Dirac bilinears transform in the Lorentz indices as,
- $\bar{\psi}\psi$ scalar
- $\bar{\psi}\gamma^\mu\psi$ vector
- $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor
- $\bar{\psi}\gamma^{\mu}\gamma^5\psi$ axial vector
- $\bar{\psi}\gamma^5\psi$ pseudoscalar
The scalar as the simplest example,
$\bar{\psi}\psi$ is invariant under Lorentz transformations and is hence a scalar (source)
When referred to as a scalar does that mean I can arbitrarily boost and treat the quantity as a scalar (i.e. freely move the quantity in within terms) or does that just mean that the quantity transforms without any change in rank but cannot be moved about as a number.
Example with a scalar: Is the following permitted?
$$ (u\bar{u})\not pu = \not pu(u\bar{u}) $$ for an arbitrary spinor $u$ and $\not p = \gamma^\mu p_\mu$ as usual.