# How to treat charge conjugation and time reversal operators for Dirac Field in representation invariant way?

Since manipulations with charge conjugation and time reversal operators involve taking complex conjugate of bispinors, most formulas are not invariant under change of representation of $\gamma$ matrices. For example in chiral representation Majorana condition takes the form $$\psi(x) = -i \gamma^2 \psi^*(x).$$ However it is known that it is much simpler in Majorana representation. I would like to know if there is a way of writing these equations down in a way which is not representation dependent. General advices on how to treat these expressions in efficient and unambigous way will also be appreciated.

• Note that I had to make an edit in the answer. I had missed the inverse sign in $B^{-1}\Psi^*$ – BoundaryGraviton Aug 11 '16 at 12:24

Consider the complete Clifford Algebra $$\Gamma^A=\brace 1,\gamma^\mu,\gamma^{\mu_1\mu_2},...,\gamma^{\mu_1...\mu_n}$$ where $d=2n$ for even spacetime dimension $d$ and $d=2n+1$ for odd spacetime dimension.
There exists a unitary matrix $C$ in any Hermitian representation(where $\gamma^{\mu\dagger}=\eta_{\mu\nu}\gamma^\nu$) such that $$(C\Gamma^A)^T=-t_rC\Gamma^A$$ where $\Gamma^A$ is a tensor of rank $r$ and $t_r=\pm1$. i.e. $C\Gamma^A$ is either symmetric or antisymmetric.
Charge conjugation of a spinor is then defined as $\Psi^C=B^{-1}\Psi^*$ where $B=it_0C\gamma^0$.
$C$ matrices in two different representations of the Clifford algebra are related as $C^\prime=(S^{T})^{-1}CS^{-1}$ where $S$ relates the two representations (I will leave this as an exercise for now).
A Majorana spinor obeys $\Psi=\Psi^C$. Charge conjugation of a matrix is defined as $M^C=B^{-1}M^*B$. The above construct is very helpful in manipulating expressions with Majorana spinors. A very good reference is Supergravity by Daniel Z Freedman and Antoine Van Proeyen.