The Four-Dimensional N=1 Super-Poincaré Algebra

In four-dimensional Lorentzian spacetime, the minimal super-Poincaré algebra has one left-handed Weyl charge $Q_\alpha$ and its Hermitian adjoint $\bar Q_{\dot\alpha}$ . Their only nonzero odd bracket is $\{Q_\alpha,\bar Q_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu$ . This normalization makes positive energy, the massless rank reduction, and closure onto translations immediately checkable.

Required background. Graded spacetime symmetry supplies the classification hypotheses and graded Jacobi identity. Dimensions, signatures, and reality explains why one Weyl charge plus its adjoint means four real supercharges.

Helpful background. Grassmann variables and Berezin integration supplies odd parameters, and continuous symmetries, generators, and charges supplies the operator convention for transformations.

Two-component convention

We use the inherited $(+---)$ metric in Lorentzian signature. Undotted indices $\alpha,\beta=1,2$ transform in $(\tfrac12,0)$ ; dotted indices $\dot\alpha,\dot\beta=1,2$ transform in $(0,\tfrac12)$ . Set

\epsilon^{12}=+1, \qquad \epsilon_{12}=-1, \qquad \psi^\alpha=\epsilon^{\alpha\beta}\psi_\beta, \qquad \psi_\alpha=\epsilon_{\alpha\beta}\psi^\beta.

The sigma matrices are

\sigma^\mu_{\alpha\dot\alpha}=(\mathbf 1,\boldsymbol\sigma), \qquad \bar\sigma^{\mu\dot\alpha\alpha}=(\mathbf 1,-\boldsymbol\sigma),

so that

\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu =2\eta^{\mu\nu}\mathbf 1.

Useful Lorentz generators are

\begin{aligned} (\sigma^{\mu\nu})_\alpha{}^\beta &=\frac14(\sigma^\mu\bar\sigma^\nu-\sigma^\nu\bar\sigma^\mu)_\alpha{}^\beta,\\ (\bar\sigma^{\mu\nu})^{\dot\alpha}{}_{\dot\beta} &=\frac14(\bar\sigma^\mu\sigma^\nu-\bar\sigma^\nu\sigma^\mu)^{\dot\alpha}{}_{\dot\beta}. \end{aligned}

No explicit matrix representation beyond the Pauli matrices is used. Complex conjugation maps an undotted spinor to a dotted one; it does not raise an undotted index. The detailed two-component algebra in this convention is the site translation of Weinberg 2000, § 25.2, pp. 29–40, whose book convention uses the opposite metric signature. The invariant round-trip checks are the rest-frame positive anticommutator and closure onto the same translation generator below.

The algebra and adjoints

Take $P_\mu$ and $M_{\mu\nu}=-M_{\nu\mu}$ Hermitian, with

\begin{aligned} [M_{\mu\nu},P_\rho] &=i(\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu),\\ [M_{\mu\nu},M_{\rho\sigma}] &=i(\eta_{\mu\sigma}M_{\nu\rho} +\eta_{\nu\rho}M_{\mu\sigma} -\eta_{\mu\rho}M_{\nu\sigma} -\eta_{\nu\sigma}M_{\mu\rho}). \end{aligned}

The odd generators obey

(Q_\alpha)^\dagger=\bar Q_{\dot\alpha}, \qquad (\bar Q_{\dot\alpha})^\dagger=Q_\alpha,

with the placement on the second equation understood through the epsilon tensors. Their brackets are

\begin{aligned} \{Q_\alpha,\bar Q_{\dot\beta}\} &=2\sigma^\mu_{\alpha\dot\beta}P_\mu,\\ \{Q_\alpha,Q_\beta\} &=0, & \{\bar Q_{\dot\alpha},\bar Q_{\dot\beta}\} &=0,\\ [P_\mu,Q_\alpha]&=0, & [P_\mu,\bar Q_{\dot\alpha}]&=0, \end{aligned}

and Lorentz covariance is expressed by

\begin{aligned} [M_{\mu\nu},Q_\alpha] &=i(\sigma_{\mu\nu})_\alpha{}^\beta Q_\beta,\\ [M_{\mu\nu},\bar Q^{\dot\alpha}] &=i(\bar\sigma_{\mu\nu})^{\dot\alpha}{}_{\dot\beta} \bar Q^{\dot\beta}. \end{aligned}

For $\mathcal N=1$ , a scalar central term in $\{Q_\alpha,Q_\beta\}$ would have the form $\epsilon_{\alpha\beta}Z$ . The anticommutator is symmetric under exchanging the complete labels, whereas $\epsilon_{\alpha\beta}$ is antisymmetric; with no second supersymmetry index to supply another antisymmetry, $Z$ must vanish. Extended supersymmetry evades this by using $Z^{IJ}=-Z^{JI}$ , as developed on extended supersymmetry and central charges.

Three Jacobi checks

The algebra is compact, but its factors are not decorative.

Translation check. The graded Jacobi identity for $(P_\rho,Q_\alpha,\bar Q_{\dot\beta})$ gives

[P_\rho,\{Q_\alpha,\bar Q_{\dot\beta}\}]=0,

which holds because translations commute. It also explains why $M_{\mu\nu}$ cannot occur on the right-hand side.

Lorentz check. Acting with $M_{\mu\nu}$ on $\{Q_\alpha,\bar Q_{\dot\beta}\}$ produces the sum of the undotted and dotted spinor actions. The sigma identity

\sigma_{\mu\nu}\sigma_\rho +\sigma_\rho\bar\sigma_{\mu\nu} =\eta_{\nu\rho}\sigma_\mu-\eta_{\mu\rho}\sigma_\nu

turns that sum into the vector transformation of $P_\rho$ . A wrong sign in either Lorentz action fails this check.

R-symmetry check. The abstract algebra admits the automorphism

Q_\alpha\longmapsto e^{-i\varphi}Q_\alpha, \qquad \bar Q_{\dot\alpha}\longmapsto e^{+i\varphi}\bar Q_{\dot\alpha}.

With a Hermitian generator $R$ this convention is $[R,Q_\alpha]=-Q_\alpha$ and $[R,\bar Q_{\dot\alpha}]=+\bar Q_{\dot\alpha}$ . The mixed anticommutator is neutral. Whether this $U(1)_R$ is a symmetry of a Lagrangian, survives anomalies, or is preserved by the vacuum is a separate dynamical question.

Positivity and the spectrum condition

For any commuting test spinor $z^\alpha$ , define $A=z^\alpha Q_\alpha$ . Then

\{A,A^\dagger\} =2z^\alpha\sigma^\mu_{\alpha\dot\beta} \bar z^{\dot\beta}P_\mu.

\{Q_\alpha,Q_\beta^\dagger\}=2m\,\delta_{\alpha\beta}.

For a future null momentum the $2\times2$ matrix $\sigma\cdot P$ has rank one, so half the complex charge components act trivially in an irreducible massless representation. These are the seeds of the massive and massless oscillator constructions on unitary supermultiplets.

Taking the trace at arbitrary momentum gives the operator identity

H=P_0=\frac14\sum_{\alpha=1}^2 \{Q_\alpha,Q_\alpha^\dagger\}\geq0.

If a normalizable state has zero energy, every term in this sum has zero expectation value, hence both $Q_\alpha$ and $Q_\alpha^\dagger$ annihilate it. The converse is immediate. This is an algebraic statement about a positive representation; it does not by itself decide whether a theory possesses such a vacuum.

Closure onto a spacetime translation

A quick field-level round trip checks the normalization. For a free chiral multiplet $(A,\psi_\alpha,F)$ , choose

\begin{aligned} \delta_\xi A&=\sqrt2\,\xi\psi,\\ \delta_\xi\psi_\alpha &=i\sqrt2(\sigma^\mu\bar\xi)_\alpha\partial_\mu A +\sqrt2\,\xi_\alpha F. \end{aligned}

Using anticommuting parameters and the conjugate transformations,

[\delta_\xi,\delta_\eta]A =a^\mu\partial_\mu A, \qquad a^\mu=2i(\xi\sigma^\mu\bar\eta-\eta\sigma^\mu\bar\xi).

The auxiliary terms cancel on the scalar. Closure on every component, including the distinction between off-shell and on-shell closure, is developed on component multiplets and closure. Here the calculation only verifies that the operator algebra and field convention generate the same translation parameter. The first interacting four-dimensional realizations were constructed by Wess and Zumino in Wess and Zumino 1974, pp. 39–50.

Common pitfalls

Mixing metric conventions inside a sigma identity. With $(+---)$ , the site uses $\sigma^\mu=(1,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(1,-\boldsymbol\sigma)$ so their symmetrized product is $+2\eta^{\mu\nu}$ . Importing a $(-+++)$ identity without its compensating sign breaks positivity or Lorentz closure.

Treating dotted indices as raised undotted indices. Dotted and undotted indices label inequivalent complex Lorentz representations. Epsilon tensors raise indices within one representation; Hermitian conjugation changes undotted to dotted.

Calling the $U(1)_R$ automorphism an exact quantum symmetry. The algebra admits it. An action, regulator, anomaly, coupling, or vacuum can preserve only a subgroup or none of it.

Check your understanding

Show directly that a supersymmetric state has nonnegative energy and that zero energy forces every supercharge to annihilate it.

Answer

Use $4H=\sum_\alpha\{Q_\alpha,Q_\alpha^\dagger\}$ . Its expectation value is a sum of squared norms, so it is nonnegative. If it vanishes, each norm $\|Q_\alpha|\Psi\rangle\|$ and $\|Q_\alpha^\dagger|\Psi\rangle\|$ vanishes separately.

References

Steven Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press (2000), § 25.2, DOI.
Julius Wess and Bruno Zumino, “Supergauge Transformations in Four Dimensions,” Nuclear Physics B 70 (1974), 39–50, DOI.