How TGD emerges from number theory?
An interesting possibility is that quantum TGD could emerge from a condition that a local version of hyper-finite factor of type II1 represented as a local version of infinite-dimensional Clifford algebra exists. The conditions are that "center or mass" degrees of freedom characterizing the position of causal diamond (CD) defined as an intersection of future and past directed light-cones separate uniquely from the "vibrational" degrees of freedom being represented in terms of octonions and that for physical states associativity holds true. The resulting local Clifford algebra would be identifiable as the local Clifford algebra of the world of classical worlds (being an analog of local gauge groups and conformal fields).
The uniqueness of M8 and M4×CP2 as well as the role of hyper-quaternionic space-time surfaces as fundamental dynamical objects indeed follow from rather weak conditions if one restricts the consideration to gamma matrices and spinors instead of assuming that M8 coordinates are hyper-octonionic as was done in the first attempts.
The importance of this result is that the Universe of Quantum TGD is mathematically completely unique: both classical and quantum dynamics follow from associativity alone.
- The unique feature of M8 and any 8-dimensional space with Minkowski signature of metric is that it is possible to have an octonionic representation of the complexified gamma matrices and of spinors. This does not require octonionic coordinates for M8. The restriction to a quaternionic plane for both gamma matrices and spinors guarantees the associativity.
- One can also consider a local variant of the octonionic Clifford algebra in M8. This algebra contains associative subalgebras for which one can assign to each point of M8 a hyper-quaternionic plane. It is natural to assume that this plane is either a tangent plane of 4-D manifold defined naturally by the induced gamma matrices defining a basis of tangent space or more generally, by modified gamma matrices defined by a variational principle (these gamma matrices do not define tangent space in general). Kähler action defines a unique candidate for the variational principle in question. Associativity condition would automatically select sub-algebras associated with 4-D hyper-quaternionic space-time surfaces.
- This vision bears a very concrete connection to quantum TGD. In the octonionic formulation of the modified Dirac equation is studied and shown to lead to a highly unique general solution ansatz for the equation working also for the matrix representation of the Clifford algebra. An open question is whether the resulting solution as such defined also solutions of the modified Dirac equation for the matrix representation of gammas. Also a possible identification for 8-dimensional counterparts of twistors as octo-twistors follows: associativity implies that these twistors are very closely related to the ordinary twistors. In TGD framework octo-twistors provide an attractive manner to get rid of the difficulties posed by massive particles for the ordinary twistor formalism.
- Associativity implies hyperquaternionic space-time surfaces (in a more general sense as usual) and this leads naturally to the notion of WCW and local Clifford algebra in this space. Number theoretic arguments imply M8-H duality. The resulting infinite-dimensional Clifford algebra would differ from von Neumann algebras in that the Clifford algebra and spinors assignable to the center of mass degrees of freedom of causal diamond CD would be expressed in terms of octonionic units although they are associative at space-time surfaces. One can therefore say that quantum TGD follows by assuming that the tangent space of the imbedding space corresponds to a classical number field with maximal dimension.
For details and background see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts .