Physics as a Generalized Number Theory

Note: Newest contributions are at the top!

 Year 2018

Increase of the dimension of extension of rationals as the emergence of a reflective level of consciousness?

in TGD framework the hierarchy of extensions of rationals defines a hierarchy of adeles and evolutionary hierarchy. What could the interpretation for the events in which the dimension of the extension of rationals increases? Galois extension is extensions of an extension with relative Galois group Gal(rel)= Gal(new)/Gal(old). Here Gal(old) is a normal subgroup of Gal(new). A highly attractive possibility is that evolutionary sequences quite generally (not only in biology) correspond to this kind of sequences of Galois extensions. The relative Galois groups in the sequence would be analogous to conserved genes, and genes could indeed correspond to Galois groups (see this). To my best understanding this corresponds to a situation in which the new polynomial Pm+n defining the new extension is a polynomial Pm having as argument the old polynomial Pn(x): Pm+n(x)=Pm(Pn(x)).

What about the interpretation at the level of conscious experience? A possible interpretation is that the quantum jump leading to an extension of an extension corresponds to an emergence of a reflective level of consciousness giving rise to a conscious experience about experience. The abstraction level of the system becomes higher as is natural since number theoretic evolution as an increase of algebraic complexity is in question.

This picture could have a counterpart also in terms of the hierarchy of inclusions of hyperfinite factors of type II1 (HFFs). The included factor M and including factor N would correspond to extensions of rationals labelled by Galois groups Gal(M) and Gal(N) having Gal(M)⊂ Gal(M) as normal subgroup so that the factor group Gal(N)/Gal(M) would be the relative Galois group for the larger extension as extension of the smaller extension. I have indeed proposed (see this) that the inclusions for which included and including factor consist of operators which are invariant under discrete subgroup of SU(2) generalizes so that all Galois groups are possible. One would have Galois confinement analogous to color confinement: the operators generating physical states could have Galois quantum numbers but the physical states would be Galois singlets.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article with the same title.

Galois groups and genes

In an article discussing a TGD inspired model for possible variations of Geff (see this) , I ended up with an old idea that subgroups of Galois group could be analogous to conserved genes in that they could be conserved in number theoretic evolution. In small variations such as above variation Galois subgroups as genes would change only a little bit. For instance, the dimension of Galois subgroup would change.

The analogy between subgoups of Galois groups and genes goes also in other direction. I have proposed long time ago that genes (or maybe even DNA codons) could be labelled by heff/h=n . This would mean that genes (or even codons) are labelled by a Galois group of Galois extension (see this) of rationals with dimension n defining the number of sheets of space-time surface as covering space. This could give a concrete dynamical and geometric meaning for the notin of gene and it might be possible some day to understand why given gene correlates with particular function. This is of course one of the big problems of biology.

One should have some kind of procedure giving rise to hierarchies of Galois groups assignable to genes. One would also like to assign to letter, codon and gene and extension of rationals and its Galois group. The natural starting point would be a sequence of so called intermediate Galois extensions EH leading from rationals or some extension K of rationals to the final extension E. Galois extension has the property that if a polynomial with coefficients in K has single root in E, also other roots are in E meaning that the polynomial with coefficients K factorizes into a product of linear polynomials. For Galois extensions the defining polynomials are irreducible so that they do not reduce to a product of polynomials.

Any sub-group H⊂ Gal(E/K)) leaves the intermediate extension EH invariant in element-wise manner as a sub-field of E (see this). Any subgroup H⊂ Gal(E/K)) defines an intermediate extension EH and subgroup H1⊂ H2⊂... define a hierarchy of extensions EH1>EH2>EH3... with decreasing dimension. The subgroups H are normal - in other words Gal(E) leaves them invariant and Gal(E)/H is group. The order |H| is the dimension of E as an extension of EH. This is a highly non-trivial piece of information. The dimension of E factorizes to a product ∏i |Hi| of dimensions for a sequence of groups Hi.

Could a sequence of DNA letters/codons somehow define a sequence of extensions? Could one assign to a given letter/codon a definite group Hi so that a sequence of letters/codons would correspond a product of some kind for these groups or should one be satisfied only with the assignment of a standard kind of extension to a letter/codon?

Irreducible polynomials define Galois extensions and one should understand what happens to an irreducible polynomial of an extension EH in a further extension to E. The degree of EH increases by a factor, which is dimension of E/EH and also the dimension of H. Is there a standard manner to construct irreducible extensions of this kind?

1. What comes into mathematically uneducated mind of physicist is the functional decomposition Pm+n(x)= Pm(Pn(x)) of polynomials assignable to sub-units (letters/codons/genes) with coefficients in K for a algebraic counterpart for the product of sub-units. Pm(Pn(x)) would be a polynomial of degree n+m in K and polynomial of degree m in EH and one could assign to a given gene a fixed polynomial obtained as an iterated function composition. Intuitively it seems clear that in the generic case Pm(Pn(x)) does not decompose to a product of lower order polynomials. One could use also polynomials assignable to codons or letters as basic units. Also polynomials of genes could be fused in the same manner.
2. If this indeed gives a Galois extension, the dimension m of the intermediate extension should be same as the order of its Galois group. Composition would be non-commutative but associative as the physical picture demands. The longer the gene, the higher the algebraic complexity would be. Could functional decomposition define the rule for who extensions and Galois groups correspond to genes? Very naively, functional decomposition in mathematical sense would correspond to composition of functions in biological sense.
3. This picture would conform with M8-M4× CP2 correspondence (see this) in which the construction of space-time surface at level of M8 reduces to the construction of zero loci of polynomials of octonions, with rational coefficients. DNA letters, codons, and genes would correspond to polynomials of this kind.
Could one say anything about the Galois groups of DNA letters?
1. Since n=heff/h serves as a kind of quantum IQ, and since molecular structures consisting of large number of particles are very complex, one could argue that n for DNA or its dark variant realized as dark proton sequences can be rather large and depend on the evolutionary level of organism and even the type of cell (neuron viz. soma cell). On the other, hand one could argue that in some sense DNA, which is often thought as information processor, could be analogous to an integrable quantum field theory and be solvable in some sense. Notice also that one can start from a background defined by given extension K of rationals and consider polynomials with coefficients in K. Under some conditions situation could be like that for rationals.
2. The simplest guess would be that the 4 DNA letters correspond to 4 non-trivial finite groups with smaller possible orders: the cyclic groups Z2,Z3 with orders 2 and 3 plus 2 finite groups of order 4 (see the table of finite groups in this). The groups of order 4 are cyclic group Z4=Z2× Z2 and Klein group Z2⊕ Z2 acting as a symmetry group of rectangle that is not square - its elements have square equal to unit element. All these 4 groups are Abelian.
3. On the other hand, polynomial equations of degree not larger than 4 can be solved exactly in the sense that one can write their roots in terms of radicals. Could there exist some kind of connection between the number 4 of DNA letters and 4 polynomials of degree less than 5 for whose roots one can write closed expressions in terms of radicals as Galois found? Could the polynomials obtained by a a repeated functional composition of the polynomials of DNA letters also have this solvability property?

This could be the case! Galois theory states that the roots of polynomial are solvable in terms of radicals if and only if the Galois group is solvable meaning that it can be constructed from abelian groups using Abelian extensions (see this).

Solvability translates to a statement that the group allows so called sub-normal series 1<G0<G1 ...<Gk=G such that Gj-1 is normal subgroup of Gj and Gj/Gj-1 is an abelian group: it is essential that the series extends to G. An equivalent condition is that the derived series is G→ G(1) → G(2) → ...→ 1 in which j+1:th group is commutator group of Gj: the essential point is that the series ends to trivial group.

If one constructs the iterated polynomials by using only the 4 polynomials with Abelian Galois groups, the intuition of physicist suggests that the solvability condition is guaranteed!

4. Wikipedia article also informs that for finite groups solvable group is a group whose composition series has only factors which are cyclic groups of prime order. Abelian groups are trivially solvable, nilpotent groups are solvable, and p-groups (having order, which is power prime) are solvable and all finite p-groups are nilpotent. This might relate to the importance of primes and their powers in TGD.

Every group with order less than 60 elements is solvable. Fourth order polynomials can have at most S4 with 24 elements as Galois groups and are thus solvable. Fifth order polynomial can have the smallest non-solvable group, which is alternating group A5 with 60 elements as Galois group and in this case is not solvable. Sn is not solvable for n>4 and by the finding that Sn as Galois group is favored by its special properties (see this). It would seem that solvable polynomials are exceptions.

A5 acts as the group of icosahedral orientation preserving isometries (rotations). Icosahedron and tetrahedron glued to it along one triangular face play a key role in TGD inspired model of bio-harmony and of genetic code (see this and this). The gluing of tetrahedron increases the number of codons from 60 to 64. The gluing of tetrahedron to icosahedron also reduces the order of isometry group to the rotations leaving the common face fixed and makes it solvable: could this explain why the ugly looking gluing of tetrahedron to icosahedron is needed? Could the smallest solvable groups and smallest non-solvable group be crucial for understanding the number theory of the genetic code.

An interesting question inspired by M8-H-duality (see this) is whether the solvability could be posed on octonionic polynomials as a condition guaranteeing that TGD is integrable theory in number theoretical sense or perhaps following from the conditions posed on the octonionic polynomials. Space-time surfaces in M8 would correspond to zero loci of real/imaginary parts (in quaternionic sense) for octonionic polynomials obtained from rational polynomials by analytic continuation. Could solvability relate to the condition guaranteeing M8 duality boiling down to the condition that the tangent spaces of space-time surface are labelled by points of CP2. This requires that tangent or normal space is associative (quaternionic) and that it contains fixed complex sub-space of octonions or perhaps more generally, there exists an integrable distribution of complex subspaces of octonions defining an analog of string world sheet.

A second attempt to understand the notions of perfectoid and its tilt

Quanta Magazine tells that 30-year old Peter Scholze is now the youngest Fields medalist due to the revolution that he launched in arithmetic geometry (see this).

Scholze's work might be interesting also from the point of view physics, at least the physics according to TGD. I have already made a attempt to understand Scholze's basic idea and to relate it to physics. About the theorems that he has proved I cannot say anything with my miserable math skills.

The notion of perfectoid

Scholze introduces first the notion of perfectoid.

1. This requires some background notions. The characteristic p for field is defined as the integer p for which px=0 for all elements x. Frobenius homomorphism (Frob familiarly) is defined as x→ xp. For a field of characteristic p Frob: x-->xp is an algebra homomorphism mapping product to product and sum to sum: this is very nice and relatively easy to show even by a layman like me.
2. Perfectoid is a field having either characteristic p=0 (reals, p-adics for instance) or for which Frob is a surjection meaning that Frob maps at least one number to a given number x.

For finite fields Frob is identity: xp=x as proved already by Fermat. For reals and p-adic number fields with characteristic p=0 Frob maps all elements to unit element and is not a surjection but this is not required now.

The tilt of the perfectoid

What Scholze introduces besides perfectoids K also what he calls tilt of the perfectoid: Kb. Kb is something between p-adic number fields and reals and leads to theorems giving totally new insights to arithemetic geometry

1. As we learned during the first student year, real numbers can be defined as Cauchy sequences of rationals converging to a real number, which can be also algebraic number or transcendental. The numbers in the tilt Kb would be this kind of sequences.
2. Scholze starts from (say) p-adic numbers and considers infinite sequence of iterates of 1/p:th roots. At given step x→ x1/p. This gives the sequences \$(x,x1/p,x1/p2,x1/p3,...) as elements of Kb. At the limit one obtains 1/p root of x.
1. For finite fields each step is trivial (xp=x) so that nothing interesting results: one has (x,x,x,x,x...).
2. For p-adic number fields the situation is non-trivial. x1/p exists as p-adic number for all p-adic numbers with unit norm having x= x0+x1p+... In the lowest order x ≈ x0 the root is just x since x is effectively an element of finite field in this approximation. One can develop the x1/pto a power series in p and continue the iteration. The sequence obtained defines the element of tilt Kb of field K, now p-adic numbers.
3. If the p-adic number x has norm pn and is therefore not unity, the root operation makes sense only if one performs an extension of p-adic numbers containing all the roots p1/pk) . These roots define one particular kind of extension of p-adic numbers and the extension is infinite-dimensional since all roots are needed. One can approximate Kb by taking only finite number iterated roots: I call these almost perfectoids as precursors of perfectoids.
3. The tilt is said to be fractal: this is easy to understand from the presence of the iterated p:th root. Each step in the sequence is like zooming. One might say that p-adic scale p becomes p:th root of itself. In TGD p-adic length scale Lp is proportional to p1/2: does the scaling mean that the p-adic length scale would defined a hierarchy of scales proportional to p1/2kp approaching the CP2 scale since the root of p approaches unity. Tilts as extensions by iterated roots would improve the length scale resolution.
One day later I got the feeling that I might have understood one more important thing about the tilt of p-adic number field: changing of the characteristic 0 of p-adic number field to characteristics p>0 of the corresponding finite field for its tilt (thanks for Ulla for the links). What could this mean?

Characteristic p (p is now the prime labelling p-adic number field) means nx=0. This property makes the mathematics of finite fields extremely simple: in the summation one need not take care of the residue as in the case of reals and p-adics. The tilt of the p-adic number field would have the same property! In the infinite sequence of the p-adic numbers coming as iterated p:th roots of starting point p-adic number one can sum each p-adic number separately. This is really cute if true!

It seems that one can formulate the arithmetics problem in the tilt where it becomes in principle as simple as in finite field with only p elements! Does the existence of solution in this case imply its existence in the case of p-adic numbers? But doesn't the situation remain the same concerning the existence of the solution in the case of rational numbers? The infinite series defining p-adic number must correspond a sequence in which binary digits repeat with some period to give a rational number: rational solution is like a periodic solution of a dynamical system whereas non-rational solution is like chaotic orbit having no periodicity? In the tilt one can also have solutions in which some iterated root of p appears: these cannot belong to rationals but to their extension by an iterated root of p.

The results of Scholze could be highly relevant for the number theoretic view about TGD in which octonionic generalization of arithematic geometry plays a key role since the points of space-time surface with coordinates in extension of rationals defining adele and also what I call cognitive representations determining the entire space-time surface if M8-H duality holds true (space-time surfaces would be analogous to roots of polynomials). Unfortunately, my technical skills in mathematics needed are hopelessly limited.

TGD inspires the question is whether the finite cutoffs of Kb - almost perfectoids - could be particularly interesting physically. At the limit of infinite dimension one would get an ideal situation not realizable physically if one believes that finite-dimensionality is basic property of extensions of p-adic numbers appearing in number theoretical quantum physics (they would related to cognitive representations in TGD). Adelic physics involves all extensions of rationals and the extensions of p-adic number fields induced by them and thus also extensions of type Kb. I have made some naive speculations about why just these extensions might be physically of a special signiticance.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry?. See also the article Could the precursors of perfectoids emerge in TGD?

About heff/h=n as the number of sheets of Galois covering

The following considerations were motivated by the observation of a very stupid mistake that I have made repeatedly in some articles about TGD. Planck constant heff/h=n corresponds naturally to the number of sheets of the covering space defined by the space-time surface.

I have however claimed that one has n=ord(G), where ord(G) is the order of the Galois group G associated with the extension of rationals assignable to the sector of "world of classical worlds" (WCW) and the dynamics of the space-time surface (what this means will be considered below).

This claim of course cannot be true since the generic point of extension G has some subgroup H leaving it invariant and one has n= ord(G)/ord(H) dividing ord(G). Equality holds true only for Abelian extensions with cyclic G. For singular points isotropy group is H1⊃ H so that ord(H1)/ord(H) sheets of the covering touch each other. I do not know how I have ended up to a conclusion, which is so obviously wrong, and how I have managed for so long to not notice my blunder.

This observation forced me to consider more precisely what the idea about Galois group acting as a number theoretic symmetry group really means at space-time level and it turned out that M8-H correspondence gives a precise meaning for this idea.

Consider first the action of Galois group (see this and this).

1. The action of Galois group leaves invariant the number theoretic norm characterizing the extension. The generic orbit of Galois group can be regarded as a discrete coset space G/H, H⊂ G. The action of Galois group is transitive for irreducible polynomials so that any two points at the orbit are G-related. For the singular points the isotropy group is larger than for generic points and the orbit is G/H1, H1⊃ H so that the number of points of the orbit divides n. Since rationals remain invariant under G, the orbit of any rational point contains only single point. The orbit of a point in the complement of rationals under G is analogous to an orbit of a point of sphere under discrete subgroup of SO(3).

n=ord(G)/ord(H) divides the order ord(G) of Galois group G. The largest possible Galois group for n-D algebraic extension is permutation group Sn. A theorem of Frobenius states that this can be achieved for n=p, p prime if there is only single pair of complex roots (see this). Prime-dimensional extensions with heff/h=p would have maximal number theoretical symmetries and could be very special physically: p-adic physics again!

2. The action of G on a point of space-time surface with imbedding space coordinates in n-D extension of rationals gives rise to an orbit containing n points except when the isotropy group leaving the point is larger than for a generic point. One therefore obtains singular covering with the sheets of the covering touching each other at singular points. Rational points are maximally singular points at which all sheets of the covering touch each other.
3. At QFT limit of TGD the n dynamically identical sheets of covering are effectively replaced with single one and this effectively replaces h with heff=n× h in the exponent of action (Planck constant is still the familiar h at the fundamental level). n is naturally the dimension of the extension and thus satisfies n≤ ord(G). n= ord(G) is satisfied only if G is cyclic group.
The challenge is to define what space-time surface as Galois covering does really mean!
1. The surface considered can be partonic 2-surface, string world sheet, space-like 3-surface at the boundary of CD, light-like orbit of partonic 2-surface, or space-time surface. What one actually has is only the data given by these discrete points having imbedding space coordinates in a given extension of rationals. One considers an extension of rationals determined by irreducible polynomial P but in p-adic context also roots of P determine finite-D extensions since ep is ordinary p-adic number.
2. Somehow this data should give rise to possibly unique continuous surface. At the level of H=M4× CP2 this is impossible unless the dynamics satisfies besides the action principle also a huge number of additional conditions reducing the initial value data ans/or boundary data to a condition that the surface contains a discrete set of algebraic points.

This condition is horribly strong, much more stringent than holography and even strong holography (SH) implied by the general coordinate invariance (GCI) in TGD framework. However, preferred extremal property at level of M4× CP2 following basically from GCI in TGD context might be equivalent with the reduction of boundary data to discrete data if M8-H correspondence is accepted. These data would be analogous to discrete data characterizing computer program so that an analogy of computationalism would emerge (see this).

One can argue that somehow the action of discrete Galois group must have a lift to a continuous flow.
1. The linear superposition of the extension in the field of rationals does not extend uniquely to a linear superposition in the field reals since the expression of real number as sum of units of extension with real coefficients is highly non-unique. Therefore the naive extension of the extension of Galois group to all points of space-time surface fails.
2. The old idea already due to Riemann is that Galois group is represented as the first homotopy group of the space. The space with homotopy group π1 has coverings for which points remain invariant under subgroup H of the homotopy group. For the universal covering the number of sheets equals to the order of π1. For the other coverings there is subgroup H⊂ π1 leaving the points invariant. For instance, for homotopy group π1(S1)= Z the subgroup is nZ and one has Z/nZ=Zp as the group of n-sheeted covering. For physical reasons its seems reasonable to restrict to finite-D Galois extensions and thus to finite homotopy groups.

π1-G correspondence would allow to lift the action of Galois group to a flow determined only up to homotopy so that this condition is far from being sufficient.

3. A stronger condition would be that π1 and therefore also G can be realized as a discrete subgroup of the isometry group of H=M4× CP2 or of M8 (M8-H correspondence) and can be lifted to continuous flow. Also this condition looks too weak to realize the required miracle. This lift is however strongly suggested by Langlands correspondence (see this).
The physically natural condition is that the preferred extremal property fixes the surface or at least space-time surface from a very small amount of data. The discrete set of algebraic points in given extension should serve as an analog of boundary data or initial value data.
1. M8-H correspondence could indeed realize this idea. At the level of M8 space-time surfaces would be algebraic varieties whereas at the level of H they would be preferred extremals of an action principle which is sum of Kähler action and minimal surface term.

They would thus satisfy partial differential equations implied by the variational principle and infinite number of gauge conditions stating that classical Noether charges vanish for a subgroup of symplectic group of δ M4+/-× CP2. For twistor lift the condition that the induced twistor structure for the 6-D surface represented as a surface in the 12-D Cartesian product of twistor spaces of M4 and CP2 reduces to twistor space of the space-time surface and is thus S2 bundle over 4-D space-time surface.

The direct map M8→ H is possible in the associative space-time regions of X4⊂ M8 with quaternionic tangent or normal space. These regions correspond to external particles arriving into causal diamond (CD). As surfaces in H they are minimal surfaces and also extremals of Kähler action and do not depend at all on coupling parameters (universality of quantum criticality realized as associativity). In non-associative regions identified as interaction regions inside CDs the dynamics depends on coupling parameters and the direct map M8→ CP2 is not possible but preferred extremal property would fix the image in the interior of CD from the boundary data at the boundaries of CD.

2. At the level of M8 the situation is very simple since space-time surfaces would correspond to zero loci for RE(P) or IM(P) (RE and IM are defined in quaternionic sense) of an octonionic polynomial P obtained from a real polynomial with coefficients having values in the field of rationals or in an extension of rationals. The extension of rationals would correspond to the extension defined by the roots of the polynomial P.

If the coefficients are not rational but belong to an extension of rationals with Galois group G0, the Galois group of the extension defined by the polynomial has G0 as normal subgroup and one can argue that the relative Galois group Grel=G/G0 takes the role of Galois group.

It seems that M8-H correspondence could allow to realize the lift of discrete data to obtain continuous space-time surfaces. The data fixing the real polynomial P and therefore also its octonionic variant are indeed discrete and correspond essentially to the roots of P.

3. One of the elegant features of this picture is that the at the level of M8 there are highly unique linear coordinates of M8 consistent with the octonionic structure so that the notion of a M8 point belonging to extension of rationals does not lead to conflict with GCI. Linear coordinate changes of M8 coordinates not respecting the property of being a number in extension of rationals would define moduli space so that GCI would be achieved.
Does this option imply the lift of G to π1 or to even a discrete subgroup of isometries is not clear. Galois group should have a representation as a discrete subgroup of isometry group in order to realize the latter condition and Langlands correspondence supports this as already noticed. Note that only a rather restricted set of Galois groups can be lifted to subgroups of SU(2) appearing in McKay correspondence and hierarchy of inclusions of hyper-finite factors of type II1 labelled by these subgroups forming so called ADE hierarchy in 1-1 correspondence with ADE type Lie groups (see this). One must notice that there are additional complexities due to the possibility of quaternionic structure which bring in the Galois group SO(3) of quaternions.

See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article with the same title.