Quaternions, octonions, and TGD
Quaternions and octonions have been lurking around for decades in hope of getting deeper role in physics but as John Baez put it: "I would not have the courage to give octonions as a research topic for a graduate student". Quaternions are algebraically a 4-D structure and this strongly suggests that space-time could be analogous to complex plane.
Classical continuous number fields reals, complex numbers, quaternions, octonions have dimensions 1, 2, 4, 8 coming in powers of 2. In TGD imbedding space is 8-D structure and brings in mind octonions. Space-time surfaces are 4-D and bring in mind quaternions. String world sheets and partonic 2-surfaces are 2-D and bring in mind complex numbers. The boundaries of string world sheets are 1-D and carry fermions and of course bring in mind real numbers. These dimensions are indeed in key role in TGD and form one part of the number theoretic vision about TGD involving p-adic numbers, classical number fields, and infinite primes.
What quaternionicity could mean?
Quaternions are non-commutative: AB is not equal to BA. Octonions are even non-associative: A(BC) is not equal to (AB)C. This is problematic and in TGD problems is turned to a victory if space-time surfaces as 4-surface in 8-D M4× CP2 are associative (or co-associative in which case normal space orthogonal to the tangent space is associative). This would be extremely attractive purely number theoretic formulation of classical dynamics.
What one means with quaternionicity of space-time is of course highly non-trivial questions. It seems however that this must be a local notion. The tangent space of space-time should have quaternionic structure in some sense.
- It is known that 4-D manifolds allow so called almost quaternionic structure: to any point of space-time one can assign three quaternionic imaginary units. Since one is speaking about geometry, imaginary quaternionic units must be represented geometrically as antisymmetric tensors and obey quaternionic multiplication table. This gives a close connection with twistors: any orientable space-time indeed allows extension to twistor space which is a structure having as "fiber space" unit sphere representing the 3 quaternionic units.
- A stronger notion is quaternionic Kähler manifold, which is also Kähler manifold - one of the quaternionic imaginary unit serves as global imaginary unit and is covariantly constant.CP2 is example of this kind of manifold. The twistor spaces associated with quaternion-Kähler manifolds are known as Fano spaces and have very nice properties making them strong candidates for the Euclidian regions of space-time surfaces obtained as deformations of so called CP2 type vacuum extremals represenging lines of generalized Feynman diagrams.
- The obvious question is whether complex analysis including notions like analytic function, Riemann surface, residue integration crucial in twistor approach to scattering amplitudes, etc... generalises to quaternions. In particular, can one generalize the notion of analytic function as a power series in z to that for quaternions q. I have made attempts but was not happy about the outcome and had given up the idea that this could allow to define associative/ co-associative space-time surface in very practically manner. It was quite a surprise to find just month or so ago that quaternions allow differential calculus and that the notion of analytic function generalises elegantly but in a slightly more general manner than I had proposed. Also the conformal invariance of string models generalises to what one might call quaternion conformal invariance. What is amusing is that the notion of quaternion analyticity had been discovered for aeons ago (see this) and I had managed to not stumble with it earlier! See this.
Octonionicity and quaternionicity in TGD
In TGD framework one can consider further notions of quaternionicity and octonionicity relying on sub-manifold geometry and induction procedure. Since the signature of the imbedding space is Minkowskian, one must replace quaternions and octonions with their complexification called often split quaternions and split octonions. For instance, Minkowski space corresponds to 4-D subspace of complexified quaternions but not to an algebra. Its tangent space generates by multiplication complexified quaternions.
The tangent space of 8-D imbedding space allows octonionic structure and one can induced (one of the keywords of TGD) this structure to space-time surface. If the induced structure is quaternionic and thus associative (A(BC)= (AB)C), space-time surface has quaternionic structure. One can consider also the option of co-associativity: now the normal space of space-time surface in M4× CP2 would be associative. Minkowskian regions of space-time surface would be associative and Euclidian regions representing elementary particles as lines of generalized Feynman diagrams would be co-associative.
Quaternionicity of space-time surface could provide purely number theoretic formulation of dynamics and the conjecture is that it gives preferred extremals of Kähler action. The reduction of classical dynamics to associativity would of course mean the deepest possible formulation of laws of classical physics that one can imagine. This notion of quaternionicity should be consistent with the quaternion-Kähler property for Euclidian space-time regions which represent lines of generalized Feynman graphs - that is elementary particles.
Also the quaternion analyticity could make sense in TGD framework in the framework provided by the 12-D twistor space of imbedding space, which is Cartesian product of twistor spaces of M4 and CP2 which are the only twistor spaces with Kähler structure and for which the generalization of complex analysis is natural. Hence it seems that space-time in TGD sense might represent an intersection of various views about quaternionicity.
What about commutativity?: number theory in fermionic sector
Quaternions are not commutative (AB is not equal to AB in general) and one can ask could one define commutative and co-commutative surfaces of quaternionic space-time surface and their variants with Minkowski signature. This is possible.
There is also a physical motivation. The generalization of twistors to 8-D twistors starts from generalization in the tangent space M8 of CP2. Ordinary twistors are defined in terms of sigma matrices identifiable as complexified quaternionic imaginary units. One should replaced the sigma matrices with 7 sigma matrices and the obvious guess is that they represent octonions. Massless irac operator and Dirac spinors should be replaced by their octonionic variant. A further condition is that this spinor structure is equivalent with the ordinary one. This requires that it is quaternionic so that one must restrict spinors to space-time surfaces.
This is however not enough - the associativity for spinor spinor dynamics forces them to 2-D string world sheets. The reason is that spinor connection consisting of sigma matrices replaced with octonion units brings in additional potential source of non-associativity. If induced gauge fields vanish, one has associativity but not quite: induce spinor connection is still non-associative. The stronger condition that induced spinor connection vanishes requires that the CP2 projection of string world sheet is not only 1-D but geodesic circle. String world sheets would be possible only in Minkowskian regions of space-time surface and their orbit would contain naturally a light-like geodesic of imbedding space representing point-like particle.
Spinor modes would thus reside at 2-surfaces 2-D surfaces - string world sheets carrying spinors. String world sheets would in turn emerge as maximal commutative space-time regions: at which induced electroweak gauge fields producing problems with associativity vanish. The gamma matrices at string world sheets would be induced gamma matrices and super-conformal symmetry would require that string world sheets are determined by an action which is string world sheet area just as in string models. It would naturally be proportional to the inverse of Newton's constant (string tension) and the ratio hbar G/R2 of Planck length and CP2 radius squared would be fixed by quantum criticality fixing the values of all coupling strengths appearing in the action principle to be of order 10-7. String world sheets would be fundamental rather than only emerging.
I have already earlier ended up to a weaker conjecture that spinors are localized to string world sheets from the condition that electromagnetic charge is well-defined quantum number for the induced spinor fields: this requires that induced W gauge fields and perhaps even potentials vanish and in the generic case string world sheets would be 2-D. Now one ends up with a stronger condition of commutativity implying that spinors at string world sheets behave like free particles. They do not act with induce gauge fields at string world sheets but just this avoidance behavior induces this interaction implicitly! Your behavior correlates with the behavior of the person whom you try to avoid! One must add that the TGD view about generalized Feynman graphs indeed allows to have non-trivial scattering matrix based on exchange of gauge bosons although the classical interaction vanishes.
Number theoretic dimensional hierarchy of dynamics
Number theoretical vision would imply a dimensional hierarchy of dynamics involving the dimensions of classical number fields. The classical dynamics for both space-times surface and spinors would simplify enormously but would be still consistent with standard model thanks to the topological view about interaction vertices as partonic 2-surfaces representing the ends of light-like 3-surface representing parton orbits and reducing the dynamics at fermion level to braid theory. Partonic 2-surfaces could be co-commutative in the sense that their normal space inside space-time surface is commutative at each point of the partonic 2-surface. The intersections of string world sheets and partonic 2-surfaces would consist of discrete points representing fermions. The light-like lines representing intersections of string world sheets with the light-like orbits of partonic 2-surfaces would correspond to orbits of point-like fermions (tangent vector of the light-like line would correspond to hypercomplex number with vanishing norm). The space-like boundary of string world sheet would correspond to real line. Therefore dimensional hierarchy would be realized.
The dimensional hierarchy would relate closely to both the generalization conformal invariance distinguish TGD from superstring models and to twistorialization. All "must be true" conjectures (physics geometry, physics as generalized number theory, M8-H duality, TGD as almost topological QFT, generalization of twistor approach to 8-D situation and induction of twistor structure, etc...) of TGD seems to converge to single coherent conceptual framework.