Gamma matrices and projection operator question on different representations

[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.

 

[cello]

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.

It's just that in other representations the left-handed and right-handed spinors are mixed. It's super clear if you use the two-component notation. For references you can read Srednicki's QFT book.

 

[choongstring]

So even in other representations, left-handed and right-handed are defined as
(0,1) (1,0) kind of way? (although this is not a rigorous definition)

 

[cello]

So even in other representations, left-handed and right-handed are defined as
(0,1) (1,0) kind of way? (although this is not a rigorous definition)

In any representation, the chiral basis is defined as the eigenvector of the projection operator. So it won't always be like (1,0).