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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 field3Then, using ``east-coast'' notation (imaginary time) the 2-component Weyl field $\nu_L$ is defined4 as[1]:
(1) \begin{displaymath}
\nu_L = {(1+\gamma_5) \over 2} \nu_L = P_L \nu_L
\end{displaymath}

The 2-component Majorana field $\nu_M$ is defined via the following relations by which the neutrino is its own antiparticle [2]:
(2) \begin{displaymath}
\nu_M = C \nu{_M}^{\dagger T} \gamma_4 \equiv
\nu_M^c; \: \nu_M={1\over\sqrt{2}}(\nu + \nu^c)
\end{displaymath}

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

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.


next up previous
Next: The unitary equivalence of Up: The non-equivalence of Weyl Previous: Outline of the paper
Rainer Plaga 2001-08-03