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

Similarly, for
,

forms a

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

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

Classification |
Covariant Form |
no. of Components |

Scalar | 1 | |

Pseudoscalar | 1 | |

Vector | 4 | |

Axial Vector | 4 | |

Rank 2 antisymmetric tensor | 6 | |

Total |
16 |

Jim Branson 2013-04-22