^{4}× CP

_{2}emerges from some deeper mathematical structure. It seems that the series of inclusions for infinite-dimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyper-finite factors of type II

_{1}, could be this deeper mathematical structure.

** 1. Quaternions, octonions, and TGD**

TGD suggests also what I call HO-H duality. Space-time can be regarded either as surface in H or as hyper-quaternionic sub-manifold of the space HO of hyper-octonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit.

The 2-dimensional partonic surfaces X^{2} are of central importance in TGD and it seems that the inclusion sequence C in H in O (complex numbers, quaternions, octonions) somehow corresponds to the inclusion sequence X^{2} in X^{4} in H. This inspires the that that whole TGD emerges from a generalized number theory and I have already proposed arguments for how this might happen.

** 2. Number theoretic Clifford algebras**

_{1}defined by infinite-dimensional Clifford algebras is one thread in the multiple strand of number-theoretic ideas involving p-adic numbers fields and their fusion with reals along common rationals to form a generalized number system, classical number fields, hierarchy of infinite primes and integers, and von Neumann algebras and quantum groups. The new ideas allow to fuse von Neumans strand with the classical number field strand.

- The mere assumption that physical states are represented by spinor fields in the infinite-dimensional "world of classical worlds" implies the notion of infinite-dimensional Clifford algebra identifiable as generated by gamma matrices of infinite-dimensional separable Hilbert space. This algebra provides a standard representation for hyperfinite factors of type II
_{1}. - Von Neumann algebras known as hyperfinite factors of type II
_{1}are rather miraculous objects. The almost defining property is that the trace of unit operator is unity instead of infinity. This justifies the attribute hyperfinite and gives excellent hopes that the resulting quantum theory is free of infinities. These algebras are strange fractal like creatures in the sense that they can be imbedded unitarily within itself endlessly and one obtains infinite hierarchies of Jones inclusions. This means what might be called Brahman=Atman property: subsystem can represent in its state the state of the entire universe and this indeed leads to the idea that symbolic and cognitive representations are realized as Jones inclusions and that Universe is busily mimicking itself in this manner. - Classical number fields F=R,C,H,O define four Clifford algebras using infinite tensor power of 2x2 Clifford algebra M
_{2}(F) associated with 2-spinors. The tensor powers associated with R and C are straightforward to define. The non-commutativity of H with C requires Connes tensor product which by definition guarantees that left and right multiplications of tensor product M_{2}(H)×M_{2}(H) by complex numbers are equivalent. For F=O the matrix algebra is not anymore associative but this implies only interpretational problems and means a slight generalization of von Neumann algebras which as far as I know are usually assumed to be associative. Denote by Cl(F) the infinite-dimensional Clifford algebras obtained in this manner. Perhaps I should not have said "only interpretational" since the solution of these problems dictates the classical and quantum dynamics.

**3. TGD does not quite emerge from Jones inclusions for number theoretic Clifford algebras**

Physics as a generalized number theory vision suggests that TGD physics is contained by the Jones inclusion sequence Cl(C) in Cl(H) in Cl(O) induced by C in H in O. This sequence could alone explain partonic, space-time, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M^{4}× CP_{2} would emerge as a unique choice allowed by mathematical existence.

- CP
_{2}indeed emerges naturally: it labels the possible H-planes of O and this observation stimulated the emergence idea for few years ago. - Also Minkowski space M
^{4}is wanted. In particular, future lightcones are needed since the super-canonical algebra defining the second super-conformal invariance of TGD is associated with the canonical algebra of δM^{4}× CP_{2}. The generalized conformal and symplectic structures of 4-D(!) lightcone boundary are crucial element here. Ordinary Super Kac-Moody algebra assignable with lightlike 3-D causal determinants is associated with the inclusion of partonic 2-surface X^{2}to X^{4}corresponding to C in H. Imbedding space cannot be dynamical anymore since no 16-D number field exists. - The representation of space-times as surfaces of H should emerge as well as the space of configuration space spinor fields (not only spinors) defined in the space of 3-surfaces (or equivalently 4-surfaces which are generalizations of Bohr orbits).
- These surfaces should also have interpretation as hyper-quaternionic sub-manifold of hyper-octonionic 8-space HO (this would dictate the classical dynamics).

** 4. Number-theoretic localization of infinite-dimensional number theoretic Clifford algebras as a lacking piece of puzzle**
The lacking piece of the big argument is below.

- Sequences of inclusions C in H in F allow to interpret infinite-D spinors in Cl(O) as a module having quaternionic spinors Cl(H) as coefficients multiplying quantum spinors with finite quantum dimension not larger than 16: this conforms with the fact that OH spinors indeed are complex 8+8 spinors (quarks, leptons). Configuration space spinors can be seen as quantized imbedding space spinors. Infinite-dimensional Cl(H) spinors in turn can be seen as 4-D quantum spinors having CL(C) spinors as coefficients. Quantum groups emerge naturally and relate to inclusions as does also Kac-Moody algebra.
- The key idea is to extend infinite-dimensional Clifford algebras to local algebras by allowing power series in hyper-F numbers with coefficients in Cl(F). Using algebraic terminology this means a direct integral of the factors. The resulting objects are generalizations of conformal fields (or quantum fields) defined in the space of hyper-complex numbers (string orbits), hyper-quaternions (space-time surface), hyper-octonions (HO). Their argument is hyper-F number instead of z. Very natural number theoretic generalization of gamma matrix fields (generators of local Clifford algebra!) of super string model is thus in question.
- Associativity at the space-time level becomes the fundamental physical law. This requires that physical Clifford algebra is associative. For Cl(O) this means that a quaternionic plane in O parametrized by a point of CP
_{2}is selected at each point hyper-quaternionic point. For the local version of Cl(O) this means that powers of hyper-octonions in powers series are restricted to be hyperquaternions assignable to some hyper-quaternionic sub-manifold of HO (classical dynamics!). But since ordinary inclusion assigns CP_{2}point to given point of M_{4}represented by a hyper-quaternion one can regard space-time surface also as a surface of H! This means HO-H duality. Parton level emerges from the requirement of commutativity implying that partonic 2-surface correspond to commutative sub-manifolds of HO and thus also of H. - Also the super-canonical invariance comes out naturally. The point is that light like hyper-quaternions do not possess inverse so that Laurent series for local Cl(F) elements does not exist at the boundaries lightcones of M
^{4}which are thus causal determinants (note the analogy with pole of analytic function). Super-canonical algebra emerges at their boundaries and the intersections of space-time surfaces with the boundaries define a natural gauge fixing for the general coordinate invariance. Configuration space spinor fields are obtained by allowing quantum superpositions of these 3-surfaces (equivalently corresponding 4-surfaces).

** 5. Explicit general formula for S-matrix emerges also**

This picture leads also to an explicit master formula for S-matrix.

- The resulting S-matrix is consistent with the generalized duality symmetry implying that S-matrix element can be always expressed using a single diagram having single vertex from which lines identified as space-time surfaces emanate. There is analogy with effective action formalism in the sense that one proceeds in a direction reverse to that in the ordinary perturbative construction of S-matrix: from the vertex to the points defining tips of the boundaries of lightcones assignable to the incoming and outgoing particles appearing in n-point function along the "lines". It remains to be shown that the generalized duality indeed holds true: now its basic implication is used to write the master formula for S-matrix.
- Configuration space integral over the 3-surfaces appearing as vertex is involved and corresponds to bosonic degrees of freedom in super string models. It is free of divergences since the exponent of Kähler function is a nonlocal functional of 3-surface, since ill-defined metric determinant is cancelled by ill-defined Gauss determinant, and since Ricci tensor for the configuration space vanishes implying the vanishing of further divergences coming from the metric determinant. Hyper-finiteness of type II
_{1}factors (infinite-dimensional unit matrix has unit trace) is expected to imply the cancellation of the infinities in fermionic sector. - Diagrams obtained by gluing of space-time sheets along their ends at the vertex rather than stringy diagrams turn indeed be the Feynman diagrams in TGD framework as previously concluded on basis of physical and algebraic arguments. These singular four-manifolds are not real solutions of field equation but only a construct emerging naturally in the definition of S-matrix based on general coordinate invariance implying that configuration space spinor fields have same value for all Diff
^{4}related 3-surfaces along the space-time surface. S-matrix is automatically non-trivial.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "TGD: an Overview".

Matti Pitkänen
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