[Paper Review] Clifford Algebras in Physics

Michel Rausch de Traubenberg|ArXiv.org|Jun 1, 2005

Algebraic and Geometric Analysis3 citations

TL;DR

This paper demonstrates that Clifford algebras are fundamental in constructing supersymmetric extensions of the Poincaré algebra in four, ten, and eleven dimensions. By analyzing matrix representations of Clifford algebras, it establishes the existence of Majorana and Weyl spinors, and shows that supersymmetry naturally unifies bosons and fermions in equal numbers within supermultiplets, with lower-dimensional theories arising via dimensional reduction from eleven-dimensional supergravity.

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 Poincaré 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.

Motivation & Objective

• To establish Clifford algebras as the foundational mathematical framework for constructing supersymmetric extensions of the Poincaré algebra.
• To classify spinor types (Majorana, Weyl) based on space-time dimension using matrix representations of Clifford algebras.
• To demonstrate how four-, ten-, and eleven-dimensional supersymmetry arise from the same algebraic structure, with lower-dimensional theories obtained via dimensional reduction.
• To show that irreducible representations of supersymmetry algebras contain equal numbers of bosonic and fermionic degrees of freedom.
• To clarify the relationship between eleven-dimensional supergravity, type IIA/B superstrings, and four-dimensional N=8 supergravity through compactification.

Proposed method

• Use of real and complex Clifford algebras defined by anticommutation relations {e_M, e_N} = 2η_MN with metric signature (t,s).
• Matrix representation of Clifford algebras via Dirac Γ-matrices acting on 2^[d/2]-dimensional spinor spaces.
• Analysis of symmetry properties of Γ-matrices to classify Weyl and Majorana spinors based on dimension modulo 8.
• Construction of non-trivial extensions of the Poincaré algebra using spinor generators from Clifford algebra representations.
• Dimensional reduction of eleven-dimensional supersymmetry to four dimensions by compactifying extra dimensions, decomposing fields under Spin(2)×Spin(7) decomposition.
• Use of group theory and branching rules to decompose representations of Spin(9) into those of Spin(2)×Spin(7) in the context of dimensional reduction.

Experimental results

Research questions

• RQ1 How do Clifford algebras determine the existence and structure of Majorana and Weyl spinors in different space-time dimensions?
• RQ2 What is the precise algebraic mechanism by which Clifford algebras generate supersymmetric extensions of the Poincaré algebra?
• RQ3 How do four-dimensional N=8 supersymmetry and ten-dimensional type IIA/B supergravity emerge from eleven-dimensional supergravity via dimensional reduction?
• RQ4 Why do supermultiplets in supersymmetric theories contain equal numbers of bosonic and fermionic degrees of freedom?
• RQ5 What role do matrix representations of Clifford algebras play in the construction of supermultiplets in 4, 10, and 11 dimensions?

Key findings

• The 11-dimensional Clifford algebra C_{1,10} provides the algebraic foundation for eleven-dimensional supergravity, with its 16-dimensional spinor representation forming the basis of the supermultiplet.
• In four dimensions, the N=8 supersymmetry multiplet contains 1 graviton, 28 vectors, 56 spinors, and 35 scalars, with equal numbers of bosonic and fermionic degrees of freedom.
• Dimensional reduction of the 11D supermultiplet yields the 4D N=8 supergravity multiplet, with the field decomposition matching the representation content of the Spin(9) group.
• The 10-dimensional type IIA and IIB supermultiplets are derived from the 11D theory via compactification, with the IIB theory containing two left-handed spinors and two anti-self-dual four-forms.
• The 11D supermultiplet decomposes under Spin(2)×Spin(7) as 1 graviton, 7 vectors, 28 scalars, 8 gravitinos, and 58 spinors, confirming the consistency of the dimensional reduction.
• The existence of Majorana and Weyl spinors is determined by the dimension modulo 8, with Weyl spinors existing only in even dimensions and Majorana spinors in dimensions where the Clifford algebra admits real representations.