U(1)

Prerequisites: Lie Groups. Full path (3 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Phase factors $e^{i \phi(\vec x,t)}$, like they appear in quantum mechanics, are just complex numbers with amplitude $1$. Therefore, we can picture them as points on a circle with radius $1$:

This collection of all complex numbers with amplitude $1$ is what we call the group $U(1)$.

Concrete

Lie Algebra

The Lie algebra corresponding to the group U(1) is usually identified with the set of pure imaginary numbers $Im \mathbb{C} = \{ i \theta : \theta \in \mathbb{R} \}$.

Take note that the tangent space of a circle is, of course, just a copy of $\mathbb{R}$ but the isomorphic space $Im \mathbb{C}$ is more convenient because its elements can be "exponentiated" to give the elements $e^{i \theta}$ of $U(1)$.

Representations

The diagram below shows the defining representation of $U(1)$ in its upper branch and the conjugate representations of the same group in its lower branch. For a more detailed explanation of this diagram and more representations of $U(1)$ see Fun with Symmetry.

Abstract

$U(1) = \{z\in\mathbb{C} : |z| = 1\}$ is a compact, connected, abelian Lie group of dimension one. As a manifold it is the circle $S^1$, and $\mathrm{e}^{i\theta}\mapsto R(\theta)$ is an isomorphism with SO(2). The exponential map $\mathfrak{u}(1) = i\mathbb{R}\to U(1)$, $i\theta\mapsto\mathrm{e}^{i\theta}$, is surjective with kernel $2\pi i\mathbb{Z}$, so $U(1)\cong\mathbb{R}/2\pi\mathbb{Z}$ and $\pi_1(U(1)) = \mathbb{Z}$.

Representations. All irreducible unitary representations are one-dimensional, $$\rho_q(\mathrm{e}^{i\theta}) = \mathrm{e}^{iq\theta}, \qquad q\in\mathbb{Z} .$$ The integer $q$ is the charge. If the symmetry group of a theory is the compact group $U(1)$ and not its cover $\mathbb{R}$, all charges are integer multiples of a smallest unit. The decomposition of $L^2(U(1))$ into the $\rho_q$ is the Fourier series.

Gauge theory. A $U(1)$ gauge field is a connection on a principal $U(1)$-bundle $P\to M$. Locally it is a real one-form $A$ (the $\mathfrak{u}(1)$-valued form is $-iA$), and we use the convention that a field of charge $q$ has covariant derivative $D\psi = (d - iqA)\psi$. A gauge transformation $\psi\mapsto\mathrm{e}^{iq\lambda}\psi$ then requires $A\mapsto A + d\lambda$. The curvature $F = dA$ is gauge invariant since the group is abelian. Gauge-invariant information beyond $F$ sits in the holonomies $\exp\left(i q\oint_\gamma A\right)$ around closed loops, which is what the Aharonov-Bohm effect measures. On a closed surface $\Sigma$, for example a sphere around a magnetic monopole, the bundle can be non-trivial, and its first Chern number is the integer $$c_1 = \frac{1}{2\pi}\int_\Sigma F\ \in\mathbb{Z} .$$ This is the topological form of the Dirac quantisation of magnetic charge. The Hopf fibration $S^3\to S^2$ is the $U(1)$-bundle with $|c_1| = 1$. In the abelian Higgs model the vacuum manifold is a circle, and $\pi_1(U(1)) = \mathbb{Z}$ labels the winding of vortex solutions.

Relation to other groups. $U(1)$ is the maximal torus of $SU(2)$ and $SO(3)$, and $U(N) = (SU(N)\times U(1))/\mathbb{Z}_N$. In the Standard Model the electromagnetic $U(1)$ is a combination of the weak hypercharge $U(1)_Y$ and the diagonal generator $T_3$ of $SU(2)_L$.

Path to this page

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