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Indices in Dirac's spinorial representation

Physics Asked on December 27, 2020

The other day in class we derived Dirac’s equation and talked a little about its relativistic covariance. In particular, we said that the electronic wavefunction $psi(x)$ transforms as $psi'(x'(x))=R(omega)psi(x(x’))$, given a parameter-dependent matrix $R(omega)=e^{frac{i}{4}omega_{munu}Sigma^{munu}}$.

I have an issue, which is probably a misunderstanding on my part of the use of indices: in the definition of $R(omega)$ we are summing over both indices $mu$ and $nu$, so… why isn’t the quantity at the exponent (and therefore the whole exponential) a scalar?

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