12. Representations of matrix groups relevant to physics

In this lecture we focus on the Lorentz group:

\[ \operatorname{Lor} = \{ A\in\operatorname{GL}_4(\mathbb R)\colon A\eta A^\top=\eta\} \]

for \(\eta=\operatorname{diag}(-1,1,1,1)\).

An important observation: the algebra of infinitesimal generators of the Lorentz gorup is isomorphic to the tensor product of the algebra of generators for the group \(\operatorname{SU}_2\):

\[ \text{``}\operatorname{Lor} \cong (\operatorname{SU}_2,\operatorname{SU}_2)\text{''} \]

Therefore finite-dimensional irreducible representations of \(\operatorname{Lor}\) are labelled by two half-integers (spins).

We have the following low-dimension irreps:

• \(\overset{\textcolor{blue}{(\bullet,\bullet)}}{(0,0)}\): this is the trivial representation, sometimes called the scalar representation. It is one-dimensional.

• \(\overset{\textcolor{blue}{(\square,\bullet)}}{(\tfrac12,0)}\): this is a two-dimensional representation, the so-called left-handed Weyl representation. In physics it is used to describe fermions, for example neutrinos.

• \(\overset{\textcolor{blue}{(\bullet,\square)}}{(0,\tfrac12)}\): this is the corresponding two-dimensional representation of right-handed Weyl spinors.

• \(\overset{\textcolor{blue}{(\square,\square)}}{(\tfrac12,\tfrac12)}\): this is the four-dimensional fundamental representation. It is used to describe spin-one bosonic particles, for example the electromagnetic field.

• \((m,n)\): general tensor representations of the Lorentz group. They are \((m+1)\cdot(n+1)\)-dimensional.

• We can also define the so-called Dirac spinors, that are 4-dimensional and transform as the direct sum of Weyl spinors:

\[\underset{\textcolor{blue}{(\square,\bullet)\oplus(\bullet,\square)}}{(\tfrac12,0)\oplus(0,\tfrac12)}\]