Super-number fields: does physics emerge from the notion of number?

A proposal that all physics emerges from the notion of number field is made. The first guess for the number field in question would be complexified octonions for which inverse exists except at complexified light-cone boundary: this has interpretation in terms of propagation of signals with light-velocity 8-D sense. The emergence of fermions however requires super-octonions as super variant of number field. Rather surprisingly, it turns out that super-number theory makes perfect sense. One can define the inverse of super-number and also the notion of primeness makes sense and construct explicitly the super-primes associated with ordinary primes. The prediction of new number piece of theory can be argued to be a strong support for the integrity of TGD.

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

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Is the quantum leakage between different signatures of the real sectors of the complexified M8 possible?

Complexified octonions have led to a dramatic progress in the understanding of TGD. One cannot however avoid
a radical question about fundamentals.

1. The basic structure at M⁸ side consists of complexified octonions. The metric tensor for the complexified inner product for complexified octonions (no complex conjugation with respect to i for the vectors in the inner product) can be taken to have any signature (ε₁,...,ε₈), ε_i=+/- 1. By allowing some coordinates to be real and some coordinates imaginary one obtains effectively any signature from say purely Euclidian signature. What matters is that the restriction of complexified metric to the allowed sub-space is real. These sub-spaces are linear Lagrangian manifolds for Kähler form representing the commuting imaginary unit i. There is analogy with wave mechanics. Why M⁸ -actually M⁴ - should be so special real section? Why not some other signature?
   
   
2. The first observation is that the CP₂ point labelling tangent space is independent of the signature so that the problem reduces to the question why M⁴ rather than some other signature (ε₁,..,ε₄). The intersection of real subspaces with different signatures and same origin (t,r)=0 is the common sub-space with the same signature. For instance, for (1,-1,-1,-1) and (-1,-1,-1,-1) this subspace is 3-D t=0 plane sharing with CD the lower tips of CD. For (-1,1,1,1) and (1,1,1,1) the situation is same. For (1,-1,-1,-1) and (1,1,-1,-1) z=0 holds in the intersection having as common with the lower boundary of CD the boundary of 3-D light-cone. One obtains in a similar manner boundaries of 2-D and 1-D light-cones as intersections.
   
   
3. What about CDs in various signatures? For a fully Euclidian signature the counterparts for the interiors of CDs reduce to 4-D intervals t∈ [0,T] and their exteriors and thus the space-time varieties representing incoming particles reduce to pairs of points (t,r)=(0,0) and (t,r)= (T,0): it does not make sense to speak about external particles. For other signatures the external particles correspond to 4-D surfaces and dynamics makes sense. The CDs associated with the real sectors intersect at boundaries of lower dimensional CDs: these lower-dimensional boundaries are analogous to subspaces of Big Bang (BB) and Big Crunch (BC).
   
   
4. I have not found any good argument for selecting M⁴=M^1,3 as a unique signature. Should one allow also other real sections? Could the quantum numbers be transferred between sectors of different signature at BB and BC? The counterpart of Lorentz group acting as a symmetry group depends on signature and would change in the transfer. Conservation laws should be satisfied in this kind of process if it is possible. For instance, in the leakage from M⁴=M^1,3 to Mi,j, say M^2,2, the intersection would be M^1,2. Momentum components for which signature changes, should vanish if this is true. Angular momentum quantization axis normal to the plane is defined by two axis with the same signature. If the signatures of these axes are preserved, angular momentum projection in this direction should be conserved. The amplitude for the transfer would involve integral over either boundary component of the lower-dimensional CD.
   

   Final question: Could the leakage between signatures be detected as disappearance of matter for CDs in elementary particle scales or lab scales?
   
   

See the articleDoes M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

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

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What the properties of octonionic product can tell about fundamental physics?

In developing the view about M⁸-H duality reducing physics to algebraic geometry for complexified octonions at the level of M⁸, I became aware of trivial looking but amazingly profound observation about basic arithmetics of of complex, quaternion, and octonion number fields.

1. Imaginary part for the product z₁z₂ of complex numbers is
   

   Im(z₁z₂)= Im(z₁)Re(z₂)+Re(z₁)Im(z₂)
   

   and linear in Im(z₁) and Im(z₂).
   
   

2. Real part
   
   Re(z₁z₂)= Re(z₁)Re(z₂)-Im(z₁)Im(z₂).
   

   is not linear in real parts:
   
   

This generalizes to the product of octonions with Re and Im replaced by RE and IM in the decomposition to two quaternions: o= RE(o)+J IM(o), J is octonion imaginary unit not belonging to quaternionic subspace.

This extremely simple observation turns out to contain amazingly deep physics.

1. Space-time surfaces can be identified as IM(P)= loci or RE(P)=0 loci. When one takes product of two polynomials P₁P₂ the IM(P₁P₂)=0 locus as space-time surface is just the union of IM(p₁)=0 locus and IM(P₂) locus. No interaction: free particles as space-time surfaces! This picture generalizes also to rational functions R=P₁/P₂ and an their zero and infty loci.
   
   
2. For RE(P₁P₂)=0 the situation changes. One does not obtain union of RE(P₁)=0 and RE(P₂) space-time surfaces. There is interaction and most naturally this interaction generates wormhole contacts connecting the space-time surfaces (sheet) carrying fermions at the throats of the wormhole contact!
   
   

The entire elementary particle physics emerges from these two simple number theoretic properties for the product of numbers!

For details see the articleDo Riemann-Roch theorem and Atyiah-Singer index theorem have applications in TGD?.

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

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Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications to TGD?

Riemann-Roch theorem (RR) is a central piece of algebraic geometry. Atyiah-Singer index theorem is one of its generalizations relating the solution spectrum of partial differential equations and topological data. For instance, characteristic classes classifying bundles associated with Yang-Mills theories have applications in gauge theories and string models.

The advent of octonionic approach to the dynamics of space-time surfaces inspired by M⁸-H duality (see this and this) gives hopes that dynamics at the level of complexified octonionic M⁸ could reduce to algebraic equations plus criticality conditions guaranteeing associativity for space-time surfaces representing external particles, in interaction region commutativity and associativity would be broken. The complexification of octonionic M⁸ replacing norm in flat space metric with its complexification would unify various signatures for flat space metric and allow to overcome the problems due to Minkowskian signature. Wick rotation would not be a mere calculational trick.

For these reasons time might be ripe for applications of possibly existing generalization of RR to TGD framework. In the following I summarize my admittedly unprofessional understanding of RR discussing the generalization of RR for complex algebraic surfaces having real dimension 4: this is obviously interesting from TGD point of view.

I will also consider the possible interpretation of RR in TGD framework. One interesting idea is possible identification of light-like 3-surfaces and curves (string boundaries) as generalized poles and zeros with topological (but not metric) dimension one unit higher than in Euclidian signature.

Atyiah-Singer index theorem (AS) is one of the generalizations of RR and has shown its power in gauge field theories and string models as a method to deduce the dimensions of various moduli spaces for the solutions of field equations. A natural question is whether AS could be useful in TGD and whether the predictions of AS at H side could be consistent with M⁸-H duality suggesting very simple counting for the numbers of solutions at M⁸ side as coefficient combinations of polynomials in given extension of rationals satisfying criticality conditions. One can also ask whether the hierarchy of degrees n for octonion polynomials could correspond to the fractal hierarchy of generalized conformal sub-algebras with conformal weights coming as n-multiples for those for the entire algebras.

For details see the article Do Riemann-Roch theorem and Atyiah-Singer index theorem have applications to TGD? or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? of "Physics as Generalized Number Theory".

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

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About enumerative algebraic geometry in TGD framework

I wrote a brief summary about basic ideas of enumerative algebraic geometry and proposals for applications to TGD. Here is the abstract of the article.

String models and M-theory involve both algebraic and symplectic enumerative geometry. Also in adelic TGD enumerative algebraic geometry emerges. This article gives a brief summary about the basic ideas involved and suggests some applications to TGD.

1. One might want to know the number of points of sub-variety belonging to the number field defining the coefficients of the polynomials. This problem is very relevant in M⁸ formulation of TGD, where these points are carriers of sparticles. In TGD based vision about cognition they define cognitive representations as points of space-time surface, whose M⁸ coordinates can be thought of as belonging to both real number field and to extensions of various p-adic number fields induced by the extension of rationals. If these cognitive representations define the vertices of analogs of twistor Grassmann diagrams in which sparticle lines meet, one would have number theoretically universal adelic formulation of scattering amplitudes and a deep connection between fundamental physics and cognition.
   
   
2. Second kind of problem involves a set algebraic surfaces represented as zero loci for polynomials - lines and circles in the simplest situations. One must find the number of algebraic surfaces intersecting or touching the surfaces in this set. Here the notion of incidence is central. Point can be incident on line or two lines (being their intersection), line on plane, etc.. This leads to the notion of Grassmannians and flag-manifolds. In twistor Grassmannian approach algebraic geometry of Grassmannians play key role. Also in twistor Grassmannian approach to TGD algebraic geometry of Grassmannians play a key role and some aspects of this approach are discussed.
   
   
3. In string models the notion of brane leads to what might be called quantum variant of algebraic geometry in which the usual rules of algebraic geometry do not apply as such. Gromow-Witten invariants provide an example of quantum invariants allowing sharper classification of algebraic and symplectic geometries. In TGD framework M⁸-H duality suggests that the construction of scattering amplitudes at level of M⁸ reduces to a super-space analog of algebraic geometry for complexified octonions. Candidates for TGD analogs of branes emerge naturally and G-W invariants could have applications also in TGD.
   
   

In the sequel I will summarize the understanding of novice about enumerative algebraic geometry and discuss possible TGD applications. This material can be also found in earlier articles but it seemed appropriate to collect the material about enumerative algebraic geometry to a separate article.

See the article About enumerative algebraic geometry in TGD framework or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? of "Physics as generalized number theory".

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

Is dark DNA dark also in TGD sense?

I encountered a highly interesting article about "dark DNA" hitherto found in the genome of gerbils and birds, for instance in the genome of of the sand rat living in deserts. The gene called Pdxl related to the production of insulin seems to be missing as also 87 other genes surrounding it! What makes this so strange that the animal cannot survive without these genes! Products that the instructions from the missing genes would create are however detected!

According to the ordinary genetic, these genes cannot be missing but should be hidden, hence the attribute "dark" in analogy with dark matter. The dark genes contain A lot of G and C molecules and this kind of genes are not easy to detect: this might explain why the genes remain undetected.

A further interesting observation is that one part of the sand rat genome has many more mutations than found in other rodent genomes and is also GC rich. Could the mutated genes do the job of the original genes? Missing DNA are found in birds too. For instance, the gene for leptin - a hormone regulating energy balance - seems to be missing.

The finding is extremely interesting from TGD view point, where dark DNA has very concrete meaning. Dark matter at magnetic flux tubes is what makes matter living in TGD Universe. Dark variants of particles have non-standard value $h_{eff}=n\times h$ of Planck constant making possible macroscopic quantum coherence among other things. Dark matter would serve as template for ordinary matter in living systems and biochemistry could be kind of shadow of the dynamics of dark matter. What I call dark DNA would correspond to dark analogs of atomic nuclei realized as dark proton sequences with entangled proton triplet representing DNA codon. The model predicts correctly the numbers of DNA codons coding for given amino-acid in the case of vertebrate genetic code and therefore I am forced to take it very seriously (see this and this).

The chemical DNA strands would be attached to parallel dark DNA strands and the chemical representation would not be always perfect: this could explain variations of DNA. This picture inspires also the proposal that evolution is not a passive process occurring via random mutations with survivors selected by the evolutionary pressures. Rather, living system would have R&D lab as one particular department. Various variants of DNA would be tested by transcribing dark DNA to ordinary mRNA in turn translated to amino-acids to see whether the outcome survives. This experimentation might be possible in much shorter time scale than that based on random mutations. Also immune system, which is rapidly changing, could involve this kind of R&D lab.

Also dark mRNA and amino-acids could be present but dark DNA is the fundamental information carrying unit and it would be natural to transcribe it to ordinary mRNA. Of course, also dark mRNA could be be produced and translated to amino-acids and even dark amino-acids could be transformed to ordinary ones. This would however require additional machinery.

What is remarkable is that the missing DNA is indeed associated with DNA sequences with exceptionally high mutation rate. Maybe R&D lab is there! If so, the dark DNA would be dark also in TGD sense! Why GC richness should relate to this, is an interesting question.

For background see the chapter Homeopathy in many-sheeted space-time.

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

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Super-octonions, super-twistors, and twistorial construction of scattering amplitudes

For last month or so I have been writing an article about reduction of quantum classical TGD to octonionic algebraic geometry. During writing process "octonionic" has been replaced with "super-octonionic" and it has turned out that the formalism of twistor Grassmannian approach could generalize to TGD. The twistorial scattering diagrams have interpretation as cognitive representations with vertices assignable to points of space-time surface in appropriate extension of rationals.

In accordance with the vision that classical theory is exact part of quantum theory, the core part of the calculations of
scattering amplitudes would reduce to the determination of zero loci for real and imaginary parts of octonionic polynomials and finding of the points of space-time surface with M⁸ coordinates in extension of rationals. At fundamental level the physics would reduce to number theory.

A proposal for the description of interactions is discussed in the article. Here is a brief summary.

1. The surprise that RE(P)=0 and IM(P)=0 conditions have as singular solutions light-cone interior and its complement and 6-spheres S⁶(t_n) with radii t_n given by the roots of the real P(t), whose octonionic extension defines the space-time variety X⁴. The intersections X²= X⁴∩ S⁶(t_n) are tentatively identified as partonic 2-varieties defining topological interaction vertices. S⁶ and therefore also X² are doubly critical, S⁶ is also singular surface.
   

   The idea about the reduction of zero energy states to discrete cognitive representations suggests that interaction vertices at partonic varieties X² are associated with the discrete set of intersection points of the sparticle lines at light-like orbits of partonic 2-surfaces belonging to extension of rationals.
   
   

2. CDs and therefore also ZEO emerge naturally. For CDs with different origins the products of polynomials fail to commute and associate unless the CDs have tips along real (time) axis. The first option is that all CDs under observation satisfy this condition. Second option allows general CDs.
   

   The proposal is that the product ∏ P_i of polynomials associated with CDs with tips along real axis
   the condition IM(∏ P_i)=0 reduces to IM(P_i)=0 and criticality conditions guaranteeing associativity and provides a description of the external particles. Inside these CDs RE(∏ P_i)=0 does not reduce to RE(∏ P_i)=0, which automatically gives rise to geometric interactions. For general CDs the situation is more complex.
   
   

3. The possibility of super-octonionic geometry raises the hope that the twistorial construction of scattering amplitudes in N=4 SUSY generalizes to TGD in rather straightforward manner to a purely geometric construction. Functional integral over WCW would reduce to summations over polynomials with coefficients in extension of rationals and criticality conditions on the coefficients could make the summation well-defined by bringing in finite measurement resolution.
   

   If scattering diagrams are associated with discrete cognitive representations, one obtains a generalization of twistor formalism involving polygons. Super-octonions as counterparts of super gauge potentials are well-defined if octonionic 8-momenta are quaternionic. Indeed, Grassmannians have quaternionic counterparts but not octonionic ones. There are good hopes that the twistor Grassmann approach to N=4 SUSY generalizes. The core part in the calculation of the scattering diagram would reduce to the construction of octonionic 4-varieties and identifying the points belonging to the appropriate extension of rationals.
   
   

For details see the article Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: part II.

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

Articles and other material related to TGD.