In Peskin and Schroeder page 37,(adsbygoogle = window.adsbygoogle || []).push({});

a diagram illustrates how, under the active transformation, the orientation of a vector field must be rotated forward as the point of evaluation of the field is changed.

I understand that the change of the orientation of the vector field is the same idea as that of the transformation matrix mixing up the vector field components to form the new vector field components.

Now, the textbook goes on to mention that:

under 3-dimensional rotations, ##V^{i}(x) \rightarrow R^{ij}V^{j}(R^{-1}x)##

under Lorentz transformations, ##V^{\mu}(x) \rightarrow \Lambda^{\mu}_{\nu}V^{\nu}(R^{-1}x)##

I understand that because we are working in the active transformation, the vector field itself is rotated (or boosted) in space(time) and the coordinate system (or reference system) is held fixed.

Can someone explain to me in simple layman terms why we are applying the a transformation matrix on the vector field when we are using the inverse transformation matrix on the coordinates?

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# Transformations of a vector in the active viewpoint

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