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There is an equation which I have known for a long time but quite never used really. Now I have doubts I really understand it. Consider the unitary operator implementing a Lorentz transformation. Many books show the following equation for vector fields:

[tex] U(\Lambda)^{-1}A^\mu U(\Lambda)=\Lambda^\mu_{..\nu} A^\nu[/tex]

The operator U should be a matrix with the dimensions corresponding to the representation of the object being transformed. Consider the spinor case for example!

I am getting confused by this. Should not the index on A on the left side be involved in a summation with one of the indices of U?

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# Lorentz transformations and vector fields

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