A First Course in String Theory

.

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 .

~~~

…

Denoting as is misleading, because that presupposes that is directly related to the matrix .

To avoid this bug, instead, we denote as . So

…

Using the Kronecker Delta and Einstein summation notation, we have

So

In other words,

— Me@2020-11-23 04:27:13 PM

.

One defines (as a matter of notation),

and may in this notation write

Now for a subtlety. The implied summation on the right hand side of

is running over *a row index* of the matrix representing . Thus, in terms of matrices, this transformation should be thought of as the *inverse transpose* of acting on the column vector . That is, in pure matrix notation,

— Wikipedia on *Lorentz transformation*

.

So

In other words,

.

Denote as

In other words,

.

The Lorentz transformation:

.

.

.

.

Now we consider as a function of ‘s:

Since ‘s and ‘s are related by Lorentz transform, is also a function of ‘s, although indirectly.

For notational simplicity, we write as

Since is a function of ‘s, we can differentiate it with respect to ‘s.

Since

,

Therefore,

It is the same as the Lorentz transform for covariant vectors:

— Me@2020-11-23 04:27:13 PM

.

.

2020.11.24 Tuesday (c) All rights reserved by ACHK