R \(\otimes \) C \(\otimes \) H \(\otimes \) O-Valued Gravity as a Grand Unified Field Theory

  • Carlos Castro PerelmanEmail author


It is argued how R \(\otimes \) C \(\otimes \) H \(\otimes \) O-valued Gravity (real-complex-quaterno-octonionic Gravity) naturally can describe a Grand Unified Field theory of Einstein’s gravity with a Yang-Mills theory containing the Standard Model group \(SU(3) \times SU(2) \times U(1)\). In particular, it leads to a \([SU(4)]^4\) symmetry group revealing the possibility of extending the standard model by introducing additional gauge bosons, heavy quarks and leptons, and an extra fourth family of fermions. We finalize by displaying the analog of the Einstein–Hilbert action for \(\mathbf{R} \otimes \mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O}\)-valued gravity via the use of matrices, and which is based on “coloring” the graviton; i.e. by attaching internal indices to the metric \(g_{\mu \nu }\). In the most general case, U(16) arises as the isometry group, while U(8) is the isometry group in the split-octonion case.


Nonassociative Geometry Clifford algebras Quaternions Octonionic Gravity Unification Strings 



We are indebted to M. Bowers for invaluable assistance in preparing the manuscript. Special thanks to T. Smith for numerous discussions of his work, and to the referees for many suggestions to improve the manuscript.


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Authors and Affiliations

  1. 1.Center for Theoretical Studies of Physical SystemsClark Atlanta UniversityAtlantaUSA
  2. 2.Ronin InstituteMontclairUSA

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