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How can I find the similarity transformation $S$ between gamma matrices in the Dirac representation $\gamma_D$ and Majorana representation $\gamma_M$ in 4 dimensions theory?

The relation is $\gamma_M = S \gamma_D S^{-1}$

Actually, my question is about the general method of finding the similarity transformation between 2 give gamma matrices in different representation, and the given problem above is just for the sake of demonstration.

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Well, in general, the "relation" you wrote is a set of linear equations so if you know how to solve linear equations, you should be able to do it. Someone with experience will figure out that $S$ has a relatively special form, perhaps block-diagonal in a basis, and so on, so the number of equations he will solve may be reduced. Even more realistically, one may guess the right solution for simple enough cases like yours and prove that the solution works. –  Luboš Motl Sep 9 '13 at 5:40
    
See French Wikipedia. The gamma matrices $\gamma_D$ are Dirac gamma matrices. –  Trimok Sep 9 '13 at 10:59
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A completely general method: Multiply on the right by $S$ to get $\gamma_M S = S \gamma_D$. Since you already know $\gamma_M$ and $\gamma_D$ these are just linear equations for the elements of $S$ which are completely straightforward to put on a computer. –  Michael Brown Sep 9 '13 at 11:36

1 Answer 1

If you know the change of the (vector) basis, the answer is straightforward.

If you don't know the change of the (vector) basis, but only want some particular representation for the gamma matrices (for instance you want only real matrices, or only imaginary matrices), you may try for $S$ :

$$S=\frac{1}{\sqrt{2}} \begin{pmatrix} A&B\\-\epsilon B&\epsilon A \end{pmatrix}, S^{-1}=\frac{1}{\sqrt{2}}\begin{pmatrix} A&-\epsilon B\\ B&\epsilon A \end{pmatrix}$$

where $\epsilon = \pm1$, $A, B$ are $2*2$ matrices, such as $A^2= B^2=1$, and $[A, B]=0$.

For instance, you may take one of the matrix equals to $\pm \mathbb{Id}$, and the other being a Pauli matrix $\pm \sigma_i$.

For obtaining Majorana representation from Dirac representation, we may use : $\epsilon = -1, A = \mathbb{Id}, B = \sigma_y$

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