A 4-vector field A^{\mu} is in the (\frac{1}{2}, \frac{1}{2}) - representation. To see this, we note that the 4-vector field has 4 components that all transform between each other under a general Lorentz transformation, thus the vector field is in an irreducible representation. A field in the (m, n)-representation has (2m + 1)(2n + 1) components. The number 4 factors as 4 \times 1 = 2 \times 2.
Therefore, the 4-vector field has to be in either (\frac{3}{2}, 0). (\frac{1}{2}, \frac{1}{2}) or (0, \frac{3}{2}) representations. But, according to the vector addition model, the first two representations allow for J = \frac{3}{2} angular momentum, only, while the second one allows for J = 0. 1. as it should be, because, under ordinary rotations, the time component of the 4-vector behaves as a scalar (J = 0), while the spatial components behave like an ordinary vector (J = 1).
There is a dictionary that transforms the components A_{a \. \dot{a}} to the components A^{\mu}:
<br />
A^{\mu} = \sigma^{\mu}_{a \, \dot{a}} \, A_{a \, \dot{a}}<br />
where, numerically it turns out that \sigma^{\mu}_{a \, \dot{a}} = (I, \vec{\sigma}), where \vec{\sigma} is a Cartesian vector whose components are the Pauli matrices.
