# The question about multiplications of field functions and vector indices

Recently I have read following.

For the field function $\Psi (x)$ of definite integer spin $n$ the multiplication $\Psi_{a}\Psi_{b}$ refers to the components of tensor rank $2n$. By the way, we may represent $\Psi (x)$ as 4-tensor rank $n$.

For the field function $\Psi (x)$ of definite half-integer spin $n + \frac{1}{2}$ the multiplication $\Psi_{a}\Psi_{b}$ refers to the components of tensor rank $n + \frac{1}{2}$. By the way, we may represent $\Psi (x)$ as $\Psi (x) = \begin{pmatrix} \psi_{a \mu_{1}...\mu_{n}} \\ \kappa^{\dot {a}}_{\quad \mu_{1}...\mu_{n}}\end{pmatrix}$, and $\psi_{a}\kappa_{\dot {a}} \tilde {=}A_{\mu}$ (maybe, it may help).

Do these words about tensors of odd or even rank refer to the 4-tensor (i.e., the product $\Psi_{a}\Psi_{b}$ transforms as the 4-tensor rank $f$ under the Lorentz group transformations) and how to prove them (I'm interested rather in the case of half-integer spin)?

• Why is this downvoted? If there is an error, misunderstanding, etc this can be pointed out in a comment or even better in an answer. – Dilaton Mar 24 '14 at 7:57