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

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 $$\text{SO}(3)$$?

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.

• Feb 13, 2019 at 15:59

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 $$\pi:\mathfrak{g}\rightarrow\text{End}(V):X\mapsto\tilde{\pi}(X,0)$$.