Consider parity then it is no longer enough to talk about two spinors

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Ryder in chapter two of his book says that if we consider parity then it is no longer enough to talk about two spinors and so he introduces 4 spinors. Is there some postulate of Quantum physics that has to do with state trasformation under parity??
 
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Srednicki's book (available free online, google to find it) goes into great detail on the properties of two- and four-component spinors, including parity transformations.
 

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