The most recent view about massless particles and particle massivation

I have updated the chapter about massless particle and particle massivation to correspond to the recent progress in the understanding of the basic TGD. I glued below the abstract.

In this chapter the goal is to summarize the recent theoretical understanding of the spectrum of massless particles and particle massivation in TGD framework. After a summary of the recent phenomenological picture behind particle massivation the notions of number theoretical compactification and number theoretical braid are introduced and the construction of quantum TGD at parton level in terms of second quantization of modified Dirac action is described. The recent understanding of super-conformal symmetries are analyzed in detail. TGD color differs in several respect from QCD color and a detailed analysis of color partial waves associated with quark and lepton chiralities of imbedding space spinors fields is carried out with a special emphasis given to the contribution of color partial wave to mass squared of the fermion. The last sections are devoted to p-adic thermodynamics and to a model providing a formula for the modular contribution to mass squared.

Although the basic predictions of p-adic mass calculations were known almost 15 years ago, the justification of the basic assumptions from basic principles of TGD (and also the discovery of these principles!) has taken a considerable time. Particle massivation can be regarded as a generation of thermal conformal weight identified as mass squared and due to a thermal mixing of a state with vanishing conformal weight with those having higher conformal weights. The observed mass squared is not p-adic thermal expectation of mass squared but that of conformal weight so that there are no problems with Lorentz invariance.

One can imagine several microscopic mechanisms of massivation. The following proposal is the winner in the fight for survival between several competing scenarios.

1. The original observation was that the pieces of CP₂ type vacuum extremals representing elementary particles have random light-like curve as an M⁴ projection so that the average motion correspond to that of massive particle. Light-like randomness gives rise to classical Virasoro conditions. This picture generalizes since the basic dynamical objects are light-like but otherwise random 3-surfaces. Fermions are identified as light-like 3-surfaces at which the signature of induced metric of deformed CP₂ type extremals changes from Euclidian to the Minkowskian signature of the background space-time sheet. Gauge bosons and Higgs correspond to wormhole contacts with light-like throats carrying fermion and antifermion quantum numbers. Gravitons correspond to pairs of wormhole contacts bound to string like object by the fluxes connecting the wormhole contacts. The randomness of the light-like 3-surfaces and associated super-conformal symmetries justify the use of thermodynamics and the question remains why this thermodynamics can be taken to be p-adic. The proposed identification of bosons means enormous simplification in thermodynamical description since all calculations reduced to the calculations to fermion level.

2. The fundamental parton level description of TGD is based on almost topological QFT for light-like 3-surfaces. Dynamics is constrained by the requirement that CP₂ projection is for extremals of Chern-Simons action 2-dimensional and for off-shell states light-likeness is the only constraint. As a matter fact, the basic theory relies on the modified Dirac action associated with Chern-Simons action and Kähler action in the sense that the generalizes eigenmodes of C-S Dirac operator correspond to the zero modes of Kähler action localized to the light-like 3-surfaces representing partons. In this manner the data about the dynamics of Kähler action is feeded to the eigenvalue spectrum. Eigenvalues are interpreted as square roots of ground state conformal weights.

3. The symmetries respecting light-likeness property give rise to Kac-Moody type algebra and super-symplectic symmetries emerge also naturally as well as N=4 character of super-conformal invariance. The coset construction for super-symplectic Virasoro algebra and Super Kac-Moody algebra identified in physical sense as sub-algebra of former implies that the four-momenta assignable to the two algebras are identical. The interpretation is in terms of the identity of gravitational inertial masses and generalization of Equivalence Principle.

4. Instead of energy, the Super Kac-Moody Virasoro (or equivalently super-symplectic) generator L₀ (essentially mass squared) is thermalized in p-adic thermodynamics (and also in its real version assuming it exists). The fact that mass squared is thermal expectation of conformal weight guarantees Lorentz invariance. That mass squared, rather than energy, is a fundamental quantity at CP₂ length scale is also suggested by a simple dimensional argument (Planck mass squared is proportional to (^h/_2p) so that it should correspond to a generator of some Lie-algebra (Virasoro generator L₀!)).

5. By Equivalence Principle the thermal average of mass squared can be calculated either in terms of thermodynamics for either super-symplectic of Super Kac-Moody Virasoro algebra and p-adic thermodynamics is consistent with conformal invariance.

6. A long standing problem has been whether coupling to Higgs boson is needed to explain gauge boson masses. It has turned out that p-adic thermodynamics is enough. From the beginning it was clear that is that ground state conformal weight is negative. Only quite recently it became clear that the ground state conformal weight need not be a negative integer. The deviation Dh of the total ground state conformal weight from negative integer gives rise to Higgs type contribution to the thermal mass squared and dominates in case of gauge bosons for which p-adic temperature is small. In the case of fermions this contribution to the mass squared is small. Higgs vacuum expectation is naturally proportional to Dh so that the coupling to Higgs apparently causes gauge boson massivation. The interpretation is that the effective metric defined by the modified gamma matrices associated with Kähler action has Euclidian signature. This implies that the eigenvalues of the modified Dirac operator are purely imaginary and analogous to cyclotron energies so that in the first approximation smallest conformal weights are of form h = -n-1/2 and for n=0 one obtains the ground state conformal weight h=-1/2 conjectured earlier. One cannot exclude the possibility of complex eigenvalues of D_C-S.

7. There is also modular contribution to the mass squared which can be estimated using elementary particle vacuum functionals in the conformal modular degrees of freedom of the partonic 2-surface. It dominates for higher genus partonic 2-surfaces. For bosons both Virasoro and modular contributions seem to be negligible and could be due to the smallness of the p-adic temperature.

An important question concerns the justification of p-adic thermodynamics.

1. The underlying philosophy is that real number based TGD can be algebraically continued to various p-adic number fields. This gives justification for the use of p-adic thermodynamics although the mapping of p-adic thermal expectations to real counterparts is not completely unique. The physical justification for p-adic thermodynamics is effective p-adic topology characterizing the 3-surface: this is the case if real variant of light-like 3-surface has large number of common algebraic points with its p-adic counterpart obeying same algebraic equations but in different number field.

2. The most natural option is that the descriptions in terms of both real and p-adic thermodynamics make sense and are consistent. This option indeed makes since the number of generalized eigen modes of modified Dirac operator is finite. The finite number of fermionic oscillator operators implies an effective cutoff in the number conformal weights so that conformal algebras reduce to finite-dimensional algebras. The first guess would be that integer label for oscillator operators becomes a number in finite field for some prime. This means that one can calculate mass squared also by using real thermodynamics but the consistency with p-adic thermodynamics gives extremely strong number theoretical constraints on mass scale. This consistency condition allows also to solve the problem how to map a negative ground state conformal weight to its p-adic counterpart. Negative conformal weight is divided into a negative half odd integer part plus positive part Dh, and negative part corresponds as such to p-adic integer whereas positive part is mapped to p-adic number by canonical identification.

p-Adic thermodynamics is what gives to this approach its predictive power.

1. p-Adic temperature is quantized by purely number theoretical constraints (Boltzmann weight exp(-E/kT) is replaced with p^L₀/T_p, 1/T_p integer) and fermions correspond to T_p=1 whereas T_p=1/n, n > 1, seems to be the only reasonable choice for gauge bosons.

2. p-Adic thermodynamics forces to conclude that CP₂ radius is essentially the p-adic length scale R ~ L and thus of order R @ 10^3.5 �{(^h/_2p) G} and therefore roughly 10^3.5 times larger than the naive guess. Hence p-adic thermodynamics describes the mixing of states with vanishing conformal weights with their Super Kac-Moody Virasoro excitations having masses of order 10^-3.5 Planck mass.

The predictions of the general theory are consistent with the earliest mass calculations, and the earlier ad hoc parameters disappear. In particular, optimal lowest order predictions for the charged lepton masses are obtained and photon, gluon and graviton appear as essentially massless particles.

For the updated version of the chapter see Massless particles and Particle Massivation.