Sunday, 17 February 2019

The Levi-Civita Symbol

The Levi-Civita symbol, which is not a tensor, is defined as
\begin{align}
{\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ even\ permitation\ of}\ 01..(n-1)\  \\
-1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ odd\ permitation\ of}\ 01..\left(n-1\right) \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(1) \\
\end{align}Instead of $\widetilde{\epsilon }$ it is also often written as $\widehat{\varepsilon }$, which is what I usually do.

We usually only need it in three or four dimensions so it becomes much simpler:
\begin{align}
{\widehat{\varepsilon }}_{\alpha \beta \gamma \delta }=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 0123\  \\
-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 0123 \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(2) \\
\end{align}or
\begin{align}
{\widehat{\varepsilon }}_{ijk}=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }ijk \mathrm{\ is\ an\ even\ permitation\ of}\ 123\  \\
-1 & \mathrm{if\ }ijk \mathrm{\ is\ an\ odd\ permitation\ of}\ 123 \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(3) \\
\end{align}One simple thing is that if any two of the indices on $\widehat{\varepsilon }$ are the same, it vanishes.

Even in four dimensions it is fairly easy to calculate, for example
\begin{align}
{\widehat{\varepsilon }}_{1023}=-{\widehat{\varepsilon }}_{0123}=-1 & \phantom {10000}(4) \\
\end{align}It's easy to jot down the indices and determine the sign by walking indices in one direction or another. Here is an example from my real sad life