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.
- 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.
- 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.
- 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.
Instead of super-fields one would have a super variant of octonionic algebraic geometry.
- 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.
- 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).
- 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.
- 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.
There are several questions about quantum numbers.
- 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.
- 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.
- 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.
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.
- 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.
- 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 a summary of earlier postings see Latest progress in TGD.