carrying fermion number the behavior in the scattering is different. Suppose that the sparticle does not split to particle moving in the new direction and νR moving in the original direction so that also νR or νRbar carrying some massless fraction of four-momentum changes its direction of motion. One can form the spin projections with respect to the new spin axis but must drop the projection which does not conserve fermion number. Therefore the kinematics at the vertices is different. Hence N=2 SUSY with fermion number conservation is suggestive when the momentum directions of particle and νR are completely correlated.

  3. Since right-handed neutrino has no standard model couplings, p-adic thermodynamics for 4-D right-handed neutrino must correspond to a very low p-adic temperature T=1/n. This implies that the excitations with higher conformal weight are effectively absent and one would have N =1 SUSY effectively.

    The simplest assumption is that particle and sparticle correspond to the same p-adic mass scale and have degenerate masses: it is difficult to imagine any good reason for why the p-adic mass scales should differ. This should have been observed -say in decay widths of weak bosons - unless the spartners correspond to large hbar phase and therefore to dark matter . Note that for the badly broken 2-D N=2 SUSY in fermionic sector this kind of almost degeneracy cannot be excluded and I have considered an explanation for the mysterious X and Y mesons in terms of this degeneracy (see this).

  4. LHC suggests that one does not have N=1 SUSY in standard sense. Could spartners correspond to dark matter with a large value of Planck constant and same mass? Or could 4-D right-handed neutrino exists only in the Minkowskian regions where they define superpartners of many particle states rather than single particle states? Could the reason be that for CP2 type vacuum extremals modified gamma matrices vanish identically? Could this be used to argue that 4-D right-handed neutrinos cannot appear in the lines of generalize Feynman graphs which involve deformations of CP2 vacuum extremals?

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents

Yangian symmetry plays a key role in N=4 super-symmetric gauge theories. What is special in Yangian symmetry is that the algebra contains also multi-local generators. In TGD framework multi-locality would naturally correspond to that with respect to partonic 2-surfaces and string world sheets and the proposal has been that the Super-Kac-Moody algebras assignable to string worlds sheets could generalize to Yangian.

Witten has written a beautiful exposition of Yangian algebras (see this). Yangian is generated by two kinds of generators JA and QA by a repeated formation of commutators. The number of commutations tells the integer characterizing the multi-locality and provides the Yangian algebra with grading by natural numbers. Witten describes a 2-dimensional QFT like situation in which one has 2-D situation and Kac-Moody currents assignable to real axis define the Kac-Moody charges as integrals in the usual manner. It is also assumed that the gauge potentials defined by the 1-form associated with the Kac-Moody current define a flat connection:

μjAν- ∂νjAν +[jAμ,jAν]=0 .

This condition guarantees that the generators of Yangian are conserved charges. One can however consider alternative manners to obtain the conservation.

  1. The generators of first kind - call them JA - are just the conserved Kac-Moody charges. The formula is given by

    JA= ∫-∞ dxjA0(x,t) .

  2. The generators of second kind contain bi-local part. They are convolutions of generators of first kind associated with different points of string described as real axis. In the basic formula one has integration over the point of real axis.

    QA= fABC-∞ dx ∫xdy jB0(x,t)jC0(y,t)- 2∫-∞ jAxdx .

    These charges are indeed conserved if the curvature form is vanishing as a little calculation shows.

How to generalize this to the recent context?

  1. The Kac-Moody charges would be associated with the braid strands connecting two partonic 2-surfaces - Strands would be located either at the space-like 3-surfaces at the ends of the space-time surface or at light-like 3-surfaces connecting the ends. Modified Dirac equation would define Super-Kac-Moody charges as standard Noether charges. Super charges would be obtained by replacing the second quantized spinor field or its conjugate in the fermionic bilinear by particular mode of the spinor field. By replacing both spinor field and its conjugate by its mode one would obtain a conserved c-number charge corresponding to an anti-commutator of two fermionic super-charges. The convolution involving double integral is however not number theoretically attactive whereas single 1-D integrals might make sense.

  2. An encouraging observation is that the Hodge dual of the Kac-Moody current defines the analog of gauge potential and exponents of the conserved Kac-Moody charges could be identified as analogs for the non-integrable phase factors for the components of this gauge potential. This identification is precise only in the approximation that generators commute since only in this case the ordered integral P(exp(i∫ Adx)) reduces to P(exp(i∫ Adx)).Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization implying that Abelian approximation works. This conforms with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.

    This would make possible a direct identification of Kac-Moody symmetries in terms of gauge symmetries. For isometries one would obtain color gauge potentials and the analogs of gauge potentials for graviton field (in TGD framework the contraction with M4 vierbein would transform tensor field to 4 vector fields). For Kac-Moody generators corresponding to holonomies one would obtain electroweak gauge potentials. Note that super-charges would give rise to a collection of spartners of gauge potentials automatically. One would obtain a badly broken SUSY with very large value of N defined by the number of spinor modes as indeed speculated earlier (see this).

  3. The condition that the gauge field defined by 1-forms associated with the Kac-Moody currents are trivial looks unphysical since it would give rise to the analog of topological QFT with gauge potentials defined by the Kac-Moody charges. For the duals of Kac-Moody currents defining gauge potentials only covariant divergence vanishes implying
    that curvature form is

    Fαβ= εαβ [jμ, jμ] ,

    so that the situation does not reduce to topological QFT unless the induced metric is diagonal. This is not the case in general for string world sheets.

  4. It seems however that there is no need to assume that jμ defines a flat connection. Witten mentions that although the discretization in the definition of JA does not seem to be possible, it makes sense for QA in the case of G=SU(N) for any representation of G. For general G and its general representation there exists no satisfactory definition of Q. For certain representations, such as the fundamental representation of SU(N), the definition of QA is especially simple. One just takes the bi-local part of the previous formula:

    QA= fABCi<jJBiJCj .

    What is remarkable that in this formula the summation need not refer to a discretized point of braid but to braid strands ordered by the label i by requiring that they form a connected polygon. Therefore the definition of JA could be just as above.

  5. This brings strongly in mind the interpretation in terms of twistor diagrams. Yangian would be identified as the algebra generated by the logarithms of non-integrable phase factors in Abelian approximation assigned with pairs of partonic 2-surfaces defined in terms of Kac-Moody currents assigned with the modified Dirac action. Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization. This would fit nicely with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.

The resulting algebra satisfies the basic commutation relations



plus the rather complex Serre relations described in Witten's article).

The connection between Kac-Moody symmetries and gauge symmetries is suggestive and in this case it would be realized in terms of 2-D Hodge duality. Also finite measurement resolution realized in the sense that the points at the ends of given braid strand are regarded to be effectively infinitesimally close so that the gauge algebra is effectively Abelian is essential. Yangian symmetry is crucial for the success of the twistor approach. Zero energy ontology implies that generalized Feynman diagrams contain only massless partonic 2-surfaces with propagators defined by longitudinal momentum components defined in terms of M2⊂ M4 characterizing given causal diamond. There there are excellent hopes that twistor approach applies also in TGD framework. Note that also the conformal transformations of M4 might allow Yangian variants.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equationof "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Sunday, July 15, 2012

The recent vision about preferred extremals and solutions of the modified Dirac equation

The understanding of preferred exrremals of preferred extremals of Kähler action and solutions of the modified Dirac equation has increased dramatically during last months and I have been busily deducing the consequences for the quantum TGD. This process led also to a new chapter of the book about physics as WCW geometry. I attach below the extended abstract of the chapter.

During years several approaches to what preferred extremals of Kähler action and solutions of the modified Dirac equation could be have been proposed and the challenge is to see whether at least some of these approaches are consistent with each other. It is good to list various approaches first.

  1. For preferred extremals generalization of conformal invariance to 4-D situation is very attractive approach and leads to concrete conditions formally similar to those encountered in string model. In particular, Einstein's equations with cosmological constant follow as consistency conditions and field equations reduce to a purely algebraic statements analogous to those appearing in equations for minimal surfaces if one assumes that space-time surface has Hermitian structure or its Minkowskian variant Hamilton-Jacobi structure. The older approach based on basic heuristics for massless equations, on effective 3-dimensionality, and weak form of electric magnetic duality, and Beltrami flows is also promising. An alternative approach is inspired by number theoretical considerations and identifies space-time surfaces as associative or co-associative sub-manifolds of octonionic imbedding space.

    The basic step of progress was the realization that the known extremals of Kähler action - certainly limiting cases of more general extremals - can be deformed to more general extremals having interpretation as preferred extremals.

    1. The generalization boils down to the condition that field equations reduce to the condition that the traces Tr(THk) for the product of energy momentum tensor and second fundamental form vanish. In string models energy momentum tensor corresponds to metric and one obtains minimal surface equations. The equations reduce to purely algebraic conditions stating that T and Hk have no common components. Complex structure of string world sheet makes this possible.

      Stringy conditions for metric stating gzz=gz*z*=0 generalize. The condition that field equations reduce to Tr(THk)=0 requires that the terms involving Kähler gauge current in field equations vanish. This is achieved if Einstein's equations hold true. The conditions guaranteeing the vanishing of the trace in turn boil down to the existence of Hermitian structure in the case of Euclidian signature and to the existence of its analog - Hamilton-Jacobi structure - for Minkowskian signature. These conditions state that certain components of the induced metric vanish in complex coordinates or Hamilton-Jacobi coordinates.

      In string model the replacement of the imbedding space coordinate variables with quantized ones allows to interpret the conditions on metric as Virasoro conditions. In the recent case generalization of classical Virasoro conditions to four-dimensional ones would be in question. An interesting question is whether quantization of these conditions could make sense also in TGD framework at least as a useful trick to deduce information about quantum states in WCW degrees of freedom.

      The interpretation of the extended algebra as Yangian suggested previously to act as a generalization of conformal algebra in TGD Universe is attractive. There is also the conjecture that preferred extremals could be interpreted as quaternionic of co-quaternionic 4-surface of the octonionic imbedding space with octonionic representation of the gamma matrices defining the notion of tangent space quanternionicity.

  2. There are also several approaches for solving the modified Dirac equation. The most promising approach is assumes that the solutions are restricted on 2-D stringy world sheets and/or partonic 2-surfaces. This strange looking view is a rather natural consequence of both strong form of holography and of number theoretic vision, and also follows from the notion of finite measurement resolution having discretization at partonic 2-surfaces as a geometric correlate. The conditions stating that electric charge is conserved for preferred extremals is an alternative very promising approach. One expects that stringy approach based on 4-D generalization of conformal invariance or its 2-D variant at 2-D preferred surfaces should also allow to understand the modified Dirac equation. In accordance with the earlier conjecture, all modes of the modified Dirac operator generate badly broken super-symmetries. Right-handed neutrino allows also holomorphic modes delocalized at entire space-time surface and the delocalization inside Euclidian region defining the line of generalized Feynman diagram is a good candidate for the right-handed neutrino generating the least broken super-symmetry. This super-symmetry seems however to differ from the ordinary one in that νR is expected to behave like a passive spectator in the scattering.

The question whether these various approaches are mutually consistent is discussed. It indeed turns out that the approach based on the conservation of electric charge leads under rather general assumptions to the proposal that solutions of the modified Dirac equation are localized on 2-dimensional string world sheets and/or partonic 2-surfaces. Einstein's equations are satisfied for the preferred extremals and this implies that the earlier proposal for the realization of Equivalence Principle is not needed. This leads to a considerable progress in the understanding of super Virasoro representations for super-symplectic and super-Kac-Moody algebra. In particular, the proposal is that super-Kac-Moody currents assignable to string world sheets define duals of gauge potentials and their generalization for gravitons: in the approximation that gauge group is Abelian - motivated by the notion of finite measurement resolution - the exponents for the sum of KM charges would define non-integrable phase factors. One can also identify Yangian as the algebra generated by these charges. The approach allows also to understand the special role of the right handed neutrino in SUSY according to TGD.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "TGD as Infinite-Dimensional Geometry" or the article with the same title.

Saturday, July 14, 2012

Is it really Higgs?

The attitudes to the discovery are becoming more and more realistic and it is indeed transforming from "Higgs discovery" to "discovery". I of course feel empathy for those who have spent their professional career by doing calculations with Higgs: it is not pleasant to find that something totally different might be in question. Even laymen are probably realizing that that 125 GeV mass is not a prediction of string theories or standard SUSY: string theorists have successfully predicted all the masses of Higgs candidates that have appeared during last years(at least 165 GeV, 115 GeV, 1145 GeV, and 125 GeV) but always only after the rumor. Usually this is called post-diction, not a prediction. In the latest New Scientist the problems are acknowledged and summarized: congratulations for New Scientist.

For most decay channels the rates differ from standard model predictions considerably (see this). In particular, gamma gamma decay rate is about three times too high and tau lepton pairs are not produced at all. This is very very alarming since Higgs should couple to leptons with coupling proportional to its mass.

It is becoming clear that it is not standard model Higgs. People have begun to talk about "Higgs like" state since nothing else they do not have because technicolor scenario is experimentally excluded.

SUSY enthusiasts claim that it is SUSY Higgs but there are no indications for SUSY. Lubos is getting desperate and is already claiming experimentalists for in-honesty in the interpretation of data! Lubos has got also irritated about certain bloggers who refuse to admit that the new particle is there. I have not seen any blogger refusing to admit that it is not here. Maybe Lubos meant that the bloggers in question are not convinced about the Higgsy character of the new particle but failed to distinguish between "Higgs" and "spinless particle" as so many times earlier. I do not even dare to imagine that it is me who should be blamed for the irritated mood of Lubos since at least officially Lubos has not shown any signal of being aware of my humble blog existence (behaving just as a brahmin of science is expected to behave towards the pariah class);-).

The most natural - albeit not the only possible - TGD identification is as a pion-like state. This would mean that it is pseudo-scalar: also SUSY predicts pseudo-scalar as one of the several Higgses.

The basic predictions of TGD scenario deserve to be summarized.

  1. Also two charged and one neutral companion of pseudoscalar should exist. This is because pseudoscalar is for expected to be replaced by imbedding space vector having only CP2 components (4) forming electroweak triplet and singled just as ew gauge bosons do. But what authority could force the experimentalists to search for the decays of this kind of states?

  2. ATLAS and CMS see their Higgs candidates at slightly different masses: mass difference is about 1 GeV. Could this mean that the predicted two neutral states contribute and have been already observed! This could also explain the too large decay rate to two gammas (WW decay rate would be more or less same as for standard model and only the second neutral state would contribute to it).

    One can however counter-argue that ordinary pion has no neutral companion of same mass. In hadronic sigma model it has scalar companion with which it forms 1+3 multiplet of SO(4), the tangent space group of CP2 reducing to SU(2)L×U(1) identifiable as U(2) ⊂ SU(3) in the concrete representaton of pion states. Could one think that this is the case also now and sigma develops vacuum expectation analogous to that of Higgs determining most of the couplings just as in sigma model for ordinary hadrons? The problem is that the neutral component should be scalar.

    Could one get rid of the additional sigma state? CP2 allows two geodesic spheres and the second one is homologically non-trivial allowing SO(3) as isometries instead of U(2). In this case one would have naturally SO(3) triplet instead of 3+1 and no sigma boson. For the four kaon like state one would have 3+1 naturally. As proposed in an earlier posting this could distinguish between pion-like and kaon-like multiplets. What is genuinely new that strong isospin groups U(2) and SO(3) would reduce to subgroups of color group in spinor representation.

  3. If there is pion-like state there, it is pseudo-scalar: this might become clear during this year. SUSY people would identify it as one of the SUSY Higgses.

  4. Pion-like states consist of "scaled up" quarks of M89 hadron physics and they prefer to decay to hadrons. Lepton pairs are produced only in higher order via box diagrams with W pair as vertical sides and quark line and lepton line as horizontal sides. This explains why tau pairs are not observed. This is very important point.

    The fastest decays could take place to two gluons of M89 hadron physics transforming to ordinary gluons in turn decaying to quarks and producing jets.

  5. The simplest option is that effective action for decays to weak gauge bosons is instanton action assignable to axial current anomaly. WW production rate is consistent with standard Higgs and this fixes the coefficient of the instanton term if one assumes that electroweak symmetry is not broken so that γ, Z, and W would have different coefficients. I would be happy if I had a graduate student as a slave doing the calculations.

  6. Associated production of bbar +W has been observed as predicted. In TGD bbar would correspond to decay to two gluons annihilating to quark pair. Light quark pairs would be produced much more than in Higgs decays where Higgs-quark coupling is proportional to quark mass. But again: who would force experimenters to search for such non-Higgsian signatures since for Higgs the rates would be quite to slow.

What experimenters have to say about these predictions after year is interesting. The discovery of charged partners would destroy the Higgs interpretation. My meager and perhaps too optimistic hope is that they could be discovered by accident.