A First Course in String Theory

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2.3 Lorentz transformations, derivatives, and quantum operators.

(b) Show that the objects transform under Lorentz transformations in the same way as the considered in (a) do. Thus, partial derivatives with respect to conventional upper-index coordinates behave as a four-vector with lower indices – as reflected by writing it as .

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The Lorentz transformation:

Lowering the indices to create covariant vectors:

In matrix form, covariant vectors are represented by row vectors:

Change the subject:

With , we have:

Now we lower the indices in order to find the Lorentz transformation for the covariant components:

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— Me@2020-07-21 10:46:32 AM

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