^{8}and M

^{4}.

**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)I_{4}, 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)=o^{2} 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)=o^{2} having double root at origin.

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

- P(o) reduces along each light-ray of δ M
^{8}_{+}to the same real valued polynomial P(t) of a real variable t apart from a multiplicative unit E= (1+in)/2 satisfying E^{2}=E. Here n is purely octonion-imaginary unit vector defining the direction of the light-ray.

IM(P)=0 corresponds to quaterniocity. If the E

^{4}(M^{8}= M^{4}× E^{4}) projection is vanishing, there is no additional condition. 4-D light-cones M^{4}_{+/-}are obtained as solutions of IM(P)=0. Note that M^{4}_{+/-}can correspond to any quaternionic subspace.

If the light-like ray has a non-vanishing projection to E

^{4}, one must have P(t)=0. The solutions form a collection of 6-spheres labelled by the roots t_{n}of P(t)=0. 6-spheres are not associative.

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

- P(o) reduces along each light-ray of δ M
- Suppose now that P(t) is shifted to P
_{1}(t)=P(t)-c, c a real number. Also now M^{4}_{+/-}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.

^{8}

_{+}and light-likeness of its points played a central role. What about the interior of 8-D CD?

- The natural expectation is that in the interior of CD one obtains a 4-D variety X
^{4}. For IM(P)=0 the outcome would be union of X^{4}with M^{4}_{+}and the set of 6-spheres for IM(P)=0. 4-D variety would intersect M^{4}_{+}in a discrete set of points and the 6-spheres along 2-D varieties X^{2}. The higher the degree of P, the larger the number of 6-spheres and these 2-varieties.

- For RE(P)=0 X
^{4}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.

- 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.

^{8}.

- Could M
^{4}_{+}(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 X^{2}perhaps identifiable as pre-images of partonic 2-surfaces and maybe string world sheets? Could the interactions between space-time varieties X^{4}_{i}assignable with different CDs be describable by regarding 6-spheres as bridges between X^{4}_{i}having only a discrete set of common points. Could one say that X^{2}_{i}interact via the 6-sphere somehow. Note however that 6-spheres are not dynamical.

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

- 6-spheres in IM(P)=0 case do not have image under M
^{8}-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 CP_{2}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.

- 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.

- 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 M
^{4}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.

- 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(P
_{i}^{k}))=0 for i=1,2, k=1,4. Along line octonionic coordinate reduces effectively to real coordinate since one has E^{2}=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: t_{2}= t_{1}-k. Therefore one has common zero for real polynomials RE(P_{1}^{k}(t)) and RE(P_{2}^{k}(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 o

^{2}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?

- 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.

- 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.

^{8}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 qq^{c}_{i} 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 M^{4} 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.

- One manner to obtain CDs naturally is that the polynomials are sums P(t)= ∑
_{k}P_{k}(o) of products of form P_{k}(o) =P_{1,k}(o)P_{2,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 M^{4}_{+}(0) and M^{4}_{-}(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.

- If one has nontrivial sum over the products, one obtains a connected 4-variety due the interaction terms. One has also as special solutions M
^{4}_{+/-}and the 6-spheres associated with the zeros P(t) or equivalently P_{1}(t_{1})== P(t), t_{1}=T-t vanishing at the upper tip of CD. The causal diamond M^{4}_{+}(0)∩ M^{4}_{-}(T) belongs to the intersection.

**Remark**: Also the union M^{4}_{-}(0)∪ M^{4}_{+}(T) past and future directed light-cones belongs to the intersection but the latter is not considered in the proposed physical interpretation.

- The time values defined by the roots t
_{n}of P(t) define a sequence of 6-spheres intersecting 4-D CD along 3-balls at times t_{n}. These time slices of CD must be physically somehow special. Space-time variety intersects 6-spheres along 2-varieties X^{2}_{n}at times t_{n}. The varieties X^{2}_{n}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 t

_{n}. The higher the evolutionary level in algebraic sense is, the higher the degree of the polynomial P(t), the number of t_{n}, and more complex the algebraic numbers t_{n}. P(t) would be coded by the values of interaction times t_{n}. 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 t_{n}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?

- 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.

- 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)=P
_{1,k0}(o)P_{2,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 t^{2}(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}P

_{k}(o-T

_{k}) 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 M^{4}/M^{8}. 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 M^{8}:s with different origins and formulate the interaction at M^{8} 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 M^{8}?

- Could the intersection of space-time varieties with zero loci for RE(P
_{i}) and IM(P_{i}) define the loci of interaction. As already found, the 6-D spheres S^{6}with radii t_{n}given by the zeros of P(t) are universal and have interpretation as t=t_{n}snapshots of 7-D spherical light front.

The 2-D intersections X

^{2}of 4-D space-time variety X^{4}with S^{6}would define natural candidates for the intersections and might allow interpretation as pre-images of partonic 2-surfaces. X^{2}would be the contact of X^{4}with S^{6}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.

- The following picture is suggestive. Consider two space-time varieties X
^{4}_{i}, 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 X^{2}_{i}= X^{4}_{i}∩ S^{6}_{i}are labelled by time values t=t_{i,ni}.

Assume that there is tube-like 3-surface X

^{3}_{1,2}connecting X^{2}_{1}and X^{2}_{2}. The union X^{2}_{1}∩ X^{2}_{2}of partonic 2-surfaces must be homologically trivial in order to define a boundary of 3-surface X^{3}_{1,2}. The surfaces X^{2}_{i}must therefore have opposite homology charges. X^{3}_{1,2}would be pre-image of a wormhole contact connecting different space-time sheets to which the CDs are assigned.

The 6-spheres S

^{6}_{i}intersect along 4-D surface X^{4}_{1,2}= S^{6}_{1}∩ S^{6}_{2}in M^{8}. One should have X^{3}_{1,2}⊂ X^{4}_{1,2}and X^{3}_{1,2}should be non-critical but associative and therefore 3-D. This surface should allow a realization as a zero locus of RE(P_{1,2}(u)) or IM(P_{1,2}(u)) and belong to X^{4}_{1,2}. One would not have manifold-topology. Rather, one could speak of two 4-D branes X^{4}_{i}(3-branes) connected by a 3-D brane X^{3}_{1,2}(2-brane). Two 2 parallel 4-planes joint by a 1-D curve is the lower-dimensional analogy. The interaction would be instantaneous inside X^{4}_{i}.

- 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 M

^{8}-H correspondence should map all the factors of (M^{8})^{n}to the same M^{4}× CP_{2}by a kind of diagonal projection.

- 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 X
^{1}_{i}with S^{k}_{i+1}for the pair of space-time sheets assignable to different CDs? Could the image in H of the structure formed by {X^{2}_{1},X^{2}_{2}, S^{6}_{1}, S^{6}_{2}} under M^{8}-H correspondences be wormhole contact.

**5. Summary**

All big pieces of quantum TGD are now tightly interlinked.

- The notion of causal diamond (CD) and therefore also ZEO can be now regarded as a consequence of the number theoretic vision and M
^{8}-H correspondence, which is also understood physically.

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

- 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.

- 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.

- 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= t
_{n}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.

^{8}-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M

^{8}-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M

^{8}-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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