About the physical interpretation of ramified primes in TGD framework

Adelic physics corresponds to a hierarchy of extensions of rationals inducing extensions of p-adic number fields and the proposal is that ramified primes of extension correspond to preferred p-adic primes.

1. Adelic physics suggests that prime p and quite generally, all preferred p-adic primes, could correspond to ramified primes for the extension of rationals defining the adele. Ramified prime divides discriminant D(P) of the irreducible polynomial P (monic polynomial with rational coefficients) defining the extension (see this).
   

   Discriminant D(P) of polynomial whose, roots give rise to extension of rationals, is essentially the resultant Res(P,P') for P and its derivative P' defined as the determinant of so called Sylvester polynomial (see this). D(P) is proportional to the product of differences r_i-r_j, i≠ j the roots of p and vanishes if there are two identical roots.
   

   Remark: For second order polynomials P(x)=x²+bx+c one has D= b²-4c.
   
   

2. Ramified primes divide D. Since the matrix defining Res(P,P') is a polynomial of coefficients of p of order 2n-1, the size of ramified primes is bounded and their number is finite. The larger coefficients P(x) has, the larger the value of ramified prime can be. Small discriminant means small ramified primes so that polynomials having nearly degenerate roots have also small ramifying primes. Galois ramification is of special interest: for them all primes of extension in the decomposition of p appear as same power. For instance, the polynomial P(x)=x²+p has discriminant D=-4p so that primes 2 and p are ramified primes.
   
   
3. What does ramification mean algebraically? The ring O(K)/(p) of integers of the extension K modulo p=π_i^e_i can be written as product ∏_i O/π_i^e_i (see this). If p is ramified, one has e_i>1 for at least one i. Therefore there is at least one nilpotent element in O(K)/(p).
   
   

Could one interpret nilpotency quantum physically?

1. For Galois extensions one has e_i=e>1 for ramified primes. e divides the dimension of extension. For the quadratic extensions ramified primes have e=2. Quadratic extensions are fundamental extensions - kind of conserved genes -, whose further extensions give rise to physically relevant extensions.
   

   On the other hand, fermionic oscillator operators and Grassmann number used to describe fermions "classically" are nilpotent. Could they correspond to nilpotent elements of order e_i=e=2 in O(K)/(p)? Fermions are building bricks of all elementary particles in TGD. Could this number theoretic analogy for the fermionic statistics have a deeper meaning?
   
   

2. What about ramified primes with e_i=e>2? Could they correspond to para-statistics (see this) or braid statistics (see this)?
   

   Both parabosonic and parafermionic fields of order n have the representation Ψ=∑_i=1ⁿ Ψ_i. For parafermion field one has {Ψ_i(x),Ψ_i(y)}=0 and [Ψ_i(x),Ψ_j(y)]=0, i≠ j, when x and y have space-like separation. For parabosons the roles of commutator and anti-commutator are changed.
   

   The states containing N identical parafermions are described by a representation of symmetric group S_N with N rows with at most e columns (anti-symmetrization). For states containing N identical parabosons one has N columns and at most e rows. For parafermions the wave function is symmetric in horizontal direction but the modes are different so that Bose-Einstein condensation is not possible.
   

   For parafermion of order n operator ∑_i=1ⁿ Ψ_i one has (∑_i=1ⁿ Ψ_i)ⁿ= ∏ Ψ₁ Ψ₂...Ψ_n and higher powers vanish so that one would have e-nilpotency. Therefore the interpretation for the nilpotent elements of order e in O(K)/(p)$ in terms of parafermion of order n=e-1 might make sense.
   

   It seems impossible to build a nilpotent operator from parabosonic field Ψ= ∑_iΨ_i: the reason is that the powers Ψ_iⁿ are non-vanishing for arbitrarily high values of n.
   
   

3. Braid statistics differs from para-statistics and is assigned with quantum groups. It would naturally correspond to quantum phase exp(iπ/p) assignable to the exchange of particles by braid operation regarded as a homotopy permuting braid strands. Could ramified prime p would correspond to braid statistics and the index e_i=e characterizing it to para-statistics of order e-1? This possibility cannot be excluded since this exotic physics would be associated in TGD framework to dark matter assigned to algebraic extensions of rationals whose dimension n equals to h_eff/h₀.
   
   

Why the primes, which do not split maximally in given extension - the ramified ones - would be physically special?

1. Do ramified primes possess exceptional evolutionary fitness and are ramified primes present for lower-dimensional extensions present also for higher-dimensional extensions? If higher extensions are formed as extensions of already existing extensions, this is the case. Hierarchy of polynomials of polynomials would to this kind of hierarchy with ramified primes of starting point polynomials analogous to conserved genes.
   
   
2. Quadratic extensions are the simplest ones and could serve as starting point extensions. Polynomials of form x²-c are the simplest among them. Discriminant is now D= -4c.
   
   
3. Why c= M_n=2ⁿ-1 allowing p=2 and Mersenne prime p=M_n as ramified primes would be favored? Extension of rationals defined by x=2ⁿ is non-trivial for odd n and is equivalent with extension containing 2^1/2. c=M_n=2ⁿ-1 as a small deformation of c=2ⁿ gives an extension having both 2 as M_n as ramified primes.
   

   For c=M_n the number of ramified primes is smallest possible and equal to 2: why minimal number of ramified primes would give rise to a fittest extension? Why smallest number of fermionic p-adic mass scales assignable to the ramified primes would be the fittest option?
   

   The p-adic length scale corresponding ro M_n would be maximal and mass scale minimal. Could one think that other quadratic extension are unstable against transforming to Mersenne extensions with smallest p-adic mass scale?
   
   

See the appendix of the article Shnoll effect decade later or of the chapter A possible explanation of Shnoll effect of "Hyper-finite Factors and Dark Matter Hierarchy: part II" .

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

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