Charge conjugation in Dirac equation

Question

I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ . Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix , $T$= transposition, $\gamma^0= $Dirac gamma matricx = $\beta$

I need to know the significance of these equation in charge conjuration .

Answer 1

First, $U$ is surely not "any non-singular matrix". For a given basis, $U$ is almost completely determined i.e. unique. It contains $\gamma_2$ because it's derived from the only imaginary Pauli matrix.

Because of the basic Dirac algebra $$ \{ \gamma_\mu,\gamma_\nu \} = 2\cdot 1_{2\times 2} \cdot g_{\mu\nu} $$ one may see that $\gamma_0$ is Hermitian, $\gamma_0=\gamma_0^\dagger$, while the spatial ones are anti-Hermitian, $\gamma_i=-\gamma_i^\dagger$.

In your identity, you want to relate $\gamma^\mu$ to its transposition $\gamma^{\mu T}$. Up to the sign that depends on the spatial or temporal character of $\mu$, the transposition is the same thing as complex conjugation.

So a related problem is whether the complex conjugate matrices $\gamma^{\mu*}$ can be related to $\gamma^\mu$ by something like a conjugation. And the answer is Yes. The main fact behind the exercise is that $\sigma^2$ is the only imaginary Pauli matrix, so complex conjugation of Pauli matrices is equivalent to the conjugation by $\sigma^2$ with an extra sign. This may be easily generalized if you also include the temporal 0th component and if you use the normal basis.

You should check the identity you want to verify in a particular convenient basis, i.e. with an explicit form of the gamma matrices. The verification is most convenient if you write the gamma matrices in block form, with $2\times 2$ blocks being either multiples of Pauli matrices or the unit matrix.

In a more general representation, the Dirac gamma matrices differ from those in the particular basis you will have verified by a conjugation only, and this may only mean that $U$ is changed in the formula, but the essence of the conjugation is unchanged.

These equations are important because $C$ is related to the charge conjugation – the replacement of particles by antiparticles (e.g. exchange of electrons and positrons). Mathematically, the most important part of the charge conjugation is complex conjugation which is why we needed to express the "complex conjugate gamma matrices as some conjugations of the normal ones".

Theories with a symmetry between matter and antimatter are symmetric under C - the charge conjugation symmetry. Spinors are mapped to $\psi\to C\psi$ etc. and the only hard part of the symmetry of the Lagrangian is a step that requires you to conjugate the gamma matrices by $C$ which is why it's good that we have a way to simplify $C^{-1T} \gamma^\mu C^T$.