Foundations of Quantum Field Theory. Klaus D Rothe
suited for discussing the relativistic limit, or the case of zero mass particles. There exists another choice of basis for the Gamma matrices called the Majorana representation which is particularly suited for the case of charge self-conjugate fermions, referred to as Majorana fermions.
Spin zero, charge neutral particles are called “self-conjugate”, and are described by real fields satisfying the Klein–Gordon equation, which itself is real. In the case of “self-conjugate” spin one-half particles, the analogon is provided by the “Majorana” representation.
The Majorana representation is, however, also useful in the case where fermions and anti-fermions are distinct particles if the symmetry group in question is for instance the orthogonal group O(N) rather than the unitary group U(N). The reason is that in the Majorana representation the Dirac equation is real. We therefore discuss separately the notion of Majorana representation and Majorana fermions.
Majorana representation
There exists a choice of basis in which all Dirac matrices are purely imaginary. This is called the Majorana representation of the Gamma-matrices. They are obtained from the corresponding ones in the Dirac representation via the unitary transformation
with
and have the property
One explicitly computes
Now, the Dirac wave function and its charge conjugate (in the Dirac and Weyl representations) are related by (see Chapter 7, Eq. (7.48)),
Applying the unitary operator (4.65) to both sides of the equation, we obtain for the Dirac wave function in the Majorana representation
where
We thus conclude that
Summarizing we have in the Dirac, Weyl and Majorana representations
where
Majorana spinors
Applying the unitary operator (4.65) to the Dirac spinors in the Dirac representation (4.21) (we use c−1 = iσ2)
we obtain for the corresponding spinors in the Majorana representation
Recalling that
we see from (4.70) and (4.71) that
In the case of Majorana fermions, it is convenient to redefine the phase of the Dirac spinors via the replacements
For this new choice of phase, Eq. (4.72) is replaced by
Self-conjugate Dirac fields
Fields describing fermions which are their own anti-particles are said to be self-conjugate and have the Fourier representation
with the property (4.73). These fields are real,
and play the role of real scalar fields in the case of spin 1/2 fields.
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1P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford 1958).
2Here and in what follows: in formulae which hold generally, without reference to a particular basis such as the Dirac representation, we omit the subscript D.
3The reason for introducing c will become clear in Chapter 7, Eq. (7.15).
4The minus sign is a consequence of our Dirac scalar product.
5We follow in general the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.
6Substituting (4.27) into (4.28) yields
7H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publications, Inc. New York, 1931).
8See Chapter 5 for this terminology.
9Note that
where S is given by (4.40). This agrees with the usual V − A coupling of neutrinos in the weak interactions.
Chapter 5
The Free Maxwell Field
As is well known, the electromagnetic field can be interpreted on the quantum level as a flux of quanta, called photons. In fact, this interpretation first arose in connection with Planck’s formula describing the spectrum of black-body radiation. As Maxwell’s equations show, these quanta propagate with the velocity of light in all inertial frames, so there exists no rest frame we can associate with them. Photons can thus be viewed as “massless” particles. According to our discussion in Section 6 of Chapter