Sunday, March 29, 2009

Are light-like loop momenta consistent with unitarity?

The original version of this note was written before it had become clear that the replacement of momenta rotating in loops with massless momenta - suggested by the expressibility of the momenta of internal lines in terms of twistors - very probably implies very bad UV behavior for general Feynman diagrams if one assumes only fermion lines. This unless symmetries lead to miraculous cancellation of divergences so that the idea does not look promising.

One can however consider the possibility of a modification of gauge theory with self energy diagrams excluded. A rough check shows that gauge boson vertices are very probably UV divergent due to the presence of momentum factors in all three three-vertices. One can however wonder whether a bootstrap approach -very natural in TGD framework and strongly encouraged by the success of twistorial unitary cut method in gauge theories - taking fermion-boson couplings as fundamental couplings and purely bosonic couplings as radiatively generated ones could lead to UV and IR finite theory by allowing only light-like loop momenta. In this case gauge theory dynamics would be emergent and resulting bosonic couplings could have form factors with IR and UV behaviors allowing finiteness of loops constructed from them. The TGD inspired physical motivation for this view is that bosons are identified as bound states of fermion and antifermion at opposite wormhole throats so that bosonic n-vertex would correspond to the decay of bosons to fermion pairs in the loop.

The following argument shows that one could achieve unitarity also in this framework by simple further assumption about the proposed rules. As a matter fact, so called generalized cut rules combined with twistor approach allow to construct massless S-matrix using just cut rules and dispersion relation from tree diagrams without any need for the calculation of loop integrals and this approach might apply also in TGD framework where one does not have any Feynman rules in the ordinary sense of the word.

If massless negative energy states are allowed in loops the analog of S-matrix constructed using modified Feynman rules might not be unitary. It is also questionable whether one can exclude negative energy particles from the final states. In TGD framework also physical picture allows to challenge these basic rules.

  1. In negative energy ontology S-matrix is replaced with M-matrix representing time-like entanglement coefficients between positive and negative energy parts of the zero energy state. M-matrix need not be unitary. The proposal is however that M-matrix decomposes into a product of square root of a positive definite diagonal density matrix and unitary S-matrix just as Schrödinger amplitude decomposes into a product of modulus and phase. This would mean unification of statistical physics and quantum theory at fundamental level. S-matrix would be still something universal and only the density matrix would be state dependent.

  2. One of the long standing issues of TGD has been whether one should allow besides positive energy states also negative states regarded as analogs of phase conjugate laser beams to be distinguished from antiparticles which can be also seen analogs of negative energy particles. The TGD based identification of bosons as pairs of wormhole throats carrying fermion numbers is most elegant if the second wormhole carries phase conjugate fermion with negative energy. The minimal deviation from standard physics picture would allow negative energy light-like momenta in loops interpreted as phase conjugate particles and their appearance in only loops would explain why they are rare. One can however consider phase conjugate states also as incoming and final states. The interpretation would be that they result through time reflection from the lower boundaries of sub-CDs whereas negative energy parts of zero energy state correspond to reflection from the upper boundary of CD.

In standard approach unitarity conditions i(T-T) = -TT are expressed very elegantly in terms of Cutkosky rules.

  1. The difference i(T-T) corresponds to the discontinuity of the Feynman diagram in a channel in which one has N parallel lines. For instance, 2-2 scattering by boson exchange corresponds to cut for a box diagram.

  2. T is obtained by changing the sign of ε in Feynman propagators giving contribution to T so that one has 1/(p2-m2+iε)→ 1/(p2-m2-iε). The subtraction of Feynman propagators with different signs of ε in T-T gives just an integral over on mass shell states with positive energy and therefore TT.

  3. Analyticity in momentum space allows to use dispersion relations deduce also the "real" part of the amplitude so that in principle one could avoid loop calculations altogether. In twistor approach where only on mass shell momenta allow a nice description in terms of twistors the unitary cut method developed by Bern, Dixon, Dunbar and Kosower (see this) approach is very natural and generalized cuts are utilized in twistor approach heavily to deduce information about amplitudes using only tree diagrams as a starting point.

For the proposed modification one obtains standard unitarity if the intermediate lines appearing in cuts contain always only Feynman propagators. The generalized Feynman rules should guarantee this automatically. Since the goal is to construct unitary S-matrix without the constraints coming from standard QFT formalism, and since the lines allowing to cut the diagram are topologically very special, nothing hinders from posing the rule that light-like loop momenta do not appear in the lines allowing a cut. It seems however that the bad UV behavior of proposed diagrammatics based on four-fermion vertices and light-like loop momenta does not allow to construct Feynman diagrams in the proposed manner. Rather, the unitary cut method looks much more natural manner to build up S-matrix in TGD framework provided one can define tree diagrams using 4-fermion vertices. One must also remember that in the general TGD vision about the construction of M-matrix involves no local vertices. The vertices are defined in terms of N-point functions of conformal field theory and each braid strand correspond to a point where fermion emerges. Therefore local four-fermion vertex would mean giving up the non-locality due to the replacement of particles with partonic 2-surfaces containg fermions at braid strands.

Note added: It turned out that p-adic length scale hypothesis and finite measurement resolution lead to a beautiful realization of the bootstrap approach assuming only fermionic propagator and fermion boson couplings as given. The approach reduces to strict Feynman rules allowing to generated bosonic propagators and bosonic vertices radiatively. The notion of light-like loop momenta turned out to be a failure without which I had never realized that in TGD framework the only possible manner to generate bosonic propagators and vertices is by radiative corrections!

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

Thursday, March 26, 2009

Could one regard space-time surfaces as surfaces in twistor space?

Twistors are used to construct solutions of free wave equations with given spin and self-dual solutions of both YM theories and Einstein's equations . Twistor analyticity plays a key role in the construction of construction of solutions of free field equations. In General Relativity the problem of the twistor approach is that twistor space does not make sense for a general space-time metric . In TGD framework this problem disappears and one can ask how twistors could possibly help to construct preferred extremals. In particular, one can ask whether it might be possible to interpret space-time surfaces as as counterparts of surfaces - not necessarily four-dimensional - in twistor space or in some space naturally related to it. The 12-dimensional space PT×CP2 indeed emerges as a natural candidate (if something is higher dimensional, the standard association which of string theories corresponds to this dimension and F-theory does the job at this time).

1. How M4×CP2 emerges in twistor context?

The finding that CP2 emerges naturally in twistor space considerations is rather encouraging.

  1. Twistor space allows two kinds of 2-planes in complexified M4 known as α- and β-planes and assigned to twistor and its dual . This reflects the fundamental duality of the twistor geometry stating that the points Z of PT label also complex planes (CP2) of PT via the condition


    To the twistor Z one can assign via twistor equation complex α-plane, which contains only null vectors and correspond to the plane defined by the twistors intersecting at Z.

    For null twistors (5-D sub-space N of PT) satisfying ZatildeZa=0 and identifiable as the space of light-like geodesics of M4 α-plane contains single real light-ray. β-planes in turn correspond to dual twistors which define 2-D null plane CP2 in twistor space via the equation ZaWa=0 and containing the point W = tildeZ. Since all lines CP1 of CP2 intersect, also they parameterize a 2-D null plane of complexified M4. The β-planes defined by the duals of null twistors Z contain single real light-like geodesic and intersection of two CP2:s defined by two points of line of N define CP1 coding for a point of M4.

  2. The natural appearance of CP2 in twistor context suggests a concrete conjecture concerning the solutions of field equations. Light rays of M4 are in 1-1 correspondence with the 5-D space N subset P of null twistors. Compactified M4 corresponds to the real projective space PN. The dual of the null twistor Z defines 2-plane CP2 of PT.

  3. This suggests the interpretation of the counterpart of M4×CP2 as a bundle like structure with total space consisting of complex 2-planes CP2 determined by the points of N. Fiber would be CP2 and base space 5-D space of light-rays of M4. The fact that N does not allow holomorphic structure suggests that one should extend the construction to PT and restrict it to N. The twistor counterparts of space-time surfaces in T would be holomorphic surfaces of PT×CP2 or possibly of PT± (twistor analogs of lower and upper complex plane and assignable to positive and negative frequency parts of classical and quantum fields) restricted to N×CP2.

2. How to identify twistorial surfaces in PT×CP2 and how to map them to M4× CP2?

The question is whether and how one could construct the correspondence between the points of M4 and CP2 defining space-time surface from a holomorphic correspondence between points of PT and CP2 restricted to N.

  1. The basic constraints are that space-time surfaces with varying values for dimensions of M4 and CP2 projections are possible and that these surfaces should result by a restriction from PT× CP2 to N× CP2 followed by a map from N to M4 either by selecting some points from the light ray or by identifying entire light rays or their portions as sub-manifolds of X4.

  2. Quantum classical correspondence would suggest that surfaces holomorphic only in PT+ or PT- should be used so that one could say that positive and negative energy states have space-time correlates. This would mean an analogy with the construction of positive and negative energy solutions of free massless fields. The corresponding space-time surfaces would emerge from the lower and upper light-like boundaries of the causal diamond CD.

  3. A rather general approach is based on an assignment of a sub-manifold of CP2 to each light ray in PT+/- in holomorphic manner that is by n equations of form

    Fi12,Z)=0 , i=1,...,n≤ 2.

    The dimension of this kind of surface in PT× CP2 is D=10-2n and equals to 6, 8 or 10 so that a connection or at least analogy with M-theory and branes is suggestive. For n=0 entire CP2 is assigned with the point Z (CP2 type vacuum extremals with constant M4 coordinates): this is obviously a trivial case. For n=1 8-D manifold is obtained. In the case that Z is expressible as a function of CP2 coordinates, one could obtain CP2 type vacuum extremals or their deformations. Cosmic strings could be obtained in the case that there is no Z dependence. For n=4 discrete set of points of CP2 are assigned with Z and this would correspond to field theory limit, in particular massless extremals. If the dimension of CP2 projection for fixed Z is n, one must construct 4-n-dimensional subset of M4 for given point of CP2.

  4. If one selects a discrete subset of points from each light ray, one must consider a 4-n-dimensional subset of light rays. The selection of points of M4 must be carried out in a smooth manner in this set. The light rays of M4 with given direction can be parameterized by the points of light-cone boundary having a possible interpretation as a surface from which the light rays emerge (boundary of CD).

  5. One could also select entire light rays of portions of them. In this case a 4-n-1-dimensional subset of light rays must be selected. This option could be relevant for the simplest massless extremals representing propagation along light-like geodesics (in a more general case the first option must be considered). The selection of the subset of light rays could correspond to a choice of 4-n-1-dimensional sub-manifold of light-cone boundary identifiable as part of the boundary of CD in this case. In this case one could worry about the intersections of selected light rays. Generically the intersections occur in a discrete set of points of H so that this problem does not seem to be acute. The lines of generalized Feynman diagrams interpreted as space-time surfaces meet at 3-D vertex surfaces and in this case one must pose the condition that CP2 projections at the 3-D vertices are identical.

  6. The use of light rays as the basic building bricks in the construction of space-time surfaces would be the space-time counterpart for the idea that light ray momentum eigen states are more fundamental than momentum eigen states.

M8-H duality is Kähler isometry in the sense that both induced metric and induced Kähler form are identical in M8 and M4× CP2 representations of the space-time surface. In the recent case this would mean that the metric induced to the space-time surface by the selection of the subset of light-rays in N and subsets of points at them has the same property. This might be true trivially in the recent case.

3. How to code the basic parameters of preferred extremals in terms of twistors?

One can proceed by trying to code what is known about preferred extremals to the twistor language.

  1. A very large class of preferred extremals assigns to a given point of X4 two light-like vectors U and V of M4 and two polarization vectors defining the tangent vectors of the coordinate lines of Hamilton-Jacobi coordinates of M4. As already noticed, given null-twistor defines via λ and tildeμ two light-like directions V and U and twistor equation defines M4 coordinate m apart from a shift in the direction of V. The polarization vectors εi in turn can be defined in terms of U and V. λ=μ corresponds to a degenerate case in which U and V are conjugate light-like vectors in plane M2 and polarization vector is also light-like. This could correspond to the situation for CP2 type vacuum extremals. For the simplest massless extremals light-like vector U is constant and the solution depends on U and transverse polarization ε vector only. More generally, massless extremals depend only on two M4 coordinates defined by U coordinate and the coordinate varying in the direction of local polarization vector ε.

  2. Integrable distribution of these light-like vectors and polarization vectors is required. This means that these vectors are gradients of corresponding Hamilton-Jacobi coordinate variables. This poses conditions on the selection of the subset of light rays and the selection of M4 points at them. Hyper-quaternionic and co-hyper-quaternionic surfaces of M8 are also defined by fixing an integrable distribution of 4-D tangent planes, which are parameterized by points of CP2 provided one can assign to the tangent plane M2(x) either as a sub-space or via the assignment of light-like tangent vector of x.
  3. Positive (negative) helicity polarization vector can be constructed by taking besides λ arbitrary spinor μa and defining

    εaa'= λa tildeμa'/[tildeλ,tildeμ] ,

    [tildeλ,tilde&mu]= εa'b'λa'μb'

    for negative helicity and

    εaa'= μa tildeλa'/<λ,μ> ,
    <lambda;&mu >= εabλaμb

    for positive helicity. Real polarization vectors correspond to sums and differences of these vectors. In the recent case a natural identification of μ would be as the second light-like vector defining point of m. One should select one light-like vector and one real polarization vector at each point and find the corresponding Hamilton-Jacobi coordinates. These vectors could also code for directions of tangents of coordinate curves in transversal degrees of freedom.

The proposed construction seems to be consistent with the proposed lifting of preferred extremals representable as a graph of some map M4→ CP2 to surfaces in twistor space. What was done in one variant of the construction was to assign to the light-like tangent vectors U and V spinors tildeμ and λ assuming that twistor equation gives the M4 projection m of the point of X4(X3l). This is the inverse of the process carried out in the recent construction and would give CP2 coordinates as functions of the twistor variable in a 4-D subset of N determined by the lifting of the space-time surface. The facts that tangent vectors U and V are determined only apart from overall scaling factor and the fact that twistor is determined up to a phase, imply that projective twistor space PT is in question. This excludes the interpretation of the phase of the twistor as a local Kähler magnetic flux. The next steps would be extension to entire N and a further continuation to holomorphic field in PT or PT±.

To summarize, although these arguments are far from final or convincing and are bound to reflect my own rather meager understanding of twistors, they encourage to think that twistors are indeed natural approach in TGD framework. If the recent picture is correct, they code only for a distribution of tangent vectors of M4 projection and one must select both a subset of light rays and a set of M4 points from each light-ray in order to construct the space-time surface. What remains open is how to solve the integrability conditions and show that solutions of field equations are in question. The possibility to characterize preferred extremal property in terms of holomorphy and integrability conditions would mean analogy with both free field equations in M4 and minimal surfaces. For known extremals holomorphy in fact guarantees the extremal property.

4. Hyper-quaternionic and co-hyper-quaternionic surfaces and twistor duality

In TGD framework space-time surface decomposes into two kinds of regions corresponding to hyper-quaternionic and co-hyper-quaternionic regions of the space-time surface in M8 (hyper-quaternionic regions were considered in preceding arguments). The regions of space-time with M4 (Euclidian) signature of metric are identified tentatively as the counterparts of hyper-quaternionic (co-hyper-quaternionic) space-time regions. Pieces CP2 type vacuum extremals representing generalized Feynman diagrams and having light-like random curve as M4 projection represent the basic example here. Also these space-time regions should have any twistorial counterpart and one can indeed assign to M4 projection of CP2 type vacuum extremal a spinor λ as its tangent vector and spinor μ via twistor equation once M4 projection is known.

The first guess would the correspondence hyper-quaternionic ↔ α and co-hyper-quaternionic ↔ β. Previous arguments in turn suggest that hyper-quaternionic space-time surfaces are mapped to surfaces for which two null twistors are assigned with given point of M4 whereas co-hyper-quaternionic space-time surfaces are mapped to the surfaces for which only single twistor corresponds to a given M4 point.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

Wednesday, March 18, 2009

Duetting guitarist's brains fire to the same beat

Just a link to a page reporting the finding that duetting guitarist's brains fire to the same beat.

This finding supports TGD based on the notion of collective levels of consciousness predicting coherent gene expression at the level of population and syncronization of EEGs. I have suggested that it might be a good idea to test this prediction. Of course, no one has taken this seriously but this accidental discovery does the job!

For TGD inspired theory of consciousness and quantum biology see my homepage.

Tuesday, March 17, 2009

Is CDF anomaly real or not?

Tommaso Dorigo told about in this posting DZERO refutes CDF’s multimuon signal… Or does it ? about refutation of dimuons signal reported by CDF collaboration and took a critical attitude towards the claim of D0 collaboration.

Lubos in turn wrote a highly emotional posting D0 debunks the lepton jets of CDF. The problem of Lubos is that he cannot avoid strong negative emotions which spoil his ability to make rational judgements. At this time the highly emotional tone was probably because Tommaso demonstrated in the debate raised by CDF finding that Lubos was simply wrong in his strongly ad hominem argument challenging the professional skills of CDF collaboration. Some people do not seem to learn that scientific debate is not a bloody rhetoric battle but exchange of ideas meant to gain new understanding.

If the findings of CDF were true they would provide a support for the prediction of TGD made already 1990 that leptons have color excitations. There is evidence for the color excitations of electron already from seventies but they have been put under the rug since standard model does not allow them: for instance, intermediate gauge boson decay widths do not allow new light particles in the conceptual framework of standard model. For year ago also evidence for excitations of muons emerged and CDF giving support for colored excitations of tau lepton was the last link in the chain. I refer to earlier postings such as this.

I take the results of CDF seriously, not as an experimentalist, but because the findings fit nicely a much more general predicted pattern having independent support from several anomalies. Basic findings of brain science is that we perceive what our world model allows us to perceive and the history of science is a documentary about this. Even most obvious facts are denied if they are in conflict with beliefs. Let us however hope that the finding of CDF will not suffer the fate of other similar anomalies and that more testing will be carried out.

For details and background see the updated chapter The recent status of leptohadron physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Is Higgs really needed and does it exist?

The mass range containing the Higgs mass is becoming narrower and narrower (see the postings of Tommas Dorigo and Lubos Motl), and one cannot avoid the question whether the Higgs really exists. This issue remains far from decided also in TGD framework where also the question whether Higgs is needed at all to explain the massivation of gauge bosons must be raised.

  1. My long-held belief was that Higgs does not exist. One motivation for this belief was that there is no really nice space-time correlate for the Higgs field. Higgs should correspond to M4 scalar and CP2 vector but one cannot identify any natural candidate for Higgs field in the geometry of CP2. The trace of CP2 part of the second fundamental form could be considered as a candidate but depends on second derivatives of the imbedding space coordinates. Its counterpart for Kähler action would be the covariant divergence of the vector defined by modified gamma matrices and this vanishes identically.

  2. For a long time I believed that p-adic thermodynamics is not able to describe realistically gauge boson massivation and the group theoretical expression for the mass ratio of W and Z gauge bosons led to the cautious conclusion that Higgs is needed and generates a coherent state and that the ordinary Higgs mechanism has TGD counterpart. This field theoretic description is of course purely phenomenological in TGD framework and whether it extends to a microscopic description is far from clear.

  3. The identification of bosons in terms of wormhole contacts having fermion and antifermion at their light-like throats allowed a construction of also Higgs like particle. One can estimate its mass by p-adic thermodynamics using the existing bounds to determine the p-adic length scale in question: p≈2k, k=94, is the best guess and gives mH=129 GeV, which is consistent with the experimental constraints. Higgs expectation cannot however contribute to fermion masses if fermions are identified as CP2 type vacuum extremals topologically condensed to single space-time sheet so that there can be only one wormhole throat present. This would mean that Higgs condensate -whatever it means in precise sense- is topologically impossible in fermionic sector. p-Adic thermodynamics for fermions allows only a very small Higgs contribution to the mass so that this is not a problem.

  4. The next step was the realization that the deviation of the ground state conformal weights from half integer values could give rise to Higgs type contribution to both fermion and boson mass. Furthermore, the contribution to the ground state conformal weight corresponds to the modulus squared for the generalized eigenvalue λ of the modified Dirac operator D. This picture suggests a microscopic description of gauge boson masses and the Weinberg angle determining W/Z mass ratio can be expressed in terms of the generalized eigenvalues of D. Higgs could be still present and if it generates vacuum expectation (characterizing coherent state) its value should be expressible in terms of the generalized eigenvalues of modified Dirac operator. The causal relation between Higgs and massivation would not however be what it is generally believed to be.

The massivation of Z0 and generation of longitudinal polarizations are the problems, which should be understood in detail before one can take seriously in TGD inspired microscopic description.

  1. The presence of an axial part in the decomposition of gauge bosons to fermion-antifermion pairs located at the throats of the wormhole contact should explain the massivation of intermediate gauge bosons and the absence of it the masslessness of photon, gluon, and gravitons.

  2. One can understand the massivation of W bosons in terms of the differences of the generalized eigenvalues of the modified Dirac operator. In the case of W bosons fermions have different charges so that the generalized eigenvalues of the modified Dirac operator differ and their difference gives rise to a non-vanishing mass. Both transverse and longitudinal polarizations are in the same position as they should be.

  3. The problem is how Z0 boson can generate mass. For Z0 the fermions for transverse polarizations should have in a good approximation same spectrum generalized eigenvalues so that the mass would vanish unless fermion and anti-fermion correspond to different eigenvalues for some reason for Z0. The requirement that the photon and Z0 states are orthogonal to each other might require different eigen values. If fermion and antifermion in both Z0 and photon correspond to the same eigen mode of the modified Dirac operator, their inner product is proportional to the trace of the charge matrices given by Tr(Qem(I3L+sin2W)Qem), which is non-vanishing in general. For different eigenmodes in the case of Z0 the states would be trivially orthogonal.

  4. Gauge bosons must allow also longitudinal polarization states. The fact that the modes associated with wormhole throats are different in the case of Z0 could allow also longitudinal polarizations. The state would have the structure bar(Ψ)- (D-D) QZΨ+, D= pkγk. This state does not vanish for intermediate gauge bosons since the action of pkγk to the two modes of the induced spinor field is different and ordinary Dirac equation is not true induced spinor fields. For photon and gluons the state would vanish.

  5. In the standard approach the gradient of Higgs field is transformed to a longitudinal polarization of massive gauge bosons. It is not clear whether this kind of idea makes sense at all microscopically in TGD framework. The point is that Higgs as a particle corresponds to a superposition of fermion-antifermion pairs with opposite M4 chiralities whereas the longitudinal part corresponds to pairs with same M4 chiralities. Hence the idea about the gradient of Higgs field transforming to the longitudinal part of gauge boson need not make sense in TGD framework although Higgs can quite well exist.

To sum up, these arguments could be seen as a support for the possibility that Higgs is not needed at all in particle massivation in TGD Universe but leave open the question whether Higgs exists as particle and possibly develops coherent state.

For details and background see the updated chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Could one lift Feynman diagrams to twistor space?

In previous posting I already considered the question how TGD could be lifted from 8-D M4×CP2 to 8-D twistor space with motivation coming from N=4 super-conformal invariance requiring that the target space in which strings live has metric signature (4,4). In the articles of Witten and Nima Arkani-Hamed and collaborators the possibility of twistor diagrammatics is considered. This inspired a crazy morning hour speculation, which I will represent since I find it difficult to imagine what I could still lose in this crazy and cruel world;-).

  1. The arguments start from ordinary momentum space perturbation theory. The amplitudes for the scattering of massless particles are expressed in terms of twistors after which one performs twistor Fourier transform obtaining amazingly simple expressions for the amplitudes. For instance, the 4-pt one loop amplitude in N=4 SYM is extremely simple in twistor space having only values '1' and '0' in twistor space and vanishes for generic momenta.

  2. Also IR divergences are absent in twistor transform of the scattering amplitude but are generated by the transform to the momentum space. Since plane waves are replaced with light rays, it is not surprising that the IR divergences coming from transversal degrees of freedom are absent. Interestingly, TGD description of massless particles as wormhole throats connecting two massless extremals extends ideal light-ray to massless extremal having finite transversal thickness so that IR cutoff emerges purely dynamically.

  3. This approach fails at the level of loops unless one just uses the already calculated loops. The challenge would be a generalization of the ordinary perturbation theory so that loops could be calculated in twistor space formulation.

The vision about lifting TGD from 8-D M4×CP2 to 8-D twistor space suggests that it should be possible to lift also ordinary M4 propagators to propagators to twistor space. The first problem is that the momenta of massive virtual particles do not allow any obvious unique representation in terms of twistors. Second problem relates to massive incoming momenta necessarily encountered in stringy picture even if one forgets massivation of light states by p-adic thermodynamics.

1. Could one restrict loop momenta to be light-like?

Could one somehow circumvent the first problem, say by bravely modifying the notion of loop? Could this modification even allow to get rid of UV divergences? The following argument- which might well be one of those arguments which come and go- suggests that this is the case and that also the second problem could be circumvented.

  1. At the level of tree diagrams representing 2-particle scattering of massless particles by particle exchange there are no problems. The propagator involves the difference of the two massless momenta and makes sense in twistor space.

  2. Twistor picture poses very strong constraint on the notion of loop: loop momenta must be expressible in terms of light-like momenta. This is achieved if only massless momenta rotate in loops so that one can express the momenta appearing in internal lines in terms of the incoming momenta and massless loop momenta and therefore express propagators in terms of incoming twistors and virtual twistor. For instance, for self energy diagram involving two N-vertices the incoming light-like momentum would decompose to N-1 light-like momenta plus one off mass shell momentum expressible in terms of light-like momenta. The odd ball momentum can be assigned to any of the internal lines of the vertex.

  3. The four-dimensional loop momentum integral would reduce to 3-dimensional integral over light-cone boundary in momentum space (over both future and past directed light-cones as it seems). The integral over the phase of the loop twistor is not needed unless it appears in the vertices. Since the only mass scale associated with the loop momentum is μ= 0, there are excellent hopes of getting rid of UV divergences. In terms of conformal invariance this kind definition of loop looks extremely natural and in the case of M-matrix unitarity constraint cannot be used to argue that ordinary loops are the only possibility.

The detailed rules possibly realizing this idea are based on the idea of using different propagators. The retarded and advanced Feynman propagators can be assigned naturally to the outgoing and incoming particles and represent classical propagation from the lower or upper light-like boundary of CD. The difference of retarded and advanced propagator -G- has in its Fourier expansion only light-like momenta, which means also classical propagation. The interpretation is that virtual particle energies to either time direction from sub-CD as on mass shell particle. Feynman propagator in turn is essentially quantal and can be used for the loop momenta, which are not light-like (also advanced or retarded propagator could be considered).

One can check whether these rules give finite results by calculating some simplest loops. For detailed computations see this. It is easy to check what these rules would give in the general case assuming only fermions and couplings involving no gradients. The basic observation is that due to the vanishing of ki2 in the denominators of the ordinary Feynman propagators give at worst a contribution behaving like k0 whereas loop lines give a contribution behaving as k. The integral over the light-cone gives k2 for each loop. For ordinary loops one has 1/k behavior. Hence the amplitude diverges as

μ3L ,

where L is the number of light-like loops. For ordinary loops with I the number of internal lines one would have

μ3L-I .

Therefore UV behavior becomes worse than for ordinary Feynman propagators in the general case.

Unfortunately (or fortunately?) I made direct checks for scalar couplings of fermions first and in this case the simplest fermionic self-energy loop vanishes for on shell particle. Also box and polygon diagrams are finite for scalars if external momenta are on mass shell but not for off mass shell momenta. The optimistic hope was that ki→ -ki symmetry could cancel the integral for large loop momenta more generally with positive and negative energies giving compensating contributions for large loop energies.

One can however consider the possibility of a modification of gauge theory by taking fermionic propagator and fermion boson couplings as fundamental ones (only fermions appear as fundamental particles in TGD framework). Both gauge boson propagators and couplings would be generated radiatively. The TGD inspired physical motivation for this view is that bosons are identified as bound states of fermion and antifermion at opposite wormhole throats. The idea is that bosons propagate only by decaying to a pair of fermion and antifermion and then fusing back. Bosonic n-vertex would correspond to the decay of bosons to fermion-antifermion pairs in the loop. In this manner one would completely avoid the introduction of bosonic propagators as primary structures.

p-Adic length scale hypothesis and zero energy ontology suggests a fractal structure for the fermionic propagator implying that it contains a scale factor depending on the p-adic length scale approaching to zero fast enough so that loop integrals reducing to a sum over octaves of momentum scale are finite. This approach could work for both Feynman diagrams (it does!) and their variants with light-like loop momenta (it doesn't!). This kind of description would be applied above p-adic momentum length scale defining the momentum resolution (determined by the size of the smallest CD included) so that no dramatic deviations from standard QFT picture would be predicted for ordinary Feynman propagators in the loops.

A possible manner to get over the problems would be by treating bosons as elementary particles and by allowing no self energy loops. For gauge theory couplings one would have only the modifications of standard type of loops with physical polarizations for on mass shell bosons and simple counting arguments support the view that UV divergences are absent for massless states. The replacement of loop momenta with massless ones improves also IR behavior.

2. Purely twistorial formulation of Feynman graphs with light-like loop momenta

One can also lift the 3-dimensional integral d3k/2E to an integral over twistor variables, which means that complete twistorialization of Feynman diagrams is possible if the loop integrals involve only light-like momenta (recall that this idea failed). This formulation generalizes to the case when loop momenta are massive but requires the introduction of an auxiliary twistor corresponding to momenta restricted to the preferred plane M2 subset M4 predicted by the number theoretical compactification and hierarchy of Planck constants.

  1. It is convenient to introduce double cylindrical coordinates λi= ρi exp(i(φ+/- ψ)) in twistor space. The integration over overall phase φ gives only a 2π factor since ordinary Feynman amplitude has no dependence on this variable so that the non-redundant variables are ρ12,ψ.

  2. The condition is that the integral measure d4uX of the spinor space with a suitable weight function X is equivalent with the measure d3k/2E in cylindrical coordinates. This gives

    d4u X= dφd3k/2E

    when the integrand does not depend on φ.

  3. In cylindrical coordinates this gives

    1ρ212dψ Xδ(U-kz)δ(V-kx)δ(W-kz)= 1 ,

    U= (ρ1222)/2 ,

    V= ρ1ρ2cos(ψ)/2 ,

    W= ρ1ρ2sin(ψ)/2 .

    Here the functions U, V, and W are obtained form the representations of kz,kx,ky in terms of spinor and its conjugate.

  4. Taking U,V,W as integration variables one has

    1ρ2[∂(ρ12,ψ)/∂ (U,V,W)]×X = 1 .

  5. The calculation of the Jacobian gives X= (ρ1222)/4= E/2 so that one has the equivalence

    (1/4π)d4u ↔ d3k/2E .

  6. Similar lifting can be carried out for the integration measure defined at light-cone boundary in M4. If the integrations in generalized Feynman diagrams are over amplitudes depending on light-like momenta and coordinates of the light-like boundaries of CDs in given length scales coming as Tn= 2nT0 or Tp=pT0 the integrals of momentum space and light-one can be transformed to integrals over twistor space in given length scale. Twistorialization requirement obviously gives a justification for the basic assumption of zero ontology that all transition amplitudes can be formulated in terms of data at the intersections of light-like 3-surfaces with the boundaries of CDs.

  7. It should be emphasized that there is no need to keep the phase angle φ as a redundant variable is the interpretation as Kähler magnetic flux is accepted. In fact, Kähler magnetic fluxes are expected to appear as zero modes define external parameters in the amplitudes.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

Sunday, March 15, 2009

Twistors, N=4 superconformal strings, and TGD

Twistors - a notion discovered by Penrose - have provided a fresh approach to the construction of perturbative scattering amplitudes in Yang-Mills theories and in N=4 supersymmetric Yang-Mills theory. This approach was pioneered by Witten. The latest step in the progress was the proposal by Nima Arkani-Hamed and collaborators that super Yang Mills and super gravity amplitudes might be formulated in twistor space possessing real metric signature (4,4).

The problem is that a space with this metric signature does not conform with the standard view about causality. The challenge is to find a physical interpretation consistent with the metric signature of Minkowski space: somehow M4 or at least light-cone boundary should be mapped to twistor space. The (2,2) resp. (4,4) signature of the metric of the target space is a problem of also N=2 resp. N=4 super-conformal string theories, and N=4 super-conformal string theory could be relevant for quantum TGD. The identification of the target space of N=4 theory as twistor space T looks natural since it has metric with the required real signature(4,4).

Number theoretical compactification implies dual slicings of the space-time surface to string world sheets and partonic 2-surfaces. Finite measurement resolution reduces light-like 3-surfaces to braids defining boundaries of string world sheets. String model in T is obtained if one can lift the string world sheets from CD×CP2 to T (CD denotes causal diamond defined as intersection of future and past directed light-cones). It turns out that this is possible and one can also find an interpretation for the phases associated with the spinors defining the twistor.

1. General remarks

Some remarks are in order before considering detailed proposal for how to achieve this goal.

  1. Penrose ends up with the notion of twistor by expressing Pauli-Lubanski vector and four-momentum vector of massless particle in terms of two spinors and their conjugates. Twistor Z consists of a pair (μa, λa') of spinors in representations (1/2,0) and (0,1/2) of Lorentz group. The hermitian matrix defined by the tensor product of λa and its conjugate characterizes the four-momentum of massless particle in the representation paσa using Pauli's sigma matrices. μa characterizes the angular angular momentum of the particle: spin is given by s=ZαZbarα. The representation is not unique since λa is fixed only apart from a phase factor, which might be called "twist". The phases of two spinors are completely correlated.

  2. This interpretation is not equivalent with that discussed mostly in Witten's paper. Two-component spinors replace light-like momentum also in this approach as a kinematic variable and a phase factor emerges as an additional kinematic variable. Scattering amplitudes are therefore not functions of momenta and polarizations but of a spinor, its conjugate defining light-like momentum, and helicity having values ±1. Fourier transform with respect to spinor or its conjugate gives scattering amplitude as a function of a twistor variable.In Minkowski space with Lorentz signature the momentum as kinematic variable is replaced with spinor and its conjugate and spinor is defined apart from a phase factor. In the article of both Witten and Nima and collaborators the signature of Minkowski space is taken to be (2,2) so that the situation changes dramatically. Light rays assignable to twistors are 2-D light-like light-like surfaces and the spinor associated with light-like point decomposes to two independent real spinors replacing light-like momentum as a kinematic variable. The phase factor as an additional kinematic variable is replaced by a real scaling factors t and 1/t for the two spinors. Fourier transform with respect to the real spinor or its conjugate is possible and gives scattering amplitude as a function of a twistor variable. In Lorentz signature the twistor Fourier transform in this sense is not possible so one cannot replace spinor and its conjugate by a twistor.

  3. Twistor space -call it T- has Kähler metric with complex signature (2,2) and real signature (4,4) and could correspond to the target space of N=4 super-conformally symmetric string theory with strings identified as T lifts of the string world sheets. The minimum requirement is that one can assign to each point of string world sheet a twistor.

2. What twistor Fourier transform could mean in TGD framework?

The twistor transform described in Wittens' article deserves some remarks.

  1. From Witten's paper one learns that twistor-space scattering amplitudes obtained as Fourier-transforms with respect to the conjugate spinor correspond in Minkowski space correspond to incoming and outgoing states for which the wave functions are not plane waves but are located to sub-spaces of Minkowski space defined by the equation
    μa'+ xaa'&lambdaa=0.
    In a more familiar notation one has xμσμλ = μ. The solution is unique apart from the shift xμ→ xμ+ kpμ, where pμ is the light-like momentum associated with λ identified as a solution of massless Dirac equation. Clearly, twistor transform corresponds to a wave function located at light-like ray of δM4+/- and momentum eigen state is represented as a superposition of this kind of wave functions localized at parallel light rays in the direction of momentum and labeled by μ.

  2. If the equivalent of twistor Fourier transform exists in some sense in Lorentz signature, the geometric interpretation would be as a decomposition of massless plane wave to a superposition of wave functions localized to light-like rays in the direction of momentum. Uncertainty Principle does not deny the existence of this kind of wave functions. These highly singular wave functions would be labeled by momentum and one point at the light ray or equivalently (apart from the phase factor) by λa and μa defining the twistor. The wave functions would be constant at the rays and thus wave functions in a 3-dimensional sub-manifold of M4 labeling the light rays. This sub-manifold could be taken light-cone boundary as is easy to see so that the overlap of different wave functions would take place only at the tip of the light-cone. Fields in twistor space would be fields in the space of light-rays characterized by a wave vector. Since twistor Fourier transform does not work, one must invent some other manner to introduce these wave functions. Here the lifting of space-time surface to twistor space suggests itself.

The basic challenge is to assign to space-time surface or to each point of space-time surface a momentum like quantity. If this is achieved one can can assign to the point also λ and μ.

  1. One can assign to space-time sheet conserved four-momentum identifiable by quantum classical correspondence as its quantal variant. This option would fix λ to be same at each point of the space-time surface about from a possible phase factor depending on space-time point. The resulting surfaces in twistor space would be rather boring.

  2. Hamilton-Jacobi coordinates suggest the possibility of defining λ as a quantity depending on space-time point. The two light-like M4 coordinates u,v define preferred coordinates for the string world sheets Y2 appearing in the slicing of X4(X3l), and the light-like tangent vectors U and V of these curves define a pair (λ,tildeμ) of spinors defining twistor Z. The vector V defining the tangent vector of the braid strand is analogous to four-momentum. Twistor equation defines a point m of M4 apart from a shift along the light ray defined by V and the consistency implying that the construction is not mere triviality is that m corresponds to the projection of space-time point to M4 in coordinates having origin at the tip of CD. One could distinguish between negative and positive energy extremals according to whether the tip is upper or lower one. One can assign to λ and tildeμ also two polarization vectors by a standard procedure to be discussed later having identification as tangent vectors of coordinate curves of transversal Hamilton-Jacobi coordinates. This would give additional consistency conditions.

  3. In this manner space-time surface representable as a graph of a map from M4 to CP2 would be mapped to a 4-surface in twistor space apart from the non-uniqueness related to the phase factor of λ. Also various field quantities, in particular induced spinor fields at space-time surface, could be lifted to fields restricted to a 4-dimensional surface of the twistor space so that the classical dynamics in twistor space would be induced from that in imbedding space.

  4. This mapping would induce also a mapping of the string world sheets Y2 Ì PM4(X4(X3l)) to twistor space. V would determine λ and U -taking the role of light-cone point m - would determine tildeμ in terms of the twistor equation. 2-surfaces in twistor space would be defined as images of the 2-D string world sheets if the integrability of the distribution for (U,V) pairs implies the integrability of (λ,tildeμ) pairs.

  5. Twistor scattering amplitude would describe the scattering of a set of incoming light-rays to a set of outgoing light-rays so that the non-locality of interactions is obvious. Discretization of partonic 2-surfaces to discrete point sets would indeed suggest wave functions localized at light-like rays going through the braid points at the ends of X3l as a proper basis so that problems with Uncertainty Principle would be overcome. The incoming and outgoing twistor braid points would be determined by M4 projections of the braid points at the ends of X3l. By quantum classical correspondence the conservation law of classical four-momentum defined would apply to the total classical four-momentum although for individual braid strands classical four-momenta would not conserved. The interpretation would be in terms of interactions. The orbits of stringy curves connecting braid points wold give string like objects in T required by N=4 super-conformal field theory.

3. Could one define the phase factor of the twistor uniquely?

The proposed construction says nothing about the phase of the spinors assigned to the tangent vectors V and U. One can consider two possible interpretations.

  1. Since the tangent vectors U and V are determined only apart from over all scaling the phase indeterminacy could be interpreted by saying that projective twistors are in question.

  2. If one can fix the absolute magnitude of U and V -say by fixing the scale of Hamilton Jacobi coordinates by some physical argument- then the map is to twistors and one should be able to fix the phase.

It turns out that the twistor formulation of field equations taking into account also CP2 degrees of freedom to be discussed latter favors the first option. The reason why the following argument deserves a consideration is that it would force braid picture and thus replacement of space-time sheets by string world sheets in twistor formulation.

  1. The phase of the spinor λa associated with the light-like four-momentum and light-like point of δM4+/- should represent genuine physical information giving the twistor its "twist". Algebraically twist corresponds to a U(1) rotation along closed orbit with a physical significance, possibly a gauge rotation. Since the induced CP2 Kähler form plays a central role in the construction of quantum TGD, the "twist" could correspond to the non-integrable phase factor defined as the exponent of Kähler magnetic flux (to achieve symplectic invariance and thus zero mode property) through an area bounded by some closed curve assignable with the point of braid strand at X2. Both CP2 and δM4+/- Kähler forms define fluxes of this kind so that two kinds of phase factors are available. CP2 flux however looks a more natural choice.

  2. The symplectic triangulation defined by CP2 Kähler form allows to identify the closed curve as the triangle defined by the nearest three vertices to which the braid point is connected by edges. Since each point of X4(X3l) belongs to a unique partonic 2-surface X2, this identification can be made for the braid strands contained by any light-like 3-surface Y3l parallel to X3l so that phase factors can be assigned to all points of string world sheets having braid strands as their ends. One cannot assign phases to all points of X4(X3l). The exponent of this phase factor is proportional to the coupling of Kähler gauge potential to fermion and distinguishes between quarks and leptons.

  3. The phase factor associated with the light-like four-momentum defined by V could be identified as the non-integrable phase factor defined by -say- CP2 Kähler form and would give the phase of λ1. The basic condition relating μ to λ would fix the phase of μ. Note that the phases of the twistors are symplectic invariants and not subject to quantum fluctuations in the sense that they would contribute to the line element of the metric of the world of classical worlds. This conforms with the interpretation as kinematical variables.

  4. Rather remarkably, this construction can assign the non-integrable phase factor only to the points of the number theoretic braid for each Y3l parallel to X3l so that one obtains only a union of string world sheets in T rather than lifting of the entire X4(X3l) to T. The phases of the twistors would code for non-local information about space-time surface coded by the tangent space of X4(X3l) at the points of stringy curves.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

Friday, March 13, 2009

New bounds on Higgs mass

CDF and D0 team have managed to pose further limit on the range for Higgs masses: see the postings of Tommas Dorigo and Lubos Motl.

With 90 per cent confidence level the Higgs is excluded in the range 157-181 GeV, which limits Higgs boson mass either to the narrow interval 181-185 GeV or to the interval 114-157 GeV. The earlier data taking all data except that from LEP II and Tevatron favor mass around 80 GeV. If also LEPT II and Tevatron data are inclued the favored mass range 115-135 GeV.

I have already earlier described TGD prediction for Higgs mass from p-adic thermodynamics. The free parameter in TGD calculation is p-adic mass scale coming as half octaves. One must consider the possibility that Higgs might appear with several mass scales and the inconsistency of mass determinations indeed encourages to do this. In this case the TGD predictions for the masses would be 89 GeV and 129 GeV. Lubos articulates in his posting that "a very weak excess of confidence may favor a Higgs near 130 GeV", which happens to be the TGD prediction. We - with me strongly included - are living interesting times!

Just after media had taught us that finding Higgs was for what LHC was born, we learn that it might be actually Tevatron, which wins the race for discovering the Higgs since LHC is tailor made to find Higgs with higher mass scale. There is however no reason to think that LHC was built in vain. Entire M89 hadron physics with overall hadronic mass scale by a factor 512 higher than for standard hadron physics is patiently waiting for its discoverers. Let us hope that it will be discovered. Let us however add that this is not easy since the experimenters -at least officially- have no idea about its existence. Professional scientists refuse to listen - officially at least- the "predictions" of some pathetic academically teased crackpot theorist without slightest academic credentials;-).

Tuesday, March 10, 2009

The last updating

I have been involved with a heavy updating of the books about basic quantum TGD during last three months. This kind of massive cleaning up procedures are unavoidable and seem to become unavoidable with a period of about 5 years. I am now 58 so that I can estimate that not too many updatings are left. Should I be relieved? Or sad for the shortness of the professional life span?

I dare claim that this endless cleaning is not a mere exotic form of cleaning neurosis. My working style is that of a light-hearted jazz musician and this produces a lot of stuff which does not represent eternal truths, and it is better to throw away this stuff away in order to not totally confuse the potential reader (in the beginning of the cleaning operation and seeing what has happened in my household I really hope that no such reader exists! When everything shines again I hope that that my friend might exist after all!.

The progress has been especially jazzy during last five years as several new visions about what TGD might be have seen the daylight. Mention only zero energy ontology, the notion of finite measurement resolution, the role of hyper-finite factors of type II1, the hierarchy of Planck constants, the construction of configuration space geometry in terms of second quantized induced spinor fields, number theoretic compactification,...). These ideas are now converging to an overall view in which various approaches to quantum TGD (physics as infinite dimensional geometry, physics as generalized number theory, physics from number theoretical universality, physics from finite measurement resolution implying effective discretization, TGD as almost topological QFT) neatly fuse together to single coherent overall view.

What is so fine in this cleaning up process that it forces to read all the stuff written during years and critically estimate the internal consistency or lack of it. I will never get rid of the feeling of deep shame than an age old archeological remnant which should have been destroyed for aeons ago creates in me. It is difficult to tolerate the childish enthusiasm of those older copies of me talking what now seems to me total non-sense.

But there is also a reward from all of this pain and trouble. New beautiful connections emerge and arguments and concepts become more precise. It is also wonderful to feel that you really might have something to give to the human kind. It might not be comparable to the Fantasie Impromptu of Chopin, but it is not totally worthless. May be it reduces just to the message that I did my best.

I also learn how incredibly tortuous the path to truth is and that it is good for ego to learn how fragile the most convincing looking argument is, how many different variants it can evolve to depending on what one means with basic concepts, and that the most difficult part in science is finding the correct interpretation. Without it you cannot write the rules. Some of us have a really good luck, and are able to do it during their lifetime and become heros. They can be however be sure that practically no one bothers to go through the same difficult path to really understand the origin of the rules.

In any case, all this work has not been in vain. I feel that I have good justifications for saying that Quantum TGD is a wonderful child full of vigor and energy and exists more and more intensively also as a mathematical theory. In the following I try to sum up some highlights about what has happened during last months. I hope that I find time to write something also the What's New sections of seven books about quantum TGD.

Number theoretical compactification as a bottle neck notion

The detailed formulation of the notion of "number theoretic compactification" (or M8H duality) stating that TGD allows equivalent formulations in terms of 4-surfaces of 8-D Minkowski space M8 (hyper-octonions) and H=M4× CP2 is responsible for everything else that has taken place during last months.

Number theoretical compactification makes strong predictions about the structure of preferred extremals of Kähler action consistent with the known extremals. The slicing of preferred extremals by stringy world sheets and their partonic duals is the basic prediction so that dimensional reduction gives string model type theory. A related prediction is a slicing by light-like 3-surfaces parallel to the fundamental light-like 3-surface X3l at which the signature of the induced metric changes: X3l carries elementary particle quantum numbers.

Finite measurement resolution replacing effectively light-like 3-surfaces with braids replaces space-time surfaces with collections of string world sheets. Note that the strings connecting braid points at partonic 2-surfaces are like strings connecting branes. The string model in question however differs in many respects from string model and the string tension -essentially density of Kähler action per unit length - does not equal to the inverse of gravitational constant.

Construction of configuration space geometry and spinor structure in terms of second quantized induced spinor fields

Thanks to the input from number theoretical compactification, the construction of the configuration space geometry and spinor structure in terms of second quantized induced spinor fields is now relatively well understood. Second quantization and configuration space geometry are in very intimate relationship and explicit formulas for configuration space Kähler function can be written. Even an explicit formula for Kähler coupling strength revealing its number theoretic anatomy is possible.

  1. The key idea is that the Dirac determinant for the modified Dirac operator defined assignable to some action defining the dynamics of space-time surface or perhaps 3-surface. The replacement of induced gamma matrices with modified gamma matrices guarantees super-conformal symmetry. The basic question is "Which action?". For five years ago I would have answered "Kähler action" without hesitation but one of the basic blunders of the last years was the attempt to reduce the entire physics to Chern-Simons action for induced Kähler gauge potential. The motivation was TGD as an almost topological QFT formulated in terms of modified Dirac action associated with C-S action and localized to the light-like 3-surfaces. Step by step I realized that the correct formulation must involve the modified Dirac operator associated with Kähler action, which indeed allows also the almost topological QFT formulation in terms of holography for preferred extremals.

  2. For a moment I thought that Kähler action is enough. It however seems that it must be complexified by adding imaginary instanton term for a preferred extremal and defines exponent of Kähler function with a phase defined by instanton term added. By its topological character instanton term does not induce imaginary part to the Kähler metric but induces Chern-Simons action at light-like 3-surfaces representing particles. This would realize the long sought CP breaking at the fundamental level explaing matter antimatter asymmetry and hadronic CP breaking in TGD Universe. Also time reversal asymmetry is implied and becomes quite explicit in the sketched generalized Feynman rules.

  3. If instanton term is absent there is only finite number of eigenmodes and the physical interpretation of the situation is very transparent: fermions in electroweak magnetic fields with regions in which induced Kähler form is non-vanishing forming natural units allowing finite number of analogs of cyclotron stats. Second quantization allows to satisfy anticommutation at only finite number of points of partonic 2-surface so that the notion of braid as a correlate for finite measurement resolution would emerge automatically. Dirac determinant reduces to a product of finite number of generalized eigenvalues and everything is nice. This picture is especially attractive from the point of view of number theoretical universality.

  4. If instanton term is allowed, infinite number of conformal excitations assignable to the strings connecting braid points are possible. In this case the Dirac determinant can be defined by standard zeta function regularization reducing to that for Riemann Zeta but it is questionable whether this option is number theoretically universal. It is not yet clear whether one must allow conformal excitations in the definition of Dirac determinant or not or whether the two definitions might give rise to same configuration space metric (but not same Kähler function since a real part of a holomorphic function of configuration space coordinates can distinguish between them!). More generally, the independence on conformal cutoff would have interpretation as renormalization group invariance of the configuration space metric.

  5. The generalized eigenvalues of the modified Dirac operator relate closely to Higgs mechanism. It however turned out that Higgs vacuum expectation does not cause massivation of gauge bosons. Rather the Higgs expectation value in boson state is expressible in terms of this kind of eigenvalues for gauge boson giving directly the ground state contribution to the mass of fermion (gauge boson is bound state of fermion and antifermion at the opposite throats of a wormhole contact). The ground state contribution to fermion mass would be small since p-adic thermodynamic contribution from the conformal excitations would dominate over the small ground state contribution. This also leads to a formula of Weinberg angle in terms of the generalized eigenvalues. Quite generally, the view about what causes what in particle massivation is drastically modified.

Space-time correlate of quantum criticality and the identification of preferred extremals

The geometric properties of preferred extremals are fixed to a high degree by number theoretic compactification. This is not quite enough. A good candidate for the additional field equations satisfied by the preferred extremals of Kähler action is revealed by the study of the modified Dirac equation (this result could have been deduced for more than decade ago!).

  1. The Noether currents associated with Kähler action involve modified gamma matrices, which are contractions of the vector field associated with the first variation of Kähler action with ordinary gamma matrices. These currents are conserved only if the second variation of Kähler action vanishes. This is quite a strong condition. It is satisfied trivially by vacuum extremals but might be too strong for general extremals.

  2. A weaker condition is that only the second variations associated with the dynamical symmetries vanish. This would give a hierarchy of criticalities beginning from that for vacuum extremals in which case all second variations vanish identically. Thus field equations alone would imply the basic vision that TGD Universe is a Universe at the edge: it would not be needed as an additional postulate. A generalization of Thom's catastrophe theory would be in question: systems would live only at the edges of catastrophe graph defined by the V shaped boundary of cusp in the simplest situation.

  3. This at-the-edge property has also several other aspects. There would be also criticality with respect to phase transitions changing Planck constant very important in TGD inspired quantum biology. As also the number theoretical criticality with respect to quantum jumps transforming p-adic and real space-time sheets to each other and assigned with the formation of cognitive representations and realization of intentional actions in TGD inspired theory of consciousness. Number theoretical would be distinct from number theoretical universality. Only those surfaces whose mathematical representations can be interpret both in terms of real and p-adic numbers would be analogous to rationals common to all number fields and would represent number theoretical criticality.

Finite measurement resolution and number theoretic braids

Finite measurement resolution has the notion of number theoretic braid as a space-time correlate. This concept is now rather well-understood.

  1. The basic assumption is that the braids must be definable in purely physical terms: one cannot pick up braid points just randomly as a mathematician armed with selection axiom would do. For instance, braid points could be identified as points of partonic two-surface at which induced Kähler field strength has extremum, as intersections of M4 and CP2 projections with with 2-D critical manifolds associated with the criticality with respect to the phase transition changing Planck constant. There is also a much more general definition inspired by the hierarchy of symplectic triangulations which can be realized in terms of quantization of Kähler magnetic fluxes and extrema of induced Kähler field strength. What is the precise rule characterizing allowed rules defining braids is not quite clear yet. This definition would allow an infinite hierarchy of conformal cutoffs in terms of symplectic triangulations with the resulting cutoff conformal algebras realized in terms of finite number of fermionic oscillator operators assignable to the braid points.

  2. Finite measurement resolution reduces the light-like 3-surfaces to braids and space-time surfaces to strings and the infinite-dimensional world of classical worlds reduces to a finite-dimensional space (δ M4+/-× CP2)n/Sn consisting of n braid points at partonic 2-surface. In the similar manner the space of configuration space spinor fields modulo finite measurement resolution reduces to a finite-dimensional space. This means enormous simplification at the mathematical level. There is a strong temptation to believe that the Clifford algebra in question can be regarded as a coset space of infinite-dimensional hyper-finite factors of type II1 N and M, where N subset M defines the measurement resolution, and that this algebra could be regarded as a quantum Clifford algebra for the nonstandard values of Planck constant.

Super-conformal symmetries and the structure of the world of classical worlds

The understanding of super-conformal symmetries is now much more detailed than before and I have deleted an impressive collection of wrong and not-even-wrong statements.

  1. It seems now clear that the coset construction for super-symplectic Virasoro algebra and Kac-Moody algebra realizes Equivalence Principle at quantum level. The space-time correlate for Equivalence Principle follows from the stringy picture. General Relativistic form of Equivalence Principle holds only in long length scales, not for a cosmic string like objects for instance. This resolves the basic poorly understood issues which have plagued the understanding of GRT-TGD correspondence and allows to throw away a lot of trash.

  2. The understanding of the detailed structure of the configuration space has improved considerably. Configuration space is union of symmetric spaces over zero modes identified as coset spaces and the challenge is to understand what this statement might mean.
    1. The values of the induced Kähler field strength for the partonic 2-surface defines the most important zero modes meaning that dynamics of induced Kähler field is completely classical. Coset construction has its counterpart at the level of configuration space as a union of coset spaces. The symmetric space associated with a given induced Kähler form correspond to the orbit of a symplectic group.
    2. Symplectic group can be made local with respect to the partonic 2-surface - or rather with the coordinate defined by the value of induced Kähler field strength taking the role of complex coordinate in conformal field theories. Kac-Moody sub-algebra defined at light-like 3-surface, whose elements vanish at the partonic 2-surfaces defining its ends, acts as a gauge algebra defining further zero modes.
    3. Quantum fluctuating degrees of freedom correspond to the coset space defined by the symplectic algebra and by the sub-Kac-Moody algebra. Note that the entire Kac-Moody algebra appears in the coset construction and p-adic mass calculations whereas only the sub-algebra appeas in the coset space construction.
  3. The identification of induced Kähler form of X2 as purely classical field means that configuration space functional integral is only over the fluctuations contributing to the induced metric metric of X2. Therefore at the configuration space level the only quantum fluctuating degrees of freedom are purely gravitational. Besides this present are fermionic degrees of freedom, modular degrees of freedom, other zero modes (Kac-Moody algebra), and topological degrees of freedom.

About the construction of M-matrix

The toughest challenge of TGD has been the construction of TGD counterpart of S-matrix - M-matrix as I call it. The understanding of the generalized Feynman rules is now rather detailed and the notion of finite measurement resolution gives excellent hopes about calculational rules making possible practical calculations.

  1. The first fundamental element is zero energy ontology allowing to identify M-matrix as time-like entanglement coefficients between positive and negative energy parts of zero energy state (counterpart of physical event) assignable to the light-like boundaries of causal diamond identified as intersection of future and past directed light-cones defining the basic piece of the world of classical worlds. There is entire hierarchy of CDs within CDs and this allow to understand p-adic coupling constant evolution in terms of finite measurement resolution defined by the size of smallest CD included.

  2. Second basic notion is generalized Feynman diagram identified as light-like 3-surface or equivalently as region of space-time with Euclidian signature of metric accompanying the light-like 3-surface. Euclidian regions would represent particles and Minkowskian regions classical fields. The conformal symmetries and stringy picture implied by the finite measurement resolution suggest strongly stringy Feynman rules.

  3. A very powerful form of General Coordinate Invariance would be the condition that one can deduce configuration space metric by using any light-like 3-surface in the slicing of space-time surface to light-like 3-surfaces parallel to the surface X3l at which the signature of the induced metric changes. Invariance would not mean invariance of Kähler function but only that of Kähler metric. This condition should pose extremely powerful constraints on the form of various expressions appearing in generalized Feynman diagrammatics.

  4. Vertices correspond to partonic 2-surfaces and n-points functions of an N=4 conformal field theory in which second quantized induced spinor fields are the fundamental fields. The TGD based interpretation of N=4 for algebra is now well-understood and reflects directly the basic symmetries of TGD. Discretization implied by the number theoretic braids implies a huge simplification of the situation and mean stringy theory at space-time level.

  5. Propagators assigned with light-like 3-surfaces connecting vertices should be stringy. The problem is how to obtain conformal excitations propagating along strings connecting braid points as zero modes of the modified Dirac operator. These excitations with non-vanishing conformal weight necessarily break the effective 2-dimensionality of 3-surfaces and thus holography. In the proposed - and yet admittedly speculative - picture about the properties of preferred extremals the only possible manner to obtain this breaking seems to be complexification of Kähler action by adding to it as imaginary part the CP breaking instanton action. The "only" in the preceding sentence should be taken with a grain of salt since the implications of number theoretical compactification for the geometry of preferred extremals are not completely understood.

  6. Besides implying CP breaking and the breaking of time reversal symmetry, the instanton term would break the effective 2-dimensionality of 3-surfaces (holography) and would give rise to stringy propagation of fermions whereas at the space-time level effective 2-dimensionality seem to prevail apart from the non-determinism of Kähler action. One can speak of a radiative generation of kinetic and mass terms in stringy propagator. The classical non-determinism of Kähler action would be responsible for generating the analogs of self energy vertices and break the effective 2-dimensionality of 3-surfaces. This conforms with what one might expect. Note that only the conformal excitations of induced spinor field would break the exact holography.

Monday, March 09, 2009

Einstein's equations and second variation of volume element

Lubos had an interesting posting about how Jacobsen has derived Einstein's equations from thermodynamical considerations as kind of equations of state. This has been actually one the basic ideas of quantum TGD, where Einstein's equations do not make sense as microscopic field equations. The argument involves approximate Poincare invariance, Equivalence principle, and proportionality of entropy to area (dS = kdA) so that the result is perhaps not a complete surprise.

One starts from an expression for the variation of the area element dA for certain kind of variations in direction of light-like Killing vector field and ends up with Einstein's equations. Ricci tensor creeps in via the variation of dA expressible in terms of the analog of geodesic deviation involving curvature tensor in its expression. Since geodesic equation involves first variation of metric, the equation of geodesic deviation involves its second variation expressible in terms of curvature tensor.

The result raises the question whether it makes sense to quantize Einstein Hilbert action and in light of quantum TGD the worry is justified. In TGD (and also in string models) Einstein's equations result in long length scale approximation whereas in short length scales stringy description provides the space-time correlate for Equivalence Principle. In fact in TGD framework Equivalence Principle at fundamental level reduces to a coset construction for two super-conformal algebras: super-symplectic and super Kac-Moody. The four-momenta associated with these algebras correspond to inertial and gravitational four-momenta.

In the following I will consider different -more than 10 year old - argument implying that empty space vacuum equations state the vanishing of first and second variation of the volume element in freely falling coordinate system and will show how the argument implies empty space vacuum equations in the "world of classical worlds". I also show that empty space Einstein equations at space-time level allow interpretation in terms of criticality of volume element - perhaps serving as a correlate for vacuum criticality of TGD Universe. I also demonstrate how one can derive non-empty space Einstein equations in TGD Universe and consider the interpretation.

1. Vacuum Einstein's equations from the vanishing of the second variation of volume element in freely falling frame

The argument of Jacobsen leads to interesting considerations related to the second variation of the metric given in terms of Ricci tensor. In TGD framework the challenge is to deduce a good argument for why Einstein's equations hold true in long length scales and reading the posting of Lubos led to an idea how one might understand the content of these equations geometrically.

  1. The first variation of the metric determinant gives rise to

    δ g1/2 = ∂μg1/2dxμ propto g1/2 Cρρμdxμ.

    Here Cρμν denotes Christoffel symbol.

    The possibility to find coordinates for which this variation vanishes at given point of space-time realizes Equivalence Principle locally.

  2. Second variation of the metric determinant gives rise to the quantity

    δ2 g1/2= ∂μνg1/2dxμdxν = g1/2Rμνdxμdxν.

    The vanishing of the second variation gives Einstein's equations in empty space. Einstein's empty space equations state that the second variation of the metric determinant vanishes in freely moving frame. The 4-volume element is critical in this frame.

2. The world of classical worlds satisfies vacuum Einstein equations

In quantum TGD this observation about second variation of metric led for two decades ago to Einstein's vacuum equations for the Kähler metric for the space of light-like 3-surfaces ("world of classical worlds"), which is deduced to be a union of constant curvature spaces labeled by zero modes of the metric. The argument is very simple. The functional integration over configuration space degrees of freedom (union of constant curvature spaces a priori: Rij=kgij) involves second variation of the metric determinant. The functional integral over small deformations of 3-surface involves also second variation of the volume element Ög. The propagator for small deformations around 3-surface is contravariant metric for Kähler metric and is contracted with Rij = lgij to give the infinite-dimensional trace gijRij = lD=l×∞. The result is infinite unless Rij=0 holds. Vacuum Einstein's equations must therefore hold true in the world of classical worlds.

4. Non-vacuum Einstein's equations: light-like projection of four-momentum projection is proportional to second variation of four-volume in that direction

An interesting question is whether Einstein's equations in non-empty space-time could be obtained by generalizing this argument. The question is what interpretation one should give to the quantity


at a given point of space-time.

  1. If one restricts the consideration to variations for which dxm is of form kme, where k is light-like vector, one obtains a situation similar to used by Jacobsen in his argument. In this case one can consider the component dPk of four-momentum in direction of k associated with 3-dimensional coordinate volume element dV3=d3x. It is given by dPk= g41/2TμνkμkνdV3 .

  2. Assume that dPk is proportional to the second variation of the volume element in the deformation dxm =εkm, which means pushing of the volume element in the direction of k in second order approximation:

    (d2g41/2/dε2)dV3= (∂2g41/2/∂ xμ∂ xν) kμkνg41/2dV3= Rμνkμkνg41/2 dV3 .

    By light-likeness of kμ one can replace Rμν by Gμν and add also gμν for light-like vector kμ to obtain covariant conservation of four-momentum. Einstein's equations with cosmological term are obtained.

That light-like vectors play a key role in these arguments is interesting from TGD point of view since light-like 3-surfaces are fundamental objects of TGD Universe.

5. The interpretation of non-vacuum Einstein's equations as breaking of maximal quantum criticality in TGD framework

What could be the interpretation of the result in TGD framework.

  1. In TGD one assigns to the small deformations of vacuum extremals average four-momentum densities (over ensemble of small deformations), which satisfy Einstein's equations. It looks rather natural to assume that statistical quantities are expressible in terms of the purely geometric gravitational energy momentum tensor of vacuum extremal (which as such is not physical). The question why the projections of four-momentum to light-like directions should be proportional to the second variation of 4-D metric determinant.

  2. A possible explanation is the quantum criticality of quantum TGD. For induced spinor fields the modified Dirac equation gives rise to conserved Noether currents only if the second variation of Kähler action vanishes. The reason is that the modified gamma matrices are contractions of the first variation of Kähler action with ordinary gamma matrices.

  3. A weaker condition is that the vanishing occurs only for a subset of deformations representing dynamical symmetries. This would give rise to an infinite hierarchy of increasingly critical systems and generalization of Thom's catastrophe theory would result. The simplest system would live at the V shaped graph of cusp catastrophe: just at the verge of phase transition between the two phases.

  4. Vacuum extremals are maximally quantum critical since both the first and second variation of Kähler action vanishes identically. For the small deformations second variation could be non-vanishing and probably is. Could it be that vacuum Einstein equations would give gravitational correlate of the quantum criticality as the criticality of the four-volume element in the local freely falling frame. Non-vacuum Einstein equations would characterize the reduction of the criticality due to the presence of matter implying also the breaking of dynamical symmetries (symplectic transformations of CP2 and diffeomorphisms of M4 for vacuum extremals).

For the recent updated view about the relationship between General Relativity and TGD see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".