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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As a recent example, see
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[PubMed]
The fallacy in the argument was pointed out in
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[PubMed]
6.
We use the notation that whenever an array is enclosed in square brackets, each entry should be thought of as a block of length 2, that is, a matrix for square arrays and a column for a column array.
7.
Even the requirement of all-imaginary matrices does not specify the representation of Eq. (12) uniquely. Other representations with the same property can be obtained by any interchange of the matrices for , , and that are given in Eq. (12), with the option of changing the overall sign of any number of them.
8.
We take the convention . To avoid any possible confusion, we use neither the antisymmetric tensor nor the components of with lower indices.
9.
This statement, by itself, only implies that left-chiral and right-chiral fields fall into different irreducible representations. It does not preclude the possibility that either of these can be further reduced. We show in Sec. VI that the chiral fields are irreducible.
10.
Such identities can be verified by considering all possible cases. They can also be derived from the basic properties of Dirac matrices in Eqs. (9) and (10). See, for example,
P. B.
Pal
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,” arXiv:physics/0703214.11.
Derived independently by
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2011
American Association of Physics Teachers
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