SO(2)

Prerequisites: Lie Groups. Full path (3 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Think of the hand of a clock that can turn by any angle, not only in whole minutes. Each possible turn is one element of SO(2). Doing one turn after another just adds the angles, so the order does not matter, and turning by a full circle brings everything back to where it started. The group therefore looks like a circle itself: one point on the circle for each angle.

SO(2) is the symmetry group of anything that looks the same after any rotation about a fixed point in a plane, such as a circle or a round disc. In physics it appears whenever a problem has an axis of symmetry, and the conserved quantity that comes with it is the angular momentum about that axis.

The Lie group $SO(2)$ describes all possible 2D rotations. The group is one dimensional, that is, it has only one parameter: the rotation angle.

The group $SO(2)$ is isomorphic to $U(1)$.

Concrete

Representations

The diagram below shows the (2-dimensional) defining representation of $SO(2)$ in its upper branch and a 4-dimensional, reducible representation of the same group in the lower branch. For a more detailed explanation of this diagram and representations of other Lie groups see Fun with Symmetry.

Abstract

$SO(2) = \{R(\theta) : \theta\in\mathbb{R}/2\pi\mathbb{Z}\}$ with $$R(\theta) = \begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix} = \mathrm{e}^{\theta J}, \qquad J = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}.$$ It is a compact, connected, abelian Lie group, diffeomorphic to the circle $S^1$. The map $R(\theta)\mapsto\mathrm{e}^{i\theta}$ is the isomorphism with $U(1)$. Its Lie algebra $\mathfrak{so}(2)\cong\mathbb{R}$ is one-dimensional with vanishing bracket. The fundamental group is $\pi_1(SO(2)) = \mathbb{Z}$ and the universal cover is $\mathbb{R}\to SO(2)$, $\theta\mapsto R(\theta)$, with kernel $2\pi\mathbb{Z}$.

Representations. Since the group is abelian and compact, every complex irreducible representation is one-dimensional and unitary. They are the characters $$\chi_n(\theta) = \mathrm{e}^{in\theta}, \qquad n\in\mathbb{Z},$$ and the condition $\chi(2\pi) = 1$ is what forces $n$ to be an integer. Over $\mathbb{C}$ the defining representation splits as $\chi_{1}\oplus\chi_{-1}$, with eigenvectors $(1,\mp i)^T$ of $J$. Over $\mathbb{R}$ the irreducible representations are the trivial one and the two-dimensional ones $\theta\mapsto R(n\theta)$, $n\geq 1$. The decomposition $L^2(S^1) = \widehat\bigoplus_{n\in\mathbb{Z}}\mathbb{C}\,\mathrm{e}^{in\theta}$ given by the Peter-Weyl theorem is the Fourier series.

In quantum mechanics the generator $-i\hbar\,\partial_\theta$ of rotations about an axis has eigenvalues $n\hbar$, which is the quantisation of the angular momentum component along that axis. The universal cover $\mathbb{R}$ has the unitary representations $\theta\mapsto\mathrm{e}^{i\alpha\theta}$ for every real $\alpha$. They are projective representations of $SO(2)$. For the rotation group of the plane in $2+1$ dimensions this allows spin that is neither integer nor half-integer, which is the group-theoretic basis for anyons.

Path to this page

1. Linear Algebra
2. Group Theory
3. Lie Groups