Lorentz transformations and vector fields

[Giuseppe Lacagnina]

Hi Everyone.

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:

$$U(\Lambda)^{-1}A^\mu U(\Lambda)=\Lambda^\mu_{..\nu} A^\nu$$

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?

 

[DelcrossA]

I believe that the LHS is just the generic notation that $$A^\mu$$ is undergoing a symmetry transformation. That is U just represents a certain symmetry group. In order to perform the transformation itself, you must choose a representation for that group, which in the vector representation of the Lorentz group is $$\Lambda^\mu_{..\nu}$$. It only makes sense for a representation to have indices because that is an actual matrix.

My jargon may be off, but that is the way I understand it.