Spinorial Algebra

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,

φ_i : R → R^d_L, i = 1,…, d_L —– (1)

ψ_α^ˆ : R → R^d_R, i = 1,…, d_R —– (2)

where the one-dimensional dependence reminds us that we restrict our attention to mechanics. The free vector spaces generated by {φ_i}_i=1^dL and {ψ_α^ˆ}_α^ˆdR are respectively V_L and V_R, isomorphic to R^d_L and R^d_R. For matrix representations in the following, the two integers are restricted to the case d_L = d_R = d. Four different linear mappings can act on V_L and V_R

M_L : V_L → V_R, M_R : V_R → V_L

U_L : V_L → V_L, U_R : V_R → V_R —– (3)

with linear map space dimensions

dimM_L = dimM_R = d_Rd_L = d²,

dimU_L = d_L² = d², dimU_R = d_R² = 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

L_I ∈ {M_L}, I = 1,…, N

R_K ∈ {M_R}, K = 1,…, N —– (5)

such that ∀ I, K = 1,…, N, we have

L_I ◦ R_K + L_K ◦ R_I = −2δ_IKI_VR

R_I ◦ L_K + R_K ◦ L_I = −2δ_IKI_VL —– (6)

where I_VL and I_VR are identity maps on V_L and V_R. 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 V_L and V_R with euclidean inner products ⟨·,·⟩_VL and ⟨·,·⟩_VR, respectively, the generators satisfy the property

⟨φ, R_I(ψ)⟩V_L = −⟨L_I(φ), ψ⟩V_R, ∀ (φ, ψ) ∈ V_L ⊕ V_R —— (7)

This condition relates L_I to the hermitian conjugate of R_I, namely R_I^†, defined as usual by

⟨φ, R_I(ψ)⟩_VL = ⟨R_I^†(φ), ψ⟩_VR —– (8)

such that

R_I^† = R_I^t = −L_I —– (9)

The role of {U_L} and {U_R} maps is to connect different representations once a set of generators defined by conditions (6) and (7) has been chosen. Notice that (R_IL_J)_i^j ∈ U_L and (L_IR_J)_αˆ^βˆ ∈ U_R. Let us consider A ∈ {U_L} and B ∈ {U_R} such that

A : φ → φ′ = Aφ

B : ψ → ψ′ = Bψ —– (10)

with V_L as an example,

⟨φ, R_I(ψ)⟩_VL → ⟨Aφ, R_I B(ψ)⟩_VL

= ⟨φ,A^† R_I B(ψ)⟩_VL

= ⟨φ, R_I^′ (ψ)⟩_VL —– (11)

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

L_I → L_I^‘ = B^†L_IA

R_I → R_I^′ = A^†R_IB —– (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 {L_I, R_I} ∼ {L_I^‘, R_I^′} iff ∃ A ∈ {U_L} and B ∈ {U_R} such that L′ = B^†L_IA and R′ = A^†R_IB.

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

δ_εφ_i = iε^I(R_I)_i^αˆψ^αˆ

δ_εψ_αˆ = −ε^I(L_I)_αˆⁱ∂_τφ_i —– (13)

where the real-valued fields {φ_i}_i=1^dL and {ψ_αˆ}_αˆ=1^dR can be interpreted as bosonic and fermionic respectively. The fermionic nature attributed to the V_R elements implies that M_L and M_R generators, together with supersymmetry transformation parameters ε^I, anticommute among themselves. Introducing the d_L + d_R dimensional space V_L ⊕ V_R with vectors

Ψ = (^ψ _φ) —– (14)

(13) reads

δ_ε(Ψ) = (^iεRψ _εL∂τφ) —– (15)

such that

[δ_ε1, δ_ε2]Ψ = iε₁^Iε₂^J (^{R_IL_J∂_τφ} _LIRJ∂τψ) – iε₂^Jε₁^I (^{R_JL_I∂_τφ} _LJRI∂τψ) = – 2iε₁^Iε₂^I∂_τΨ —– (16)

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

δ_QΨ := δ_εΨ = iε^IQ_IΨ —– (17)

which is equivalent to writing

δ_Qφ_i = i(ε^IQ_Iψ)_i

δ_Qψ_αˆ = i(ε^IQ_Iφ)_αˆ —– (18)

with

Q1 = (0LIH RI0) —– (19)

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

{Q_I, Q_J} = − 2iδ_IJH —– (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,

δ_Qξ_αˆ = ε^I(L_I)_αˆⁱF_i

δ_QF_i = − iε^I(R_I)_i^αˆ∂_τξ_αˆ —– (21)

Advertisement