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^{dL}, i = 1,…, d_{L} —– (1)

ψ_{αˆ} : R → R^{dR}, 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_{αˆ}}_{α}^{ˆdR }are respectively V_{L} and V_{R}, isomorphic to R^{dL} and R^{dR}. 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_{R}d_{L} = d^{2},

dimU_{L} = d_{L2} = d^{2}, dimU_{R} = d_{R2} = d^{2} —– (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_{IK}I_{VR}

R_{I} ◦ L_{K} + R_{K} ◦ L_{I} = −2δ_{IK}I_{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_{I}L_{J})_{i}^{j} ∈ U_{L} and (L_{I}R_{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_{I}A

R_{I} → R_{I}^{′} = A^{†}R_{I}B —– (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_{I}A and R′ = A^{†}R_{I}B.

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

δ_{ε}φ_{i} = iε^{I}(R_{I})_{i}^{αˆ}ψ^{αˆ}

δ_{ε}ψ_{αˆ} = −ε^{I}(L_{I})_{αˆ}^{i}∂_{τ}φ_{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 = iε_{1}^{I}ε_{2}^{J} (^{RILJ∂τφ} _{LIRJ∂τψ}) – iε_{2}^{J}ε_{1}^{I} (^{RJLI∂τφ} _{LJRI∂τψ}) = – 2iε_{1}^{I}ε_{2}^{I}∂_{τ}Ψ —– (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_{Q}Ψ := δ_{ε}Ψ = iε^{I}Q_{I}Ψ —– (17)

which is equivalent to writing

δ_{Q}φ_{i} = i(ε^{I}Q_{I}ψ)_{i}

δ_{Q}ψ_{αˆ} = i(ε^{I}Q_{I}φ)_{αˆ} —– (18)

with

Q_{1} = (^{0}_{LIH} ^{RI}_{0}) —– (19)

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

{Q_{I}, Q_{J}} = − 2iδ_{IJ}H —– (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_{Q}ξ_{αˆ} = ε^{I}(L_{I})_{αˆ}^{i}F_{i}

δ_{Q}F_{i} = − iε^{I}(R_{I})_{i}^{αˆ}∂_{τ}ξ_{αˆ} —– (21)