Jordan algebras and TGD

Lawrence B. Crowell had an FB posting related to never ending blackhole wars mentioning Jordan algbras. Jordan algebra are certainly a nice algebraic structure. If I understand correctly, LBC believes that Jordan at the level of operator algebras of quantum could lead to a new quantum theory. Jordan algebra has a product which is simply the symmetrized product of operators (AB+BA)/2 (see this). It was proposed to solve the ordering problems of operator formalism of quantum theory. There are however heavy interpretational problems if one wants to interpret operator algebra in Hilbert space as Jordan algebra.

1. One starts from algebra of observables as in the operator formalism plagued by the normal ordering problems implying non-uniqueness. In quantum field theories this approach leads to a problem with infinities.
2. Situation changes if the observables form a Lie algebra. This allows to get rid of the normal ordering difficulties and makes sense also in infinite-D case. In TGD, the “world of classical worlds” (WCW) for Bohr orbits of particles as 3-surfaces satisfying holography= holomorphy principle has geometry only if it has infinite-D group of isometries. In the case of loop spaces this fixes the geometry completely and the Lie algebra is infinite-dimensional meaning infinite number of observables. In TGD the same is expected to be true.
3. The locality of linear superposition for Jordan algebras would suggest to me to ask whether could make sense to assign Jordan algebra structure with the tangent space of an infinite-D manifold rather than the Hilbert space of quantum states. In TGD this space would be the “world of classical worlds” (WCW), whose tangent space has the structure of Hilbert space.
4. Jordan algebra is non-associative and there seems to be no interpretation in terms of symmetries. The author of the above article suggests that Jordan algebras define what he calls generalized projective geometry.
5. Jordan algebra approach forces to give up lineary of quantum theory and to replace it with something called local linearity. I am really worried.

What about the situation in TGD?

1. The tangent space of WCW can be given Hilbert space structure. Could Jordan algebra be assigned with the tangent space geometry of WCW rather than the space of quantum states and be a useful companion for the Lie algebra structure, say in the case of WCW and its infinite-D Lie algebra of symmetries?
2. In TGD super-conformal/super-symplectic symmetries imply that one can assign to the Lie-algebra generators of isometries of WCW super-counterparts carrying fermion number as WCW gamma matrices and their hermitian conjugates defining WCW spinor structure. Super generators as elements of Jordan algebra are contractions of isometry generators with gamma matrices. This I discovered already 35 years ago: (see this, this, and this) .
3. The WCW gamma matrices satisfy anticommutation relations and anticommutator corresponds to the Jordan product giving the WCW metric. Jordan algebra structure (Clifford algebra structure) would be associated with the fermionic sector of the state space and Lie-algebra structure with its bosonic sector.
4. This means geometrization of supersymmetries: no superspace is introduced and no Majorana spinors are needed. WCW gamma matrices can be expressed as linear superpositions of the fermionic oscillator operators for the second quantized free spinor fields of H=M⁴×CP₂: all problems related to the quantization of fermions appearing in QFTs in curved baskgrounds are avoided.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.