## Bilinear Covariants

We have seen that the constant matrices can be used to make a conserved vector current that transforms correctly under Lorentz transformations. With 4 by 4 matrices, we should be able to make up to 16 components. The vector above represents 4 of those.

The Dirac spinor is transformed by the matrix . This implies that transforms according to the equation. Looking at the two transformations, we can write the inverse transformation. This also holds for . From this we can quickly get that is invariant under Lorentz transformations and hence is a scalar. Repeating the argument for we have according to our derivation of the transformations . Under the parity transformation the spacial components of the vector change sign and the fourth component doesn't. It transforms like a Lorentz vector under parity.

Similarly, for , forms a rank 2 (antisymmetric) tensor.

We now have 1+4+6 components for the scalar, vector and rank 2 antisymmetric tensor. To get an axial vector and a pseudoscalar, we define the product of all gamma matrices. which obviously anticommutes with all the gamma matrices. For rotations and boosts, commutes with since it commutes with the pair of gamma matrices. For a parity inversion, it anticommutes with . Therefore its easy to show that transforms like a pseudoscalar and transforms like an axial vector. This now brings our total to 16 components of bilinear (in the spinor) covariants. Note that things like is just a constant times another antisymmetric tensor element, so its nothing new.

 Classification Covariant Form no. of Components Scalar 1 Pseudoscalar 1 Vector 4 Axial Vector 4 Rank 2 antisymmetric tensor 6 Total 16
The matrices can be used along with Dirac spinors to make a Lorentz scalar, pseudoscalar, vector, axial vector and rank 2 tensor. This is the complete set of covariants, which of course could be used together to make up Lagrangians for physical quantities. All sixteen quantities defined satisfy .

Jim Branson 2013-04-22