Higgs and p-adic mass calculations

In the earlier blog posting I told that the boundary condition for the Kähler-Dirac equation is massless Dirac equation with the analog of Higgs term added. The interpretation in terms of Higgs mechanism however fails since the term can be also tachyonic. Intriguingly, p-adic mass calculations require the ground state conformal weight to be negative half odd integer. This raises the question whether the the boundary condition for Kähler-Dirac equation could be equivalent for the mass formulation given by the condition that the scaling generator L₀ annihilates the physical states for Super Virasoro representations. This equivalence is suggested by quantum classical correspondence.

If this is the case, the two mass shell conditions would be equivalent. This possibility is discussed more precisely below.

1. The boundary condition for K-D equation reads as

   (p^kγ_k+ Γⁿ)Ψ=0 .

   (p^kγ_k is algebraic Dirac operator in Minkowski space and Γⁿ is Kähler Dirac gamma matrix defined as contraction of the canonical energy momentum current of Kähler action with imbedding space gamma matrices.

   Mass shell condition corresponds to the vanishing of the square of the algebraic Dirac operator and should be equivalent with the mass shell condition given by the vanishing of the action of L₀:

   p^kp_k== p²= m₀²× (h_gr +n) ==m_{n² .}

   m₀ is CP₂ mass scale dictated by CP₂ size scale and analogous to that given by string tension. m₀ is evaluated for the standard value of Planck constant. h_gr is ground state conformal weight and n is the conformal weight assignable to the Super-Virasoro generators creating the state.

   p-Adic mass calculations require that h_gr is negative and half odd integer valued so that ground state would be tachyonic. The first principle explanation for this could be the presence of Minkowskian time-like contribution in Γⁿ coming from the canonical momentum density for Kähler action. One cannot exclude a p-adically small deviation of h_gr from the negative half odd integer value proportional to at least second power of prime p perhaps assignable to Higgs like contribution or contribution of string like objects assignable to elementary particle.

2. One can decompose Γⁿ to M⁴ and CP₂ parts corresponding to the contractions of the canonical momentum density with M⁴ and CP₂ gamma matrices respectively:

   Γⁿ = Tⁿ(M⁴)+ Tⁿ(CP₂) .

   Tⁿ(M⁴) involves M⁴ gamma matrices is determined by the energy momentum tensor T_K of Kähler action determined by its imbedding space variation coming from the induced metric. Tⁿ(CP₂) involves CP₂ gamma matrices and is sum coming from the imbedding space variations coming from a variation with respect to the induced metric and induced Kähler form. M⁴ and CP₂ contributions are orthogonal to each other as imbedding space vectors.

3. The square of the mass boundary condition gives

   (p+Tⁿ(M⁴))^k(p+Tⁿ(M⁴))_k +T^nk(CP₂)Tⁿ_k(CP₂)=0 .

   This condition can be simplified if one assumes that the direction of classical energy momentum density vector T^nk(M⁴) is same as four momentum p^k. This assumption is motivated by quantum classical correspondence. This would give

   Tⁿk(M⁴)= α (x) p^k .

   The coeffcient α can depend on the position along string.

4. With these assumptions the condition reads

   (1+α)² p² +T^nk(CP₂)Tⁿ_k(CP₂)=0 .

   and gives

   T^nk(CP₂)Tⁿ_k(CP₂)/(1+α)²=- m_n² .

   where m_n² is the mass squared associated with the state as given by the vanishing of L₀ action on the state.

   In coordinate changes the left hand changes in position dependent manner but the change of the factor α compensates the change of T²(CP₂) term so that the condition is general coordinate invariant statement.

5. Combiging this with the mass shell condition coming from the vanishing of the action of L₀ gives

   T^nk(CP₂)Tⁿ_k(CP₂)= -(1+α)²m₀²(h_gr+n) .

   One can solve α from this condition:

   α=+/- S/M_n -1 , S²k== - T^nk(CP₂)Tⁿ_k(CP₂) (≥ 0) .

6. The interpretation of the effective metric defined by the Kähler-Dirac gamma matrices has been a longstanding problem. It seems that the g_effⁿⁿ of this metric appears naturally if one assumes that Super-Virasoro conditions for L₀ is equivalent with that given by the boundary condition for Kähler-Dirac equation.