Abstract
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.
Highlights
A reduced number of free parameters by enlarging the local symmetry group
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
In this paper we present a workable example of a unified local symmetry group of the above kind, along with a corresponding toy model, where the above type gauge fields with “exotic” charges, necessary for a gauge-spacetime type symmetry unification, can be transformed out from the Lagrangian
Summary
Whenever some particle field theory has a classical field theory limit, one has a firm mathematical handle on the notion of its symmetry generators: the generators of the continuous symmetries of the theory are smooth vector fields on some kind of a total space of fields of the theory, which respect certain mathematical structures associated to the model. As discussed in [9] and the appendix A, the Lie algebra of the super-Poincaré group can be considered as an example to the Levi-Mal’cev decomposition, with a non-abelian, but two-step nilradical. In model building one often invokes a Yang-Mills-like kinetic Lagrangian term, with the requirement that all gauge fields propagate This requirement is satisfied if and only if the Lie algebra of the internal group has an invariant, non-degenerate scalar product. An important sub-class of quadratic Lie algebras are the reductive ones, admitting faithful finite dimensional completely reducible representations, which are most commonly used in model building, and are always direct sums of copies of u(1) and of simple Lie algebras. A quadratic Lie algebra is compact if its invariant scalar product is positive definite These are always reductive, and the Standard Model internal Lie algebra u(1)⊕su(2)⊕su(3) provides an example. Due to the O’Raifeartaigh theorem, these have to carry a nontrivial nilradical, if they are indecomposable (unified)
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