The definition of a Majorana-neutrino field

For definitness I first discuss the quantitative definition of a ``Majorana neutrino field''. In this I follow the standard literature. Let $\nu$ be a 4-component Dirac neurino field<sup>3</sup>Then, using ``east-coast'' notation (imaginary time) the 2-component Weyl field $\nu_L$ is defined⁴ as[1]:

$$\nu_L = {(1+\gamma_5) \over 2} \nu_L = P_L \nu_L$$ (1)

The 2-component Majorana field $\nu_M$ is defined via the following relations by which the neutrino is its own antiparticle [2]:

$$\nu_M = C \nu{_M}^{\dagger T} \gamma_4 \equiv \nu_M^c; \: \nu_M={1\over\sqrt{2}}(\nu + \nu^c)$$ (2)

Here C is the charge conjugation matrix, $\dagger$ the hermitian conjugate and T a transpose acting only on the spinor, $^c$ symbolizes charge conjugation and a conventional ``creation phase factor''[3] was set to 1 ⁵. The definition in eq.(2) after the semicolon defines the field normalization and it can be easily shown to be the one that fulfills the usual field-anticommutation axioms of quantum-field theory:

$$\left[ \nu_M(\vec{x},t),\eta(\vec{x}^{\prime},t) \right]_+= i\delta(\vec{x}-\vec{x}^{\prime})$$ (3)

where $\eta$ is the field which is canonical conjugate to $\nu_M$. This field normalisation (with the factor 1/$\sqrt{2}$) is generally used in the literature[3,8,9,10,11] ⁶.
Clearly the conditions eq.(1) and eq.(2) are mathematically mutually exclusive; a Weyl particle can never be its own antiparticle.