Comments about p-adic mass calculations

I have been reformulating basic quantum TGD using partonic formulation based on light-like 3-surfaces 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. Super-canonical Virasoro algebra involves only generators L_n, n<0, and creates tachyonic ground states required by p-adic 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. Super-canonical generators and Kac-Moody generators applied to this state give massless ground state and p-adic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The non-determinism of almost topological parton dynamics partially justifies the use of p-adic thermodynamics.

The hypothesis that the commutator of super-canonical 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 p-adic thermodynamics

The consistency with p-adic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the super-conformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence p-adic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S²subset CP₂ for CP₂ degrees of freedom and S² subset δ M⁴_+/- for M⁴ 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.

1. 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)× E² isometries and to SU(2)_L× U(1) electro-weak holonomy algebra having only spinor realization. SO(3) holonomy is identifiable as the spinor counterpart of SO(3) rotation. If E² can be counted as a single sector rather than two (SO(2)subset SO(3) acts as rotations in E² sector) the number of sectors is indeed 5.

2. 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 CP₂ Dirac operator correspond to definite values of conformal weight in p-adic mass calculations. It is indeed possible to assign to the center of mass of partonic 2-surface X² CP₂ partial waves correlating strongly with the net electro-weak quantum numbers of the parton so that the assignment of ground state conformal weight to CP₂ partial waves makes sense. In the case of M⁴ 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 light-cone containing the particle and serving the role of argument of N-point function at the level of particle S-matrix.

3. The additivity of conformal weight means additivity of mass squared at parton level and this has been indeed used in p-adic mass calculations. This implies the conditions

   (∑_i p_i)²= ∑_i m_i²

   The assumption p_i²= m_i² 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,II² = m_i² , -∑_i p_i,perp² +2∑_i,j p_i· p_j=0 .

   The masses would be reduced in bound states: m_i²→ m_i²-(p_T²)_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.

4. Single particle conformal weights can have also imaginary part and if only sums y=∑_kn_ky_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. p-Adic thermodynamics suggest that the measured value of y is a p-adic 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 p-adic 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 p-adic mass calculations and the decay rate is of same order of magnitude as mass. If y is of form pⁿy_k for massless state then a decay width of order Γ≈ p^(n-1)/2m 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 "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" contains a more detailed about the topic.