Spinorial Algebra

Pascal-g1

Superspace is to supersymmetry as Minkowski space is to the Lorentz group. Superspace provides the most natural geometrical setting in which to describe supersymmetrical theories. Almost no physicist would utilize the component of Lorentz four-vectors or higher rank tensor to describe relativistic physics.

In a field theory, boson and fermions are to be regarded as diffeomorphisms generating two different vector spaces; the supersymmetry generators are nothing but sets of linear maps between these spaces. We can thus include a supersymmetric theory in a more general geometrical framework defining the collection of diffeomorphisms,

φi : R → RdL, i = 1,…, dL —– (1)

ψαˆ : R → RdR, i = 1,…, dR —– (2)

where the one-dimensional dependence reminds us that we restrict our attention to mechanics. The free vector spaces generated by {φi}i=1dL and {ψαˆ}αˆdR are respectively VL and VR, isomorphic to RdL and RdR. For matrix representations in the following, the two integers are restricted to the case dL = dR = d. Four different linear mappings can act on VL and VR

ML : VL → VR, MR : VR → VL

UL : VL → VL, UR : VR → VR —– (3)

with linear map space dimensions

dimML = dimMR = dRdL = d2,

dimUL = dL2 = d2, dimUR = dR2 = d2 —– (4)

as a consequence of linearity. To relate this construction to a general real (≡ GR) algebraic structure of dimension d and rank N denoted by GR(d,N), two more requirements need to be added.

Defining the generators of GR(d,N) as the family of N + N linear maps

LI ∈ {ML}, I = 1,…, N

RK ∈ {MR}, K = 1,…, N —– (5)

such that ∀ I, K = 1,…, N, we have

LI ◦ RK + LK ◦ RI = −2δIKIVR

RI ◦ LK + RK ◦ LI = −2δIKIVL —– (6)

where IVL and IVR are identity maps on VL and VR. Equations (6) will later be embedded into a Clifford algebra but one point has to be emphasized, we are working with real objects.

After equipping VL and VR with euclidean inner products ⟨·,·⟩VL and ⟨·,·⟩VR, respectively, the generators satisfy the property

⟨φ, RI(ψ)⟩VL = −⟨LI(φ), ψ⟩VR, ∀ (φ, ψ) ∈ VL ⊕ VR —— (7)

This condition relates LI to the hermitian conjugate of RI, namely RI, defined as usual by

⟨φ, RI(ψ)⟩VL = ⟨RI(φ), ψ⟩VR —– (8)

such that

RI = RIt = −LI —– (9)

The role of {UL} and {UR} maps is to connect different representations once a set of generators defined by conditions (6) and (7) has been chosen. Notice that (RILJ)ij ∈ UL and (LIRJ)αˆβˆ ∈ UR. Let us consider A ∈ {UL} and B ∈ {UR} such that

A : φ → φ′ = Aφ

B : ψ → ψ′ = Bψ —– (10)

with Vas an example,

⟨φ, RI(ψ)⟩VL → ⟨Aφ, RI B(ψ)⟩VL

= ⟨φ,A RI B(ψ)⟩VL

= ⟨φ, RI (ψ)⟩VL —– (11)

so a change of representation transforms the generators in the following manner:

LI → LI = BLIA

RI → RI = ARIB —– (12)

In general (6) and (7) do not identify a unique set of generators. Thus, an equivalence relation has to be defined on the space of possible sets of generators, say {LI, RI} ∼ {LI, RI} iff ∃ A ∈ {UL} and B ∈ {UR} such that L′ = BLIA and R′ = ARIB.

Moving on to how supersymmetry is born, we consider the manner in which algebraic derivations are defined by

δεφi = iεI(RI)iαˆψαˆ

δεψαˆ = −εI(LI)αˆiτφi —– (13)

where the real-valued fields {φi}i=1dL and {ψαˆ}αˆ=1dR can be interpreted as bosonic and fermionic respectively. The fermionic nature attributed to the VR elements implies that ML and MR generators, together with supersymmetry transformation parameters εI, anticommute among themselves. Introducing the dL + dR dimensional space VL ⊕ VR with vectors

Ψ = (ψ φ) —– (14)

(13) reads

δε(Ψ) = (iεRψ εL∂τφ) —– (15)

such that

ε1, δε2]Ψ = iε1Iε2J (RILJτφ LIRJτψ) – iε2Jε1I (RJLIτφ LJRIτψ) = – 2iε1Iε2IτΨ —– (16)

utilizing that we have classical anticommuting parameters and that (6) holds. From (16) it is clear that δε acts as a supersymmetry generator, so that we can set

δQΨ := δεΨ = iεIQIΨ —– (17)

which is equivalent to writing

δQφi = i(εIQIψ)i

δQψαˆ = i(εIQIφ)αˆ —– (18)

with

Q1 = (0LIH RI0) —– (19)

where H = i∂τ. As a consequence of (16) a familiar anticommutation relation appears

{QI, QJ} = − 2iδIJH —– (20)

confirming that we are about to recognize supersymmetry, and once this is achieved, we can associate to the algebraic derivations (13), the variations defining the scalar supermultiplets. However, the choice (13) is not unique, for this is where we could have a spinorial one,

δQξαˆ = εI(LI)αˆiFi

δQFi = − iεI(RI)iαˆτξαˆ —– (21)