About quantum arithmetics

Holomorphy= holography vision reduces the gravitation as geometry to gravitations as algebraic geometry and leads to exact general solution of geometric field equations as local algebraic equations for the roots and poles rational functions and possibly also their inverses.

The function pairs f=(f₁,f₂): H→ C² define a function field with respect to element-wise sum and multiplication. This is also true for the function pairs g=(g₁,g₂): C²→ C². Now functional composition _º is an additional operation. This raises the question whether ordinary arithmetics and p-adic arithmetics might have functional counterparts.

Functional (quantum) counterparts of integers, rational and algebraic numbers

Do the notions of integers, rationals and algebraic numbers generalize so that one could speak of their functional or quantum counterparts? Here the category theoretical approach suggesting that degree of the polynomial defines a morphism from quantum objects to ordinary objects leads to a unique identification of the quantum objects.

1. For maps g: C²→ C², both the ordinary element-wise product and functional composition _º define natural products. The element-wise product does not respect polynomial irreducibility as an analog of primeness for the product of polynomials. Degree is multiplicative in _º. In the sum, call it +_e, the degree should be additive. This leads to the identification of +_e as an elementwise product. One can identify neutral element 1_º of _º as 1_º=Id and the neutral element 0_e of +_e as ordinary unit 0_e=1. This is a somewhat unexpected conclusion.

   The inverse of g with respect to _º corresponds to g^-1 for _º, which is a many-valued algebraic function and to 1/g for +_e. The maps g, which do not allow decomposition g= h_º i, can be identified as functional primes and have prime degree. If one restricts the product and sum to g₁ (say), the degree of a functional prime g corresponds to an ordinary prime. These functional integers/rationals can be mapped to integers by a morphism mapping their degree to integer/rational. f is a functional prime with respect to _º if it does not allow a decomposition f= g_º h. One can construct integers as products of functional primes.

2. The non-commutativity of _º could be seen as a problem. The fact that the maps g act like operators suggest that for the functional primes g_p the primes in the product commute. Since g is analogous to an operator, this can be interpreted as a generalization of commutativity as a condition for the simultaneous measurability of observables.
3. One can also define functional polynomials P(X), quantum polynomials, using these operations. In the terms p_nº Xⁿ p_n and g should commute and the sum ∑_e p_nXⁿ corresponds to +_e. The zeros of functional polynomials satisfy the condition P(X)=0_e=1 and give as solutions roots X_k as functional algebraic numbers. The fundamental theorem of algebra generalizes at least formally if X_k and X commute. The roots have representations as space-time surfaces. One can also define functional discriminant D as the _º product of root differences X_k-_e X_l, with -_e identified as element-wise division.

About the notion of functional primeness

There are two cases to consider corresponding to f and g. Consider first the pairs (f₁,f₂): H→ C².

1. Primeness could mean that f does not have a composition f=g_º h. Second notion of primeness is based on irreducibility, which states that f does not reduce to an elementwise product of f= g× h. Concerning the definition of powers of functional primes in this case, a possible problem is that the power (f₁ⁿ,f₂ⁿ) defines the same surface as (f₁,f₂) as a root with n-fold degeneracy. Irreducibility eliminates this problem but does not allow defining the analog of p-adic numbers using (f₁ⁿ,f₂ⁿ) as analog of pⁿ.

2. Since there are 3 complex coordinates of H, f_i are labelled by 3 ordinary primes p_r(f_i), r=1,2,3, rather than single prime p. By the earlier physical argument related to cosmological constant one could assume f₂ fixed, and restrict the consideration to f₁. Every functional p-adic number, in particular functional prime, corresponds to its own ramified primes. The simplest functional would correspond to (f₁,f₂)=(0,0) (could this be interpreted as stating the analog of mod ~p=0 condition).

3. The degrees for the product of polynomial pairs (P₁,P₂) and (Q₁,Q₂) are additive. In the sum, the degree of the sum is not larger than the larger degree and it can happen that the highest powers sum up to zero so that the degree is smaller. This reminds of the properties of non-Archimedean norm for the p-adic numbers. The zero element defines the entire H as a root and the unit element does not define any space-time surface as a root.

Also the pairs (g₁,g₂) can be functional primes, both with respect to powers defined by element-wise product and functional composition _º.

1. The ordinary sum is the first guess for the sum operation in this case. Category theoretical thinking however suggests that the element-wise product corresponds to sum, call it +_e. In this operation degree is additive so that products and +_e sums can be mapped to ordinary integers. The functional p-adic number in this case would correspond to an elementwise product ∏ X_{n º} P_pⁿ, where X_n is a polynomial with degree smaller than p defining a reducible polynomial.
2. A natural additional assumption is that the coefficient polynomials X_n commute with each other and P_p. This is natural since the X_n and P_p act like operators and in quantum theory a complete set of commuting observables is a natural notion. This motivates the term quantum p-adics. The space-time surface is a disjoint union of space-time surfaces assignable to the factors X_{k º} P_p^k _º f. In quantum theory, quantum superpositions of these surfaces are realized. If the surface associated with X_{k º} P_p^k _º f is so large that it cannot be realized inside the CD, it is effectively absent from the pinary expansion. Therefore the size of the CD defines a pinary cutoff.

The notion of functional p-adics

What about functional p-adics?

1. The functional powers gpº k of prime polynomials gp define analogs of powers of p-adic primes and one can define a functional generalization of p-adic numbers as quantum p-adics. The coefficients Xk in Xkº gpk are polynomials with degree smaller than p. The first idea which pops up in mind is that ordinary sum of these powers is in question. What is however required is the sum +e so that the roots are disjoint unions of the roots of the +e summands Xkº gpk. The disjointness corresponds to the fact that cognition can be said to be an analysis decomposing the system into pieces.
2. Large powers of prime appearing in p-adic numbers must approach 0_e with respect to the p-adic norm so that g_Pⁿ must effectively approach Id with respect to _º. Intuitively, a large n in g_Pⁿ corresponds to a long p-adic length scale. For large n, g_Pⁿ cannot be realized as a space-time surface in a fixed CD. This would prevent their representation and they would correspond to 0_e and Id. During the sequence of SSFRs the size of CD increases and for some critical SSFRs a new power can emerge to the quantum p-adic.

The very inspiring discussions with Robert Paster, who advocates the importance of universal Witt Vectors (UWVs) and Witt polynomials (see this) in the modelling of the brain, forced me to consider Witt vectors as something more than a technical tool. As the special case Witt vectors code for p-adic number fields.

1. Both the product and sum of ordinary p-adic numbers require memory digits and are therefore technically problematic. This is the case also for the functional p-adics. Witten polynomials solve this problem by reducing the product and sum purely digit-wise operations.
2. Universal Witt vectors and polynomials can be assigned to any commutative ring R, not only p-adic integers. Witt vectors X_n define sequences of elements of a ring R and Universal Witt polynomials W_n(X₁,X₂,...,X_n) define a sequence of polynomials of order n. In the case of p-adic number field X_n correspond to the pinary digit of power pⁿ and can be regarded as elements of finite field F_p,n, which can be also mapped to phase factors exp(ik 2π/p). The motivation for Witt polynomials is that the multiplication and sum of p-adic numbers can be done in a component-wise manner for Witt polynomials whereas for pinary digits sum and product affect the higher pinary digits in the sum and product.
3. In the general case, the Witt polynomial as a polynomial of several variables can be written as W_n(X₀,X₁,...)=∑_d|n d X_d^n/d, where d is a divisor of n, with 1 and n included. For p-adic numbers n is power of p and the factors d are powers of p. X_d are analogous to elements of a finite field G_p,n as coefficients of powers of p.

Witt polynomials are characterized by their roots, and the TGD view about space-time surfaces both as generalized numbers and representations of ordinary numbers, inspires the idea how the roots of for suitably identified Witt polynomials could be represented as space-time surfaces in the TGD framework. This would give a representation of generalized p-adic numbers as space-time surfaces making the arithmetics very simple. Whether this representation is equivalent with the direct representation of p-adic number as surfaces, is not clear.

Could the prime polynomial pairs (g₁,g₂): C²→ C² and (f₁,f₂): H=M⁴× CP₂→ C² (perhaps states of pure, non-reflective awareness) characterized by ordinary primes give rise to functional p-adic numbers represented in terms of space-time surfaces such that these primes could correspond to ordinary p-adic primes?

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter with the same title.

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.