Abstract
Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie super group OSpC(1|2) acting on Minkowski (3,1|4)-dimensional super space essentially by Hermitean conjugation. The invariant quadratic form is x1x2−xx¯+ϕψ+ϕ¯ψ¯ in Wigner's coordinates x1,x2∈R and x∈C, where ϕ,ψ are Dirac fermions. The extended action is linear in the super space variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie super group with its Lie algebra. This extension of Wigner's representation opens the door to studying geometry in super hyperbolic three-space with this metric, initiated here with foundations on super geodesics, triangles and tetrahedra, and culminating in a proof of divergence of the volume of a typical ideal tetrahedron and a discussion of the non-zero fermionic correction to the Schläfli formula.
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