Differential calculus on Jordan algebras and Jordan modules

Exceptional quantum geometry and particle physics II

“…There is an important step remaining which is to make the relation with the theory of Jordan modules in particular with the theory of J 8 3 -modules. This is necessary for instance in order to use the differential calculus on Jordan algebras and Jordan modules developed in [13] and in [5] to formulate the relevant dynamical theory and to get an interpretation of the Higgs sector. This is also of course important for the problems of classification.…”

Spectral Noncommutative Geometry Standard Model and all that

“…Such a formulation would be very interesting from the gravitational point of view. Indeed, in the zeta-function-based spectral Lagrangian the only allowed gravitational term would be the Weyl tensor square, what is nothing but a classical Lagrangian of conformal gravity 30 . This higher derivative gravitational model is renormalizable by 30 In general, the zeta-based scale-invariant spectral Lagrangian may contain φ 2 R and φ 4 terms, where power counting, and, what is also important, PT-symmetric.…”

“…Indeed, in the zeta-function-based spectral Lagrangian the only allowed gravitational term would be the Weyl tensor square, what is nothing but a classical Lagrangian of conformal gravity 30 . This higher derivative gravitational model is renormalizable by 30 In general, the zeta-based scale-invariant spectral Lagrangian may contain φ 2 R and φ 4 terms, where power counting, and, what is also important, PT-symmetric. Generally speaking the higher-derivative theories are non unitary, however, a presence of the PT-symmetry allows to quantise it in a unitary way [102], using the PT-symmetry Bender-Mannheim formalism [21,101], see also [8,[106][107][108] for the recent progress PT-symmetric studies in particular in the context of the noncommutative geometry [109].…”

“…Unfortunately in this context we have no explanation for the presence of three generations, with identical quantum numbers, except for the different masses of the fermions. An extension of the model involving the Jordan algebra of Hermitean octonionic matrices might give an explanation, as discussed in [30,63].…”

Connections on Jordan modules and particle physics

“…As we shall review in the following sections, in [8] the author shows how these two missing features could be obtained by allowing Jordan algebras to play the role of quantum observables in the Standard Model. In the last section we shall present some recent development in the theory (which are given with full details in [3]), in particular we will show the explicit description of the theory of connections for modules over Jordan algebras which will be equivalent, once provided a certain physical interpretation, to present the admissible potentials of the theory. Moreover, we characterized flat connections which are known in quantum field theories to be in one to one correspondence with vacua states.…”