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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.

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

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