Nilpotent symmetries as a mechanism for Grand Unification

• Published in: The journal of high energy physics Vol. 2021; no. 5; pp. 1 - 39
• Main Authors: Andersson, Lars, László, András, Ruba, Błażej
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-05-2021; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Dirac equation; Elementary Particles; Extended Supersymmetry; Gauge Symmetry; Generators; Gluing; Group theory; High energy physics; Invariants; Lie groups; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity; Relativity Theory; Space-Time Symmetries; Spacetime; String Theory; Supersymmetry; Symmetry; Theorems; Gauge Symmetry; Space-Time Symmetries; Extended Supersymmetry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP05(2021)240

A
bstract
In the classic Coleman-Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive
semi
-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact (U(1)) component, a Lorentz (SL(2
,
ℂ)) component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman-Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.