Wigner Classification

This is a long-ish post (apologies), but it consists of several steps. We’re working in the mostly-pluses metric signature. So what’s the roadmap?

Basically, a particle in quantum field theory is a unitary irreducible representation of the Poincare group, characterized by a half-integer spin $j$ and a non-negative real number “mass” $m\geq0$. How will we see this?

First, we examine the irreducible representations of the “infinitesimal” proper, orthochronous Lorentz group, and find in 3+1 dimensions it is isomorphic (as a Lie algebra) to two copies of su(2). This lets us classify the “species” of particles by spin.

Next, we look at the Poincare symmetries (i.e., Lorentz boosts plus spacetime translations). We find its unitary irreducible representations are parametrized by half-integer “spin” and non-negative mass-squared.

Lie Algebras

Lorentz Algebra

We only care about proper orthochronous Lorentz transformations, i.e., the subgroup $\Lambda\in\mathrm{ISO}(1,3)\subset\mathrm{O}(1,3)$ such that $\det(\Lambda)=+1$ and ${\Lambda^{0}}_{0}=+1$. We can write any element of this group as \begin{equation} {\Lambda^{\mu}}_{\nu} = {\exp\left(\frac{-\mathrm{i}}{2}\omega_{\kappa\lambda}M^{\kappa\lambda}\right)^{\mu}}_{\nu} \end{equation} where $\omega_{\kappa\lambda}=-\omega_{\lambda\kappa}$ are “rotation angles” (real constants parametrizing the symmetry). The indices are not spacetime indices, but refer to the Lorentz symmetry, and $M^{\kappa\lambda}$ is an indexed family of matrices (fix a value of $\kappa$ and $\lambda$, and you get a 4-by-4 matrix). Specifically, we have \begin{equation} (M^{\kappa\lambda})_{\mu\nu} = \mathrm{i}(\delta^{\kappa}_{\mu}\delta^{\lambda}_{\nu}-\delta^{\kappa}_{\nu}\delta^{\lambda}_{\mu}) \end{equation} which satisfy the Lorentz algebra’s commutator relationship: \begin{equation} [M^{\kappa\lambda},M^{\rho\sigma}]=\mathrm{i}(\eta^{\lambda\rho}M^{\kappa\sigma} -\eta^{\kappa\rho}M^{\lambda\sigma}-\eta^{\lambda\sigma}M^{\kappa\rho}+\eta^{\kappa\sigma}M^{\lambda\rho}). \end{equation}

Remark 1. We can work with $M^{[\kappa\lambda]}=(M^{\kappa\lambda}-M^{\lambda\kappa})/2$ instead of $M^{\kappa\lambda}$, since the $\omega_{\kappa\lambda}$ parameters form an antisymmetric matrix. We denote the antisymmetric matrix obtained from $M$ by $\widetilde{M}$.

Remark 2. We’re interested in representations of a Lie algebra, which typically examines a Lie algebra morphism from our particular algebra to endomorphisms on a vector space. Lets consider a few we’ll be working with.

Example 1. The representation with $\widetilde{M}=0$ is the trivial representation.

Example 2. If we consider the Dirac Gamma matrices $\{\gamma^{\mu},\gamma^{\nu}\}^{\alpha}_{\beta}=2\eta^{\mu\nu}\delta^{\alpha}_{\beta}$, then the Dirac representation is given by \begin{equation} \widetilde{M}^{\kappa\lambda}=\gamma^{\kappa\lambda}=\frac{\mathrm{i}}{4}[\gamma^{\kappa},\gamma^{\lambda}]. \end{equation}

Isomorphic to two copies of su(2)

We rewrite the generators in a non-covariant basis. (This will make classifying all the irreducible representations easier.) The generators of the Lorentz algebra are \begin{equation} L^{i} = \frac{1}{2}\epsilon^{ijk}M_{jk} \end{equation} for spatial rotations, and \begin{equation} K^{i} = M^{0i} \end{equation} for Lorentz boosts. We define \begin{equation} \vec{J}_{\pm} = \frac{1}{2}(\vec{L}\pm\mathrm{i}\vec{K}). \end{equation} We see the commutation relations become \begin{equation} [J^{i}_{\pm}, J^{j}_{\pm}] = \mathrm{i}\epsilon^{ijk}J^{k}_{\pm} \end{equation} and all others vanish. But look, this is 2 copies of the Lie algebra su(2). (More precisely, it is sl(2, C).)

The punchline, however, is:

Each irreducible representation of so(1, 3) is characterized by a pair of half-integers $(j_{+}, j_{-})$.

We have a table of common “families” for particles (in most QFT textbooks, they focus on particles of spin 1 or less…because they’re renormalizable):

$(j_{+}, j_{-})$ Name of Field Dimension of Rep
(0, 0) Scalar 1
(1/2, 0) Left-handed Weyl Spinor 2
(0, 1/2) Right-handed Weyl Spinor 2
(1, 0) (Imaginary) Self-dual 2-form 3
(0, 1) (Imaginary) Anti-self-dual 2-form 3
(1/2, 1/2) Vector (gauge field) 4
(1/2, 1) Left-Handed Rarita-Schwinger field 6
(1, 1/2) Right-Handed Rarita-Schwinger field 6
(1, 1) Graviton (spin-2 field) 9

Poincare Algebra

Lets consider Lorentz transformations plus spacetime translations, i.e., the Poincare group. The algebra changes by adding a generator $P^{\mu}$ for spacetime translations. We find additional commutator relations \begin{equation} [M^{\kappa\lambda}, P^{\mu}]= \mathrm{i}(\eta^{\kappa\mu}P^{\lambda}-\eta^{\mu\lambda}P^{\kappa}) \end{equation} and \begin{equation} [P^{\mu}, P^{\nu}]=0. \end{equation}

Definition (Wigner). A “Particle” is a positive-energy unitary irreducible representation of the Poincare algebra. (End of definition)

Remark. Recall for spacetime translations, $P^{\mu}$ is the energy-momentum four-operator. So $P^{0}$ is the energy. The positive energy condition merely restricts the spectrum of $P^{0}$ to be entirely positive. (End of Remark)

Definition. The “Pauli-Lubanski Vector” is an element of the Poincare algebra defined by \begin{equation} W_{\mu} = \frac{-1}{2}\epsilon_{\mu\nu\kappa\lambda}M^{\nu\kappa}P^{\lambda} \end{equation} where $\epsilon_{0123}=-1$. (End of Definition)

Exercise 1. Work out $[W_{\mu}, P^{\nu}]$, $[W_{\mu}, M^{\kappa\lambda}]$, $[W_{\mu}, W_{\nu}]$.

Exercise 2. Prove or find a counter-example: $W^{\mu}W_{\mu}$ commutes with all elements of the Poincare algebra. (Hint: work out the commutation relations with the generators of the algebra.)

Proposition. The operators $C_{1}:=-P^{\mu}P_{\mu}$ and $C_{2}:=W^{\mu}W_{\mu}$ are Casimir operators for the Poincare algebra, i.e., they commute with all the generators. (End of Proposition)

The eigenvalues of the Casimir operator characterize a particle. The eigenvalue $m^{2}$ of $C_{1}$ is called the mass-squared.

References