How also finite fields could define fundamental number fields in Quantum TGD?

One can represent two objections against the number theoretic vision.

1. The first problem is related to the physical interpretation of the number theoretic vision. The ramified primes p_ram dividing the discriminant of the rational polynomial P have a physical interpretation as p-adic primes defining p-adic length- and mass scales.

   The problem is that without further assumptions they do not correlate at all with the degree n of P. However, physical intuition suggests that they should depend on the degree of P so that a small degree n implying a low algebraic complexity should correspond to small ramified primes. This is achieved if the coefficients of P are smaller than n and thus involve only prime factors p<n.

2. All number fields except finite fields, that is rationals and their extension, p-adic numbers and their extensions, reals, complex numbers, quaternions, and octonsions appear at the fundamental level in TGD. Could there be a manner to make also finite fields a natural part of TGD?

These problems raise the question of whether one could pose additional conditions to the polynomials P of degree n defining 4-surfaces in M⁸ with roots defining mass shells in M⁴⊂ M⁸ (complexification assumed) mapped by M⁸-H duality to space-time surfaces in H.

1. P=Q condition

One such condition was proposed here. The proposal is that infinite primes forming a hierarchy are central for quantum TGD. It is proposed that the notion of infinite prime generalizes to that of the notion of adele.

1. Infinite primes at the lowest level of the hierarchy correspond to polynomials of single variable x replaced with the product X=∏_p p of all finite primes. The coefficients of the polynomial do not have common prime divisors. At higher levels, one has polynomials of several variables satisfying analogous conditions.
2. The notion of infinite prime generalizes and one can replace the argument x with Hilbert space,group representation, or algebra and sum and product of ordinary arithmetics with direct sum ⊕ and tensor product ⊗.
3. The proposal is P=Q: at the lowest level of the hierarchy, the polynomial P(x) defining a space-time surface corresponds to an infinite prime determined by a polynomial Q(X). This would be one realization of quantum classical correspondence. This gives strong constraints to the space-time surface and one might speak of the analog of preferred extremal (PE) at the level of M⁸ but does not yet give any special role for the finite fields.
4. The infinite primes at the higher level of the hierarchies correspond to polynomials Q(x₁,x₂,...,x_k) of several variables. How to assign a polynomial of a single argument and thus a 4-surface to Q? One possibility is that one does as in the case of multiple poly-zeta and performs a multiple residue integral around the pole at infinity and obtains a finite result. The remaining polynomial would define the space-time surface.

2. Additional conditions

The speculations related to the p-adicization of ξ inspire the following questions.

1. Option I: Rational polynomial is apart from scaling a polynomial with integer coefficients having the same roots. Could it make sense to assume that the coefficients of the P(x)= Q(x) of degree n are integers divisible only by primes p<n?
2. Option II: A stronger condition would be that the integer coefficients of P=Q are smaller than n. This implies that they are divisible by primes p<n, which cannot however appear as common factors of the coefficients. One could say that the corresponding space-time sheet effectively lives in the ring Z_n instead of integers. For prime value n=p space-time sheet would effectively "live" the finite field F_p and finite fields would gain a fundamental status in the structure of TGD.

   Should one allow both signs for the coefficients as the interpretation as rationals would suggest? In this case, finite field interpretation would mean the replacement of -1 with p-1.

3. Option III: A still stronger, perhaps too strong, condition would be that only the prime factors of n appear as factors of the coefficients of P=Q. For integers n with a small number of prime divisors it is easy to find the possible coefficients. For instance, for n= p all coefficients are equal to 1 or 0!

   For n=p₁p₂, two of the coefficients can be equal to power of p₁ or p₂ if smaller than n and remaining coefficients equal to 1 or 0. For instance, n=p₁p₂ for p₁=M₁₂₇=2¹²⁷-1 and p₂=2, one coefficient could be M₁₂₇, second coefficient power of 2 smaller than 2¹²⁷ and the remaining coefficients would be equal to 1 or 0.

Option II would solve the two problems whereas Option II is un-necessarily strong.

1. For n=p, P would make sense in a finite field F_p if the second condition is true. Finite fields, which have been missing from the hierarchy of numbers fields, would find a natural place in TGD if this condition holds true!
2. The number of polynomial coefficients is n, whereas the number of primes smaller than n behaves as n/log(n). By infinite prime property, the coefficients would not contain common primes p<n. Very few polynomials could define space-time surfaces.

3. How does Option II relate to prime polynomials?

1. The degree of a composite of polynomials with orders m and n is mn so that a polynomial with prime degree p does not allow expression as a composite of polynomials of lower orders so that any polynomials with prime order is a prime polynomial. Polynomials of order m can in principle be functional composites of prime polynomials with orders, which are prime factors of m.

   Obviously, all prime polynomials cannot satisfy Option II. However, those satisfying Option II could be prime polynomials. Note that the polynomials, which have an interpretation in terms of a finite field F_p have degree p-1.

2. There are also non-prime polynomials satisfying Option II. P₁=x^m and P₂= xⁿ satisfy Option II as also the composite P= x^mn, which is however not a prime polynomial. The composite of P₁= x² and P₂= 1+x^m gives P= 1+2x^m+x^2m, which satisfies Option II but is not prime. By the symmetry B(n,k)= B(n,n-k) of binomial coefficients the composite of P₁=x^m, m>2, and P₂= 1+x^m does not satisfy the conditions.
3. Quite generally, polynomials P satisfying Option II and having degree n, which is not prime, can decompose to prime polynomials and probably do so. There the polynomial primeness and Option II do not seem to have a simple relationship.

These observations suggest the tightening of the Option II to the following condition.

All physically allowed polynomials P are functional composites of the prime polynomials of prime degree satisfying Option II. In a rather precise sense, finite fields would serve as basic building blocks of the Universe.

4. Examples of Option II

The following examples illustrate the conditions for Option II.

1. For instance, for M₁₂₇=2¹²⁷-1 assigned with electron by p-adic mass calculations one has n/log(n)∼ M₁₂₇/log(2)127 ∼ M₁₂₇/88 so that only about 12 percent of coefficients of P could differ from 0 or 1.
2. For small values of n it is easy to construct the possible polynomials P.
   1. For n=p=2 one obtains only the coefficients (p₀,p₁)⊂ {+/- 1,0},{0,+/- 1},{+/- 1,+/- 1} corresponding to P(x)∈{+/- 1,+/- x,+/- 1 +/- x}.
   2. For n=p=3, one of the coefficients is p=2 and the remaining coefficients are equal to 1 or 0. The coefficients are (p₀,p₁,p₂)⊂ {+/- 2,x,y}, {x,+/- 2,y}, {x,y,+/- 2} with x,y ∈{0,1,-1} and (p₀,p₁,p₂) with p_i∈ {0,1,-1}.

      A little calculation shows that extensions of rationals containing i, 2^1/2, i2^1/2, 3^1/2, i3^1/2, 5^1/2 (from P=x²+x-1 defining Golden Mean), and i7^1/2 are obtained.

   3. Roots of small primes appear in the Weyl groups, which are reflection groups associated with Dynkin diagrams characterizing Lie groups at Lie algebra level. The finite discrete subgroups of the rotation group SU(2) characterized extensions of hyper-finite factors of type II₁ and roots of small primes appear in the matrix elements of these groups. Could the proposed polynomials give in a natural way rise to the extensions of rationals appearing in these two cases?

The above considerations inspire further questions. Could one also allow polynomials P having coefficients in an algebraic extension of rationals? Does this bring in anything new? Could one have coefficients in an extension containing e or even root of e as perhaps the only transcendental extension defining a finite extension of p-adic numbers? The roots would be generalizations of algebraic numbers involving e and could make sense p-adically via Taylor expansion.

See the article New Ideas Related to Langlands Program viz. TGD or the chapter with the same title.

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

Articles related to TGD.