Abstract
Based on an interpretation of the quark–lepton symmetry in terms of the unimodularity of the color group SU(3) and on the existence of 3 generations, we develop an argumentation suggesting that the “finite quantum space” corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.
Highlights
It is well known that the Standard Model of particles is very successful but contains several inputs which ought to have explainations at a fundamental level
Which can be grouped in 3 generations of doublets of quark-lepton which are the columns of the following table generations quarks Q = 2/3 u leptons Q = 0 νe quarks Q = −1/3 d leptons Q = −1 e c t νμ ντ s b μ τ where Q denotes the electric charge. This is the present experimental situation which reveals a sort of “triality”. This triality combined with the above interpretation of the quark-lepton symmetry is the starting point for the following analysis which suggests to add over each spacetime point an exceptional “finite quantum space” corresponding to the exceptional Jordan algebra J38 of the hermitian 3 × 3 octonionic matrices and to take the internal spaces of the basic fermions as elements of appropriate modules over this algebra
Note the obvious fact that a finite-dimensional Euclidean Jordan algebra J which is associative is the algebra of real functions on a finite space K which can be identified to the set of characters of J, that is the set of homomorphisms χ:J →R
Summary
It is well known that the Standard Model of particles is very successful but contains several inputs which ought to have explainations at a fundamental level. It is classical (see e.g. in [35], [4], [41]) that the automorphism group of the real algebra O of octonions is the first exceptional group G2 and that SU(3) identifies with the subgroup of G2 of automorphisms which preserve a given imaginary unit of O This is directly connected to the above construction which is explained in details in Section 2; the corresponding imaginary unit of O being the imaginary unit i of C. The group SU(3) is the group of automorphisms of this algebra which preserves the structure of 4-dimensional complex vector space of C ⊕ E or, equivalently, the group of all complex automorphisms of C ⊕ E which preserve the product given by (2.8)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
