Gamma matrices projection operator

[choongstring]

Typically I understand that projection operators are defined as

$$P_-=\frac{1}{2}(1-\gamma^5)$$
$$P_+=\frac{1}{2}(1+\gamma^5)$$

where typically also the fifth gamma matrices are defined as

$$\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$$

and.. as we choose different representations the projection operators are.. sometimes in nice form where there is only one identity element however what happens when in certain representations it doesn't come out nicely like that how do I interpret which type of spinors are which chiraliity and such. .. anyways and what are some good materials. (shorter the better) on something complete on clifford algebra and it's representations and all the other things like charge conjugation and such.

 

[azntoon]

A good resource for Clifford algebras is the book "Clifford Algebras and Spinors" by H.F. Jones (second edition). As for understanding the chirality of spinors in different representations, it is important to understand what the gamma matrices are doing in each representation. The gamma matrices form a basis for the Clifford algebra, which is a vector space over the reals. In each representation, the gamma matrices will be written as some combination of the identity matrix and the Pauli matrices (or some other basis). This will determine how the projection operators P_+ and P_- act on a given spinor. For example, in the Weyl representation, the gamma matrices are given by: \gamma^0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\gamma^1=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\gamma^2=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\gamma^3=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}The projection operators in this representation are then given by: P_+=\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}P_-=\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}In this case, it is easy to see that a spinor which has a 1 in the top left component is a right-handed spinor and a spinor which has a 1 in the bottom left component is a left-handed spinor. In general, the easiest way to determine which type of spinor (right- or left-handed) a given spinor is in a given representation is to calculate the matrix product of the projection operator with the spinor. If the result is a non-zero vector, then the spinor has the corresponding chirality.