TGD diary: 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.

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 t_n 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.
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?

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.
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.
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.

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 X⁴_i assignable with different CDs be describable by regarding 6-spheres as bridges between X⁴_i having only a discrete set of common points. Could one say that X²_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 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.

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 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.
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 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(P₁^k(t)) and RE(P₂^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 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?
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.

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 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 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.

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 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.
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.
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 X²_n at times t_n. The varieties X²_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 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 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 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?

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 S⁶ 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 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.
The following picture is suggestive. Consider two space-time varieties X⁴_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 X²_i= X⁴_i∩ S⁶_i are labelled by time values t=t_{i,n_i}.

Assume that there is tube-like 3-surface X³_1,2 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 X³_1,2. The surfaces X²_i must therefore have opposite homology charges. X³_1,2 would be pre-image of a wormhole contact connecting different space-time sheets to which the CDs are assigned.

The 6-spheres S⁶_i intersect along 4-D surface X⁴_1,2= S⁶₁∩ S⁶₂ in M⁸. One should have X³_1,2⊂ X⁴_1,2 and X³_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 X⁴_1,2. One would not have manifold-topology. Rather, one could speak of two 4-D branes X⁴_i (3-branes) connected by a 3-D brane X³_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 X⁴_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 M⁸-H correspondence should map all the factors of (M⁸)ⁿ to the same M⁴× CP₂ 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 X¹_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 {X²₁,X²₂, S⁶₁, S⁶₂} under M⁸-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 M⁸-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.