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The N=1 supercharge bracket is a translation

Example

Use the complexified algebra of Complexified four-dimensional N=1 super-Poincare bracket data and The complexified N=1 super-Poincare bracket data satisfy super-Jacobi. Let $u_{1} = (1, 0), u_{2} = (0, 1)$ be the standard basis of $S_{+}$ , and let $ϕ_{1} (z_{1}, z_{2}) = z_{1}, ϕ_{2} (z_{1}, z_{2}) = z_{2}$ be the coordinate-dual basis of $S_{-}$ . Set $Q_{α} = u_{α}$ , ${\overline{Q}}_{\dot{β}} = ϕ_{β}$ , and $P_{α \dot{β}} = u_{α} \otimes ϕ_{β}$ . These four $P_{α \dot{β}}$ form a basis of the translation space, and $[Q_{α}, {\overline{Q}}_{\dot{β}}] = [{\overline{Q}}_{\dot{β}}, Q_{α}] = P_{α \dot{β}} .$ For the explicit Pauli basis and dual-basis translation elements defined in step 3.1, this becomes the familiar convention ${Q_{α}, {\overline{Q}}_{\dot{β}}} = 2 i σ_{α \dot{β}}^{μ} P_{μ}$ . The braces denote the superbracket of two odd elements, hence the anticommutator whenever the action is realized by odd elements of an associative superalgebra.

Facts & Assumptions

Given: The bracket data of the complexified N=1 super-Poincare superalgebra and the tensor-product translation space $V = S_{+} \otimes S_{-}$ .

[F1]

Complexified four-dimensional N=1 super-Poincare bracket data: $[u, ϕ] = u \otimes ϕ$ for $u \in S_{+}$ , $ϕ \in S_{-}$ , and the translation space is $V = S_{+} \otimes S_{-}$ .

[F2]

The complexified N=1 super-Poincare bracket data satisfy super-Jacobi: this bracket data define a complex Lie superalgebra.

[F3]

Associative superalgebra over the complex numbers: in a graded associative representation, the superbracket of odd operators is their anticommutator.

[F4]

The elementary tensors of two bases form the product basis of the tensor product: for free modules over a commutative ring with specified bases, the elementary tensors form a basis of their tensor product.

[F5]

Linear functionals and the algebraic dual $V^{*} = L (V, F)$ : the algebraic dual consists of all complex-linear maps from $S_{+}$ to $C$ .

[F6]

Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring: the complex field is a commutative ring.

Verification

technique · direct

1.1givenF1F4F5F6algebra

The coordinate forms $ϕ_{1}, ϕ_{2}$ are linear by [F5]. For any $ϕ \in S_{-}$ and $(z_{1}, z_{2}) \in S_{+}$ , linearity gives $ϕ (z_{1}, z_{2}) = z_{1} ϕ (u_{1}) + z_{2} ϕ (u_{2})$ , hence $ϕ = ϕ (u_{1}) ϕ_{1} + ϕ (u_{2}) ϕ_{2}$ ; evaluating at $u_{1}, u_{2}$ proves their independence. Thus $u_{1}, u_{2}$ and $ϕ_{1}, ϕ_{2}$ are bases of the two free $C$ -modules. By [F6], the scalar ring is commutative; [F4] then makes their four elementary tensors a basis of $S_{+} \otimes S_{-}$ and hence of the translation space.

2.1F1F2F3step 1.1algebra

The bracket formula in [F1] gives $[Q_{α}, {\overline{Q}}_{\dot{β}}] = P_{α \dot{β}}$ . Because both entries are odd, graded skew symmetry makes this bracket symmetric; in a graded associative representation [F3] identifies it with the anticommutator. Same-chirality odd brackets vanish and translations commute by the definition in [F1].

3.1givenF1F4step 1.1algebra∎

Let $σ^{0} = I$ , $σ^{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $σ^{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ , and $σ^{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ . Direct multiplication gives $tr (σ^{μ} σ^{ν}) = 2 δ_{μ ν}$ , so $τ^{μ} = σ^{μ} / 2$ is trace-dual to the four $σ^{ν}$ . The trace-pairing identity makes the $σ^{μ}$ linearly independent; since a $2 \times 2$ matrix has four entries, they form a basis of $M_{2} (C)$ . Regard $P = (P_{α \dot{β}})$ as a matrix with entries in $V$ , and define $P_{μ} = (2 i)^{- 1} tr (τ^{μ} P)$ . The dual-basis expansion gives $P = \sum_{μ} σ^{μ} tr (τ^{μ} P) = 2 i \sum_{μ} σ^{μ} P_{μ}$ . This is an invertible coordinate change followed by multiplication by the nonzero scalar $(2 i)^{- 1}$ , so the $P_{μ}$ form a basis of translations and the displayed normalization follows. The dotted bar on $\overline{Q}$ is a chiral label, not complex conjugation. This algebraic normalization supplies no real form, positive-energy representation, or field-theoretic off-shell closure.

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Sources

José Miguel Figueroa-O'Farrill, BUSSTEPP Lectures on Supersymmetry, §I.3 and Appendix A.2–A.3 (standard reference, not scraped)
Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.2 (standard reference, not scraped)