Thursday, August 17, 2017

Geometrization of fermions using super version of the octonionic algebraic geometry

Could the octonionic level provide an elegant description of fermions in terms of super variant of octonionic algebraic geometry? Could one even construct scattering amplitudes at the level of M8 using the variant of the twistor approach discussed earlier.

The idea about super-geometry is of course very different from the idea that fermionic statistics is realized in terms of the spinor structure of "world of classical worlds" (WCW) but M8-H duality could however map these ideas and also number theoretic and geometric vision to each other. The angel of geometry and the devil of algebra could be dual to each other.

1. Octonionic superspace

Consider now what super version of the octonionic super-space might look like.

  1. What makes octonions so nice is the octonionic triality. One has three 8-D representations: vector representation 8V, spinor representation 8s and its conjugate 8*s. The tensor products of two representations gives the third representation in the triplet. This is the completely unique feature of dimension 8 and makes octonionic physics so fascinating an option. The octonionic triality is central also in super-string models but in a different manner since of starts from 10-D situation and ends up with effectively 8-D situation for physical states.

  2. One can define super octonion as os= o +θ1 + θ2. Here o is bosonic octonionic coordinate. θi= θki Ek, where Ek are octonionic units, is Grassmann valued octonion in 8s satisfying the usual anti-commutations and θ2 transforms as 8*s. (I have already earlier considered as natural candidates for spinors in octonionic M8).

    The first interpretation is that θ1 and θ2 correspond to objects with opposite fermion numbers. If this is not the case, one could perhaps define the conjugate of super-coordinate as o*s=o* +θ*1 + θ*2. This looks however ugly.

  3. What could be the physical interpretation? One should obtain particles and antiparticles naturally as also separately conserved baryon and lepton numbers (I have also considered the identification of hadrons in terms of anyonic bound states of leptons with fractional charges).

    Quarks and leptons have different coupling to the induced Kähler form at the level of H. It seems impossible to understand this at the level of M8, where the dynamics is purely algebraic and contains no gauge couplings.

    The difference between quarks and leptons is that they allow color partial waves with triality t=+/- 1 and triality t=0. Color partial waves correspond to wave functions in the moduli space CP2 for M40 ⊃ M20. Could the distinction between quarks and leptons emerge at the level of this moduli space rather than at the fundamental octonionic level? There would be no need for gauge couplings to distinguish between quarks and leptons at the level of M8. All couplings would follow from the criticality conditions guaranteeing 4-D associativity for external particles (on mass shell states would be critical).

    If so, one would have only the super octonions os= θ1+ θ2 =θ*1 and θ1 and θ2 =θ*1 would correspond to fermions and antifermions with no differentiation to quarks or leptons. Fermion number conservation would be coded by the Grassmann algebra.

    One can imagine also other options but they have their problems. Therefore this option will be considered in the sequel.

2. Super version of octonionic algebraic geometry

Instead of super-fields one would have a super variant of octonionic algebraic geometry.

  1. Super polynomials make still sense and reduce to a sum of octonionic polynomials Pklθ1kθ2l, where the integers k and l would be tentatively identified as fermion numbers.

    One would clearly have an upper bound for k and l for given CD. Therefore these many-fermion states must correspond to fundamental particles rather than many-fermion Fock states. One would obtain bosons with non-vanishing fermion numbers if the proposed identification is correct. Octonionic algebraic geometry for single CD would describe only fundamental particles or states with bounded fermion numbers. Fundamental particles would be indeed fundamental also geometrically.

  2. I have already earlier considered the question whether the partonic 2-surfaces can carry also many-fermion states or not, and adopted the working hypothesis that fermion numbers is not larger than 1 for given wormhole throat, possibly for purely dynamical reasons. This picture however looks too limited. The many fermion states might not however propagate as ordinary particles (the proposal has been that their propagator pole corresponds to higher power of p2).

  3. The result looks somewhat disappointing at first. It would seem that the states with high fermion numbers must be described in terms of Cartesian products just like in condensed matter physics with interactions described by the proposed braney mechanism in which intersection of space-time surfaces with S6 giving analogs of partonic 2-surfaces are involved.

  4. One can also now define space-time varieties as zero loci via the conditions RE(Ps)(os)=0 or IM(Ps)(os)=0. One obtains a collection of 4-surfaces as zero loci of Pkl. One would have a correlation with between fermion content and algebraic geometry of the space-time surface unlike in the ordinary super-space approach, where the notion of the geometry remains rather formal and there is no natural coupling between fermionic content and classical geometry. At the level of H this comes from quantum classical correspondence (QCC) stating that the classical Noether charges are equal to eigenvalues of fermionic Noether charges.

3. Questions about quantum numbers

There are several questions about quantum numbers.

  1. Could octonionic super geometry code for quantum numbers of the particle states? It seems that super-octonionic polynomials multiplied by octonionic multi-spinors inside single CD can code only for the electroweak quantum numbers of fundamental particles besides their fermion and anti-fermion numbers.

    As already suggested, color corresponds to partial waves in CP2 serving as moduli space for M40⊃ M20 and quarks and leptons have different trialities. Also four-momentum and angular momentum are naturally assigned with the translational degrees for the tip of CD assignable with the fundamental particle.

    Remark: There is a funny accident that deserves to be noticed. Octonionic spinor decomposes to 1⊕ 1⊕ 3 ⊕3* under SU(3)⊂ G2. Could it be that 1⊕ 1 corresponding to real unit and preferred imaginary unit assignable to M20 correspond to color wave functions in CP2 transforming like leptons and 3+3* corresponds to wave functions transforming like quarks and antiquarks? Unfortunately, one cannot understand electroweak quantum numbers in this framework. There would be uncertainty principle allowing to measure either of these quantum numbers but not both.

  2. What about twistors in this framework? M4× CP1 as twistor space with CP1 coding for the choice of M20⊂ M40 allows projection to the usual twistor space CP3. Twistor wave functions describing spin elegantly would correspond to wave functions in the twistor space and one expects that the notion of super-twistor is well-defined also now. The 6-D twistor space SU(3)/U(2)× U(1) of CP2 would code besides the choice of M40⊃ M20 also quantization axis for color hypercharge and isospin.

  3. What about the sphere S6 serving as the moduli space for the choices of M8+? Should one have wave functions in S6 or can one restrict the consideration to single M8+? As found, one obtains S6 also as the zero locus of Im(P)=0 for some radii identifiable as values tn of time coordinates given as roots of P(t). This would be crucial for the braney description of interactions between space-time surfaces associated with different CDs.

4. Could scattering amplitudes be computed at the level of M8?

It would be extremely nice if the scattering amplitudes could be computed at the octonionic level by using a generalization of twistor approach in ZEO finding a nice justification at the level of M8. Something rather similar to N=4 twistor Grassmann approach suggests itself.

  1. In ZEO picture one would consider the situation in which the passive boundary of CD and members of state pairs at it appearing in zero energy state remain fixed during the sequence of state function reductions inducing stepwise drift of the active boundary of CD and change of states at it by unitary U-matrix at each step following by a localization in the moduli space for the positions of the active boundary.

  2. At the active boundary one would obtain quantum superposition of states corresponding to different octonionic geometries for the outgoing particles. Instead of functional integral one would have sum over discrete points of WCW. WCW coordinates would be the coefficients of polynomial P in the extension of rationals. This would give undefined result without additional constraints since rationals are a dense set of reals.

    Criticality however serves as a constraint on the coefficients of the polynomials and is expected to realize finite measurement resolution, and hopefully give a well defined finite result in the summation. Criticality for the outgoing states would realize purely number theoretically the cutoff due to finite measurement resolution and would be absolutely essential for the finiteness and well-definedness of the theory.

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

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

Articles and other material related to TGD.

Wednesday, August 16, 2017

Emergence of Zero Energy Ontology and Causal Diamonds from octonionic algebraic surface dynamics

Octonionic polynomials provide a promising approach to the understanding of zero energy ontology (ZEO) and causal diamonds (CDs) defined as intersections of future and past directed lightcones: CD makes sense both in octonionic (8-D) and quaternionic (4-D) context. Light-like boundary of CD as also light-cone emerge naturally as zeros of octonionic polynomials. This does not yet give CDs and ZEO: one should have intersection of future and past directed light-cones. The intuitive picture is that one has a hierarchy of CDs and that also the space-time surfaces inside different CDs an interact. It turns out that CDs and thus also ZEO emerge naturally both at the level of M8 and M4 .

Remark: In the sequel RE(o) and IM(o) refer to real and imaginary parts of octonions in quaternionic sense: one has o= RE(o)+IM(o)I4, where RE(o) and IM(o) are quaternions.

1. General view about solutions to RE(P)=0 and IM(P)=0 conditions

The first challenge is to understand at general level the nature of RE(P)=0 and IM(P)=0 conditions expected to give 4-D space-time surfaces as zero loci. Appendix shows explicitly for P(o)=o2 that Minkowski signature gives rise to unexpected phenomena. In the following these phenomena are shown to be completely general but not quite what one obtains for P(o)=o2 having double root at origin.

  1. Consider first the octonionic polynomials P(o) satisfying P(0)=0 restricted to the light-like boundary δ M8+ assignable to 8-D CD, where the octonionic norm of o vanishes.

    1. P(o) reduces along each light-ray of δ M8+ to the same real valued polynomial P(t) of a real variable t apart from a multiplicative unit E= (1+in)/2 satisfying E2=E. Here n is purely octonion-imaginary unit vector defining the direction of the light-ray.

      IM(P)=0 corresponds to quaterniocity. If the E4 (M8= M4× E4) projection is vanishing, there is no additional condition. 4-D light-cones M4+/- are obtained as solutions of IM(P)=0. Note that M4+/- can correspond to any quaternionic subspace.

      If the light-like ray has a non-vanishing projection to E4, one must have P(t)=0. The solutions form a collection of 6-spheres labelled by the roots tn of P(t)=0. 6-spheres are not associative.

    2. RE(PE)=0 corresponding to co-quaternionicity leads to P(t)=0 always and gives a collection of 6-spheres.

  2. Suppose now that P(t) is shifted to P1(t)=P(t)-c, c a real number. Also now M4+/- is obtained as solutions to IM(P)=0. For RE(P)=0 one obtains two conditions P(t)=0 and P(t-c)=0. The common roots define a subset of 6-spheres which for special values of c is not empty.

The above discussion was limited to δ M8+ and light-likeness of its points played a central role. What about the interior of 8-D CD?
  1. The natural expectation is that in the interior of CD one obtains a 4-D variety X4. For IM(P)=0 the outcome would be union of X4 with M4+ and the set of 6-spheres for IM(P)=0. 4-D variety would intersect M4+ in a discrete set of points and the 6-spheres along 2-D varieties X2. The higher the degree of P, the larger the number of 6-spheres and these 2-varieties.

  2. For RE(P)=0 X4 would intersect the union of 6-spheres along 2-D varieties. What comes in mind that these 2-varieties correspond in H to partonic 2-surfaces defining light-like 3-surfaces at which the induced metric is degenerate.

  3. One can consider also the situation in the complement of 8-D CD which corresponds to the complement of 4-D CD. One expects that RE(P)=0 condition is replaced with IM(P)=0 condition in the complement and RE(P)= IM(P)=0 holds true at the boundary of 4-D CD.

6-spheres and 4-D empty light-cones are special solutions of the conditions and clearly analogs of branes. Should one make the (reluctant-to-me) conclusion that they might be relevant for TGD at the level of M8.
  1. Could M4+ (or CDs as 4-D objects) and 6-spheres integrate the space-time varieties inside different 4-D CDs to single connected structure with space-time varieties glued to the 6-spheres along 2-surfaces X2 perhaps identifiable as pre-images of partonic 2-surfaces and maybe string world sheets? Could the interactions between space-time varieties X4i assignable with different CDs be describable by regarding 6-spheres as bridges between X4i having only a discrete set of common points. Could one say that X2i interact via the 6-sphere somehow. Note however that 6-spheres are not dynamical.

  2. One can also have Poincare transforms of 8-D CDs. Could the description of their interactions involve 4-D intersections of corresponding 6-spheres?

  3. 6-spheres in IM(P)=0 case do not have image under M8-H correspondence. This does not seem to be possible for RE(P)=0 either: it is not possible to map the 2-D normal space to a unique CP2 point since there is 2-D continuum of quaternionic sub-spaces containing it.

2. Some general observations about CDs

CD defines the basic geometric object in ZEO. It is good to list some basic features of CDS, which appear as both 4-D and 8-D variants.

  1. There are both 4-D and 8-D CDs defined as intersections of future and past directed light-cones with tips at say origin 0 at real point T at quaternionic or octonionic time axis. CDs can be contained inside each other. CDs form a fractal hierarchy with CDs within CDs: one can add smaller CDs with given CD in all possible manners and repeat the process for the sub-CDs. One can also allow overlapping CDs and one can ask whether CDs define the analog of covering of O so that one would have something analogous to a manifold.

  2. The boundaries of two CDs (both 4-D and 8-D) can intersect along light-like ray. For 4-D CD the image of this ray in H is light-like ray in M4 at boundary of CD. For 8-D CD the image is in general curved line and the question is whether the light-like curves representing fermion orbits at the orbits of partonic 2-surfaces could be images of these lines.


  3. The 3-surfaces at the boundaries of the two 4-D CDs are expected to have a discrete intersection since 4 + 4 conditions must be satisfied (say RE(Pik))=0 for i=1,2, k=1,4. Along line octonionic coordinate reduces effectively to real coordinate since one has E2=E for E=(1+in)/2, n octonionic unit. The origins of CDs are shifted by a light-like vector kE so that the light-like coordinates differ by a shift: t2= t1-k. Therefore one has common zero for real polynomials RE(P1k(t)) and RE(P2k(t-k)).

    Are these intersection points somehow special physically? Could they correspond to the ends of fermionic lines? Could it happen that the intersection is 1-D in some special cases? The example of o2 suggest that this might be the case. Does 1-D intersection of 3-surfaces at boundaries of 8-D CDs make possible interaction between space-time surfaces assignable to separate CDs as suggested by the proposed TGD based twistorial construction of scattering amplitudes?

  4. Both tips of CD define naturally an origin of quaternionic coordinates for D=4 and the origin of octonionic coordinates for D=8. Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be along the real line (time axis) connecting the tips of CD. Only the translations in this specified direction are symmetries preserving the commutativity and associativity of the polynomial algebra.

  5. One expects that also Lorentz boosts of 4-D CDs are relevant. Lorentz boosts leave second boundary of CD invariant and Lorentz transforms the other one. Same applies to 8-D CDs. Lorentz boosts define non-equivalent octonionic and quaternionic structures and it seems that one assume moduli spaces of them.

One can of course ask whether the still somewhat ad hoc notion of CD general enough. Should one generalize it to the analog of the polygonal diagram with light-like geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the light-like boundaries assignable to CDs but leaves open the question whether more complex structures with light-like boundaries are possible. How do the space-time surfaces associated with different quaternionic structures of M8 and with different positions of tips of CD interact?

3. The emergence of CDs

CDs are a key notion of zero energy ontology (ZEO). Could the emergence of CDs be understood in terms of singularities of octonion polynomials located at the light-like boundaries of CDs? In Minkowskian case the complex norm qqci is present in P (c is conjugation changing the sign of quaternionic unit but not that of the commuting imaginary unit i). Could this allow to blow up the singular point to a 3-D boundary of light-cone and allow to understand the emergence of causal diamonds (CDs) crucial in ZEO.

The study of the special properties for zero loci of general polynomial P(o) at light-rays of O indeed demonstrated that both 8-D land 4-D light-cones and their complements emerge naturally, and that the M4 projections of these light-cones and even of their boundaries are 4-D future - or past directed light-cones. What one should understand is how CDs as their intersections, and therefore ZEO, emerge.

  1. One manner to obtain CDs naturally is that the polynomials are sums P(t)= ∑k Pk(o) of products of form Pk(o) =P1,k(o)P2,k(o-T), where T is real octonion defining the time coordinate. Single product of this kind gives two disjoint 4-varieties inside future and past directed light-cones M4+(0) and M4-(T) for either RE(P)=0 (or IM(P)=0) condition. The complements of these cones correspond to IM(P)=0 (or RE(P)=0) condition.

  2. If one has nontrivial sum over the products, one obtains a connected 4-variety due the interaction terms. One has also as special solutions M4+/- and the 6-spheres associated with the zeros P(t) or equivalently P1(t1)== P(t), t1=T-t vanishing at the upper tip of CD. The causal diamond M4+(0)∩ M4-(T) belongs to the intersection.

    Remark: Also the union M4-(0)∪ M4+(T) past and future directed light-cones belongs to the intersection but the latter is not considered in the proposed physical interpretation.

  3. The time values defined by the roots tn of P(t) define a sequence of 6-spheres intersecting 4-D CD along 3-balls at times tn. These time slices of CD must be physically somehow special. Space-time variety intersects 6-spheres along 2-varieties X2n at times tn. The varieties X2n are perhaps identifiable as 2-D interaction vertices, pre-images of corresponding vertices in H at which the light-like orbits of partonic 2-surfaces arriving from the opposite boundaries of CD meet.

    The expectation is that in H one as generalized Feynman diagram with interaction vertices at times tn. The higher the evolutionary level in algebraic sense is, the higher the degree of the polynomial P(t), the number of tn, and more complex the algebraic numbers tn. P(t) would be coded by the values of interaction times tn. If their number is measurable, it would provide important information about the extension of rationals defining the evolutionary level. One can also hope of measuring tn with some accuracy! Octonionic dynamics would solve the roots of a polynomial! This would give a direct connection with adelic physics.

    Remark: Could corresponding construction for higher algebras obtained by Cayley-Dickson construction solve the "roots" of polynomials with larger number of variables? Or could Cartesian product of octonionic spaces perhaps needed to describe interactions of CDs with arbitrary positions of tips lead to this?

  4. Above I have considered only the interiors of light-cones. Also their complements are possible. The natural possibility is that varieties with RE(P)=0 and IM(P)=0 are glued at the boundary of CD, where RE(P)=IM(P)=0 is satisfied. The complement should contain the external (free) particles, and the natural expectation is that in this region the associativity/co-associativity conditions can be satisfied.

  5. The 4-varieties representing external particles would be glued at boundaries of CD to the interacting non-associative solution in the complement of CD. The interaction terms should be non-vanishing only inside CD so that in the exterior one would have just product P(o)=P1,k0(o)P2,k0(o-T) giving rise to a disjoint union of associative varieties representing external particles. In the interior one could have interaction terms proportional to say t2(T-t)2 vanishing at the boundaries of CD in accordance with the idea that the interactions are switched one slowly. These terms would spoil the associativity.

Remark: One can also consider sums of the products ∏k Pk(o-Tk) of n polynomials and this gives a sequence CDs intersecting at their tips. It seems that something else is required to make the picture physical.

4. How could the space-time varieties associated with different CDs interact?

The interaction of space-time surfaces inside given CD is well-defined. Sitation is not so clear for different CDs for which the choice of octonionic coordinate origin is in general different and polynomial bases for different CDs do not commute nor associate.

The intuitive expectation is that 4-D/8-D CDs can be located everywhere in M4/M8. The polynomials with different origins neither commute nor are associative. Their sum is a polynomial whose coefficients are not real. How could one avoid losing the extremely beautiful associative and commutative algebra?

It seems that one cannot form their products and sums and must form the Cartesian product of M8:s with different origins and formulate the interaction at M8 level in this framework. Note that Cayley-Dickson hierarchy does not seem to be relevant since the dimension are powers of 2 rather than multiples of 8.

Should one give up associativity and allow products (but not sums since one should give up the assumption that the coefficients of polynomials are real) of polynomials associated with different CDs as an analog for the formation of free many-particle states. One can still have separate vanishing of the polynomials in separate CDs but how could one describe their interaction?

If one does not give up associativity and commutativity, how can one describe the interactions between space-time surfaces inside different CDs at the level of M8?

  1. Could the intersection of space-time varieties with zero loci for RE(Pi) and IM(Pi) define the loci of interaction. As already found, the 6-D spheres S6 with radii tn given by the zeros of P(t) are universal and have interpretation as t=tn snapshots of 7-D spherical light front.

    The 2-D intersections X2 of 4-D space-time variety X4 with S6 would define natural candidates for the intersections and might allow interpretation as pre-images of partonic 2-surfaces. X2 would be the contact of X4 with S6 associated with second 8-D CD. Together with SH this gives hopes about an elegant description of interactions in terms of connected space-time varieties.

  2. The following picture is suggestive. Consider two space-time varieties X4i, i=1,2 associated with CDs with different origins and connected by a connected sum contact, which at the level of H corresponds to a wormhole contact connecting space-time sheets with different octonionic coordinates. The partonic 2-varieties X2i= X4i∩ S6i are labelled by time values t=ti,ni.

    Assume that there is tube-like 3-surface X31,2 connecting X21 and X22. The union X21∩ X22 of partonic 2-surfaces must be homologically trivial in order to define a boundary of 3-surface X31,2. The surfaces X2i must therefore have opposite homology charges. X31,2 would be pre-image of a wormhole contact connecting different space-time sheets to which the CDs are assigned.

    The 6-spheres S6i intersect along 4-D surface X41,2= S61∩ S62 in M8. One should have X31,2⊂ X41,2 and X31,2 should be non-critical but associative and therefore 3-D. This surface should allow a realization as a zero locus of RE(P1,2(u)) or IM(P1,2(u)) and belong to X41,2. One would not have manifold-topology. Rather, one could speak of two 4-D branes X4i (3-branes) connected by a 3-D brane X31,2 (2-brane). Two 2 parallel 4-planes joint by a 1-D curve is the lower-dimensional analogy. The interaction would be instantaneous inside X4i.


  3. The polynomials associated with different 8-D CDs do not commute nor associate. Should one allow their products so that one would still effectively have a Cartesian product of commutative and associative algebras?

    Or should one introduce Cartesian powers of O and CD:s inside these powers to describe the interaction? This would be analogous to what one does in condensed matter physics. What seems clear is that M8-H correspondence should map all the factors of (M8)n to the same M4× CP2 by a kind of diagonal projection.

  4. Partonic 2-surfaces define wormhole throats and appear in pairs if they carry monopole charges. Could one think that the above mentioned 2-surfaces are intersections of X1i with Ski+1 for the pair of space-time sheets assignable to different CDs? Could the image in H of the structure formed by {X21,X22, S61, S62} under M8-H correspondences be wormhole contact.

5. Summary

All big pieces of quantum TGD are now tightly interlinked.

  1. The notion of causal diamond (CD) and therefore also ZEO can be now regarded as a consequence of the number theoretic vision and M8-H correspondence, which is also understood physically.

  2. The hierarchy of algebraic extensions of rationals defining evolutionary hierarchy corresponds to the hierarchy of octonionic polynomials.

  3. Associative varieties for which the dynamics is critical are mapped to minimal surfaces with universal dynamics without any dependence on coupling constants as predicted by twistor lift of TGD. The 3-D associative boundaries of non-associative 4-varieties are mapped to initial values of space-time surfaces inside CDs for which there is coupling between Kähler action and volume term.

  4. Free many particle states as algebraic 4-varieties correspond to product polynomials in the complement of CD and are associative. Inside CD the addition of interaction terms vanishing at its boundaries spoils associativity and makes these varieties connected.

  5. The basic building bricks of topological scattering diagrams identified as space-time surfaces having as vertices partonic 2-surfaces emerge from the special features of the octonionic algebraic geometry predicting sequence of 3-balls as intersections of hyperplanes t= tn with CD. One can say that octonionic dynamics solves roots of the polynomial P(t) whose octonionic extension defines space-time surfaces as zero loci. Furthermore, the generic prediction is the existence of 6-spheres inside octonionic CDs having 2-D partonic 2-variety as intersection with space-time surface inside CD and interpreted as a vertex of generalized scattering diagram.

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

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

Articles and other material related to TGD.

Sunday, August 13, 2017

Are we all artists?: or what my "Great Experience" taught me about consciousness

Are all of us artists? I could immediately answer the question: we are artists - all of us. This is what my "Great Experience" taught me about consciousness before I had any theory of consciousness. I started to work systematically with the problem of consciousness - that is to write a book, which is the manner I work - only 10 years after this experience.

What I claim is that the construction of sensory mental images is not a passive process but a creation of an artwork, kind of caricature giving a representation of the sensory input optimal as far as survival is considered. This means decomposition of the sensory input to features and picking up the key features relevant for the survival.

The article with link below is a written and slightly longer version of a talk in which I told about the role of vision in sensory experience seen in the theoretical framework provided by TGD inspired theory of consciousness. I decided to tell about my "Great Experience around 1985 since it divides my life to two parts: life before and after this experience, and because this experience provided fascinating insights to consciousness and perception, not only visual, but also auditory perception and proprioception (body experience). I have told about this experience in my homepage (see this) and in some material in books and articles to be found there (for instance).

There are online books about TGD proper and two published books (Luniver and Bentham). For TGD inspired theory of consciousness and quantum biology see the online books at my homepage and the published book about consciousness and quantum biology (Lambert). The article Philosophy of Adelic Physics published by Springer explains the recent vision about the mathematics forced by consciousness theory.

For details see the article Are we all artists?: or what my "Great Experience" taught me about consciousness or the chapter Magnetospheric sensory representations of "Magnetospheric consciousness".

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

Articles and other material related to TGD.

Sunday, August 06, 2017

M8-H duality: summary and future prospects

In the following I give a brief summary about what has been done. I concentrate on M8-H duality since the most significant results are achieved here.

It is fair to say that the new view answers the following a long list of open questions.

  1. When M8-H correspondence is true (to be honest, this question emerged during this work!)? What are the explicit formulas expressing associativity of the tangent space or normal space of the 4-surface?

    The key element is the formulation in terms of complexified M8 identified in terms of octonions and restriction to M8. One loses the number field property but for polynomials ring property is enough. The level surfaces for real and imaginary parts of octonionic polynomials with real coefficients define 4-D surfaces in the generic case.

    Associativity condition is an additional condition reducing the dimension of the space-time surface unless some components of RE(P) or IM(P) are critical meaning that also their gradients vanish. This conforms with the quantum criticality of TGD and provides a concrete first principle realization for it.

  2. How this picture corresponds to twistor lift? The twistor lift of Kähler action (dimensionally reduced Kähler action in twistor space of space-time surface) one obtains two kinds of space-time regions. The regions, which are minimal surfaces and obey dynamics having no dependence on coupling constants, correspond naturally to the critical regions in M8 and H.

    There are also regions in which one does not have extremal property for both Kähler action and volume term and the dynamics depends on coupling constant at the level of H. These regions are associative only at their 3-D ends at boundaries of CD and at partonic orbits, and the associativity conditions at these 3-surfaces force the initial values to satisfy the conditions guaranteeing preferred extremal property. The non-associative space-time regions are assigned with the interiors of CDs. . The particle orbit like space-time surfaces entering to CD are critical and correspond to external particles.

  3. The surprise was that M4⊂ M8 is naturally co-associative. If associativity holds true also at the level of H, M4 ⊂ H must be associative. This is possible if M8-H duality maps tangent space in M8 to normal space in H and vice versa.

  4. The connection to the realization of the preferred extremal property in terms of gauge conditions of subalgebra of SSA is highly suggestive. Octonionic polynomials critical at the boundaries of space-time surfaces would determine by M8-H correspondence the solution to the gauge conditions and thus initial values and by holography the space-time surfaces in H.

  5. A beautiful connection between algebraic geometry and particle physics emerges. Free many-particle states as disjoint critical 4-surfaces can be described by products of corresponding polynomials satisfying criticality conditions.
    These particles enter into CD , and the non-associative and non-critical portions of the space-time surface inside CD describe the interactions. One can define the notion of interaction polynomial as a term added to the product
    of polynomials. It can vanish at the boundary of CD and forces the 4-surface to be connected inside CD. It also spoils associativity: interactions are switched on. For bound states the coefficients of interaction polynomial are such that one obtains a bound state as associative space-time surface.

  6. This picture generalizes to the level of quaternions. One can speak about 2-surfaces of space-time surface
    with commutative or co-commutative tangent space. Also these 2-surfaces would be critical. In the generic case commutativity/co-commutativity allows only 1-D curves.

    At partonic orbits defining boundaries between Minkowskian and Euclidian space-time regions inside CD the string world sheets degenerate to the 1-D orbits of point like particles at their boundaries. This conforms with the twistorial description of scattering amplitudes in terms of point like fermions.

    For critical space-time surfaces representing incoming states string world sheets are possible as commutative/co-commutative surfaces (as also partonic 2-surfaces) and serve as correlates for (long range) entaglement) assignable also to macroscopically quantum coherent system (heff/h=n hierarchy implied by adelic physics).

  7. The octonionic polynomials with real coefficients form a commutative and associative algebra allowing besides algebraic operations function composition. Space-time surfaces therefore form an algebra and WCW has algebra structure. This could be true for the entire hierarchy of Cayley-Dickson algebras, and one would have a highly non-trivial generalization of the conformal invariance and Cauchy-Riemann conditions to their n-linear counterparts at the n:th level of hierarchy with n=1,2,3,.. for complex numbers, quaternions, octonions,... One can even wonder whether TGD generalizes to this entire hierarchy!

What mathematical challenges one must meet?
  1. One should prove more rigorously that criticality is possible without the reduction of dimension of the space-time surface.

  2. One must demonstrate that SSA conditions can be true for the images of the associative regions (with 3-D or 4-D). This would obviously pose strong conditions on the values of coupling constants at the level of H.

What questions should be answered?
  1. Does associativity hold true in H for minimal surface extremals obeying universal critical dynamics? As found, the study of the known extremals supports this view.

  2. Could one construct the scattering amplitudes at the level of M8? Here the possible problems are caused by the exponents of action (Kähler action and volume term) at H side. Twistorial construction however leads to a proposal that the exponents actually cancel. This happens if the scattering amplitude can be thought as an analog of Gaussian path integral around single extremum of action and conforms with the integrability of the theory. In fact, nothing prevents from defining zero energy states in this manner! If this holds true then it might be possible to construct scattering amplitudes at the level of M8.

  3. What about coupling constants? Coupling constants make themselves visible at H side both via the vanishing conditions for Noether charges in sub-algebra of SSA and via the values of the non-vanishing Noether charges. M8-H correspondence determining the 3-D boundaries of interaction regions within CDs suggests that these couplings must emerge from the level M8 via the criticality conditions posing conditions on the coefficients of the octonionic polynomials coding for interactions.

    Could all coupling constant emerge from the criticality conditions at the level of M8? The ratio of R2/lP2 of CP2 scale and Planck length appears at H level. Also this parameter should emerge from M8-H correspondence and thus from criticality at M8 level. Physics would reduce to a generalization of the catastrophe theory of Rene Thom!

  4. There are questions related to ZEO. Is the notion of CD general enough or should one generalize it to the analog of the polygonal diagram with light-like geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the light-like boundaries assignable to CDs but leaves open the question whether more complex structures with light-like boundaries are possible. How the space-time surfaces associated with different quaternionic structures of M8 and with different positions of tips of CD interact?

  5. Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be at real line (time axis) in the octonionic sense. All CDs cannot be located along this line. How do the varieties associated with octonionic polynomials with different origins interact? The polynomials with different origins neither commute nor are associative. How could one avoid losing the extremely beautiful associative and commutative algebra? It seems that one cannot form their products and sums and must form the Cartesian product of M8:s with different origins and formulate the interaction in this framework. Could Cayley-Dickson hierarchy be necessary to describe the interactions between different CDs (note however that the dimension are powers of 2) than multiples of 8?

    Is the interaction nr well-defined only at the level of H inside CD to which these 4-D varieties arrive through the boundary of this CD? All CDs, whose tips are along light-like ray of CD boundary, share this ray. There is a common M20 shared by these CDs. Could M20 make possible the interaction. The CDs able to interact with given CD would have tips at the 3-D boundary of this CD and share common M20. These M20:s are labelled by the points of twistor sphere so that twistoriality seems to enter into the game in non-trivial manner also at the level of M8!

    This however allows interactions only between varieties with same M40. What about a more general picture in which unit octonions define 6-D sphere S6 of directions of 8-D light-rays and parameterize different quaternionic structures with fixed M20. Could the condition for interaction be that S6 coordinates are same so that M20 is shared. In this case light rays in 7-D light-cone would parameterize the allowed origins for octonionic polynomials for which the interaction of zero loci is possible. The images of these light-rays in H would be more complex. This could allow varieties with different M40 but common M20 to interact via the common light ray/M20. Somewhat similar picture involving preferred M20 for given connected part of twistor graph emerges from the construction of twistor amplitudes.

    What would be the interaction at the intersection? Light-like ray naturally defines a string like object with fermions at its ends. Could this fermionic string be transferred between space-time surfaces in the intersecting CDs. Also a branching of a string like object between space-time varieties in different CDs intersecting along this ray would be possible. This would describe stringy reaction vertex with incoming strings in different CDs. It must be admitted that this unavoidably brings in mind branes and all the nasty things that I have said about them during years!

    Or could the strange singularity behavior in Minkowski signature play a role in the interactions? In the generic situation the intersection consists of discrete points but as the study of o2 shows the surface RE(o2)=0 and IM(o2)=0 can have dimensions 5 and 6 and their intersection naively expected to consist of discrete points can be interior or exterior of light-cone. Could the zero loci of singular polynomials play a key role in the interaction and allow 4-varieties in M8 to interact by being glued to this higher than 4-D objects. 4-D space-time variety could have 2-D intersection with 6-D variety and the 6-D variety could allow the interactions to between two 4-D varieties by correlating them. Also this strongly brings in mind branes but looks less elegant that above proposal.

  6. What is the connection with Yangian symmetry, whose generalization in TGD framework is highly suggestive?

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

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

Articles and other material related to TGD.

Friday, August 04, 2017

Philosophy of Adelic Physics from Springer


The article "The philosophy of adelic physics" about adelic physics providing number theoretic formulation of TGD was published as an article in the book below published by Springer. The article is 80 pages long. See this.

Wednesday, August 02, 2017

Does M8-H duality reduce classical TGD to octonionic algebraic geometry?

I have used last month to develop a detailed vision about M8-H duality and now I dare to speak about genuine breakthrough. I attach below the abstract of the resulting article.

TGD leads to several proposals for the exact solution of field equations defining space-time surfaces as preferred extremals of twistor lift of Kähler action. So called M8-H duality is one of these approaches. The beauty of M8-H duality is that it could reduce classical TGD to algebraic geometry and would immediately provide deep insights to cognitive representation identified as sets of rational points of these surfaces.

In the sequel I shall consider the following topics.

  1. I will discuss basic notions of algebraic geometry such as algebraic variety, surface, and curve, rational point of variety central for TGD view about cognitive representation, elliptic curves and surfaces, and rational and potentially rational varieties. Also the notion of Zariski topology and Kodaira dimension are discussed briefly. I am not a mathematician and what hopefully saves me from horrible blunders is physical intuition developed during 4 decades of TGD.

  2. It will be shown how M8-H duality could reduce TGD at fundamental level to algebraic geometry. Space-time surfaces in M8 would be algebraic surfaces identified as zero loci for imaginary part IM(P) or real part RE(P) of octonionic polynomial of complexified octonionic variable oc decomposing as oc= q1c+q2c I4 and projected to a Minkowskian sub-space M8 of complexified O. Single real valued polynomial of real variable with algebraic coefficients would determine space-time surface! As proposed already earlier, spacetime surfaces would form commutative and associative algebra with addition, product and functional composition.

    One can interpret the products of polynomials as correlates for free many-particle states with interactions described by added interaction polynomial, which can vanish at boundaries of CDs thanks to the vanishing in Minkowski signature of the complexified norm qcqc* appearing in RE(P) or IM(P) caused by the quaternionic non-commutativity. This leads to the same picture as the view about preferred extremals reducing to minimal surfaces near boundaries of CD. Also zero zero energy ontology (ZEO) could emerge naturally from the failure of number field property for for quaternions at light-cone boundaries.

  3. The fundamental challenge is to prove that the octonionic polynomials with real coefficients determine associative/quaternionic surfaces as the zero loci of their imaginary/real parts in quaternionic sense. Here the intuition comes from the idea that the octonionic polynomials map from octonionic space O to second octonionic space W. Real and imaginary parts in W are quaternionic/co-quaternionic. These planes correspond to surfaces in O defined by the vanishing of real/imaginary parts, and the natural guess is that they are quaternionic/co-quaternionic, that is associative/co-associative.

    The hierarchy of notions involved is well-ordering for 1-D structures, commutativity for complex numbers, and associativity for quaternions. This suggests a generalization of Cauchy-Riemann conditions for complex analytic functions to quaternions and octonions. Cauchy Riemann conditions are linear and constant value manifolds are 1-D and thus well-ordered. Quaternionic polynomials with real coefficients define maps for which the 2-D spaces corresponding to vanishing of real/imaginary parts of the polynomial are complex/co-complex or equivalently commutative/co-commutative. Commutativity is expressed by conditions bilinear in partial derivatives. Octonionic polynomials with real coefficients define maps for which 4-D surfaces for which real/imaginary part are quaternionic/co-quaternionic, or equivalently associative/co-associative. The conditions are now 3-linear.

    In fact, all algebras obtained by Cayley-Dickson construction adding imaginary units to octonionic algebra are power associative so that polynomials with real coefficients define an associative and commutative algebra. Hence octonion analyticity and M8-H correspondence could generalize.

  4. It turns out that in the generic case associative surfaces are 3-D and are obtained by requiring that one of the coordinates RE(Y)i or IM(Y)i in the decomposition Yi=RE(Y)i +IM(Y)iI4 of the gradient of RE(P)= Y=0 with respect to the complex coordinates zik, k=1,2, of O vanishes that is critical as function of quaternionic components z1k or z2k associated with q1 and q2 in the decomposition o= q1+q2I4, call this component Xi. In the generic case this gives 3-D surface.

    In this generic case M8-H duality can map only the 3-surfaces at the boundaries of CD and light-like partonic orbits to H, and only determines the boundary conditions of the dynamics in H determined by the twistor lift of Kähler action. M8-H duality would allow to solve the gauge conditions for SSA (vanishing of infinite number of Noether charges) explicitly.

    One can also have criticality. 4-dimensionality can be achieved by posing conditions on the coefficients of the octonionic polynomial P so that the criticality conditions do not reduce the dimension: Xi would have possibly degenerate zero at space-time variety. This can allow 4-D associativity with at most 3 critical components Xi. Space-time surface would be analogous to a polynomial with a multiple root. The criticality of Xi conforms with the general vision about quantum criticality of TGD Universe and provides polynomials with universal dynamics of criticality. A generalization of Thom's catastrophe theory emerges. Criticality should be equivalent to the universal dynamics determined by the twistor lift of Kähler action in H in regions, where Kähler action and volume term decouple and dynamics does not depend on coupling constants.

    One obtains two types of space-time surfaces. Critical and associative (co-associative) surfaces can be mapped by M8-H duality to preferred critical extremals for the twistor lift of Kähler action obeying universal dynamics with no dependence on coupling constants and due to the decoupling of Kähler action and volume term: these represent external particles. M8-H duality does not apply to non-associative (non-co-associative) space-time surfaces except at 3-D boundary surfaces. These regions correspond to interaction regions in which Kähler action and volume term couple and coupling constants make themselves visible in the dynamics. M8-H duality determines boundary conditions.

  5. Cognitive representations are identified as sets of rational points for algebraic surfaces with "active" points containing fermion. The representations are discussed at both M8- and H level. Rational points would be now associated with 4-D algebraic varieties in 8-D space. General conjectures from algebraic geometry support the vision that these sets are concentrated at lower-dimensional algebraic varieties such as string world sheets and partonic 2-surfaces and their 3-D orbits, which can be also identified as singularities of these surfaces.

  6. Some aspects related to homology charge (Kähler magnetic charge) and genus-generation correspondence are discussed. Both are central in the proposed model of elementary particles and it is interesting to see whether the picture is internally consistent and how algebraic surface property affects the situation. Also possible problems related to heff/h=n hierarchy realized in terms of n-fold coverings of space-time surfaces are discussed from this perspective.

In order to get more perspective I add an FB response relating to this.

Octonions and quaternions are 20 year old part of TGD: one of the three threads in physics as generalized number theory vision. Second vision is quantum physics as geometry of WCW. The question has been how to fuse geometric and number theory visions. Algebraic geometry woul do it since it is both geometry and algebra and it has been also part of TGD but only now I realized how to get acceess to its enormous power.

Even the proposal discussed now about the algebra of octonionic polynomials with real coefficients was made about two decades ago but only now I managed to formulate it in detail. Here the general wisdom gained from adelic physics helped enormously. I dare say that classical TGD at the most fundamental level is solved exactly.

From the point of pure mathematics the generalization of complex analyticity and linear Cauchy Riemann conditions to multilinear variants for quaternions, octonions and even for the entire hierarchy of algebras obtained by Cayley-Dickson construction is a real breakthrough. Consider only the enormous importance of complex analyticity in mathematics and physics, including string models. I do not believe that this generalization has been discovered: otherwise it would be key part of mathematical physics. Quaternionic and octonionic analyticities will certainly mean huge evolution in mathematics. I had never ended to these discoveries without TGD: TGD forced them.

At these moments I feel deep sadness when knowing that the communication of these results to collegues is impossible in practice. This stupid professional arrogance is something which I find very difficult to accept even after 4 decades. I feel that when society pays a monthly salary for a person for being a scientists, he should feel that his duty is to be keenly aware what is happening in his field. When some idiot proudly tells that he reads only prestigious journals, I get really angry.

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

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

Articles and other material related to TGD.