Wigner's Little Groups

Prerequisites: Poincare Group, Representation Theory. Full path (10 pages)

• Intuitive
• Concrete
• Abstract
• History

Intuitive

In special relativity a particle is described by its energy and momentum. Lorentz transformations (rotations and boosts) change the momentum, but some of them leave a given momentum unchanged. These form the little group of that momentum. Wigner's insight was that the little group decides what kind of internal states, such as spin or polarization, a particle can have.

For a massive particle we can go to its rest frame. Boosts would set the particle in motion, so the transformations that leave it at rest are the ordinary rotations in three dimensions. The internal states of the particle must therefore form a representation of the rotation group, and this is exactly what spin is: a particle has twice its spin plus one states, which rotations mix among each other.

A massless particle has no rest frame, because it always moves at the speed of light. The transformations that leave its momentum unchanged are rotations about its direction of motion, together with a less obvious set of combined boosts and rotations. For all known massless particles the second set acts trivially, and only rotations about the direction of motion matter. A rotation about this axis can only multiply a state by a phase, so a massless particle has a single internal label, its helicity, which is the spin component along the direction of motion. This explains why a photon has only two polarization states (helicity plus one and minus one) although a massive spin-1 particle has three.

Together with the mass, the representation of the little group thus classifies the possible one-particle states that are compatible with special relativity.

Concrete

The little group is a subgroup of the Poincare group which leaves the four-momentum of a given particle invariant. The little groups for massive and massless particles are locally isomorphic to the three-dimensional rotation group and the two-dimensional euclidean group respectively. GAUGE TRANSFORMATIONS AS LORENTZ-BOOSTED ROTATIONS by D. HAN et. al.

• A good explanation can be found at page 8 in https://indico.cern.ch/event/544849/contributions/2214530/attachments/1301154/1942518/yskim.pdf
• Another great introduction is section 2 here https://arxiv.org/pdf/1709.04891.pdf
• A good discussion can also be found in Groups, Physics, and Geometry by Gilmore

Abstract

We use the metric signature $(+,-,-,-)$, so $p^2 = m^2$ for a particle of mass $m$. The symmetry group of relativistic quantum mechanics is the universal cover of the proper orthochronous Poincaré group, $\widetilde{\mathcal{P}} = \mathbb{R}^{1,3}\rtimes SL(2,\mathbb{C})$. By Wigner's theorem, symmetries act on the Hilbert space by (projective) unitary representations, and irreducible ones describe elementary particles.

Because the translation subgroup is abelian and normal, a unitary irreducible representation decomposes over the joint spectrum of the momentum operators $P^\mu$. The Lorentz group permutes the momenta, so the spectrum is a single orbit $\mathcal{O}$ of $SL(2,\mathbb{C})$ acting on momentum space. For a reference momentum $k\in\mathcal{O}$ the little group (stabilizer) is

$$G_k = \{\Lambda\in SL(2,\mathbb{C}) : \Lambda k = k\} .$$

Mackey's theory of induced representations states that unitary irreducible representations of $\widetilde{\mathcal{P}}$ are in one-to-one correspondence with pairs $(\mathcal{O}, D)$, where $D$ is a unitary irreducible representation of $G_k$. Concretely, choose for each $p\in\mathcal{O}$ a standard Lorentz transformation $L(p)$ with $L(p)k = p$, and define the Wigner rotation $W(\Lambda,p) = L(\Lambda p)^{-1}\Lambda L(p)\in G_k$. With states normalised as $\langle p',\sigma'|p,\sigma\rangle = \delta_{\sigma'\sigma}\delta^3(\vec p'-\vec p)$,

$$U(\Lambda)\,|p,\sigma\rangle = \sqrt{\frac{(\Lambda p)^0}{p^0}}\ \sum_{\sigma'} D_{\sigma'\sigma}\big(W(\Lambda,p)\big)\,|\Lambda p,\sigma'\rangle , \qquad U(a)\,|p,\sigma\rangle = e^{ip\cdot a}|p,\sigma\rangle ,$$

where the sign in the translation phase is a convention.

The orbits and little groups are:

• $p^2 = m^2>0$, $p^0>0$: $k = (m,0,0,0)$, $G_k = SU(2)$, the double cover of $SO(3)$. Its irreducible representations are labelled by spin $s\in\frac{1}{2}\mathbb{Z}_{\geq0}$, of dimension $2s+1$. The Casimirs are $P^2 = m^2$ and $W^2 = -m^2 s(s+1)$, with $W^\mu = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}P_\nu M_{\rho\sigma}$ the Pauli-Lubanski vector.
• $p^2 = 0$, $p^0>0$: $k = (E,0,0,E)$, $G_k$ is the double cover of $ISO(2)$, the Euclidean group of the plane. It is generated by the rotation $J_3$ about the direction of motion and two commuting combinations of $J_1, J_2, K_1, K_2$ that act like translations of the plane. If these act trivially, $D$ is one-dimensional, $D = e^{ih\theta}$ for a rotation by $\theta$, with helicity $h\in\frac{1}{2}\mathbb{Z}$. Each helicity gives a separate irreducible representation; parity, when it is a symmetry, relates $h$ and $-h$. If the translation-like generators act non-trivially, one obtains the infinite-dimensional "continuous spin" representations, which are not used for known particles.
• $p = 0$: $G_k = SL(2,\mathbb{C})$ itself, describing the vacuum (trivial representation) and other zero-momentum representations.
• $p^2<0$: $G_k$ covers $SO(1,2)$; these tachyonic representations do not appear in physical theories with a stable vacuum.

The negative-energy orbits $p^0<0$ have the same little groups and are excluded by the spectrum condition. The statement in the quote above, that the little groups are locally isomorphic to $SO(3)$ and $ISO(2)$, refers to the first two cases.

History

https://arxiv.org/pdf/hep-th/9512151.pdf

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Newton's Second Law
5. Vector Calculus
6. Newtonian Mechanics
7. Representation Theory
8. Special Relativity
9. Lorentz Group
10. Poincare Group