Problem 2.3b1

A First Course in String Theory

2.3 Lorentz transformations, derivatives, and quantum operators.

(b) Show that the objects $\displaystyle{\frac{\partial}{\partial x^\mu}}$ transform under Lorentz transformations in the same way as the $\displaystyle{a_\mu}$ considered in (a) do. Thus, partial derivatives with respect to conventional upper-index coordinates $\displaystyle{x^\mu}$ behave as a four-vector with lower indices – as reflected by writing it as $\displaystyle{\partial_\mu}$ .

~~~

$\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\ \frac{\partial}{\partial (x')^\mu} &= \frac{\partial x^\nu}{\partial (x')^\mu} \frac{\partial}{\partial x^\nu} \\ &= \frac{\partial x^0}{\partial (x')^\mu} \frac{\partial}{\partial x^0} + \frac{\partial x^1}{\partial (x')^\mu} \frac{\partial}{\partial x^1} + \frac{\partial x^2}{\partial (x')^\mu} \frac{\partial}{\partial x^2} + \frac{\partial x^3}{\partial (x')^\mu} \frac{\partial}{\partial x^3} \\ \end{aligned}}$

The Lorentz transformation:

$\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\ \end{aligned}}$

Lowering the indices to create covariant vectors:

$\displaystyle{ \begin{aligned} x_\mu &= \eta_{\mu \nu} x^\nu \\ \end{aligned}}$

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

$\displaystyle{ \begin{aligned} \left[ x_\mu \right] &= \left( [\eta_{\mu \nu}] [x^\nu] \right)^T \\ \end{aligned}}$

Change the subject:

$\displaystyle{ \begin{aligned} \left[ x_\mu \right]^T &= [\eta_{\mu \nu}] [x^\nu] \\ [\eta_{\mu \nu}] [x^\nu] &= \left[ x_\mu \right]^T \\ [x^\nu] &= [\eta_{\mu \nu}]^{-1} \left[ x_\mu \right]^T \\ \end{aligned}}$

With $\displaystyle{ \begin{aligned} \eta^{\mu \nu} &\stackrel{\text{\tiny def}}{=} \left[ \eta_{\mu \nu} \right]^{-1} \\ \end{aligned}}$ , we have:

$\displaystyle{ \begin{aligned} \left[ x^\nu \right] &= \left[ \eta^{\mu \nu} \right] \left[ x_\mu \right]^T \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} x^\nu &= x_\mu \eta^{\mu \nu} \\ \end{aligned}}$

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

$\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\ \eta^{\rho \mu} x_\rho &= L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\ x_\rho &= \eta_{\rho \mu} L^\mu_{~\nu} \eta^{\sigma \nu} x_\sigma \\ \end{aligned}}$

…

— Me@2020-07-21 10:46:32 AM

Physics Town

continues

Problem 2.3b1

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