Four-Spinor Reference Sheets

Some facts about 4-spinors listed and discussed. None, well perhaps some, of the work is original. However, locating formulas in other places has proved a timeconsuming process in which one must always worry that the formulas found in any given source assume the other metric ( I use {−1,−1,−1,+1}) or assume some other unexpected preconditions. Here I list some formulas valid in general representations first, then formulas using a chiral representation are displayed, and finally formulas in a special reference frame (the rest frame of the ‘current’ j) in the chiral representation are listed. Some numerical and algebraic exercises are provided. 1 General Representation We can use any four complex numbers as the components of a 4-spinor in a given representation, ψ = col{a+ bi, c + di, e+ fi, g + hi}, where ‘col’ indicates a column matrix and the eight numbers a...h are real. The 4-spinor generates four real-valued vectors: two light-like, one time-like and one space-like. These may be defined using the gamma matrices of the representation as follows: j ≡ ψγψ ; a ≡ ψγγψ ; r ≡ ψγ (