Advanced Quantum Mechanics Schwabl · Problem 11.2

Quantization of Relativistic Fields

Physics — Quantum Mechanics · majorana-representation-dirac-equation

Problem

In a Majorana representation of the Dirac equation, the

\gamma

matrices -- indicated here by the subscript

M

for Majorana -- are purely imaginary,

\gamma_M^{\mu\,*} = -\gamma_M^\mu\,, \quad \mu = 0, 1, 2, 3.

A special Majorana representation is given by the unitary transformation

\gamma_M^\mu = U\gamma^\mu U^\dagger

with

U = U^\dagger = U^{-1} = \frac{1}{\sqrt{2}}\gamma^0\left(\mathbb{1} + \gamma^2\right)

. (a) Show that

\gamma_M^0 = \gamma^0\gamma^2 = \begin{pmatrix} 0 & \sigma^2 \\ \sigma^2 & 0 \end{pmatrix}, \qquad \gamma_M^1 = \gamma^2\gamma^1 = \begin{pmatrix} \mathrm{i}\sigma^3 & 0 \\ 0 & \mathrm{i}\sigma^3 \end{pmatrix}

\gamma_M^2 = -\gamma^2 = \begin{pmatrix} 0 & -\sigma^2 \\ \sigma^2 & 0 \end{pmatrix}, \qquad \gamma_M^3 = \gamma^2\gamma^3 = \begin{pmatrix} -\mathrm{i}\sigma^1 & 0 \\ 0 & -\mathrm{i}\sigma^1 \end{pmatrix}.

(b) In Eq. (11.3.14

'

) it was shown that, in a Majorana representation, the charge conjugation transformation (apart from an arbitrary phase factor) has the form

\psi_M^C = \psi_M^*

. Show that application of the transformation

U

to Eq. (11.3.7

'

)

\psi^C = \mathrm{i}\gamma^2\psi

leads to

\psi_M^C = -\mathrm{i}\psi_M\,.