Graded Spacetime Symmetry and the Supersymmetry Theorems

In the massive setting of the Haag–Łopuszański–Sohnius theorem—four-dimensional Lorentzian particle physics with no massless particles or long-range forces, and with the additional assumptions stated below—the nontrivial finite-dimensional graded extension of spacetime symmetry has super-Poincaré form. Its odd generators are Weyl spinors $Q^I_\alpha$ and their adjoints; their mixed anticommutator gives momentum, while extended supersymmetry can also admit scalar central charges. This classifies possible visible global S-matrix symmetries. It neither constructs an interacting QFT nor applies unchanged to all-massless, non-particle, curved-background, gauge-redundant, or extended-object settings.

Required background. Lie groups, Lie algebras, and adjoint maps supplies Jacobi identities and representations, while Lorentz fields and Poincaré particles supplies the Lorentz and little-group classification used below.

Helpful background. Projective actions and central extensions distinguishes a central extension from an ordinary internal generator. Poincaré covariance and the spectrum condition explains the positive-energy input.

Graded operators and the Jacobi identity

Abstractly, a Lie superalgebra is a $\mathbb Z_2$-graded vector space $\mathfrak g=\mathfrak g_{\bar0}\oplus\mathfrak g_{\bar1}$ with a graded-antisymmetric bracket satisfying the graded Jacobi identity. In an associative operator representation, the bracket of homogeneous operators is realized as

$$[X,Y\}=XY-(-1)^{\lvert X\rvert\lvert Y\rvert}YX.$$

Thus

$$[X,Y\}=-(-1)^{\lvert X\rvert\lvert Y\rvert}[Y,X\}.$$

The bracket is a commutator unless both entries are odd, in which case it is an anticommutator. Associativity of the represented operators implies

$$(-1)^{\lvert X\rvert\lvert Z\rvert}[X,[Y,Z\}\} +(-1)^{\lvert Y\rvert\lvert X\rvert}[Y,[Z,X\}\} +(-1)^{\lvert Z\rvert\lvert Y\rvert}[Z,[X,Y\}\}=0.$$

Two specializations will be used repeatedly. If $T$ is even and $Q,R,S$ are odd, then

$$\begin{aligned} [T,\{Q,R\}]&=\{[T,Q],R\}+\{Q,[T,R]\},\\ [Q,\{R,S\}]+[R,\{S,Q\}]+[S,\{Q,R\}]&=0. \end{aligned}$$

The even part contains spacetime and internal generators. An odd generator reverses fermion parity, $(-1)^FQ(-1)^F=-Q$. Lorentz covariance makes the finite set of odd generators a sum of finite-dimensional Lorentz representations. Positivity constrains $\{Q,Q^\dagger\}$; the Jacobi identities then constrain every candidate bracket. Because charges are generally unbounded, these operator equations are understood on a common invariant dense domain. The graded construction and its sign conventions are developed in Weinberg 2000, § 25.1, pp. 25–29.

Two theorem settings, not one slogan

The Coleman–Mandula and Haag–Łopuszański–Sohnius results are related, but their hypotheses should not be merged into an informal “relativistic QFT” assumption.

Coleman–Mandula. In four-dimensional Minkowski space, let a connected group of unitary S-matrix symmetries contain the Poincaré group locally, preserve the one-particle subspace, act on multiparticle states as the sum of its one-particle actions, and commute with scattering. If the particle spectrum and amplitudes also satisfy the conditions below, the connected bosonic symmetry group is locally a direct product of the Poincaré group and an internal group. The theorem is a statement about the connected symmetry algebra; it does not classify discrete symmetries or the global topology of the group. The five conditions are stated explicitly in Coleman and Mandula 1967, pp. 1251–1252.

Haag–Łopuszański–Sohnius. Replace the ordinary Lie algebra by a $\mathbb Z_2$-graded algebra on the physical Hilbert space. In the original paper’s complete massive analysis, the theory has no zero-mass particles or long-range forces; a symmetry generator commutes with the S-matrix, acts additively on incoming multiparticle states, connects appropriate one-particle mass multiplets through a regular finite-order momentum kernel, and belongs to a finite particle-type setting. The even subalgebra is constrained by Coleman–Mandula. Nonzero odd generators are then spinors, commute with translations, and close with their adjoints on momentum; same-chirality brackets may supply scalar charges that are central in the full symmetry superalgebra. The original scope and assumptions are explicit in Haag, Łopuszański, and Sohnius 1975, §§ 1–2, pp. 257–261, and the massive algebra is derived in Haag, Łopuszański, and Sohnius 1975, §§ 3–4, pp. 262–269.

The following source-tagged table separates the inputs rather than treating both papers as one identical theorem.

Source tag	Input	Role in the conclusion or failure mode
CM definition and condition 1	A connected unitary symmetry group contains Poincaré locally, maps one-particle states to one-particle states, acts additively on multiparticle states, and commutes with $S$	Without this particle-S-matrix notion of symmetry, the conclusion does not address the proposed transformation
CM condition 2	Particle types furnish positive-energy Poincaré representations, and only finitely many types occur below each fixed mass	An accumulating or tensionless tower can evade the finite-dimensional step; an infinite tower whose masses grow without accumulation does not fail this condition merely by being infinite
CM conditions 3–4	Generic two-particle states scatter, apart from isolated energies, and elastic amplitudes are analytic in the stated physical domain	Free theories, two-dimensional exceptional kinematics, or singular amplitudes can evade the continuation argument
CM condition 5	Generator kernels are distributions of the required regularity	Pathological or genuinely non-particle actions need a different theorem
HLS massive scope	No zero-mass particles or long-range forces; finitely many particle types in each mass multiplet	Infrared sectors, vacuum degeneracy, and all-massless systems require a separate analysis
HLS generator properties	The generator acts on the physical Hilbert space, commutes with $S$, is additive on incoming states, and has the finite-order kernel property	Higher-form actions, boundary algebras, or non-additive charges need not define the same symmetry problem
Positive representation	The physical inner product is positive and the spectrum is future-directed	$\{Q,Q^\dagger\}$ need not be positive on an indefinite or nonunitary state space, so the spinor and normalization argument can fail

Locality check. Microscopic locality is not a separate item in Coleman and Mandula’s five-condition statement. In HLS, locality of the induced field transformation is one sufficient route to additivity and the finite-order kernel property; the paper also explains that the last property can follow from S-matrix invariance, additivity, and nontrivial scattering. Therefore “nonlocal” is not by itself a diagnosis. One must name the actual particle, kernel, additivity, analyticity, or scattering premise that fails.

Massless check. Coleman–Mandula’s listed conditions do not literally say “mass gap,” but the complete original HLS result is explicitly the massive, no-long-range-force branch. Its all-massless discussion assumes away infrared problems and vacuum degeneracy and permits a conformal enlargement. In a charged massless theory, an ordinary exclusive particle S-matrix may fail to exist without inclusive or dressed states. A careful modern formulation and the relevant infrared warnings are given in Weinberg 2000, Appendix 24.B, pp. 12–22.

Why the odd generators must be spinors

Consider the massive four-dimensional branch, for which the even algebra has the Coleman–Mandula form. A finite set of odd charges decomposes into irreducible Lorentz representations $(A,B)$. The Hermitian adjoint of $(A,B)$ transforms as $(B,A)$, so the product of a charge with its adjoint contains a highest diagonal representation

$$(A+B,A+B).$$

This component cannot be discarded by choosing a structure constant to vanish. A highest component is proportional to an operator anticommutator $\{q,q^\dagger\}$. On the common invariant domain,

$$\langle\psi|\{q,q^\dagger\}|\psi\rangle =\lVert q|\psi\rangle\rVert^2 +\lVert q^\dagger|\psi\rangle\rVert^2.$$

If that operator vanished, positivity would force both $q$ and $q^\dagger$ to vanish; Lorentz raising and lowering would then make the entire $(A,B)$ charge vanish. Thus every nonzero odd representation requires a nonzero even generator in $(A+B,A+B)$.

The bosonic theorem permits only

• a vector $(\tfrac12,\tfrac12)$ for $P_\mu$;
• $(1,0)\oplus(0,1)$ for $M_{\mu\nu}$; and
• scalars $(0,0)$ for internal charges.

The largest available diagonal representation is therefore $(\tfrac12,\tfrac12)$, and $A+B\leq\tfrac12$. The remaining scalar possibility $(0,0)$ would reverse fermion parity without changing one-particle spin, contradicting the conventional spin–statistics grading of the physical particle Hilbert space. The nonzero possibilities are $(\tfrac12,0)$ and $(0,\tfrac12)$. This is a proof sketch conditional on the bosonic theorem; the Clebsch–Gordan projection is given explicitly in Weinberg 2000, § 25.2, Eqs. (25.2.12)–(25.2.14), pp. 31–32.

Reconstructing the canonical brackets

Let $Q^I_\alpha$ be independent left-handed charges, with $I=1,\ldots,\mathcal N$, and let $\bar Q_{\dot\alpha I}=(Q^I_\alpha)^\dagger$. The mixed product is a Lorentz vector. In the massive Coleman–Mandula setting, momentum is the only even vector generator, so initially

$$\{Q^I_\alpha,\bar Q_{\dot\beta J}\} =2\sigma^\mu_{\alpha\dot\beta}P_\mu N^I{}_J.$$

Hermitian conjugation makes $N$ Hermitian. In a positive-energy massive sector, positivity of arbitrary linear combinations of the independent charges makes $N$ positive definite after any null generator has been removed. A basis change using its positive square root sets $N^I{}_J=\delta^I{}_J$. This is the normalization step, not an extra physical symmetry.

Translations have not yet been assumed to commute with $Q$. Lorentz covariance and the absence of higher-spin odd generators leave the ansatz

$$[P_\mu,Q^I_\alpha] =(\sigma_\mu)_{\alpha\dot\alpha} K^I{}_J\bar Q^{\dot\alpha J}.$$

Its adjoint fixes $[P_\mu,\bar Q]$. The $(P_\mu,P_\nu,Q)$ Jacobi identity contains

$$(K K^\dagger)^I{}_J (\sigma_\mu\bar\sigma_\nu-\sigma_\nu\bar\sigma_\mu)_\alpha{}^\beta Q^J_\beta.$$

Translations commute, whereas the displayed Lorentz tensor is not identically zero. Independence of the charges gives $K K^\dagger=0$, and hence

$$[P_\mu,Q^I_\alpha]=[P_\mu,\bar Q_{\dot\alpha I}]=0.$$

The same-chirality product decomposes as

$$(\tfrac12,0)\otimes(\tfrac12,0)=(1,0)\oplus(0,0).$$

The scalar part is $\epsilon_{\alpha\beta}Z^{IJ}$. Symmetry under exchange of the complete labels $(I,\alpha)\leftrightarrow(J,\beta)$ then requires $Z^{IJ}=-Z^{JI}$. The $(1,0)$ slot could appear to admit a term $(\sigma^{\mu\nu})_{\alpha\beta}M_{\mu\nu}$. It is excluded by the $(P_\rho,Q_\alpha,Q_\beta)$ identity:

$$[P_\rho,\{Q^I_\alpha,Q^J_\beta\}]=0,$$

while no nonzero Lorentz-generator combination commutes with every $P_\rho$. A distinct tensorial surface charge can occupy the same Lorentz slot only after leaving the point-particle theorem setting.

The canonically normalized point-particle algebra is therefore

$$\begin{aligned} \{Q^I_\alpha,\bar Q_{\dot\beta J}\} &=2\sigma^\mu_{\alpha\dot\beta}P_\mu\,\delta^I{}_J,\\ \{Q^I_\alpha,Q^J_\beta\} &=2\epsilon_{\alpha\beta}Z^{IJ},\\ [P_\mu,Q^I_\alpha]&=0, \end{aligned}$$

The Jacobi identities now establish centrality in stages:

1. $(P,Q,Q)$ gives $[P_\mu,Z^{IJ}]=0$.
2. $(Q,Q,\bar Q)$, together with $[P,Q]=0$, gives $[Z^{IJ},\bar Q]=0$; its adjoint and the corresponding identities give $[Z,Q]=0$.
3. Applying these relations to further Jacobi identities gives $[Z,Z']=0$.

This proves centrality in the supertranslation algebra. To include every genuine internal S-matrix generator $T_A$, note that the $Z$‘s span an invariant Abelian ideal of the compact-semisimple-plus-Abelian internal algebra supplied by Coleman–Mandula. Such an ideal lies in the Abelian center, so $[T_A,Z]=0$. The full argument appears in Weinberg 2000, § 25.2, Eqs. (25.2.21)–(25.2.29), pp. 35–36.

This statement must be distinguished from basis covariance. At $Z=0$, the canonical brackets admit $U(\mathcal N)$ automorphisms $Q\mapsto UQ$. Across the family of centrally extended brackets,

$$Z\longmapsto UZU^{\mathsf T}.$$

A fixed nonzero numerical charge matrix retains only its stabilizer. A genuine HLS internal generator, by contrast, commutes with the central-charge operators. The exact two-component convention is fixed on the four-dimensional N=1 algebra, while basis covariance, fixed-charge stabilizers, and tensorial extensions are developed on the extended algebra page.

Dimension and signature are new inputs

The original HLS theorem is four-dimensional. Moving to another dimension or signature requires recomputing the real spinor modules, the symmetry of the admissible bilinears, and the Lorentz tensors that can occur in $\{Q,Q\}$. Nahm supplies a cross-dimensional algebraic classification Nahm 1978, pp. 149–166, but that classification is not an existence theorem for interacting QFTs.

The dimension-by-dimension figure and semantic table compare the Lorentzian spinor packages, real supercharge counts, R-symmetry families, and representative extension channels. Read them as a spinor-and-algebra census, not as evidence that every entry has an interacting realization.

Boundaries and genuine loopholes

The theorem’s conclusion should be weakened exactly when its inputs are weakened:

All-massless and conformal branches. In the restricted all-massless case treated by HLS, and only after infrared problems and vacuum degeneracy are set aside, $D$ and $K_\mu$ can enlarge the even algebra and conformal supercharges $S$ can accompany $Q$ Haag, Łopuszański, and Sohnius 1975, § 5, pp. 269–272. This is a superconformal branch of the classification. A modern CFT without an ordinary particle S-matrix is instead outside the theorem’s starting problem; its bounded continuation is superconformal algebras and shortening.

Infrared and long-range sectors. Exclusive charged-particle S-matrix elements may vanish or fail to exist in a massless gauge theory. Inclusive observables, dressed states, or asymptotic charges define different objects; one must first prove that the theorem’s symmetry and analyticity inputs survive that replacement.

Extended objects and boundaries. Strings, walls, and branes can carry tensorial surface charges. In a translation-invariant bulk or isolated-object sector these may commute with translations while transforming under Lorentz transformations, so they are not central in the full super-Poincaré algebra. A physical boundary preserves only tangential translations. These sectors do not furnish the same finite point-particle one-particle action assumed by HLS; the precise charge distinctions are made on the extended algebra page.

Two-dimensional integrability, free theories, and accumulating towers. Two-dimensional scattering has no generic scattering angle, integrable models may possess infinitely many conserved charges, and a free S-matrix violates nontriviality. An infinite spectrum alone is not enough to evade particle finiteness: a tower with masses tending to infinity may leave only finitely many states below each fixed mass. A tensionless or accumulating tower can fail that condition.

Gauge and local supersymmetry. Gauge transformations are redundancies, not ordinary global S-matrix generators. Local supersymmetry in supergravity therefore poses a different classification problem, and gravity also changes the meaning of energy and asymptotic observables.

Spontaneous breaking. The local current algebra and Ward identities can remain meaningful even when the vacuum is not annihilated by $Q$. In an infinite-volume broken phase, however, the global charge need not act as a well-defined operator within one vacuum sector. The unbroken-particle-multiplet degeneracy argument must not be imported without checking that representation-theoretic premise.

BRST and twisted scalar charges. A BRST differential is an odd Lorentz scalar on a gauge-fixed complex, often before passage to the positive physical Hilbert space; it is not a fermionic particle S-matrix symmetry of the HLS kind. See the gauge-fixed BRST complex. A scalar supercharge obtained by a topological twist is scalar under a new diagonal rotation group built from Lorentz and R-symmetry generators, not under the original Lorentz group; see topological and holomorphic twists.

Algebra versus QFT existence. Passing Lorentz covariance and Jacobi identities proves only that a candidate algebra is allowed. One must still construct operators on a common domain, a positive-energy representation, local observables, and—if dynamics is claimed—an anomaly-free interacting QFT. The theorem-first status of stronger existence claims belongs to theorem-first claim records.

A worked theorem check

Suppose someone proposes an additional irreducible odd charge

$$S_{(\alpha\beta)\dot\gamma} \quad\text{in}\quad (1,\tfrac12)$$

as a global symmetry of a massive four-dimensional particle S-matrix. Its adjoint transforms in $(\tfrac12,1)$, so $\{S,S^\dagger\}$ contains the diagonal representation $(\tfrac32,\tfrac32)$. Positivity makes that highest component nonzero unless $S$ vanishes. Coleman–Mandula supplies no even generator in $(\tfrac32,\tfrac32)$, so the proposed charge is excluded under the stated hypotheses.

This conclusion has a precise ceiling. It does not exclude a gauge-fixed BRST operator, a twisted scalar charge, an asymptotic symmetry, or an extended-object charge, because each changes the symmetry problem or one of its inputs. Nor does it show that an allowed Weyl-spinor charge is realized by a QFT.

A reproducible consistency check

Given a proposed odd charge $\mathcal Q$, proceed in this order:

1. Name the theorem or classification being invoked, including spacetime dimension, signature, massive or massless branch, asymptotic-state setting, and real form.
2. Check the source-specific particle, finiteness, scattering, analyticity, kernel, infrared, and vacuum assumptions.
3. Determine the Lorentz representation, adjoint, fermion-parity action, and common operator domain of $\mathcal Q$.
4. Decompose $\{\mathcal Q,\mathcal Q^\dagger\}$ and use positivity to test every required even representation.
5. Reconstruct the allowed brackets and check $(P,Q,Q)$, $(P,P,Q)$, $(M,Q,Q)$, and $(T,Q,Q)$ explicitly.
6. Distinguish a scalar central operator, an algebra automorphism, a fixed-charge stabilizer, and a Lorentz-tensor surface charge.
7. State the strongest surviving conclusion: an allowed algebra, a positive representation, and an interacting local QFT realization are different achievements.

Common pitfalls

Calling supersymmetry an exception with no qualifications. It is the graded extension allowed after replacing an ordinary Lie algebra by a Lie superalgebra under a related assumption set. It does not invalidate Coleman–Mandula’s conclusion about ordinary bosonic generators.

Treating every extra term as a central charge. A true central charge commutes with Lorentz transformations. A $p$-form brane charge transforms as a tensor and is central only in the narrower supertranslation sense.

Using “infinite tower” as a theorem loophole. The Coleman–Mandula condition concerns the number of particle types below each fixed mass, not the total number at arbitrarily high mass.

Calling BRST or a twisted charge a counterexample. BRST acts on a gauge complex, while twisting changes the rotation group. Neither is the Lorentz-scalar odd particle symmetry excluded in the HLS proof.

Inferring dynamics from closure. A symbolic Jacobi check cannot establish locality, unitarity, anomaly freedom, or the existence of an interacting S-matrix.

Exercises

1. Excluding the Lorentz generator

Suppose the same-chirality bracket contains

$$\{Q_\alpha,Q_\beta\} \supset c(\sigma^{\mu\nu})_{\alpha\beta}M_{\mu\nu}.$$

Use the graded Jacobi identity to show that $c=0$ once $[P_\rho,Q_\alpha]=0$.

Solution

For one even and two odd entries, the Jacobi identity gives

$$[P_\rho,\{Q_\alpha,Q_\beta\}] =\{[P_\rho,Q_\alpha],Q_\beta\} +\{Q_\alpha,[P_\rho,Q_\beta]\}=0.$$

The proposed term instead gives

$$c(\sigma^{\mu\nu})_{\alpha\beta} [P_\rho,M_{\mu\nu}],$$

which is a nonzero linear combination of momenta for generic indices unless $c=0$. A distinct tensorial surface charge is not ruled out by this calculation if the point-particle theorem assumptions have already been abandoned.

2. Why translations commute with the supercharges

Start from

$$[P_\mu,Q^I_\alpha] =(\sigma_\mu)_{\alpha\dot\alpha} K^I{}_J\bar Q^{\dot\alpha J}.$$

Show that the $(P_\mu,P_\nu,Q)$ Jacobi identity forces $K=0$ in a positive representation.

Solution

Hermitian conjugation gives the corresponding commutator of $P_\nu$ with $\bar Q$. Substitution into

$$[P_\mu,[P_\nu,Q]]-[P_\nu,[P_\mu,Q]]=0$$

produces

$$(K K^\dagger)^I{}_J (\sigma_\nu\bar\sigma_\mu-\sigma_\mu\bar\sigma_\nu)_\alpha{}^\beta Q^J_\beta=0.$$

The sigma-matrix commutator is not identically zero, and the $Q^J$ are independent. Hence $K K^\dagger=0$. For any vector $v$, $v^\dagger K K^\dagger v=\lVert K^\dagger v\rVert^2$, so $K^\dagger=0$ and therefore $K=0$.

3. Centrality versus an R-automorphism

Starting from $\{Q^I_\alpha,Q^J_\beta\}=2\epsilon_{\alpha\beta}Z^{IJ}$ and the canonical mixed bracket, identify what the Jacobi identities prove about $Z$. Then explain why $Z\mapsto UZU^{\mathsf T}$ under a $U(\mathcal N)$ basis change does not contradict centrality.

Solution

The $(P,Q,Q)$ identity gives $[P,Z]=0$. The $(Q,Q,\bar Q)$ identity and $[P,Q]=0$ give $[Z,\bar Q]=0$; adjoints and the corresponding identities give $[Z,Q]=0$. Further Jacobi identities yield $[Z,Z']=0$. The $Z$‘s are therefore central in the supertranslation algebra. In the HLS setting they form an invariant Abelian ideal of the compact internal algebra, so they also commute with genuine internal S-matrix generators.

A unitary change of supercharge basis compares presentations of the family of brackets and acts by congruence, $Z\mapsto UZU^{\mathsf T}$. It need not preserve one chosen numerical matrix. The automorphisms of a fixed nonzero charge sector are only the stabilizer of that matrix. Centrality concerns commutators inside a fixed symmetry algebra; basis covariance concerns how its presentation changes.

4. Diagnose four apparent counterexamples

For each setting, state why the standard massive HLS conclusion cannot simply be quoted: (a) a free four-dimensional theory, (b) a two-dimensional integrable model, (c) a CFT with no particle S-matrix, and (d) a tensionless limit with infinitely many massless higher-spin states. Contrast (d) with a massive tower having only finitely many states below every fixed mass.

Solution

In (a), nontrivial generic scattering fails, so accidental higher symmetries are not excluded. In (b), the generic angular scattering and analyticity argument used in four dimensions is unavailable, and infinitely many conserved charges may also invalidate the finite symmetry setting. In (c), there is no ordinary particle S-matrix on which the theorem’s generators act; superconformal representation theory is the appropriate problem. In (d), particle finiteness fails because infinitely many species occur below a fixed mass, here already at zero mass, and the massive/no-long-range-force HLS branch also fails. By contrast, a tower whose masses tend to infinity can contain only finitely many species below each fixed mass; its infinitude alone is not a Coleman–Mandula loophole. Other premises must be checked before drawing a conclusion.

References

• Sidney Coleman and Jeffrey Mandula, “All Possible Symmetries of the S Matrix,” Physical Review 159 (1967), 1251–1256, DOI.
• Rudolf Haag, Jan T. Łopuszański, and Martin F. Sohnius, “All Possible Generators of Supersymmetries of the S-Matrix,” Nuclear Physics B 88 (1975), 257–274, DOI.
• Werner Nahm, “Supersymmetries and Their Representations,” Nuclear Physics B 135 (1978), 149–166, DOI.
• Steven Weinberg, The Quantum Theory of Fields, Volume III: Supersymmetry, Cambridge University Press, 2000, Chapters 24–25 and 32, DOI.