Super-Poincaré algebra

In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part. In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part. The Poincaré algebra describes the isometries of Minkowski spacetime. From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed 2 {displaystyle 2} and 2 ¯ {displaystyle {overline {2}}} . Taking their tensor product, one obtains 2 ⊗ 2 ¯ = 3 ⊕ 1 {displaystyle 2otimes {overline {2}}=3oplus 1} ; such decompositions of tensor products of representations into direct sums is given by the Littlewood-Richardson rule. Normally, one treats such a decomposition as relating to specific particles: so, for example, the pion, which is a chiral vector particle, is composed of a quark-anti-quark pair. However, one could also identify 3 ⊕ 1 {displaystyle 3oplus 1} with Minkowski spacetime itself. This leads to a natural question: if Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry be extended to the fundamental representation? Well, it can: this is exactly the super-Poincaré algebra. There is a corresponding experimental question: if we live in the adjoint representation, then where is the fundamental representation hiding? This is the program of supersymmetry, which has not been found experimentally. The super-Poincaré algebra was first proposed in the context of the Haag–Łopuszański–Sohnius theorem, as a means of avoiding the conclusions of the Coleman–Mandula theorem. That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum. However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers. The proposal was to consider a supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries. The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation: and all other anti-commutation relations between the Qs and Ps vanish. In the above expression P μ {displaystyle P_{mu }} are the generators of translation and σ μ {displaystyle sigma ^{mu }} are the Pauli matrices. The index α {displaystyle alpha } runs over the values α = 1 , 2. {displaystyle alpha =1,2.} A dot is used over the index β ˙ {displaystyle {dot {eta }}} to remind that this index transforms according to the inequivalent conjugate spinor representation; one must never accidentally contract these two types of indexes. The Pauli matrices can be considered to be a direct manifestation of the Littlewood-Richardson rule mentioned before: they indicate how the tensor product 2 ⊗ 2 ¯ {displaystyle 2otimes {overline {2}}} of the two spinors can be re-expressed as a vector. The index μ {displaystyle mu } of course ranges over the space-time dimensions μ = 0 , 1 , 2 , 3. {displaystyle mu =0,1,2,3.} It is convenient to work with Dirac spinors instead of Weyl spinors; a Dirac spinor can be thought of as an element of 2 ⊕ 2 ¯ {displaystyle 2oplus {overline {2}}} ; it has four components. The Dirac matrices are thus also four-dimensional, and can be expressed as direct sums of the Pauli matrices. The tensor product then gives an algebraic relation to the Minkowski metric g μ ν {displaystyle g^{mu u }} which is expressed as: