Quantum theory Axiomatic and algebraic quantum field theory Theorem

Haag–Łopuszański–Sohnius theorem

Builds on Coleman–Mandula theorem Lie superalgebra Poincaré group S-matrix

Statement

Consider a relativistic quantum theory in four spacetime dimensions with massive particles, satisfying the hypotheses of the Coleman–Mandula theorem. Allow the symmetries of the S-matrix to be generated by a Lie superalgebra $\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1$ instead of a Lie algebra. Then the following hold.

The even part $\mathfrak{g}_0$ is the direct sum of the Lie algebra of the Poincaré group and an internal symmetry algebra.
The odd part $\mathfrak{g}_1$ is spanned by $N$ spinor charges $Q^A_\alpha$ and their adjoints $\bar Q_{\dot\alpha A}$ , with $A = 1, \dots, N$ . The $Q^A_\alpha$ transform as $(\tfrac12, 0)$ and the $\bar Q_{\dot\alpha A}$ as $(0, \tfrac12)$ under the Lorentz group.
With a standard normalization, $\{Q^A_\alpha, \bar Q_{\dot\beta B}\} = 2 \delta^A_B \sigma^\mu_{\alpha\dot\beta} P_\mu$ , where $P_\mu$ is the momentum and $\sigma^\mu = (1, \sigma^1, \sigma^2, \sigma^3)$ .
$\{Q^A_\alpha, Q^B_\beta\} = \epsilon_{\alpha\beta} Z^{AB}$ , where the central charges $Z^{AB} = -Z^{BA}$ commute with all of $\mathfrak{g}$ .
The internal symmetries that act nontrivially on the supercharges form a subalgebra of $\mathfrak{u}(N)$ , the R-symmetry.

Remarks

The proof applies the Coleman–Mandula theorem to the even part. Then it classifies the representations of the Lorentz group in which odd generators can transform: higher spinor representations would give charges whose anticommutators violate the Coleman–Mandula restrictions. The anticommutator of $Q$ and $\bar Q$ is a positive operator, which is why the energy is nonnegative in a supersymmetric theory.

Example: for $N = 1$ there are no central charges, and the superalgebra is the $N = 1$ super-Poincaré algebra. If all particles are massless, the symmetry superalgebra can be larger, a superconformal algebra.

Going deeper

For a student

The theorem is why four-dimensional supersymmetry is classified by the number $N$ of supercharges. Theories with $N = 1$ , $N = 2$ and $N = 4$ supersymmetry are studied a lot, and central charges appear in the BPS bounds of $N = 2$ theories such as Seiberg–Witten theory. The theorem relies on the S-matrix, so it says nothing directly about theories without massive particles and nontrivial scattering.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

Cross product concept
Determinant concept
General linear group object
Group concept
Hermitian matrix concept
Identity matrix object
Lie algebra concept
Lie group concept
Lie superalgebra concept
Matrix definition
Möbius transformation concept
Quadratic form concept
Real projective space object
Riemann sphere concept
S-matrix concept
Semidirect product concept
Signature of a quadratic form concept
Translation concept
Vector space definition
Ring definition
Square matrix definition
Transpose definition
Linear map definition
Unit of a ring definition
Conjugate transpose definition
Multilinear map definition
Orthogonal matrix definition
Division ring definition
Alternating multilinear map definition
Orthogonal group object
Unitary matrix definition
The quaternions example
Determinant definition
Unitary group object
Indefinite orthogonal group example
Special linear group object
Special orthogonal group object
Special unitary group SU(n) example
Rotation group SO(3) example
SU(2) example
SL(2,C) example
Lorentz group example
Poincaré group example
Coleman–Mandula theorem statement
Haag–Łopuszański–Sohnius theorem statement

Tag	`WKMW5H0GJZ`
URI	`m2://WKMW5H0GJZ`
Kind	statement, theorem
Writing	W0 stub
MSC	81T60, 81T05, 81U20
Status	This copy of the site has no build data. Run `m2 status WKMW5H0GJZ`.
About	Supersymmetry
Uses	Coleman–Mandula theorem, Lie superalgebra, Poincaré group, S-matrix
Same as	Wikidata Q586051 Wikipedia Haag–Łopuszański–Sohnius theorem
Changed	3 November 2026