The first problem is that if the 10-bein group defines the gauge group, the gauge group for a Minkowskian signature of X is non-compact variant of SO(10), which is the group of isometries for the space of M_{10} with Euclidean signature. In gauge theories non-compactness of the gauge group implies the loss of unitarity. Weinstein admits that his proposal works only in the Euclidean case.

Second problem is posed by the general coordinate invariance. General coordinate transformations do not induce a mere gauge transformation of the matrix of M_{10} as they should. This could mean severe difficulties in the realization of the general coordinate invariance.

In the TGD framework, one of the challenges is the more precise definition of the QFT limit of TGD. In this article I will consider a variant of Weinstein's theory obtained by replacing H=M^{4}× CP_{2} with M^{4}× S^{n} as a possible manner to approach the problem. For n=9 and n=10 one obtains SO(n+1) as maximal isometry group and holonomy group. It turns out that one can obtain standard model symmetries but the predicted number of fermion families turns out to be wrong. In TGD fermion families have a topological explanation. M can be replaced by a sphere S^{n}, and n=10 gives 4 generations and n=8 and n=9 2 generations. For larger values of n the number generations increases exponentially. Whether the QFT model could serve as a phenomenological description of the family replication phenomenon remains open.

In this article, I will consider a variant of Weinstein's theory obtained by replacing H=M^{4}× CP_{2} with M^{4}× S^{n}. For n=9 and n=10 one obtains SO(n+1) as maximal isometry group and holonomy group. It turns out that one can obtain standard model symmetries but the predicted number of fermion families turns out to be wrong. In TGD fermion families have a topological explanation. M can be replaced by a sphere S^{n}, and n=10 gives 4 generations and n=8 and n=9 2 generations. For larger values of n the number generations increases exponentially. Whether the QFT model could serve as a phenomenological description of the family replication phenomenon remains open.

See the article Could a TGD analog of Weinstein's proposal help to define the QFT limit of TGD? or the chapter with the same title.

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.