Supersymmetry: Supercharges and Supermultiplets

1. Supercharges as Quantum Operators

The most direct way to understand supercharges is to see them in action. In a quantum theory, they are operators that change the character of a state, turning bosons into fermions and vice-versa. Their defining algebraic relation links them directly to the Hamiltonian.

To build intuition, we can start with a simple quantum mechanical model.

This toy model reveals the three essential features of quantum supersymmetry:

1. Supercharges $Q$ and $Q^\dagger$ are fermionic operators: They obey $\{Q, Q\} = \{Q^\dagger, Q^\dagger\} = 0$ which implies $Q^2 = (Q^\dagger)^2 = 0$.
2. Supersymmetry relates to dynamics: The Hamiltonian is generated by the supercharges, $H \propto \{Q, Q^\dagger\}$.
3. Physical Consequences: This structure guarantees that energy is positive semi-definite and that any state with non-zero energy $E$ has a degenerate "superpartner" state with the same energy. States with zero energy (vacuum states) are special: they must be annihilated by all supercharges, $Q|0\rangle = Q^\dagger|0\rangle = 0$.

Now, let's promote these ideas to a 4-dimensional relativistic quantum field theory. The supercharges $Q$ become spinor operators. Given your familiarity with the 2-component spinor formalism, we can write the N=1 Poincaré superalgebra directly. It consists of the Poincaré algebra generators ($P_\mu, M_{\mu\nu}$) plus a pair of Weyl spinor supercharges, $Q_\alpha$ and its conjugate $\bar{Q}_{\dot{\alpha}}$.

The crucial anti-commutation relation is:

This is the 4D relativistic version of $H \propto \{Q, Q^\dagger\}$. Since $P_0$ is the Hamiltonian, this algebra again guarantees that the energy in any unitary supersymmetric theory is non-negative. A state has zero energy if and only if it is annihilated by all supercharges, `$Q_{\alpha }|\psi ⟩={\bar{Q}}_{\dot{\alpha }}|\psi ⟩=0$. This property is central to many modern applications of supersymmetry.

2. Constructing On-Shell Representations

Now that we have defined the supercharges as operators and established their algebra, how do we find the particle states that can exist in such a theory? We are looking for the irreducible representations of the Poincaré superalgebra. The procedure is a beautiful extension of Wigner's method of induced representations for the Poincaré group.

Let's briefly recall Wigner's method:

1. Choose a reference momentum $p^\mu$ for a particle of a given mass $m$ (either $m>0$ or $m=0$).
2. Identify the "little group"—the subgroup of the Lorentz group that leaves $p^\mu$ invariant. For $m>0$, this is $SU(2)$; for $m=0$, it's the double cover of $ISO(2)$.
3. Find the irreducible representations of the little group. These are labeled by spin $s$ for the massive case and helicity $\lambda$ for the massless case.
4. The particle state is then defined by its momentum, mass, and this little group representation.

To extend this to the Poincaré superalgebra, we simply note that the supercharges $Q_\alpha$ and $\bar{Q}_{\dot{\alpha}}$ commute with the momentum operator $P_\mu$. This means they also belong to the algebra that stabilizes the reference momentum $p^\mu$. The "little group" is thus extended into a "little supergroup," and we must find its irreducible representations.

Massless Representations

Let's start with the massless case ($m^2=0$). A convenient reference momentum is $p^\mu = (E, 0, 0, E)$ with $E>0$.

As you read, the key steps are:

1. In the chosen frame, the algebra $\{Q_\alpha, \bar{Q}_{\dot{\beta}}\} = 2(\sigma \cdot p)_{\alpha\dot{\beta}}$ simplifies dramatically. It forces $Q_1=0$ and leaves a simple fermionic oscillator algebra for the other component: $\{Q_2, \bar{Q}_{\dot{2}}\} \propto E$.
2. The supercharges $Q_2$ and $\bar{Q}_{\dot{2}}$ act as helicity-shifting operators: one raises helicity by $1/2$ and the other lowers it by $1/2$.
3. An irreducible representation (a "supermultiplet") is built by choosing a starting state with a given helicity $\lambda$ that is annihilated by the lowering operator, and then acting with the raising operator once.
4. This procedure gives a multiplet containing two states, with helicities $\lambda$ and $\lambda+1/2$.

This gives rise to the fundamental massless $N=1$ supermultiplets:

• Chiral Multiplet (or Wess-Zumino Multiplet): Starting with helicity $\lambda=0$ gives states with helicities $\{0, 1/2\}$. A physical theory must be CPT-invariant, which requires including the conjugate multiplet with helicities $\{0, -1/2\}$. Together, this describes the on-shell degrees of freedom of a complex scalar field and a Majorana fermion.
• Vector Multiplet (or Gauge Multiplet): Starting with helicity $\lambda=1/2$ gives states with helicities $\{1/2, 1\}$. The CPT conjugate has helicities $\{-1, -1/2\}$. This describes the on-shell degrees of freedom of a gauge boson (like a photon, helicities $\pm 1$), and its superpartner, a Majorana fermion called a "gaugino" (helicities $\pm 1/2$).

Massive Representations

Now for the massive case ($m^2>0$). We can go to the particle's rest frame, $p^\mu = (m, 0, 0, 0)$.

The construction for massive particles is slightly different:

1. In the rest frame, the algebra $\{Q_\alpha, \bar{Q}_{\dot{\beta}}\} = 2m \delta_{\alpha\dot{\beta}}$ simplifies to the algebra of two independent fermionic oscillators, $q_1$ and $q_2$.
2. The "ground state" $|\Omega\rangle$ is annihilated by both $q_1$ and $q_2$. Unlike the massless case, this state is itself a representation of the massive little group, $SU(2)$, so it has a definite spin $s$.
3. The full multiplet is built by acting on $|\Omega\rangle$ with the creation operators $q_1^\dagger$ and $q_2^\dagger$: $|\Omega\rangle, q_1^\dagger|\Omega\rangle, q_2^\dagger|\Omega\rangle, q_1^\dagger q_2^\dagger|\Omega\rangle$. This gives a total of $4$ states (counting spin degeneracy).
4. The spins of the resulting states are found using Clebsch-Gordan addition, since the $q^\dagger$ operators themselves transform as a spin-1/2 doublet under $SU(2)$.

The simplest example is the Massive Scalar Multiplet:

• Start with a spin $s=0$ state $|\Omega\rangle$.
• Acting with $q_\alpha^\dagger$ (a spin-1/2 operator) gives a spin-1/2 state ($0 \otimes 1/2 = 1/2$). There are two such states.
• Acting with $q_1^\dagger q_2^\dagger$ (a spin-0 operator) gives another spin-0 state.
• The multiplet contains two spin-0 states and one spin-1/2 state (with two spin projections). This matches the degrees of freedom of two real scalar fields and one Dirac fermion.

Conclusion

In this lesson, we have laid the quantum foundations of supersymmetry. We have moved from the classical picture of transformations to a quantum framework of operators acting on states.

Key Takeaways:

• Supercharges $Q_\alpha$ and $\bar{Q}_{\dot{\alpha}}$ are fermionic, spinorial quantum operators.
• The fundamental algebra $\{Q, \bar{Q}\} \sim P$ ensures that energy is non-negative in any supersymmetric theory.
• On-shell particle states form irreducible representations of the Poincaré superalgebra, called supermultiplets, where all particles have the same mass.
• A supermultiplet contains an equal number of on-shell bosonic and fermionic degrees of freedom.
• The structure of these multiplets is found by extending Wigner's method of induced representations to the "little supergroup," leading to different structures for massive and massless particles.

Next Lesson Preview:
We have focused on supersymmetry in the context of the Poincaré group (global SUSY). Our main goal in this course is to study Conformal Field Theory. In the next lesson, we will combine the symmetries of supersymmetry and conformal symmetry to construct the superconformal algebra. We will begin with the 2D case, which is crucial for string theory, and uncover its rich structure, including the famous Neveu-Schwarz (NS) and Ramond (R) sectors.