Clifford Algebras in Physics

TL;DR

This paper explores the mathematical structure of Clifford algebras, their role in constructing spinors, and their application in formulating supersymmetric extensions of the Poincaré algebra across various space-time dimensions.

Contribution

It provides a detailed analysis of Clifford algebra representations, spinor types, and their use in building supersymmetric theories in multiple dimensions, highlighting the connections between different supersymmetries.

Findings

Clifford algebras are identified as matrix algebras.

Spinor types depend on space-time dimension, e.g., Majorana or Weyl.

Supersymmetric extensions of Poincaré algebra are constructed and analyzed.

Abstract

We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The precise structure of these matrices gives rise to the type of spinors one is able to construct in a given space-time dimension: Majorana or Weyl. Properties of spinors are also studied. We finally show how Clifford algebras enable us to construct supersymmetric extensions of the Poincar\'e algebra. A special attention to the four, ten and eleven-dimensional space-times is given. We then study the representations of the considered supersymmetric algebras and show that representation spaces contain an equal number of bosons and fermions. Supersymmetry turns out to be a symmetry which mixes non-trivially the bosons and the fermions since one multiplet contains bosons and fermions together. We also show how supersymmetry in four and ten dimensions are related to eleven dimensional supersymmetry by compactification or dimensional reduction.