What's new inPhysics as a Generalized Number TheoryNote: 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 P_{m+n} defining the new extension is a polynomial P_{m} having as argument the old polynomial P_{n}(x): P_{m+n}(x)=P_{m}(P_{n}(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 II_{1} (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 M^{8}-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 G_{eff} (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 h_{eff}/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 E^{H} 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 E^{H} invariant in element-wise manner as a sub-field of E (see this). Any subgroup H⊂ Gal(E/K)) defines an intermediate extension E^{H} and subgroup H_{1}⊂ H_{2}⊂... define a hierarchy of extensions E^{H1}>E^{H2}>E^{H3}... 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 E^{H}. This is a highly non-trivial piece of information. The dimension of E factorizes to a product ∏_{i} |H_{i}| of dimensions for a sequence of groups H_{i}. Could a sequence of DNA letters/codons somehow define a sequence of extensions? Could one assign to a given letter/codon a definite group H_{i} 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 E^{H} in a further extension to E. The degree of E^{H} increases by a factor, which is dimension of E/E^{H} and also the dimension of H. Is there a standard manner to construct irreducible extensions of this kind?
See the chapter Does M^{8}-H duality reduce classical TGD to octonionic algebraic geometry?. See also the article Is the hierarchy of Planck constants behind the reported variation of Newton's constant?. |
A second attempt to understand the notions of perfectoid and its tiltQuanta 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.
The tilt of the perfectoid What Scholze introduces besides perfectoids K also what he calls tilt of the perfectoid: K_{b}. K_{b} is something between p-adic number fields and reals and leads to theorems giving totally new insights to arithemetic geometry
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 M^{8}-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 K_{b} - 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 K_{b}. I have made some naive speculations about why just these extensions might be physically of a special signiticance. See the chapter Does M^{8}-H duality reduce classical TGD to octonionic algebraic geometry?. See also the article Could the precursors of perfectoids emerge in TGD? |
About h_{eff}/h=n as the number of sheets of Galois coveringThe following considerations were motivated by the observation of a very stupid mistake that I have made repeatedly in some articles about TGD. Planck constant h_{eff}/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 H_{1}⊃ H so that ord(H_{1})/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 M^{8}-H correspondence gives a precise meaning for this idea. Consider first the action of Galois group (see this and this).
See the chapter Does M^{8}-H duality reduce classical TGD to octonionic algebraic geometry? or the article with the same title. |