12.19.01 · quantum / supersymmetry

The supersymmetry algebra: Coleman-Mandula, Haag-Lopuszanski-Sohnius, and the graded extension of Poincare

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Anchor (Master): Weinberg, S., *The Quantum Theory of Fields, Vol. 3: Supersymmetry* (Cambridge, 2000), Ch. 24 (the Coleman-Mandula theorem; the Haag-Lopuszanski-Sohnius theorem; the structure of the SUSY algebra; central charges; the BPS bound and positivity of the energy); Coleman, S. & Mandula, J., *Phys. Rev.* 159, 1251 (1967); Haag, R., Lopuszanski, J. T. & Sohnius, M., *Nucl. Phys. B* 88, 257 (1975)

Intuition Beginner

Symmetries are the deep bookkeeping of physics. Some symmetries move things around in space and time: you can slide a laboratory, rotate it, or set it gliding at steady speed, and the laws of motion stay the same. Other symmetries rotate one kind of particle into another, like electric charge mixing up the members of a family. For a long time these two kinds of symmetry seemed to live in separate worlds, and a famous result said they had to: a theorem proved by Sidney Coleman and Jeffrey Mandula in 1967 showed that, in any ordinary theory of scattering particles, the space-and-time symmetries and the family symmetries can never genuinely mix. They simply sit side by side.

Supersymmetry is the one loophole. The Coleman-Mandula result quietly assumed that all symmetry operations behave like ordinary numbers: do one and then another, and the order does not matter for combining them. But the world has two species of particle. There are bosons, the force-carriers and the things that pile up, and there are fermions, the matter particles that refuse to share a state. A symmetry that turns a boson into a fermion has to be built from a stranger kind of quantity, one where order matters with a minus sign. Allowing that one new ingredient reopens the door.

Once you allow it, something remarkable happens. Apply the boson-into-fermion move, then apply it again, and you do not get back where you started: you get a shift in space and time. The symmetry that swaps the two species, applied twice, is motion itself. Supersymmetry is the bridge that ties the particle-type world to the spacetime world, the one bridge the no-go theorem could not block.

Visual Beginner

Picture two ladders standing side by side. The left ladder holds the bosons of a theory, the right ladder holds the fermions. Each rung is a particle state. The supersymmetry operation is an arrow that reaches across from a rung on one ladder to the matching rung on the other: a boson state and its fermion partner, paired forever at the same height.

The height of each rung is the energy of the state. Because the cross-arrow links a boson to a fermion of the same height, the two partners must have the same mass. That is the sharpest prediction of unbroken supersymmetry: every particle comes with a partner of equal mass but opposite species. We do not see such partners in nature, which tells us that if supersymmetry exists at all, it must be hidden, with the partners pushed to higher rungs we have not yet reached.

The curved arrow in the picture carries the punchline. Follow the cross-arrow over and back, and you do not return to the same rung; you climb. Two boson-to-fermion-to-boson moves add up to a step along the ladder, and a step along the ladder is a motion through spacetime. The pairing arrow and the climbing arrow are the same structure seen twice.

Worked example Beginner

Count the partners in the simplest supersymmetric package, the one a free massless particle of spin one-half lives in. The rule we will use is the pairing rule from the picture: bosons and fermions come in matched sets with equal numbers of independent states.

Step 1. Start with the fermion. A massless spin one-half particle has two independent spin states, often called "spin up along the motion" and "spin down along the motion." So the fermion side of the package has two states.

Step 2. Demand a boson partner with the same count. The pairing rule says the boson side must also carry two independent states, at the same mass, which here is zero.

Step 3. Identify the boson. A massless particle that carries exactly two independent states, and is a boson, is naturally described by one complex number at each point, which is the same as two real numbers. This is a complex scalar field: a spinless particle together with its antiparticle.

Step 4. Read off the package. The smallest supersymmetric family with a spin one-half particle is one fermion of spin one-half paired with one complex scalar. Two fermion states, two boson states, equal masses. This matched set is called a chiral multiplet, and it is the building block of supersymmetric matter.

What this shows: supersymmetry never gives you a lone particle. It hands you a balanced set, and the balance is a strict count of states, boson against fermion. The same counting, done for a force-carrier of spin one, pairs it with a fermion of spin one-half and gives the other basic package.

Check your understanding Beginner

Formal definition Intermediate+

Conventions box (read before any formula below). Two incompatible-looking conventions dominate the literature; this unit fixes one and records the dictionary.

Metric signature. Weinberg, Vol. 3, uses mostly-plus $η_{μν} = diag (-, +, +, +)$ . Wess-Bagger also print $η = diag (-, +, +, +)$ in their 2nd edition (despite the folklore "Wess-Bagger is mostly-minus" — that refers to older printings). Many model-building references (Martin's Primer) use mostly-minus $η = diag (+, -, -, -)$ . We adopt mostly-plus $η = diag (-, +, +, +)$ to match Weinberg and Wess-Bagger. To convert a mostly-minus formula, send $P_{μ} \to - P_{μ}$ on lowered indices and flip the sign of every $η_{μν}$ contraction.

Two-component (van der Waerden) spinors. Left-handed Weyl spinors carry an undotted index $ψ_{a}$ , $a = 1, 2$ , transforming in the $(\frac{1}{2}, 0)$ rep of $S L (2, C)$ ; right-handed carry a dotted index $\overset{χ}{ˉ}_{\overset{a}{˙}}$ , $\overset{a}{˙} = 1, 2$ , in $(0, \frac{1}{2})$ . Indices are raised and lowered with the antisymmetric $ϵ$ : $ψ^{a} = ϵ^{ab} ψ_{b}$ , $ψ_{a} = ϵ_{ab} ψ^{b}$ , with $ϵ^{12} = - ϵ_{12} = 1$ , and likewise dotted. The intertwiners are $σ_{a \dot{b}}^{μ} = (1, σ^{i})$ and $\overset{σ}{ˉ}^{μ \overset{a}{˙} b} = (1, - σ^{i})$ , where $σ^{i}$ are the Pauli matrices; $σ^{μ}$ carries exactly one undotted and one dotted index, the hallmark of a $(\frac{1}{2}, \frac{1}{2})$ object — a four-vector.

Supercharge normalisation. We pin ${Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} = 2 σ_{a \dot{b}}^{μ} P_{μ}$ (Wess-Bagger / Weinberg factor of $2$ ). Some texts absorb the $2$ into $Q$ ; if you meet ${Q, \overset{ˉ}{Q}} = σ^{μ} P_{μ}$ , rescale $Q \to 2 Q$ .

The Poincare algebra is generated by the translations $P_{μ}$ and the Lorentz generators $M_{μν} = - M_{ν μ}$ , with $[P_{μ}, P_{ν}] = 0$ , $[M_{μν}, P_{ρ}] = i (η_{ν ρ} P_{μ} - η_{μ ρ} P_{ν})$ , and the standard $[M, M]$ relations. Its unitary irreducible representations are the one-particle states classified by mass and spin, the content of 07.07.06, with $P^{2} = - M^{2}$ (mostly-plus) and the Pauli-Lubanski Casimir fixing the spin, in the manner of 07.06.10.

Definition (the $N = 1$ super-Poincare algebra). Adjoin to the Poincare generators a pair of fermionic (anticommuting) supercharges $Q_{a}$ ( $a = 1, 2$ , transforming as $(\frac{1}{2}, 0)$ ) and their conjugates $\overset{ˉ}{Q}_{\overset{a}{˙}}$ ( $(0, \frac{1}{2})$ ), subject to

{Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} = 2 σ_{a \dot{b}}^{μ} P_{μ}, {Q_{a}, Q_{b}} = 0, {\overset{ˉ}{Q}_{\overset{a}{˙}}, \overset{ˉ}{Q}_{\dot{b}}} = 0,

[Q_{a}, P_{μ}] = 0, [Q_{a}, M_{μν}] = (σ_{μν})_{a}^{b} Q_{b},

where $σ_{μν} = \frac{1}{4} (σ_{μ} \overset{σ}{ˉ}_{ν} - σ_{ν} \overset{σ}{ˉ}_{μ})$ generates the spinor rep of the Lorentz group. The relation $[Q, M]$ states only that $Q_{a}$ rotates as a spinor; the relation $[Q, P] = 0$ states that the supercharge is translation-invariant. The content is concentrated in the anticommutator ${Q, \overset{ˉ}{Q}} = 2 σ^{μ} P_{μ}$ : the (anti)square of the fermionic symmetry is the generator of spacetime translations.

This is a $Z_{2}$ -graded Lie algebra (a Lie superalgebra): the even part is the Poincare algebra, the odd part is spanned by $Q_{a}, \overset{ˉ}{Q}_{\overset{a}{˙}}$ , and the bracket is a commutator between even elements but an anticommutator between two odd elements. The graded structure is the substrate developed in 07.06.27.

Definition (extended $N > 1$ supersymmetry with central charges). Replace the single supercharge by $N$ copies $Q_{a}^{I}$ , $I = 1, \dots, N$ . The most general closure compatible with the Lorentz and translation structure is

{Q_{a}^{I}, \overset{ˉ}{Q}_{\dot{b}}^{J}} = 2 σ_{a \dot{b}}^{μ} P_{μ} δ^{I J}, {Q_{a}^{I}, Q_{b}^{J}} = 2 ϵ_{ab} Z^{I J},

where the central charges $Z^{I J} = - Z^{J I}$ are antisymmetric in the internal indices, commute with every generator of the algebra (hence "central"), and are Lorentz scalars. For $N = 1$ antisymmetry forces $Z = 0$ , recovering the minimal algebra. The eigenvalues of $Z^{I J}$ control the BPS bound $M \geq ∣ Z ∣$ developed in the next section.

Counterexamples to common slips

The supercharges are not generators of an internal symmetry that commutes with the Lorentz group: $[Q_{a}, M_{μν}] \neq = 0$ . A supercharge carries a spinor index and rotates under the Lorentz group; an object that commuted with all of $M_{μν}$ would be a Lorentz scalar and could not close on $P_{μ}$ .
The central charge is not an internal-symmetry generator either. It is central — it commutes with everything — so it cannot be a Lorentz generator (those do not commute among themselves) nor a translation (those would spoil antisymmetry). It is a genuinely new bosonic invariant available only when $N > 1$ .
"Supersymmetry evades Coleman-Mandula by mixing spacetime and internal symmetry" is right only with the graded caveat. The Coleman-Mandula theorem constrains ordinary (commutator) Lie algebras of bosonic charges; it says nothing about anticommuting charges, which is precisely the gap Haag-Lopuszanski-Sohnius fills.

Key theorem with proof Intermediate+

Theorem (Haag-Lopuszanski-Sohnius, 1975). Under the Coleman-Mandula hypotheses (an interacting analytic S-matrix, a mass gap, finitely many particle types below any mass) but with the bosonic-generator assumption relaxed to admit fermionic charges, the most general graded Lie algebra of symmetries of the S-matrix is the super-Poincare algebra: the Poincare algebra, a (possibly absent) internal bosonic symmetry, and $N$ spinor supercharges $Q_{a}^{I}$ obeying

{Q_{a}^{I}, \overset{ˉ}{Q}_{\dot{b}}^{J}} = 2 σ_{a \dot{b}}^{μ} P_{μ} δ^{I J}, {Q_{a}^{I}, Q_{b}^{J}} = 2 ϵ_{ab} Z^{I J},

with the central charges $Z^{I J}$ antisymmetric and commuting with all generators. No higher-spin (vector, tensor) conserved supercharge is allowed.

Proof (structure of the argument). First, the fermionic charges must transform under the Lorentz group in some finite-dimensional representation; the Coleman-Mandula machinery, applied to the bosonic charges obtained by anticommuting two fermionic ones, restricts these representations sharply. Suppose a supercharge $Q$ sat in the $(\frac{j}{2}, 0)$ representation with $j \geq 2$ (spin one or higher). Then the anticommutator ${Q, \overset{ˉ}{Q}}$ would be a bosonic conserved charge transforming as a Lorentz tensor of rank $\geq 2$ beyond a vector. By the Coleman-Mandula theorem such a bosonic conserved tensor charge forces the S-matrix to be that of a free theory (its conservation over-constrains every scattering angle). Excluding the free case leaves only the lowest possibilities: the supercharge must sit in $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ , a two-component Weyl spinor and its conjugate.

Given $Q_{a} \in (\frac{1}{2}, 0)$ and $\overset{ˉ}{Q}_{\overset{a}{˙}} \in (0, \frac{1}{2})$ , the anticommutator ${Q_{a}, \overset{ˉ}{Q}_{\dot{b}}}$ is a bosonic conserved charge in $(\frac{1}{2}, 0) \otimes (0, \frac{1}{2}) = (\frac{1}{2}, \frac{1}{2})$ , a Lorentz four-vector. The only conserved four-vector charge that Coleman-Mandula permits is the translation generator $P_{μ}$ (any other would again mix in a forbidden way with momenta and collapse the S-matrix). Hence ${Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} \propto σ_{a \dot{b}}^{μ} P_{μ}$ ; the proportionality constant is fixed to $2$ by the chosen normalisation. The like-chirality anticommutator ${Q_{a}^{I}, Q_{b}^{J}}$ lies in $(\frac{1}{2}, 0) \otimes (\frac{1}{2}, 0) = (1, 0) \oplus (0, 0)$ ; the $(1, 0)$ self-dual tensor part must vanish (no conserved antisymmetric tensor charge survives Coleman-Mandula), leaving only the Lorentz-scalar $(0, 0)$ piece $ϵ_{ab} Z^{I J}$ . The Jacobi identities then force $Z^{I J}$ to commute with every generator (central) and to be antisymmetric in $I, J$ because $ϵ_{ab}$ is antisymmetric and the overall anticommutator is symmetric under $(I, a) \leftrightarrow (J, b)$ . $□$

Bridge. This builds toward the representation theory of the next units — the supermultiplets whose Bose-Fermi degeneracy is dictated entirely by the anticommutator just derived — and the very same algebra appears again in 12.19.02, where the relation ${Q, \overset{ˉ}{Q}} = 2 σ^{μ} P_{μ}$ is realised geometrically as a translation in the anticommuting directions of superspace. The foundational reason supersymmetry exists at all is that the Coleman-Mandula no-go theorem assumed commutators, and this is exactly the loophole the graded bracket opens: putting these together, the spinorial supercharge is the unique conserved fermionic charge an interacting S-matrix can carry, and its anticommutator is forced to be the translation generator and nothing else. The central insight is that the structure ${Q, \overset{ˉ}{Q}} = 2 σ^{μ} P_{μ}$ is not an assumption but a theorem — the Lorentz representation theory of 07.07.09 leaves no other option — and this generalises the Poincare classification of 07.07.06 from particles to multiplets of particles. The bridge is the recognition that one algebraic relation simultaneously fixes the spin content of every superpartner, the equality of their masses, and the positivity bound on the energy.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Take the trace of the $N = 1$ anticommutator. Using $σ_{a \dot{b}}^{μ} \overset{σ}{ˉ}_{μ}^{\dot{b} a} = 2 η^{μ}_{μ}$ -type identities, show that the operator $\sum_{a} {Q_{a}, \overset{ˉ}{Q}_{\overset{a}{˙}}}$ (suitably contracted) is proportional to $P^{0}$ , the energy, and comment on its sign.

Hint

Contract the spinor indices with $\overset{σ}{ˉ}_{0}$ . Use $tr σ^{μ} \overset{σ}{ˉ}^{ν} = 2 η^{μν}$ and that $σ^{0} = 1$ .

Answer

Contracting with $\overset{σ}{ˉ}_{0}^{\dot{b} a} = δ^{\dot{b} a}$ (since $σ^{0} = 1$ ) gives $\sum_{a} {Q_{a}, \overset{ˉ}{Q}_{\overset{a}{˙}}}_{diag} = 2 tr (σ^{μ}) P_{μ}$ on the relevant contraction; more carefully, $\sum_{a} {Q_{a}, (Q_{a})^{†}} = 2 (σ^{0})_{a \overset{a}{˙}} P^{0} \cdot (\dots) = 4 P^{0}$ after using $\overset{ˉ}{Q}_{\overset{a}{˙}} = (Q_{a})^{†}$ and $σ^{0} = 1$ . Because the left side is a sum of ${Q_{a}, Q_{a}^{†}} = Q_{a} Q_{a}^{†} + Q_{a}^{†} Q_{a}$ , each term is a positive operator, so $P^{0} \geq 0$ : the energy is non-negative in any supersymmetric theory. Rubric: full credit for $\propto P^{0}$ and the positivity/non-negative-energy conclusion.

Exercise 4 (medium, short-answer).

Explain why a supercharge transforming in the spin-one $(1, 0)$ representation of the Lorentz group is forbidden.

Hint

Anticommute two such charges and ask what bosonic tensor charge results, then invoke Coleman-Mandula.

Answer

If a fermionic charge sat in $(1, 0)$ (spin one), its anticommutator with the conjugate would be a conserved bosonic charge transforming as a Lorentz tensor of rank higher than a four-vector. The Coleman-Mandula theorem forbids any conserved tensor charge beyond $P_{μ}$ and $M_{μν}$ (and Lorentz scalars): such a charge constrains every scattering angle and forces the S-matrix to be free (no scattering). To have an interacting theory with genuine scattering, the supercharge must sit in the lowest spinor representation $(\frac{1}{2}, 0)$ . Rubric: full credit for "higher-tensor conserved charge forces free S-matrix."

Exercise 5 (medium, symbolic).

For extended supersymmetry, show that the central charge $Z^{I J}$ must be antisymmetric in $I, J$ .

Hint

The anticommutator ${Q_{a}^{I}, Q_{b}^{J}}$ is symmetric under simultaneous exchange of $(I, a)$ with $(J, b)$ . The Lorentz-scalar part carries $ϵ_{ab}$ .

Answer

The anticommutator is symmetric: ${Q_{a}^{I}, Q_{b}^{J}} = {Q_{b}^{J}, Q_{a}^{I}}$ , i.e. symmetric under $(I, a) \leftrightarrow (J, b)$ . The Lorentz-scalar piece is written $ϵ_{ab} Z^{I J}$ , and $ϵ_{ab} = - ϵ_{ba}$ is antisymmetric in the spinor indices. For the product $ϵ_{ab} Z^{I J}$ to be symmetric under the joint exchange, the internal factor must absorb the sign: $Z^{I J} = - Z^{J I}$ . Hence the central charge is antisymmetric, and for $N = 1$ (a single value of $I$ ) it vanishes identically. Rubric: full credit for tying the antisymmetry of $Z$ to the antisymmetry of $ϵ_{ab}$ and the symmetry of the anticommutator.

Exercise 6 (hard, short-answer).

Derive the BPS bound $M \geq ∣ Z ∣$ schematically for an $N = 2$ algebra by exhibiting the supercharge combination whose square is a positive operator equal to $M - ∣ Z ∣$ (up to normalisation) in the rest frame.

Hint

In the rest frame $P_{μ} = (M, 0)$ . Form the Hermitian combinations $a_{a} = Q_{a}^{1} \pm ϵ_{ab} (\overset{ˉ}{Q}^{2})^{b}$ and compute ${a, a^{†}}$ .

Answer

In the rest frame ${Q_{a}^{I}, \overset{ˉ}{Q}_{\dot{b}}^{J}} = 2 M δ_{a \dot{b}} δ^{I J}$ and ${Q_{a}^{I}, Q_{b}^{J}} = 2 ϵ_{ab} Z^{I J}$ with $Z^{12} = Z$ . Define $a_{a} = \frac{1}{2} (Q_{a}^{1} + ϵ_{ab} (\overset{ˉ}{Q}^{2})^{b})$ . Then ${a_{a}, a_{a}^{†}} = \frac{1}{2} ({Q^{1}, \overset{ˉ}{Q}^{1}} + {Q^{2}, \overset{ˉ}{Q}^{2}} + cross terms) \propto (M - ∣ Z ∣)$ after using the central-charge relation, and similarly the orthogonal combination gives $M + ∣ Z ∣$ . Since ${a, a^{†}} = a a^{†} + a^{†} a$ is a sum of positive operators, its expectation value is non-negative, forcing $M - ∣ Z ∣ \geq 0$ , i.e. $M \geq ∣ Z ∣$ . Equality holds when $a ∣ ψ ⟩ = 0$ , i.e. when half the supercharges annihilate the state — a BPS (short) multiplet. Rubric: full credit for the positive Hermitian combination and the conclusion $M \geq ∣ Z ∣$ with the saturation condition.

Exercise 7 (hard, short-answer).

A massless representation of the $N = 1$ algebra has fewer states than a massive one. Using $P_{μ} = (E, 0, 0, E)$ for a massless state, show that one of the two supercharge components must be represented by zero, halving the multiplet.

Hint

Evaluate ${Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} = 2 σ_{a \dot{b}}^{μ} P_{μ} = 2 (E σ^{0} + E σ^{3})_{a \dot{b}}$ and look at the matrix structure.

Answer

With $P_{μ} = (E, 0, 0, E)$ , $σ^{μ} P_{μ} = E (σ^{0} + σ^{3}) = E (2000)$ . So ${Q_{1}, \overset{ˉ}{Q}_{\dot{1}}} = 4 E$ but ${Q_{2}, \overset{ˉ}{Q}_{\dot{2}}} = 0$ . The vanishing anticommutator ${Q_{2}, Q_{2}^{†}} = 0$ on a positive-norm Hilbert space forces $Q_{2} = \overset{ˉ}{Q}_{\dot{2}} = 0$ as operators on physical states (a zero-norm operator annihilates everything). Only the single raising/lowering pair $(Q_{1}, \overset{ˉ}{Q}_{\dot{1}})$ acts as a nonzero operator, generating a two-state multiplet: a Clifford module on one fermionic oscillator. Hence massless multiplets are shorter (two states) than massive ones (which use both oscillators, four states). Rubric: full credit for the $(2000)$ structure and the conclusion that one supercharge is null (represented by zero).

Advanced results Master

The single anticommutator ${Q, \overset{ˉ}{Q}} = 2 σ^{μ} P_{μ}$ controls three deep structural facts: the energy is positive, the central charges bound the mass, and the representations split into long and short multiplets. The following assemble them.

Proposition (positivity of the energy). In any supersymmetric quantum field theory the Hamiltonian is a non-negative operator, $H \geq 0$ , and a state has exactly zero energy if and only if it is annihilated by all supercharges (a supersymmetric vacuum).

This is the field-theory sharpening of Exercise 3. Because $H = P^{0} = \frac{1}{4} \sum_{a} {Q_{a}, Q_{a}^{†}}$ is a sum of operators of the form $Q Q^{†} + Q^{†} Q$ , every expectation value $⟨ ψ ∣ H ∣ ψ ⟩ = \frac{1}{4} \sum_{a} (∥ Q_{a} ∣ ψ ⟩ ∥^{2} + ∥ Q_{a}^{†} ∣ ψ ⟩ ∥^{2}) \geq 0$ . Vanishing energy forces $Q_{a} ∣ ψ ⟩ = Q_{a}^{†} ∣ ψ ⟩ = 0$ for all $a$ . This single fact organises the theory of spontaneous supersymmetry breaking: supersymmetry is unbroken precisely when a zero-energy vacuum exists, and the order parameter for breaking is the vacuum energy itself — the subject taken up in 12.19.05.

Proposition (the BPS bound and short multiplets). For the extended algebra with central charge eigenvalues $Z_{n}$ , every physical state satisfies $M \geq ∣ Z_{n} ∣$ for each $n$ . States saturating the bound, $M = ∣ Z_{n} ∣$ , are annihilated by a fraction of the supercharges and lie in shortened (BPS) multiplets with fewer states than a generic massive multiplet.

The bound follows from positivity of the rest-frame supercharge anticommutator, diagonalised by forming the Hermitian combinations of Exercise 6. Its consequences are far-reaching. BPS states are stable — they are the lightest states carrying their central charge, so they cannot decay — and their mass is fixed exactly by $∣ Z ∣$ , a quantity often protected from quantum corrections by holomorphy. This exactness makes BPS states the rigid scaffolding on which strong-coupling and duality arguments are built: the central charge of 12.19.06 and the monopole/dyon spectra of extended gauge theories are computed by saturating this bound. The field-theory BPS bound here is the central-charge cousin of the energetic Bogomolny bound of 03.07.07, but its origin is the supersymmetry algebra rather than a completion of squares in a static energy functional.

Proposition (multiplet structure from a Clifford algebra). Fixing the momentum, the supercharges that act as nonzero operators form a Clifford algebra of fermionic creation and annihilation operators; the irreducible representation is the corresponding Fock space, automatically containing equal numbers of bosonic and fermionic states.

In the rest frame of a massive $N = 1$ state, the rescaled $a_{a} = Q_{a} / 2 M$ satisfy ${a_{a}, a_{b}^{†}} = δ_{ab}$ , ${a_{a}, a_{b}} = 0$ : two fermionic oscillators. Building the Fock space on a Clifford vacuum $∣Ω ⟩$ of spin $j_{0}$ gives $2^{2} = 4$ states with spins $(j_{0}, j_{0} \pm \frac{1}{2}, j_{0})$ , half bosonic and half fermionic. For massless states one oscillator decouples (Exercise 7), halving the count. The equality of bosonic and fermionic state counts in every multiplet is not imposed; it is forced by the fermionic-oscillator structure, and it is the algebraic origin of the Bose-Fermi degeneracy that the next unit makes physical.

Synthesis. Putting these together, the one relation ${Q, \overset{ˉ}{Q}} = 2 σ^{μ} P_{μ}$ is the foundational reason for every structural feature of supersymmetry at once: the central insight is that the supercharge anticommutator is the energy-momentum operator, so its manifest positivity makes $H \geq 0$ , its rest-frame eigenvalues make $M \geq ∣ Z ∣$ , and its oscillator structure makes every multiplet a balanced Clifford module. This is exactly the pattern by which an algebra dictates a spectrum: the Wigner classification of 07.07.06 read particle content off the Poincare Casimirs, and the super-Poincare algebra generalises that reading from single particles to whole supermultiplets. The BPS bound is dual to the positivity of the energy — one bounds mass from below by a charge, the other bounds energy from below by zero, and both are the same statement that a square is non-negative. The whole edifice builds toward the dynamical questions of supersymmetry breaking and duality: the vacuum energy as the breaking order parameter appears again in 12.19.05, and the central charge as the holomorphically protected mass appears again in 12.19.06, so the algebra derived here is the rigid skeleton on which the entire physics of the subject hangs.

Full proof set Master

Proposition (the anticommutator is forced to be $P_{μ}$ , representation-theoretic proof). Let $Q_{a}$ be a conserved fermionic charge in $(\frac{1}{2}, 0)$ and $\overset{ˉ}{Q}_{\overset{a}{˙}}$ its conjugate in $(0, \frac{1}{2})$ in an interacting theory satisfying the Coleman-Mandula hypotheses. Then ${Q_{a}, \overset{ˉ}{Q}_{\dot{b}}} = c σ_{a \dot{b}}^{μ} P_{μ}$ for a constant $c$ .

Proof. The anticommutator $B_{a \dot{b}} := {Q_{a}, \overset{ˉ}{Q}_{\dot{b}}}$ is a bosonic operator, conserved because each factor is, and it transforms in $(\frac{1}{2}, 0) \otimes (0, \frac{1}{2}) = (\frac{1}{2}, \frac{1}{2})$ , the four-vector representation. Write $B_{a \dot{b}} = σ_{a \dot{b}}^{μ} B_{μ}$ , defining a conserved Lorentz four-vector charge $B_{μ}$ . The Coleman-Mandula theorem (under its analyticity and mass-gap hypotheses) states that the only conserved four-vector charges in an interacting theory are the translation generators $P_{μ}$ themselves; any further conserved vector charge would fail to commute with $P_{μ}$ in a way that forces the scattering matrix to act as the identity on all but a measure-zero set of momenta, i.e. a free theory. Excluding the free case, $B_{μ} = c P_{μ}$ for a single constant $c$ . Positivity of $B_{a \dot{b}}$ as a sum $Q \overset{ˉ}{Q} + \overset{ˉ}{Q} Q$ of a positive operator and the requirement that $B_{0} = c P^{0}$ be a positive operator fix $c > 0$ ; rescaling $Q$ sets $c = 2$ . $□$

Proposition (centrality of $Z^{I J}$ , Jacobi-identity proof). In the extended algebra the Lorentz-scalar charge $Z^{I J}$ appearing in ${Q_{a}^{I}, Q_{b}^{J}} = 2 ϵ_{ab} Z^{I J}$ commutes with all generators of the super-Poincare algebra.

Proof. That $Z^{I J}$ is a Lorentz scalar follows because the $(1, 0)$ part of $(\frac{1}{2}, 0) \otimes (\frac{1}{2}, 0)$ would be a conserved antisymmetric tensor charge, forbidden by Coleman-Mandula, leaving only the $(0, 0)$ part $ϵ_{ab} Z^{I J}$ ; being a Lorentz scalar, $[Z^{I J}, M_{μν}] = 0$ . For translations, the (graded) Jacobi identity on $(P_{μ}, Q_{a}^{I}, Q_{b}^{J})$ reads $[P_{μ}, {Q_{a}^{I}, Q_{b}^{J}}] = {[P_{μ}, Q_{a}^{I}], Q_{b}^{J}} + {Q_{a}^{I}, [P_{μ}, Q_{b}^{J}]}$ . The right-hand side vanishes because $[P_{μ}, Q] = 0$ , so $[P_{μ}, 2 ϵ_{ab} Z^{I J}] = 0$ and $Z$ commutes with $P_{μ}$ . For commutation with the supercharges, the Jacobi identity on $(Q_{c}^{K}, Q_{a}^{I}, Q_{b}^{J})$ together with $[P, Q] = 0$ and the established $[Z, P] = [Z, M] = 0$ forces $[Z^{I J}, Q_{c}^{K}] = 0$ (the would-be right-hand side is a Lorentz-covariant combination of $Z$ and $Q$ that must vanish by index counting and antisymmetry). Finally $Z$ commutes with any internal symmetry generator $T_{r}$ up to a structure-constant rotation $[T_{r}, Z^{I J}] = (t_{r})^{I J}_{K L} Z^{K L}$ ; choosing the basis in which the antisymmetric $Z^{I J}$ is block-diagonal (skew-eigenvalue form) makes the relevant combination central. Hence $Z^{I J}$ commutes with the full algebra. $□$

Proposition (BPS bound, full positivity proof). For the $N = 2$ algebra in the rest frame, every state satisfies $M \geq ∣ Z ∣$ , with equality iff half the supercharges annihilate it.

Proof. In the rest frame $P_{μ} = (M, 0)$ , so ${Q_{a}^{I}, \overset{ˉ}{Q}_{\dot{b}}^{J}} = 2 M δ_{a \dot{b}} δ^{I J}$ and ${Q_{a}^{I}, Q_{b}^{J}} = 2 ϵ_{ab} ϵ^{I J} Z$ (taking $Z^{I J} = ϵ^{I J} Z$ with $Z$ real after a phase rotation). Define $A_{a} = \frac{1}{2} (Q_{a}^{1} + ϵ_{ab} ϵ^{ab} (\overset{ˉ}{Q}^{2})^{\dot{b}})$ and the orthogonal $B_{a}$ with a relative sign. A direct computation of ${A_{a}, A_{a}^{†}}$ using the two anticommutators gives ${A_{a}, A_{a}^{†}} = 2 (M - ∣ Z ∣) δ_{aa}$ summed over $a$ , and ${B_{a}, B_{a}^{†}} = 2 (M + ∣ Z ∣) δ_{aa}$ . Each left-hand side is $A A^{†} + A^{†} A$ , a sum of positive semi-definite operators, so its expectation value in any normalised state is $\geq 0$ . Therefore $M - ∣ Z ∣ \geq 0$ and $M + ∣ Z ∣ \geq 0$ ; the binding constraint is $M \geq ∣ Z ∣$ . Equality $M = ∣ Z ∣$ holds iff $⟨ ψ ∣ {A_{a}, A_{a}^{†}} ∣ ψ ⟩ = 0$ , which forces $A_{a} ∣ ψ ⟩ = A_{a}^{†} ∣ ψ ⟩ = 0$ : half of the supercharge combinations annihilate the state. The annihilated half does not act, so the Clifford module is built on one oscillator instead of two, giving a short (BPS) multiplet with $2$ rather than $4$ states (per pair). $□$

Connections Master

Lie superalgebras 07.06.27. The super-Poincare algebra is the physical instance of the abstract $Z_{2}$ -graded Lie algebra developed in 07.06.27: the even part is Poincare, the odd part is the supercharges, and the graded antisymmetry plus super-Jacobi identity of 07.06.27 are exactly the consistency conditions that the anticommutators and the Jacobi-identity proofs above must satisfy. Where 07.06.27 supplies the algebraic skeleton and the Kac classification, this unit supplies the unique skeleton selected by a relativistic S-matrix. (Co-produced this wave; referenced here, not as a prerequisite.)
Wigner's classification of the Poincare group 07.07.06. Wigner read off the one-particle spectrum from the Casimirs $P^{2}$ and $W^{2}$ of the Poincare algebra; the super-Poincare algebra extends this. Because $[Q, P_{μ}] = 0$ , the supercharges commute with $P^{2}$ , so an entire supermultiplet shares one mass — the Wigner mass label is now a label on a multiplet. The shortening of massless multiplets derived above is the supersymmetric refinement of Wigner's distinct treatment of massive versus massless little groups.
Lorentz / $S L (2, C)$ representations 07.07.09. The two-component spinor calculus of 07.07.09 — the $(\frac{1}{2}, 0)$ and $(0, \frac{1}{2})$ Weyl spinors, the $σ^{μ}$ intertwiner carrying one dotted and one undotted index — is the exact apparatus that pins the supercharge to a spinor and forces ${Q, \overset{ˉ}{Q}} \in (\frac{1}{2}, \frac{1}{2})$ . The representation-theoretic proof that the anticommutator must equal $P_{μ}$ is, at bottom, the Clebsch-Gordan decomposition $(\frac{1}{2}, 0) \otimes (0, \frac{1}{2}) = (\frac{1}{2}, \frac{1}{2})$ of 07.07.09.
Casimir element 07.06.10. The Casimir machinery of 07.06.10 labels Poincare irreps by $P^{2} = - M^{2}$ and the Pauli-Lubanski invariant; in the super-Poincare algebra $P^{2}$ remains a Casimir (it commutes with $Q$ ), but the spin Casimir does not, which is precisely why a single multiplet contains several spins. The central charge $Z$ is a new Casimir of the extended algebra, and the BPS bound is the statement relating two Casimirs, $M$ and $∣ Z ∣$ .

Historical & philosophical context Master

Sidney Coleman and Jeffrey Mandula proved in 1967 that, under broad assumptions about an interacting analytic S-matrix with a mass gap, any continuous symmetry of the scattering matrix must be a direct product of the Poincare group with an internal symmetry group commuting with it — spacetime and internal symmetries cannot be unified (Coleman & Mandula, Phys. Rev. 159, 1251, 1967) ^{[source pending]}. The theorem appeared to close, permanently, the dream of a deeper symmetry tying the geometry of spacetime to the quantum numbers of particles. Its hidden assumption — that all symmetry generators are bosonic, combining under ordinary commutators — went unremarked, because no other possibility seemed physical.

The loophole was opened from two directions. Yuri Golfand and Evgeny Likhtman in the Soviet Union (1971), and Dmitri Volkov and Vladimir Akulov (1972), independently introduced fermionic symmetry generators, while Julius Wess and Bruno Zumino (1974) constructed the first four-dimensional supersymmetric field theory in the West, bringing the idea to wide attention. The decisive structural statement came from Rudolf Haag, Jan Lopuszanski, and Martin Sohnius (Nucl. Phys. B 88, 257, 1975) ^{[source pending]}, who proved that once fermionic generators are admitted, the most general graded symmetry algebra of an interacting S-matrix is the super-Poincare algebra — with spinor supercharges, the anticommutator closing on the momentum, and (for extended supersymmetry) antisymmetric central charges. Where Coleman-Mandula was a no-go theorem, Haag-Lopuszanski-Sohnius was a uniqueness theorem: supersymmetry is not one option among many but the only consistent way to unify spacetime and internal symmetry. Weinberg's Vol. 3 (Cambridge, 2000) ^{[source pending]} gives the canonical modern derivation, including the BPS bound from positivity. The philosophical lesson is sharp: a no-go theorem is only as strong as its least-examined hypothesis, and the entire subject of supersymmetry — and with it much of the architecture of modern theoretical physics — lives in the single word "bosonic" that Coleman and Mandula did not know they had assumed.

Bibliography Master

@article{ColemanMandula1967,
  author  = {Coleman, Sidney and Mandula, Jeffrey},
  title   = {All Possible Symmetries of the $S$ Matrix},
  journal = {Phys. Rev.},
  volume  = {159},
  year    = {1967},
  pages   = {1251--1256}
}

@article{HaagLopuszanskiSohnius1975,
  author  = {Haag, Rudolf and Lopusza\'nski, Jan T. and Sohnius, Martin},
  title   = {All Possible Generators of Supersymmetries of the $S$ Matrix},
  journal = {Nucl. Phys. B},
  volume  = {88},
  year    = {1975},
  pages   = {257--274}
}

@book{WeinbergQTFVol3,
  author    = {Weinberg, Steven},
  title     = {The Quantum Theory of Fields, Vol. 3: Supersymmetry},
  publisher = {Cambridge University Press},
  year      = {2000}
}

@book{WessBagger1992,
  author    = {Wess, Julius and Bagger, Jonathan},
  title     = {Supersymmetry and Supergravity},
  edition   = {2nd},
  publisher = {Princeton University Press},
  year      = {1992}
}

@article{Sohnius1985,
  author  = {Sohnius, Martin F.},
  title   = {Introducing Supersymmetry},
  journal = {Phys. Rept.},
  volume  = {128},
  year    = {1985},
  pages   = {39--204}
}

@article{GolfandLikhtman1971,
  author  = {Golfand, Yuri A. and Likhtman, Evgeny P.},
  title   = {Extension of the Algebra of Poincar\'e Group Generators and Violation of $P$ Invariance},
  journal = {JETP Lett.},
  volume  = {13},
  year    = {1971},
  pages   = {323--326}
}

@article{WessZumino1974,
  author  = {Wess, Julius and Zumino, Bruno},
  title   = {Supergauge Transformations in Four Dimensions},
  journal = {Nucl. Phys. B},
  volume  = {70},
  year    = {1974},
  pages   = {39--50}
}

@article{VolkovAkulov1973,
  author  = {Volkov, Dmitri V. and Akulov, Vladimir P.},
  title   = {Is the Neutrino a Goldstone Particle?},
  journal = {Phys. Lett. B},
  volume  = {46},
  year    = {1973},
  pages   = {109--110}
}

Prerequisites

07.07.06
07.07.09
07.06.10

Tier anchors

beginner: Tong, D., *Supersymmetric Field Theory* (DAMTP Lecture Notes, 2022), §1 (why supersymmetry, the algebra in words); Bilal, A., *Introduction to Supersymmetry*, arXiv:hep-th/0101055, §1–2 (Coleman-Mandula in plain language, the graded loophole); Zee, A., *Quantum Field Theory in a Nutshell*, 2nd ed. (Princeton, 2010), §VIII.4 (a first pass at the SUSY idea, bosons and fermions paired)
intermediate: Wess, J. & Bagger, J., *Supersymmetry and Supergravity*, 2nd ed. (Princeton, 1992), Chs. 1–2 (two-component conventions, the N=1 algebra, multiplets); Martin, S. P., *A Supersymmetry Primer*, arXiv:hep-ph/9709356, §3 (the algebra and its representations for model-builders); Sohnius, M. F., *Introducing Supersymmetry*, Phys. Rept. 128, 39 (1985), §2–4 (the algebra, central charges, and the BPS bound, from a co-discoverer)
master: Weinberg, S., *The Quantum Theory of Fields, Vol. 3: Supersymmetry* (Cambridge, 2000), Ch. 24 (the Coleman-Mandula theorem; the Haag-Lopuszanski-Sohnius theorem; the structure of the SUSY algebra; central charges; the BPS bound and positivity of the energy); Coleman, S. & Mandula, J., *Phys. Rev.* 159, 1251 (1967); Haag, R., Lopuszanski, J. T. & Sohnius, M., *Nucl. Phys. B* 88, 257 (1975)

References

Coleman, S. & Mandula, J. — All possible symmetries of the S matrix · Phys. Rev. 159, 1251 (1967) — under the assumptions (an interacting analytic S-matrix, a mass gap, finitely many particle types below any mass, and symmetry generators that are Lie-algebra-valued bosonic charges) the most general symmetry Lie algebra of the S-matrix is the direct product of the Poincare algebra with an internal symmetry algebra; spacetime and internal symmetries cannot mix in any way beyond this direct product
Haag, R., Lopuszanski, J. T. & Sohnius, M. — All possible generators of supersymmetries of the S matrix · Nucl. Phys. B 88, 257 (1975) — relaxing the Coleman-Mandula restriction to bosonic generators by admitting fermionic (graded, anticommuting) charges, the most general graded Lie algebra of S-matrix symmetries is the super-Poincare algebra with N spinor supercharges Q^I_a; the anticommutator closes on the translation generator, {Q^I_a, Qbar^J_bdot} = 2 sigma^mu_{a bdot} P_mu delta^{IJ}, and the extended algebra admits antisymmetric central charges Z^{IJ} commuting with everything
Weinberg, S. — The Quantum Theory of Fields, Vol. 3: Supersymmetry (Cambridge, 2000) · Ch. 24 §24.1 (the Coleman-Mandula theorem and its assumptions); §24.2 (the Haag-Lopuszanski-Sohnius theorem, the unique graded extension, the spinor supercharges); §24.3 (extended supersymmetry, the central charges Z^{IJ}, and the derivation of the BPS bound M >= |Z| from positivity of the supercharge anticommutator on physical states)
Wess, J. & Bagger, J. — Supersymmetry and Supergravity, 2nd ed. (Princeton, 1992) · Chs. 1–2 — the two-component van der Waerden spinor conventions (mostly-minus metric eta = diag(-,+,+,+) in their printing, epsilon^{12} = -epsilon_{12} = 1, sigma^mu = (1, sigma^i), bar-sigma^mu = (1, -sigma^i)), the N=1 algebra with the normalisation {Q_a, Qbar_bdot} = 2 sigma^mu_{a bdot} P_mu, and the construction of the massless and massive multiplets
Sohnius, M. F. — Introducing Supersymmetry · Phys. Rept. 128, 39 (1985), §2–4 — a co-discoverer's exposition of the graded algebra, the role of the central charges in extended supersymmetry, and the saturation of the BPS bound by states annihilated by half the supercharges (short multiplets)

Estimated time

beginner: 20m
intermediate: 48m
master: 85m