Are representations of $\text{SL}(2,\mathbb{C})$ indexed by one half-integer or two?

Question

I am very confused by this. In Hall's book on Lie theory, he states that the representations of $\text{sl}(2,\mathbb{C})$ are indexed by a half-integer. This is the usual result for $\text{su}(2)$ in non-relativistic quantum mechanics. This is the case since the complexification $\text{su}(2)_\mathbb{C}=\text{sl}(2,\mathbb{C})$. I am guessing then that to find representations of $\text{SU}(2)$ one is interested in exponentiating representations of $\text{su}(2)_\mathbb{C}$. In quantum field theory, however, it is claimed that the representations of $\text{sl}(2,\mathbb{C})$ are indexed by two half-integers. I believe this is because $\text{sl}(2,\mathbb{C})$ is here seen as a real Lie algebra and thus has twice the dimension. How does this relate to us now considering the double cover $\text{SL}(2,\mathbb{C})$ of the orthochronous proper Lorentz group $L_+^\uparrow$ rather than the double cover $\text{SU}(2)$ of the proper rotation group $\text{SO}(3)$?

Answer 1

1. The Lie group $SL(2,\mathbb{C})$, viewed as a complex Lie group, has irreducible representations of complex dimension $2j+1$ classified by a single half-integer $j\in\frac{1}{2}\mathbb{N}_0$.

2. The Lie group $SL(2,\mathbb{C})$ is the double-cover of the restricted Lorentz group $G:=SO^+(1,3;\mathbb{R})$. The latter is naturally viewed as a real Lie group in physics.

3. Its complexification $G_{\mathbb{C}}=SO(1,3;\mathbb{C})$ has double cover $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$, whose irreducible representations are classified by two half-integers (since there are now a product of two $SL(2,\mathbb{C})$ groups). See e.g. this & this Phys.SE posts.

4. A representation of the complexification $G_{\mathbb{C}}$ is also a representation of the restricted Lorentz group $G$. Conversely, any physically relevant representation of $G$ is expected on physical grounds to be a representation of $G_{\mathbb{C}}$ by analyticity.

Answer 2

In quantum theory (and thus in the classical field theory which will ultimately be physically relevant through quantization), we are only interested in projective representations of the orthochronous Lorentz group $L_+^\uparrow$ on complex vector spaces. Equivalently, we are interested in the representations of the double cover $\text{SL}(2,\mathbb{C})$. Being a simply connected Lie group, these are in one to one correspondence via the exponential map to the representations of its Lie algebra $\text{sl}(2,\mathbb{C})$. Coming from a Lie group, this Lie algebra is to be regarded as real. We are thus interested in the real linear representations of $\text{sl}(2,\mathbb{C})$. These are on one to one correspondence to the complex linear representations of the complexification $\text{sl}(2,\mathbb{C})_\mathbb{C}$.

Indeed, if one has a real linear representation $\pi:\mathfrak{g}\rightarrow\text{End}(V)$, an extension to the complexification is readily available via $\tilde{\pi}:\mathfrak{g}_\mathbb{C}\rightarrow\text{End}(V):(X_1,X_2)\mapsto\pi(X_1)+i\pi(X_2)$. On the other hand, a complex linear representation $\tilde{\pi}:\mathfrak{g}_\mathbb{C}\rightarrow\text{End}(V)$ induces a real linear one $\pi:\mathfrak{g}\rightarrow\text{End}(V):X\mapsto\tilde{\pi}(X,0)$.