We discuss the Dirac, Majorana, and Weyl fermion fields. The definitions and motivations for introducing each kind of field is discussed, along with the connections between them. It is pointed out that these definitions have to do with the proper Lorentz group and not with any discrete symmetry. The action of discrete symmetries, such as charge conjugation and CP on various types of fermion fields, which are particularly important for Majorana fermions, is also clarified.
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A 2011 study studied this question.