Problem 2.3b1

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A First Course in String Theory

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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}.

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\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}}

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— Me@2020-07-21 10:46:32 AM

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2020.07.22 Wednesday (c) All rights reserved by ACHK