Lorentz transformations and vector fields

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SUMMARY

The discussion centers on the application of Lorentz transformations in the context of vector fields, specifically the equation U(Λ)⁻¹A^μU(Λ) = Λ^μ_{..ν}A^ν. The unitary operator U represents a symmetry transformation, and its matrix dimensions correspond to the representation of the object being transformed, such as spinors. The confusion arises regarding the summation of indices on the left-hand side, which is clarified by understanding that A^μ is undergoing a symmetry transformation, necessitating a specific representation of the Lorentz group.

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Giuseppe Lacagnina
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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?
 
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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.
 
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