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Section 5.8 Unitary representations of the Poincare group

In the previous section we studied finite-dimensional representations of the Lorentz group. As we saw, because the Lorentz group is non-compact, all those representations are not unitary, since there is not non-trivial unitary finite-dimensional representation. But in quantum mechanics and in particle physics in general, we are interested in unitary representations. The reason is that the representations act on the Hilbert space of states, i.e. they tell us how wave-functions transform. Since the norm square of the wavefunction gives the probability that the system is observed in a given state, we want the representations to preserve the norm square, or, in other words, to preserve the inner product. That is, we want unitary representations.

Moreover, the symmetry group of spacetime is larger than the Lorentz group: it also includes translations. This is the so-called Poincare group (also called “inhomogeneous Lorentz group” in older references). Putting this together, in particle physics we can in fact identify particles by how they transform under the group of symmetries. So we think of particles as irreducible unitary representations of the Poincare group. More precisely, since wave-functions are only defined up to scale, we allow irreducible unitary projective representations. This is what we study in this section.

Subsection 5.8.1 The Poincare group

Let us start by recalling the main features of the Poincare group. The Poincare group acts on a four-dimensional spacetime with coordinates \(x^\mu\) as

\begin{equation*} x^\mu \mapsto (x')^\mu = \sum_{\nu=1}^4 L^\mu_\nu x^\nu + a^\mu, \end{equation*}

where \(L^\mu_\nu\) are the components of a Lorentz transformation. Thus the Poincare group includes both Lorentz transformations and translations.

We can extract the Lie algebra of the Poincare group as usual. It is \(10\)-dimensional. A convenient choice of basis uses the generators \(J_{\mu \nu} = - J_{\nu \mu}\) for the Lorentz transformations, and four generators \(P_\mu\) for the translations. The commutation relations can be written as

\begin{align*} [P_\mu, P_\nu] =\amp 0,\\ [J_{\mu \nu}, P_\rho] =\amp -i (\eta_{\mu \rho} P_\nu - \eta_{\nu \rho} P_\mu),\\ [J_{\mu \nu}, J_{\rho \sigma}] =\amp -i(\eta_{\mu \rho} M_{\nu \sigma} - \eta_{\mu \sigma} M_{\nu \rho} - \eta_{\nu \rho} M_{\mu \sigma} + \eta_{\nu \sigma} M_{\mu \rho}), \end{align*}

where \(\eta_{\mu \nu}\) are the components of the Minkowski metric (i.e. the diagonal matrix \(\text{diag}(-1,1,1,1)\)).

Remark 5.8.1.

The generators \(J_{\mu \nu}\) for Lorentz transformations are connected to the generators of rotation \(J_i\) and boosts \(K_i\) introduced in (5.7.6) by

\begin{equation*} J_i = \frac{1}{2}\sum_{m,n=1}^3 \epsilon_{i m n} J^{m n}, \qquad K_i = J_{i 0}, \end{equation*}

where we raised indices using the Minkowksi metric as usual.

Subsection 5.8.2 Wigner's classification

The goal of this section is to classify unitary representations of the Poincare group. First, since the group is non-compact, we know that the only finite-dimensional unitary representation is the trivial representation. So all non-trivial unitary representations are infinite-dimensional.

The classification of unitary representations of the Poincare group is known as Wigner's classification, since it was originally achieved in Wigner's famous paper “On unitary representations of the inhomogeneous Lorentz group”. This is a very nice paper in fact, have a look!

More precisely, Wigner was interested in representations that are physical, and may correspond to physical particles. Thus, what he classified are irreducible unitary representations that have non-negative, real, mass (\(m \geq 0\)) - negative mass representations are not expected to be physical. We will see in a second how mass arises from the point of view of representations of the Poincare group.

The method that he used is the method of induced representations, also known as the method of little groups in physics. In this approach, one constructs the representations of a group starting from the representations of a subgroup. We will not explain how the method of induced representations works in these notes, but will simply go through the main steps in the context of the Poincare group. We will be very hand-wavy, but I suppose that's ok. For more information one could for instance refer to these lecture notes.

The starting point is to consider the subgroup of the Poincare group consisting of translations. From the point of view of Lie algebras, we consider the subspace generated by the translation generators \(P_\mu\text{.}\) Those generators commute, and hence can be simultaneously diagonalized. We pick an eigenvector \(|p \rangle\text{,}\) with (real) eigenvalues \(p_\mu\text{:}\)

\begin{equation*} P_\mu |p \rangle = p_\mu |p \rangle. \end{equation*}
Remark 5.8.2.

We note in passing that the Poincare algebra has two Casimir invariants. One of them is \(P^2 = \sum_{\mu=1}^4 P_\mu P^\mu,\) and the other is constructed as \(W^2 = \sum_{\mu=1}^4 W_\mu W^\mu\) with

\begin{equation*} W_\sigma = - \frac{1}{2} \sum_{\mu,\nu,\sigma=1}^4 \epsilon_{\mu \nu \rho \sigma} J^{\mu \nu} P^\sigma. \end{equation*}

\(W_\sigma\) is known as the Pauli-Lubanski pseudovector. In any case, since those are Casimir invariants, we can label the irreducible representations by the eigenvalues of these operators. In particular, the eigenvalue of \(P^2\) on the state \(| p \rangle\) is \(- m^2 := \sum_{\mu=1}^4 p_\mu p^\mu.\) What this means is that the “mass squared” \(m^2\) is one of the label for representations of the Poincare group. Thus mass appears naturally in this context; it is one of the label that specifies the representations of the Poincare group.

In any case, going back to the classification, we fix an eigenvector \(|p \rangle\text{,}\) with eigenvalues \(p_\mu\text{.}\) The idea of the method of little groups is to now consider the subgroup of the Poincare group that leaves \(p_\mu\) invariant: this is called the little group (in mathematics, it is called the “stabilizer subgroup”). The idea of the method is to induce unitary irreducible representations of the whole group from the unitary irreducible representations of the little group. In other words, we classify the states with a fixed momentum \(p_\mu\) in terms of the unitary irreducible representations of the little group.

Concretely, we need to separate the classification into three cases, depending on the choice of starting eigenvector \(|p \rangle\text{:}\)

  1. Positive mass \(m >0\text{,}\) in which case we choose \(|p \rangle\) to have eigenvalues \(p_\mu = (m, 0, 0, 0)\) (that is, we use Lorentz transformations to bring ourselves to the rest frame of the particle);
  2. Zero mass \(m = 0\) but with non-zero \(p_\mu\text{,}\) in which case we choose \(|p \rangle\) to have eigenvalues \(p_\mu = (p,0,0,p)\text{,}\) \(p >0\text{;}\)
  3. Zero mass \(m=0\) and zero eigenvalues \(p_\mu = (0,0,0,0)\text{.}\)

The next step is to determine the little group for each of these cases, and classify the irreducible representations of the little groups.

Subsubsection 5.8.2.1 Zero mass and zero momentum

Let us start with the case \(m=0\) and \(p_\mu = (0,0,0,0)\text{.}\) In this case, the little group that leaves \(p_\mu\) fixed is the whole Lorentz group. But we know that its only finite-dimensional unitary representation is the trivial representation. Thus what this case corresponds to is the trivial representation for the Poincare group. We call a state that transforms according to that representation the vacuum, since it is invariant under all symmetries of the Poincare group.

Subsubsection 5.8.2.2 Positive mass

Next, let us consider the case where \(m > 0\) and we are in the rest frame of the particle, that is, \(p_\mu = (m,0,0,0)\text{.}\) In this case, the little group is the subgroup of the Poincare group that leaves \((m,0,0,0)\) invariant. It turns out that this is given by the group of rotations in three-dimensional space, that is, \(SO(3)\text{.}\)

Nice! Because we already know the irreducible unitary projective representations of \(SO(3)\text{.}\) We know that there is an infinite family of them, indexed by a non-negative half-integer \(j\) called the spin, and with dimensions \(2j+1\text{.}\) We know that when \(j \in \mathbb{Z}\) we get an ordinary representation, while when \(j \notin \mathbb{Z}\) we get a spin representation.

Thus, massive particles are classified by an irreducible representation of \(SO(3)\text{,}\) identified by its spin \(j\text{,}\) which is half-integer.

Subsubsection 5.8.2.3 Zero mass with non-zero \(p_\mu\)

Let us now do the case of massless particles, but with non-zero \(p_\mu\text{.}\) We choose a frame such that \(p_\mu = (p,0,0,p)\text{.}\)

The little group here that fixes \((p,0,0,p)\) is not so obvious to see. But it turns out to be given by the special Euclidean group \(SE(2)\text{,}\) which consists of rotations and translations in two dimensions (with Euclidean signature). Thus, to complete Wigner's classification, we need to understand the unitary representations of \(SE(2)\text{.}\)

The Lie algebra \(\mathfrak{se}(2)\) associated to the special Euclidean group \(SE(2)\) is three-dimensional. One generator \(J\) is the generator of infinitesimal rotations, with the two other generators \(P_1\) and \(P_2\) being generators of infinitesimal translations. The commutation relations are:

\begin{equation*} [J,P_1] = i P_2, \qquad [J,P_2] = -i P_1, \qquad [P_1, P_2] = 0. \end{equation*}

We are interested in irreducible unitary representations of \(SE(2)\text{.}\)

Since \(SE(2)\) is non-compact, all its non-trivial unitary representations are infinite-dimensional. But they come in two types: so-called “finite spin representations” and “continuous spin representations”.

The easiest way to construct representations of \(SE(2)\) is to in fact use the method of induced representations again. In this case, we consider the subspace of the Lie algebra generated by translations, and pick an eigenstate \(|k\rangle\) with eigenvalues \(k_i\) under translations:

\begin{equation*} P_i |k \rangle = k_i |k \rangle, \qquad i=1,2. \end{equation*}

We then consider the two cases with either \(k_i = 0\) or \(k_i \neq 0\text{.}\)

In the \(k_i=0\) case, the little group is the group of rotations \(SO(2)\text{.}\) Thus irreducible unitary representations of \(SE(2)\) in this case are induced from the unitary irreducible representations of \(SO(2)\text{,}\) which are all one-dimensional, and indexed by an integer \(h\) which we call the helicity. In fact, since we also allow spin representations (more precisely, the little group should have been the double cover of \(SE(2)\)), we allow the helicity \(h\) to be a half-integer.

In the case where \(k_i \neq 0\text{,}\) we choose a frame such that \(k_i = (k, 0)\text{.}\) The little group is \(SO(1)\text{,}\) which is trivial. In fact, this case gives rise to so-called “continuous spin” (or “infinite spin”) representations. There are two of them here. We will not comment further on this, since those do not appear to be physical.

Subsubsection 5.8.2.4 The end result of Wigner's classification

The end result of the classification is rather remarkable. What it says is that unitary irreducible representations of the Poincare group, which we call “particles” in physics, are indexed by two parameters: a continuous parameter \(m\text{,}\) that we call the “mass of the particle”, and a discrete parameter \(j \in \frac{1}{2} \mathbb{Z}\) or \(h \in \frac{1}{2} \mathbb{Z}\text{,}\) which we call the “spin” or “helicity” of a particle. More precisely, focusing on the case of non-negative mass, particles come in two kinds (we omit the vacuum case here and the continuous spin representations):

  • Massive particles, which are indexed by a positive real number \(m >0\) (the mass), and a half-integer \(j\) (the spin).
  • Massless particles, which are indexed by a zero mass (\(m=0\)), and a half-integer \(h\) (the helicity).

Isn't that great? This is why it makes sense to talk about the mass and spin (or helicity) of particles: that's because those index the unitary irreducible representations of the Poincare group! Moreover, the distinction between massive and massless particles which is crucial in physics appear naturally here in terms of representations. Beautiful!

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