Tuesday, September 19, 2017

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.

Articles and other material related to TGD.

Saturday, September 16, 2017

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), εi=+/- 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=M1,3 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=M1,3 to Mi,j, say M2,2, the intersection would be M1,2. 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.

Articles and other material related to TGD.

Thursday, September 14, 2017

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.

Articles and other material related to TGD.

Wednesday, September 13, 2017

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.

Articles and other material related to TGD.

Friday, September 08, 2017

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.