1. Background and motivations for the idea
M8-H-duality (H= M4× CP2) states that space-time surfaces defined as 4-D roots of complexified octonionic polynomials so that they have quaternionic normal space, can be mapped to 4-surfaces in H (see this, this, and this).
The octonionic polynomials are obtained by algebraic continuation of ordinary real polynomials with rational coefficients although one can also consider algebraic coefficients.
This construction makes sense also for analytic functions with rational (or algebraic) coefficients. For the twistor lift of TGD, cosmological constant Λ emerges via the coefficient of a volume term of the action containing also Kähler action (see this). This leads to an action consisting of Kähler action with both CP2 and M4 terms having very interesting and physically attractive properties, such as spin glass degeneracy. Λ=0 would correspond to an infinite volume limit making the QFT description possible as an approximate description. Also the thermodynamic limit could correspond to this limit (see this).
Irreducible polynomials of rational coefficients give rise to algebraic extensions characterized by the Galois group and these notions are central in adelic vision (see this).
I do not know of any deep reason preventing analytic functions with rational Taylor coefficients. These would make possible transcendental extensions. For instance, the product ∏p(ex-p) for some set of primes p would give as roots transcendental numbers log(p). The Galois group would be however trivial although the extension is infinite. Second example is provided by trigonometric functions sin(x) and cos(x) with roots coming as multiples of nπ and (2n+1)π/2. This might be necessary in order to have Fourier analysis. The translations by a multiple of π for x act permuted roots but do not leave rational numbers rational so that the interpretation as a Galois group is not possible so that also now Galois group would be trivial.
A long standing question has been whether these analytic functions could be regarded as polynomials of infinite order by posing some conditions to the Taylor coefficients. If so, one might hope that the notion of Galois group could make sense also now, and one would obtain a unified view about transcendental extensions of rationals.
- For polynomials as roots of octonionic polynomials space-time surfaces are finite and located inside finite-sized causal diamond (CD).
In the TGD Universe cosmological constant Λ depends on the p-adic length scale and approaches zero at infinite length scale. At the Λ=0 limit, which corresponds also to QFT and thermodynamical limits, space-time surfaces would have infinite size. Only Kähler action with M4 and CP2 parts and having ground state degeneracy analogous to spin glass degeneracy would be present.
- The octonionic algebraic continuations of analytic functions with rational coefficients and subject to restrictions guaranteeing that the notion of prime function makes sense, would define space-time surfaces as their roots.
- Prime analytic functions defining space-time surfaces would in some sense be polynomials of infinite degree and could be even characterized by the Galois group. For real polynomials complex conjugations for the roots is certainly this kind of symmetry.
These functions should have Taylor series at origin, which is a special point for octonionic polynomials with rational (or perhaps even algebraic) coefficients. The selection of origin as a preferred point relates directly to the condition eliminating possible problems due to the loss of associativity and commutativity.
The prime property is possible only if the set of these polynomials fails to have a field property (so that the inverse of any element would be well-defined) since for fields one does not have the notion of prime. The field property is lost if the allowed functions vanish at origin so that one cannot have a Taylor series at origin and the inverse diverges at origin.
The vanishing at origin guarantees that the functional composite f○g of f and g has the roots of g. Roots are inherited as algebraical complexity as a kind of evolution increases. In TGD inspired biology, the roots of polynomials are analogous to genes and the conservation of roots in the function composition would be analogous to the conservation of genes.
Could one make anything concrete about this idea? What kind of functions f could serve as analogs of polynomials of infinite degree with transcendental roots. Could any analytic function with rational coefficients vanishing at origin have a possibly unique decomposition to prime analytic functions?
- Suppose that the analytic prime decomposes to a product over monomials x-xi with transcendental roots xi such that the Taylor series has rational coefficients. This requires an infinite Taylor series.
- One obtains an infinite number of conditions. Each power xn in f has a rational coefficient fn equal to the sum over all possible products ∏k=1n xik of n transcendental roots xik. This gives an infinite number of conditions and each condition involves an infinite number of roots. If the number N of transcendental roots is finite as it is for polynomials, each term involves a finite number of products and the conditions imply that the roots are algebraic. The number of transcendental roots must therefore be infinite. At least formally, these conditions make sense.
- The sums of products are generalized symmetric functions of transcendental roots and should have rational values equal to xn. This generalizes the corresponding condition for ordinary polynomials. Symmetric functions for Sn have Sn as a group of symmetries. For a Galois extension of a polynomial of order n, the Galois group is a subgroup of Sn. This suggests that the Galois group is a subgroup of S∞. S∞ has the simple A∞ of even permutations as a subgroup . The simple groups are analogs of primes for finite groups and one can hope that this is true for infinite and discrete groups (see this).
- Consider a general decomposition of the polynomial of an infinite order to a product of monomials with roots spanning the possibly transcendental extension. Could a suitable representation of extension as an infinite polynomial allow rational coefficients fn for the function ∑ fn xn defined by the infinite product?
- fn is the sum over all possible products of roots obtained by dropping n different roots from the product of all roots which should be finite and equal to one for the generalization of monic polynomials. Therefore there is an infinite sum of terms, which are inverses of finite products and therefore transcendental but one can hope that the infinite number of the summands allows the rationality condition to be satisfied.
3. Profinite groups and Galois extensions as inverse limits
Infinite groups indeed appear as Galois groups of infinite extensions. Absolute Galois groups, say Galois groups of algebraic numbers, provide the basic example.
- There exists a natural topology, known as Krull topology, which turns Galois group to a profinite group (totally disconnected, Hausdorff topological group) (see this), which is also Stone space (see this).
- Profinite groups are not countably infinite but are effectively finite just as hyper-finite factors of type II1 are finite-dimensional: they appear naturally in the TGD framework. Profinite groups have Haar measure giving them a finite volume. Profinite groups behave in many respects like finite groups (compact groups also behave in this manner as far representations are considered). Profiniteness is possessed by products, closed subgroups, and the coset groups associated with the closed normal subgroups.
- Every profinite infinite group is a Galois group for an infinite extension for some field K but one cannot control which field K is realized for a given profinite group. Additive p-adics groups and their products appear as Galois groups of an infinite extension for some field K. The Galois theory of infinite field extensions involves profinite groups obtained as Galois groups for the inverse limits of finite field extensions ..Fn→ Fn+1→ .
- This kind of iterated extensions are of special interest in the TGD framework and an infinite extension would be obtained at the limit (see this). The naive expectation is that the polynomial of infinite degree is a limit of a composite ...Pn○ Pn-1..○ P1 of rational polynomials. The number of infinite extensions obtained in this manner would be infinite.
An interesting question is under what conditions the limiting infinite polynomial exists as an analytic function and whether the Taylor coefficients are rational or in some extension of rationals. The naive intuition is that the inverse limit preserves rationality.
- The identification as the iterate ...Pn○ Pn-1○ P1 is indeed suggestive. Infinite cyclic extension defined at the limit by the polynomial xN, N=∞, to be discussed below, has this kind of interpretation. The Galois group of this kind of extension is however not simple.
Remark: The polynomials in question satisfying P(0)=0 are not irreducible: the composite of N polynomials has xN as a factor and has 0 as N-fold root. The origin of octonionic M8 appears as an isolated root.
- Is the infinite-D extension obtained as an inverse limit transcendental or algebraic? In the TGD framework the condition that the polynomial P1○ P2 has the roots of P1 as roots implies the loss of the field property of analytic functions making the notion of analytic prime possible. The roots of the infinite polynomial contain all roots of finite polynomials appearing in the sequences. This would suggest that the extension is not transcendental. Giving up the property Pi(0)=0 also leads to a loss of root inheritance.
4. Could infinite extensions of rationals with a simple Galois group exist?
Simple Galois groups have no normal subgroups and are of special interest as the building bricks of extensions by functional composition of polynomials. The infinite Galois groups obtained as inverse limit have however an infinite hierarchy of normal subgroups and simple argument suggests that the extensions are algebraic. Could infinite-D transcendental extensions defined by an analytic function with rational coefficients and with a simple infinite Galois group, exist?
A∞ is simple and could not be seen as an inverse limit. Also the groups PSLn(K) are infinite discrete groups for K=Z or Q. A further example is provided by Tarski monster groups (see this) having only cyclic groups Zp for a fixed p as subgroups and existing or p>1075. For these primes, there is a continuum of these monsters.
If the inverse limit is essential for profiniteness for infinite groups, then simple infinite groups are excluded as Galois groups. Indeed, the topology of an infinite simple group G cannot be profinite. The Krull topology has as a basis for open sets all cosets of normal subgroups H of finite index (the number of cosets gH is finite). Simple group has no normal subgroups except a trivial group consisting of a unit element and the group itself. The only open sets would be the empty set and G itself.
In fact, there is also a theorem stating that every Galois group is profinite (see this). All finite groups are profinite in discrete topology. This theorem however excludes infinite simple Galois groups. If one allows only polynomials with P(0)=0, the conservation of algebraic roots suggests that infinite polynomials with transcendental roots are not possible.
The condition for the failure of the field property however leaves the iterates of polynomials for which only the highest polynomial in the infinite sequence of functional compositions vanishes at origin. These infinite polynomials could have transcendental roots.
In the following two examples are consider to test whether the notion of a polynomial of infinite order might work.
5.1 Cyclic extension as an example
The natural question is whether the transcendental roots be regarded as limits of roots for a polynomial with rational coefficients at the limit when the degree N approaches infinity. The above arguments suggest that the limits involve an infinite function composition.
Consider as an example cyclic extension defined by a polynomial XN, which can be regarded as a composite of polynomials xpi for ∏ pi=N. This is perhaps the simplest possible extension than one can imagine.
- The roots are now powers of roots of unity. The notion of the root of unity as ei2π/N does not make sense at the limit N→ ∞. One can however consider the roots ei2π M/N and its powers such that the limit M/N → α is irrational. The powers of ei nα give a dense subset of the circle S1 consisting of irrational points. Note that one obtains an infinite number of extensions labelled by irrational values of α.
- The polynomial should correspond to the limit PN(x)=xN-1, N→ ∞. For each finite value of N, one has PN(x)= ∏n=1N(x-Un)-1, U= ei2π/N . The reduction to P=xN-1 follows from the vanishing of all terms involving lower powers of x than xN.
- If these conditions hold true at the limit N→ ∞, one obtains the same result. The coefficient of xN equals to 1 trivially. The coefficient of xN-1 is the sum over all roots and should vanish. This is also assumed in Fourier analysis ∑n eiα n=0 for irrational α. For α=0 the sum equals to N=∞ identified as Dirac delta function. The lower terms give conditions expected to reduce to this condition. This can be explicitly checked for the coefficient f1.
- The Galois group is in this case the cyclic group U∞,α defined by the powers of Uα.
The iterations of polynomials define an N→ ∞ limit, which can be handled mathematically whereas for an arbitrary sequence of polynomials in the functional composition it is difficult to say anything about the possible emergence of transcendental roots. Note however that the LimN→ ∞ (1+1/N)N=e shows that transcendentals can appear as limits of rationals. I have considered iterations of polynomials and approach to chaos from the point of view of M8-H duality in (see this).
Consider polynomials PN= QN○ R, where R with Q(0)=0 is fixed polynomial and QN=Q○N is the N:th iterate of some irreducible polynomial Q with Q(0)=0 and dQ/dz(0)=0. Origin is a fixed critical point of Q and the attractor towards which the points in the attractor basin of origin end up in the iteration and become roots of P∞ and are roots at this limit. For the real points in the intersection of the positive real axis and attractor basin are roots at this limit so that one has a continuum of roots. The set of roots consists of a continuous segment [0,T) and a discrete set coming from the Julia set defining the boundary of the attractor basin.
Profiniteness suggests an interpretation of this set in terms of p-adic topology or a product of a subset p-adic topologies somehow determined by the number theoretic properties of Q. p-Adic number fields are indeed profinite and as additive groups can act as infinite Galois groups permuting the zeros. The action of p-adic translations could indeed leave the basin of attraction invariant.
In the TGD framework these roots correspond to values of M4 time (or actually energy!) in M8 mapped to actual time values in H by M8-H duality. I have referred to them as "very special moments in the life of self" with a motivation coming from TGD inspired theory of consciousness (see this, this). One might perhaps say that at this limit subjective time consisting of these moments becomes continuous in the interval [0,T].
See the article Does the notion of polynomial of infinite order make sense? or the chapter About the role of Galois groups in TGD framework.
For a summary of earlier postings see Latest progress in TGD.