Comments about padic mass calculations.
I have been reformulating basic quantum TGD using partonic formulation based on lightlike 3surfaces identifiable as parton orbits. This provides a precise and rigorous identification of various conformal symmetries which have been previously identified as mathematical necessities. Also concrete geometric picture emerges by using quantum classical correspondence. This kind of reformulation of course means that some stuff appears to be obsolete or simply wrong.
1. About the construction of physical states
The previous construction of physical states was still far from complete and involved erraneous elements. The partonic picture confirms however the basic vision. Supercanonical Virasoro algebra involves only generators L_{n}, n<0, and creates tachyonic ground states required by padic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by L_{n}, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Supercanonical generators and KacMoody generators applied to this state give massless ground state and padic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The nondeterminism of almost topological parton dynamics partially justifies the use of padic thermodynamics.
The hypothesis that the commutator of supercanonical and SKM algebras annihilates physical states seems attractive and would define the analog of Dirac equation in the world of classical worlds and eliminate large number of exotic states.
2. Consistency with padic thermodynamics
The consistency with padic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the superconformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence padic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S^{2}subset CP_{2} for CP_{2} degrees of freedom and S^{2} subset δ M^{4}_{+/} for M^{4} degrees of freedom so that a rather concrete picture in terms of orbits of harmonic oscillator would result.
There are some questions which pop up in mind immediately.
 The most crucial consistency test is the requirement that the number of SKM sectors is N=5 to yield realistic mass spectrum. The SKM sectors correspond to SU(3)× SO(3)× E^{2} isometries and to SU(2)_{L}× U(1) electroweak holonomy algebra having only spinor realization. SO(3) holonomy is identifiable as the spinor counterpart of SO(3) rotation. If E^{2} can be counted as a single sector rather than two (SO(2)subset SO(3) acts as rotations in E^{2} sector) the number of sectors is indeed 5.
 Why mass squared corresponds to the thermal expectation value of the net conformal weight? As already explained this option is forced among other things by Lorentz invariance but it is not possible to provide a really satisfactory answer to this question yet. The coefficient of proportionality can be however deduced from the observation that the mass squared values for CP_{2} Dirac operator correspond to definite values of conformal weight in padic mass calculations. It is indeed possible to assign to the center of mass of partonic 2surface X^{2} CP_{2} partial waves correlating strongly with the net electroweak quantum numbers of the parton so that the assignment of ground state conformal weight to CP_{2} partial waves makes sense. In the case of M^{4} degrees of freedom it is not possible to talk about momentum eigen states since translations take parton out of δ H_{+/} so that momentum must be assigned with the tip of the lightcone containing the particle
and serving the role of argument of Npoint function at the level of particle Smatrix.
 The additivity of conformal weight means additivity of mass squared at parton level and this has been indeed used in padic mass calculations. This implies the conditions
(∑_{i} p_{i})^{2}= ∑_{i} m_{i}^{2}
The assumption p_{i}^{2}= m_{i}^{2} makes sense only for massless partons moving collinearly. In the QCD based model of hadrons only longitudinal momenta and transverse momentum squared are used as labels of parton states, which would suggest that one has
p_{i,II}^{2} = m_{i}^{2} , ∑_{i} p_{i,perp}^{2} +2∑_{i,j} p_{i}· p_{j}=0 .
The masses would be reduced in bound states: m_{i}^{2}→ m_{i}^{2}(p_{T}^{2})_{i}. This could explain why massive quarks can behave as nearly massless quarks inside hadrons. Conduction electrons in graphene
behave as massless particles and dark electrons forming hadron like bound states (say Cooper pairs) could be in question.
 Single particle conformal weights can have also imaginary part and if only sums y=∑_{k}n_{k}y_{k}, n_{k}≥ 0, are allowed, y is always rather sizable in the scale for conformal weights. The question is what complex mass squared means physically. Complex conformal weights have been assigned with an inherent time orientation distinguishing positive energy particle from negative energy antiparticle (in particular, phase conjugate photons from ordinary photons). This suggests an interpretation of y in terms of a decay width. pAdic thermodynamics suggest that the measured value of y is a padic thermal average. This makes sense if the values of y_{k} are algebraic (or perhaps even rational) numbers as the sharpening of Riemann Hypothesis states and the number theoretically universal definition of Dirac determinant requires. The simplest possibility is that y does not depend on the thermal excitation so that the decay width would be characterized by the massless state alone. Perhaps a more reasonable option is that y characterizes the decay rates for massive excitations and is in principle calculable.
For instance, if a massless state characterized by padic prime p has y=p× s y_{k}, where s is the denominator of rational valued y_{k}=r/s, the lowest order contribution to the decay width is proportional to 1/p by the basic rules of padic mass calculations and the decay rate is of same order of magnitude as mass. If y is of form p^{n}y_{k} for massless state then a decay width of order Γ≈ p^{(n1)/2}m results. For electron n should be rather large. This argument generalizes trivially to the case in which massless state has vanishing value of y.
The chapter Massless states and Particle Massivation of "Topological Geometrodynamics: Overall View" contains a more detailed about the topic.
