nLab unitary representation of the super Poincaré group

Idea

A super-unitary representation of the super Poincaré group (supersymmetry)

The analog in superalgebra/supergeometry of a unitary representation of the Poincaré group.

An irreducible representation of the super Poincaré Lie algebra is called a super multiplet. This is in general itself a super vector space which contains an ordinary irreducible unitary representation of the Poincaré group. Via the Wigner classification of fundamental particles with Poincaré irreps, this may be identified with a bosonic relativistic particle of some mass – together with the images of these “bosonic” elements under the odd generators: the superpartners of the bosonic particles.

Related concepts

• Deligne's theorem on tensor categories

• spin representation

• BPS state

• supersymmetry

• Wigner classification

  • unitary representation of the Poincaré group

• Doplicher-Roberts reconstruction theorem

• quantum field theory

References

Reviews and lecture notes include

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.4.1 in Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Veeravalli Varadarajan, Unitary representations of super Lie groups (pdf)

• Daniel Freed, lecture 6 of Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)

• Daniel Freed, Lecture 3 of Five lectures on supersymmetry

• Antoine Van Proeyen, Tools for supersymmetry (arXiv:hep-th/9910030)

The classification is due to

• Werner Nahm, Supersymmetries and their representations, Nucl. Phys. B135 (1978) 149

The analogous discussion generalized to parasupersymmetry? is in

• A. Nikitin, V. Tretynyk, Irreducible representations of the Poincaré parasuperalgebra (pdf)

Expositional slides:

• Valerie Domcke, Supermultiplets (pdf) (in German)