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 and 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).

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.