Supersymmetry

Remarks

Example: the $N = 1$ supersymmetry algebra in four dimensions. In two-component notation the supercharges are $Q_\alpha$ and $\bar Q_{\dot\alpha}$, with $\alpha, \dot\alpha \in \set{1, 2}$, and the only non-zero brackets among them are $$\set{Q_\alpha, \bar Q_{\dot\beta}} = 2 \sigma^\mu_{\alpha\dot\beta} P_\mu,$$ where $\sigma^\mu = (I_2, \sigma^1, \sigma^2, \sigma^3)$ uses the Pauli matrices. The supercharges $Q_\alpha$ and $\bar Q_{\dot\alpha}$ form the two spinor representations of $\mathfrak{sl}(2, \C)$, together one real (Majorana) spinor, so $N = 1$, and their anticommutators are translations, so clauses 2 and 3 hold.

Non-example: a Lie superalgebra whose odd part is a vector representation of the Lorentz algebra fails clause 2. By the spin–statistics connection, odd generators of a spacetime symmetry must have half-integer spin.

Consequences: in a unitary representation the energy is non-negative, because $H$ is a sum of terms $Q Q^\dagger + Q^\dagger Q \ge 0$. In each irreducible representation with $P^2 > 0$, all states have the same mass, and the bosonic and fermionic states have equal numbers. Supersymmetry in a theory that includes gravity is supergravity.