Irreducible Representations of $\mathrm{U}(1)$

A representation $\rho$ of a group $G$ on a vector space $V$ always has the subspace $\{0\}$ and $V$ itself as invariant subspaces. Here a subspace $W\subset V$ is invariant means that $\rho(g)W\subset W$ for every $g\in G$. if $\rho$ has no other invariant subspaces, we say that it is irreducible.

Theorem. If $G$ is compact, every representation of $G$ is equivalent to a direct sum of irreducible representations.

This theorem is important for physicists since most Lie groups that are important in physics are compact. The theorem also says that if $G$ is compact, irreducible representations are the building blocks of other representations.

Example. For each $n\in\mathbb{Z}$, define $\rho_n:\mathrm{U}(1)\longrightarrow\mathrm{GL}(1,\mathbb{C})$ by
$$\rho_n(e^{i\theta})v=e^{i n\theta}v.$$
Then each $\rho_n$ is irreducible since $\mathbb{C}$ has no nontrivial vector subspaces. What is really important about this example is that any complex 1-dimensional representation is equivalent to $\rho_n$ for some $n\in\mathbb{Z}$. (Prove this!)

Schur’s Lemma. Let $\rho: G\longrightarrow\mathrm{GL}(V)$ be an irreducible complex representation and let $\phi: V\longrightarrow V$ an interwining map of $V$ with itself (i.e. $\phi(\rho(g)v)=\rho(g)(\phi(v))$ for all $g\in G$, $v\in V$). Then $\phi=\lambda I$ for some $\lambda\in\mathbb{C}$.

Suppose the group $G$ is abelian and $g\in G$. Then
for all $g’\in G$ , $v\in V$. Since $\rho(g)$ is an interwining map of $V$ with itself, $\rho(g)$ is a scalar multiple of $I$ by Schur’s Lemma. So every subspace of $V$ is invariant and hence $\rho$ is 1-dimensional. This means that any irreducible representation of $\mathrm{U}(1)$ is equivalent to one the $\rho_n$. Since $\mathrm{U}(1)$ is compact, any finite dim representation of $\mathrm{U}(1)$ is given as a direct sum of the $\rho_n$.

In quantum mechanics the electric charge of a particle is assumed to be a (integer) multiple of a certain unit charge $q$ i.e. charge is quantized. (This was indeed the case as confirmed by experiments.) In terms of representation, this means that a particle with charge $nq$ transforms according to $\rho_n$ of $\mathrm{U}(1)$. If we move a particle of charge $nq$ around a loop $\gamma$ in spacetime, its wave function is multiplied by a certain phase
$$e^{-\frac{i}{\hbar}nq\oint_\gamma A}\in\mathrm{U}(1)$$
where $A$ is the vector potential or more generally a connection as
$$\rho_n\left(e^{-\frac{i}{\hbar}q\oint_\gamma A}\right)v=e^{-\frac{i}{\hbar}nq\oint_\gamma A}v.$$

Proposition. The tensor product $\rho_n\otimes\rho_m$ is equivalent to $\rho_{n+m}$.

I will leave the proof of this proposition as an exercise. This proposition has an interesting physical implication. If we have two particles corresponding to two different representations of a group, a bound state corresponds to the tensor product of the two representations. The lectric charge of such a bound state is the sum of the charges of the constituents.


[1] John Baez, Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific 1994

[2] Brian C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Springer-Verlag 2003

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