special relativity - What is the meaning of a (1/2, 1/2) representation?

A spin-1 representation is equivalently a (1/2, 1/2) representation of the Lorentz group.

Does this mean we are summing together two irreducible representations labelled by the 'quantum number' a 1/2 each; or are they also another irreducible representation which is labelled by the quantum numbers (1/2, 1/2)?

I'm guessing the latter, as otherwise we're getting a boson (spin-1) by putting two fermions (spin-1/2) together, which doesn't on the face of it seem right; and also particles are irreducible representations.

In which case what are the 1/2's in (1/2, 1/2) mean?

Answer

The labelling of finite-dimensional irreducible representations of (the universal cover of the connected component of) the Lorentz group $\mathrm{SO}(1,3)^\uparrow $ by two half-integers $(s_1,s_2)$ arises as follows:

The complexified Lorentz algebra $\mathfrak{so}(1,3)_\mathbb{C}$ has $\mathfrak{su}(2)\times\mathfrak{su}(2)$ as a compact real form, and the finite-dimensional unitary representation theory of these algebras is equivalent, and moreover equivalent to the complex representation theory of $\mathfrak{so}(1,3)$. The actual rotation algebra $\mathfrak{su}(2)\subset\mathfrak{so}(1,3)$ of physical rotations embeds diagonally into the $\mathfrak{su}(2)\times\mathfrak{su}(2)$ algebra. Qmechanic discusses a lot of the mathematics around the relation of $\mathrm{SO}(1,3)^\uparrow$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ e.g. in this answer.

The finite-dimensional irreducible representations of $\mathfrak{su}(2)$ are well-known from quantum mechanics to be the spin representations labelled by half-integers $s$, so representing two copies of this algebra gives representations labelled by $(s_1,s_2)$. Since the true rotation algebra embeds diagonally, one says that $s_1+s_2$ is the 3-dimensional spin one associates to such a representation. However, one must note that if one looks at the $(1/2,1/2)$ representation as a representation of $\mathrm{SO}(3)$, it is not irreducible - it decomposes as the $1\oplus 0$ representation in terms of spins, or the $\mathbf{3}\oplus\mathbf{1}$ representation in terms of dimensions.

Also, since you said "particles are irreducible representations", note that these representations are finite-dimensional representations on the target space of the classical fields, not unitary representations on the Hilbert space of quantum states. There are no faithful finite-dimensional non-trivial unitary representations of the Lorentz group since the Lorentz group is not compact (i.e. the rapidity can take on all values). The classification of the unitary representations has nothing to do with the finite-dimensional $(s_1,s_2)$ representations and is called Wigner's classification.