*theory is free at parton level*as suggested by the recent view about the construction of S-matrix and by the almost topological QFT property of quantum TGD at parton level (see this and this ). Or more concretely: do partonic 2-surfaces carry only free many-fermion states or can they carry also bound states of fermions and anti-fermions identifiable as bosons?

What is known that Higgs boson corresponds naturally to a wormhole contact (see this ). The wormhole contact connects two space-time sheets with induced metric having Minkowski signature. Wormhole contact itself has an Euclidian metric signature so that there are two wormhole throats which are light-like 3-surfaces and would carry fermion and anti-fermion number in the case of Higgs. Irrespective of the identification of the remaining elementary particles MEs (massless extremals, topological light rays) would serve as space-time correlates for elementary bosons. Higgs type wormhole contacts would connect MEs to the larger space-time sheet and the coherent state of neutral Higgs would generate gauge boson mass and could contribute also to fermion mass.

The basic question is whether this identification applies also to gauge bosons (certainly not to graviton). this identification would imply quite a dramatic simplification since the theory would be free at single parton level and the only stable parton states would be fermions and anti-fermions. As will be found this identification allows to understand the dramatic difference between graviton and other gauge bosons and the weakness of gravitational coupling, gives a connection with the string picture of gravitons, and predicts that stringy states are directly relevant for nuclear and condensed matter physics as has been proposed already earlier (see this , this , and this ).

**1. Option I: Only Higgs as a wormhole contact**

The only possibility considered hitherto has been that elementary bosons correspond to partonic 2-surfaces carrying fermion-anti-fermion pair such that either fermion or anti-fermion has a non-physical polarization. For this option CP_{2} type extremals condensed on MEs and travelling with light velocity would serve as a model for both fermions and bosons. MEs are not absolutely necessary for this option. The couplings of fermions and gauge bosons to Higgs would be very similar topologically. Consider now the counter arguments.

- This option fails if the theory at partonic level is free field theory so that anti-fermions and elementary bosons cannot be identified as bound states of fermion and anti-fermion with either of them having non-physical polarization.
- Mathematically oriented mind could also argue that the asymmetry between Higgs and elementary gauge bosons is not plausible whereas asymmetry between fermions and gauge bosons is. Mathematician could continue by arguing that if wormhole contacts with net quantum numbers of Higgs boson are possible, also those with gauge boson quantum numbers are unavoidable.
- Physics oriented thinker could argue that since gauge bosons do not exhibit family replication phenomenon (having topological explanation in TGD framework) there must be a profound difference between fermions and bosons.

**2. Option II: All elementary bosons as wormhole contacts**

The hypothesis that quantum TGD reduces to a free field theory at parton level is consistent with the almost topological QFT character of the theory at this level. Hence there are good motivations for studying explicitly the consequences of this hypothesis.

**2.1 Elementary bosons must correspond to wormhole contacts if the theory is free at parton level**

Also gauge bosons could correspond to wormhole contacts connecting MEs (see this ) to larger space-time sheet and propagating with light velocity. For this option there would be no need to assume the presence of non-physical fermion or anti-fermion polarization since fermion and anti-fermion would reside at different wormhole throats. Only the definition of what it is to be non-physical would be different on the light-like 3-surfaces defining the throats.

The difference would naturally relate to the different time orientations of wormhole throats and make itself manifest via the definition of light-like operator o=x^{k} γ_{k} appearing in the generalized eigenvalue equation for the modified Dirac operator (see this and this ). For the first throat o^{k} would correspond to a light-like tangent vector t^{k} of the partonic 3-surface and for the second throat to its M^{4} dual t_{d}^{k} in a preferred rest system in M^{4} (implied by the basic construction of quantum TGD). What is nice that this picture non-asks the question whether t^{k} or t_{d}^{k} should appear in the modified Dirac operator.

Rather satisfactorily, MEs (massless extremals, topological light rays) would be necessary for the propagation of wormhole contacts so that they would naturally emerge as classical correlates of bosons. The simplest model for fermions would be as CP_{2} type extremals topologically condensed on MEs and for bosons as pieces of CP_{2} type extremals connecting ME to the larger space-time sheet. For fermions topological condensation is possible to either space-time sheet.

**2.2 Phase conjugate states and matter-antimatter asymmetry**

By fermion number conservation fermion-boson and boson-boson couplings must involve the fusion of partonic 3-surfaces along their ends identified as wormhole throats. Bosonic couplings would differ from fermionic couplings only in that the process would be 2→ 4 rather than 1→ 3 at the level of throats.

The decay of boson to an ordinary fermion pair with fermion and anti-fermion at the same space-time sheet would take place via the basic vertex at which the 2-dimensional ends of light-like 3-surfaces are identified. The sign of the boson energy would tell whether boson is ordinary boson or its phase conjugate (say phase conjugate photon of laser light) and also dictate the sign of the time orientation of fermion and anti-fermion resulting in the decay.

Also a candidate for a new kind interaction vertex emerges. The splitting of bosonic wormhole contact would generate fermion and time-reversed anti-fermion having interpretation as a phase conjugate fermion. this process cannot correspond to a decay of boson to ordinary fermion pair. The splitting process could generate matter-antimatter asymmetry in the sense that fermionic antimatter would consist dominantly of negative energy anti-fermions at space-time sheets having negative time orientation (see this and this ).

This vertex would define the fundamental interaction between matter and phase conjugate matter. Phase conjugate photons are in a key role in TGD based quantum model of living matter. this involves a model for memory as communications in time reversed direction, mechanism of intentional action involving signalling to geometric past, and mechanism of remote metabolism involving sending of negative energy photons to the energy reservoir (see this ). The splitting of wormhole contacts has been considered as a candidate for a mechanism realizing Boolean cognition in terms of "cognitive neutrino pairs" resulting in the splitting of wormhole contacts with net quantum numbers of Z^{0} boson (see this ).

**3. Graviton and other stringy states**

Fermion and anti-fermion can give rise to only single unit of spin since it is impossible to assign angular momentum with the relative motion of wormhole throats. Hence the identification of graviton as single wormhole contact is not possible. The only conclusion is that graviton must be a superposition of fermion-anti-fermion pairs and boson-anti-boson pairs with coefficients determined by the coupling of the parton to graviton. Graviton-graviton pairs might emerge in higher orders. Fermion and anti-fermion would reside at the same space-time sheet and would have a non-vanishing relative angular momentum. Also bosons could have non-vanishing relative angular momentum and Higgs bosons must indeed possess it.

Gravitons are stable if the throats of wormhole contacts carry non-vanishing gauge fluxes so that the throats of wormhole contacts are connected by flux tubes carrying the gauge flux. The mechanism producing gravitons would the splitting of partonic 2-surfaces via the basic vertex. A connection with string picture emerges with the counterpart of string identified as the flux tube connecting the wormhole throats. Gravitational constant would relate directly to the value of the string tension.

The TGD view about coupling constant evolution (see this ) predicts G proportional to L_{p}^{2} , where L_{p} is p-adic length scale, and that physical graviton corresponds to p=M_{127}=2^{127} -1. Thus graviton would have geometric size of order Compton length of electron which is something totally new from the point of view of usual Planck length scale dogmatism. In principle an entire p-adic hierarchy of gravitational forces is possible with increasing value of G.

The explanation for the small value of the gravitational coupling strength serves as a test for the proposed picture. The exchange of ordinary gauge boson involves the exchange of single CP_{2} type extremal giving the exponent of Kähler action compensated by state normalization. In the case of graviton exchange two wormhole contacts are exchanged and this gives second power for the exponent of Kähler action which is not compensated. It would be this additional exponent that would give rise to the huge reduction of gravitational coupling strength from the naive estimate G ≈ L_{p}^{2}.

Gravitons are obviously not the only stringy states. For instance, one obtains spin 1 states when the ends of string correspond to gauge boson and Higgs. Also non-vanishing electro-weak and color quantum numbers are possible and stringy states couple to elementary partons via standard couplings in this case. TGD based model for nuclei as nuclear strings having length of order L(127) (see this ) suggests that the strings with light M_{127}quark and anti-quark at their ends identifiable as companions of the ordinary graviton are responsible for the strong nuclear force instead of exchanges of ordinary mesons or color van der Waals forces.

Also the TGD based model of high T_{c} super-conductivity involves stringy states connecting the space-time sheets associated with the electrons of the exotic Cooper pair (see this and this ). Thus stringy states would play a key role in nuclear and condensed matter physics, which means a profound departure from stringy wisdom, and breakdown of the standard reductionistic picture.

**4. Spectrum of non-stringy states**

The 1-throat character of fermions is consistent with the generation-genus correspondence. The 2-throat character of bosons predicts that bosons are characterized by the genera (g_{1},g_{2}) of the wormhole throats. Note that the interpretation of fundamental fermions as wormhole contacts with second throat identified as a Fock vacuum is excluded.

The general bosonic wave-function would be expressible as a matrix M_{g1,g2} and ordinary gauge bosons would correspond to a diagonal matrix M_{g1,g2} δ_{g1,g2} as required by the absence of neutral flavor changing currents (say gluons transforming quark genera to each other). 8 new gauge bosons are predicted if one allows all 3× 3 matrices with complex entries orthonormalized with respect to trace meaning additional dynamical SU(3) symmetry. Ordinary gauge bosons would be SU(3) singlets in this sense. The existing bounds on flavor changing neutral currents give bounds on the masses of the boson octet. The 2-throat character of bosons should relate to the low value T=1/n<< 1.

If one forgets the complications due to the stringy states (including graviton), the spectrum of elementary fermions and bosons is amazingly simple and almost reduces to the spectrum of standard model. In the fermionic sector one would have fermions of standard model. By simple counting leptonic wormhole throat could carry 2^{3} =8 states corresponding to 2 polarization states, 2 charge states, and sign of lepton number giving 8+8=16 states altogether. Taking into account phase conjugates gives 16+16=32 states.

In the non-stringy boson sector one would have bound states of fermions and phase conjugate fermions. Since only two polarization states are allowed for massless states, one obtains (2+1)×(3+1)=12 states plus phase conjugates giving 12+12=24 states. The addition of color singlet states for quarks gives 48 gauge bosons with vanishing fermion number and color quantum numbers. Besides 12 electro-weak bosons and their 12 phase conjugates there are 12 exotic bosons and their 12 phase conjugates. For the exotic bosons the couplings to quarks and leptons are determined by the orthogonality of the coupling matrices of ordinary and boson states. For exotic counterparts of Wbosons and Higgs the sign of the coupling to quarks is opposite. For photon and Z^{0} also the relative magnitudes of the couplings to quarks must change. Altogether this makes 48+16+16=80 states. Gluons would result as color octet states. Family replication would extend each elementary boson state into SU(3)octet and singlet and elementary fermion states into SU(3)triplets.

**5. Higgs mechanism**

Consider next the generation of mass as a vacuum expectation value of Higgs when also gauge bosons correspond to wormhole contacts. The presence of Higgs condensate should make the simple rectilinear ME curved so that the average propagation of fields would occur with a velocity less than light velocity. Field equations allow MEs of this kind as solutions (see this ).

The finite range of interaction characterized by the gauge boson mass should correlate with the finite range for the free propagation of wormhole contacts representing bosons along corresponding ME. The finite range would result from the emission of Higgs like wormhole contacts from gauge boson like wormhole contact leading to the generation of coherent states of neutral Higgs particles. The emission would also induce non-rectilinearity of ME as a correlate for the recoil in the emission of Higgs.

For more details see the end of the chapter Hyperfinite Factors and Construction of S-matrix of "Towards S-matrix" or the chapters Construction of Elementary Particle Vacuum Functionals and Massless states and Particle Massivation of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

## 4 comments:

Thanks, Matti. Even if we have tubes in our operads, I still don't see why they should be called Higgs bosons. I take the 'no vacuum' idea very seriously. That means working with operads

beforeworrying about what their algebras might turn out to be. On the other hand, I suppose you are just defining the vacuum differently from what shows up in the operad diagrams. The question is: what are your LHC predictions for the Higgs?In principle an entire p-adic hierarchy of gravitational forces is possible with increasing value of G.OK, this sounds good. Our 'p-adic heirarchy' will most probably operate this way.

Dear Kea,

thank you for questions.

1. Question about vacuum.

One can assign to partonic 2-surfaces the space of fermionic Fock states and also bosonic states. Second quantized induced spinors of imbedding space which are generalized eigenstates of the modified Dirac operator supersymmetrically related to the Chern-Simons action for the induced Kahler form define the Fock space and fermionic time evolution with respect to the light-like coordinate (Huh!).

Fermionic vacuum is just Fock vacuum. There is also vacuum in bosonic degrees of freedom which correspond to the deformations of light-like 3-surface and Kac-Moody type algebra. Everything reduces to the construction of ground states of super-conformal representations in one-one correspondence with light particles.

Number theoretical braids emerge when one constructs S-matrix requiring number theoretical universality. One might hope the reduction of everything to some axiomatics but I think that concrete physics must be understood first.

The main point is that if the theory is indeed free at the level of fermions, one cannot construct gauge bosons as single parton states since the only states are free many-fermion-antifermion states. The solution is to assume that also gauge bosons correspond to the pairs of partons identified as light-like throats of wormhole contact having CP_2 distance of order CP_2 size. The vacuums for these partons are related by time-reversal.

2. Higgs, LHC, and TGD

The basic predictions for Higgs are following:

a) Higgs exists but can couple weakly to fermions since they give only a small correction to fermion mass with dominating contribution coming from p-adic thermo-dynamics. The coupling to fermions is proportional to the p-adic mass scale rather than mass and by a factor of order 1/100-1/1000 weaker (Kahler coupling strength appears naturally in coupling) than in standard model. This would explain part of the difficulty to detect Higgs since only bosonic channels are effective in production.

b) The upper and lower mass bounds to Higgs mass derived in standard model do not apply as long as they depend on couplings to fermions.

c) Couplings to quarks and lepton mass scale could differ by a numerical factor for which TGD suggest the ratio of Kahler charges: quark coupling would be by a factor 1/3 weaker which relates directly to em charge of quark. This might explain why the might-be-there bbar bump is lower than deduced from ttbar bump.

d) Higgs gives dominating contribution to intermediate boson mass since p-adic temperature for gauge bosons is low so that thermal mass is also low (T=1/n, n>>1).

e) The weak coupling to fermions, in particular top, combined with unitarity bound suggests strongly that there must exist new physics above intermediate boson mass scale. M_89 hadron physics is the most natural candidate for this physics. The naive scaling estimate (there are several of them and each gives different mass!) for pion of this physics is about 154 GeV. With my meager experimental wisdom I cannot exclude the possibility that the recent 160 GeV could correspond to the decays of M_89 pion. Mono-chromatic photon pairs with photon energy 76 GeV is the signature.

f) The most dramatic general almost-prediction would be a scaled up copy of ordinary hadron physics (Mersenne prime M_107) with masses scaled up by a factor 512 (roughly).

3. Gravitational hierarchy

There are good reasons to believe that scaled up copies of various "physics" (copies of ew-, QCD-, gravitational/stringy physics,... ) correspond to Mersenne primes (perhaps also Gaussian Mersennes abundant in biological length scales).

M_127 defining electron's p-adic length scale defines the largest not completely super-astronomical Mersenne length scale. Since M_107,... give rise to much weaker gravitation (G propto L_p^2) it seems that M_127 must correspond to the gravitation familiar to us.

These strings are of electron size and are very relevant for nuclear and condensed matter physics in TGD Universe which means a dramatic deparature from Planck dogmatism of string models. G is just coupling constant, and would not define a fundamental length. CP_2 size which is 10^4 times longer would however define such a length.

Best,

Matti

Hi Matti

Wormhole contact itself has an Euclidian metric signature so that there are two wormhole throats which are light-like 3-surfaces and would carry fermion and anti-fermion number in the case of Higgs.In trying to visualize your description in the simpler case of CP^1, I imagine a hyperboloid (De Sitter space dS_2) with a copy of CP^1 centered inside the hyperboloid at the intersection of the line joining the foci and its perpendicular bisector. If we conveniently make the point of intersection the origin (0,0,0), lines through the origin map points of De Sitter space to points of projective space.

The generalization of this to the hyperboloid in 5 dimensions is what Baez in using in his BF approach to SO(4,1) MacDowell-Mansouri gravity.

What has interested me as of late, are the cases CP^2, HP^2 and OP^2, as these are used to describe the moduli spaces for N=2, d=5 supergravity (see Gunaydin's hep-th/0502235). As I mentioned over at Kea's blog, the moduli spaces for extremal black holes in d=5 for KP^2 (K=R,C,H,O) are of the form M_5=Coll(KP^2)/Isom(KP^2) (Lorentz transformations modulo rotations).

In going from CP^2 to OP^2, M_5(C)=SL(3,C)/SU(3) is enhanced to M_5(O)=E6(-26)/F4, where the 11D light-cone little group SO(9) plays a role. As a projective basis for OP^2 has at least three linearly independent elements, SO(9) is the subgroup of F4 that fixes one such element. The three embeddings of SO(9) in F4 are just the ways of fixing one of the three projective basis elements. In the CP^2 case, I think SU(2)xU(1) plays the role of SO(9) for CP^2.

Dear kneemo,

thank you for your comments. I understood dimly what you saya bout moduli spaces. I hope that following helps in attempt to visualize what I mean with CP_2 type extremal of Kahler action (Maxwell action for induced Kahler form on spacetime surface X^4 in M^4xCP_2).

The wormhole contact is simply an Euclidian 4-surface obtained as a piece of what I call CP_2 type extremal.

*You can imbed CP_2 to M^4xCP_2 in canonical manner as 4-surface X^4: put M^4 coordinates constant. A more complex imbedding is obtained by taking CP_2 coordinates as coordinates of X^4 and allowing M^4 coordinates to be functions of single CP_2 coordinate s. This means that M^4 projection is 1-dimensional. Assuming further that this projection is light-like curve guarantees that the contribution to the induced metric from M^4 vanishes and vacuum extremal of Kahler action results. Internally the resulting surface is exactly like CP_2.

*This gives the condition that the tangent vector of this curve is lightlike

m_kldm^k/dsdm^l/ds=0

This condition reduces to classical Virasoro conditions when one Fourier expands m^k(s) , which is the first hint that Kac-Moody invariance is inherent to the theory. Lightlikeness of partonic 3-surfaces in the partonic formulation indeed guarantees generalized conformal symmetries as consequence of metric 2-dimensionality.

*The action density (Maxwell action for induced Kahler form) is non-vanishing although energy momentum tensor vanishes. A precise analog of self-dual U(1) instanton is in question carrying homological magnetic and electric Kahler charge of one unit. One could say that elementary particles are Kahler magnetic and electric monopoles but that Kahler magnetic flux flows in "internal degrees of freedom" rather than generating radial field visible outside.

*Similar construction works in case of M^4xS^2 if one drops the dimension of space-time surface from 4 to 2.

About your comments:

a) You mention role of CP_2 as moduli group and also octonion related symmetries. In the rather speculative number theoretical vision about TGD space-time surface X^4 can be seen either as a (co-)hyperquaterionic 4-surface of 8-D space M^8 of hyperoctonions (linear subspace of complexified octonions with Minkowski signature). CP_2 can be identified as space of (hyper-)quaternionic 4-D sub-spaces of (hyper-)octonions. Since the point of CP_2 parametrizes the tangent space of space-time surface, one can regard space-time surface also as surface of M^4xCP_2 and standard model symmetries have number theoretic origin. One can speak of "number theoretic compactification": of course, these are just dual manners to describe space-time, no real compactification occurs.

b) SU(3) plays the group of projective symmetries in consistency with the fact that A_2 Dynkin diagram characterizes 2-D complex projective space.

Matti

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