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
.
~~~
…
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
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2020.11.24 Tuesday (c) All rights reserved by ACHK