Does the Lorentz invariance for p-adic mass calculations require the p-adic mass squared values to be Teichmüller elements?

p-Adic mass calculations involve canonical identification I: x= ∑_nx_npⁿ ∑ x_np^-n mapping the p-adic values of mass squared to real numbers. The momenta p_i at the p-adic side are mapped to real momenta I(p_i) at the real side. Lorenz invariance requires I(p_i· p_j)= I(p_i)· I(p_j). The predictions for mass squared values should be Lorentz invariant. The problem is that without additional assumptions the canonical identification I does not commute with arithmetics operations.

Sums are mapped to sums and products to products only at the limit of large p-adic primes p and mass squared values, which correspond to x_n≤≤ p. The p-adic primes are indeed large: for the electron one has p= M₁₂₇=2¹²⁷-1∼ 10³⁸. In this approximation, the Lorentz invariant inner products p_i· p_j for the momenta at the p-adic side are indeed mapped to the inner products of the real images: I(p_i· p_j)= I(p_i)· I(p_j). This is however not generally true.

1. Should this failure of Lorentz invariance be accepted as being due to the approximate nature of the p-adic physics or could it be possible to modify the canonical identification? It should be also noticed that in zero energy ontology (see this), the finite size of the causal diamond (CD) (see this) reduces Lorent symmetries so that they apply only to Lorenz group leaving invariant either vertex of the CD.
2. Or could one consider something more elegant and ask under what additional conditions Lorentz invariance is respected in the sense that inner products for momenta on the p-dic side are mapped to inner products of momenta on the real side.

The so called Teichmüller elements of the p-adic number field could allow to realize exact Lorentz invariance.

1. Teichmüller elements T(x) associated with the elements of a p-adic number field satisfy x^p=x, and define therefore a finite field G_p, which is not the same as that given by p-adic integers modulo p. Teichmüller element T(x) is the same for all p-adic numbers congruent modulo p and involves an infinite series in powers of p.

   The map x→ T(x) respects arithmetics. Teichmüller elements of for the product and sum of two p-adic integers are products and sums of their Teichmüller elements: T(x₁+x₂)= T(x₁)+T(x₂) and T(x₁x₂)= T(x₁)T(x₂).

2. If the thermal mass squared is Teichmüller element, it is possible to have Lorentz invariance in the sense that the p-adic mass squared m²_p= p^kp_k defined in terms of p-adic momenta p_k is mapped to m²_R=I(m²_p) satisfying I(m²_p)= I(p^k)I(p_k). Also the inner product p₁· p₂ of p-adic momenta mapped to I(p₁· p₂)=I(p₁)· I(p₂) if the momenta are Teichmüller elements.
3. Should the mass squared value coming as a series in powers of p mapped to Teichmüller element or should it be equal to Teichmüller element?
   1. If the mass squared value is mapped to the Teichmüller element, the lowest order contribution to mass squared from p-adic thermodynamics fixes the mass squared completely. Therefore the Teichmüller element does not differ much from the p-adic mass squared predicted by p-adic thermodynamics. For the large p-adic primes assignable to elementary particles this is true.

   2. The radical option is that p-adic thermodynamics and momentum spectrum is such that it predicts that thermal mass squared values are Teichmüller elements. This would fix the p-adic thermodynamics apart from the choice of p-adic number field or its extension. Mass squared spectrum would be universal and determined by number theory. Note that the p-adic mass calculations predict that mass squared is of order O(p): this is however not a problem since one can consider the m²/p.

This would have rather dramatic physical implications.

1. If the allowed p-adic momenta are Teichmüller elements and therefore elements of G_p then also the mass squared values are Teichmüller elements. This would mean theoretical momentum quantization. This would imply Teichmüller property also for the thermal mass squared since p-adic thermodynamics in the approximation that very higher powers of p give a negligible contribution give a finite sum over Teichm\"muller elements. Number theory would predict both momentum and mass spectra and also thermal mass squared spectrum.

   What does it mean that the product of Teichmüller elements is Teichmüller element? The product xy can be written as ∑_k (xy)_k p^k, (xy)_k=∑_l x_k-ly_l. For Teichmüller elements (xy)_k has no overflow digits. This is true also for I(xy) so that I(xy)= I(x)I(y). Similar argument applies to the sum.

2. The number of possible mass squared values in p-adic thermodynamics would be equal to the p-adic prime p and the mass squared values would be determined purely number theoretically as Teichmüller representatives defining the elements of finite field G_p. The p-adic temperature (see this), which is quantized as 1/T_p=n, can have only p values 0,1,...p-1 and 1/T_p=0 corresponds to high temperature limit for which p-adic Boltzman weights are equal to 1 and the p-adic mass squared is proportional to m²= ∑ g(m) m/∑g(m), where g(m) is the degeneracy of the state with conformal weight h=m. T_p=1/(p-1) corresponds to the low temperature limit for which Boltzman weights approach rapidly zero.

See the article Could the precursors of perfectoids emerge in TGD? or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: Part III

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.