I have already asked this on the mathematics Stack exchange but I thought I'd try it here too!

The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an inner product space $V$ of dimension $n$.
So we can we write;
\begin{equation}
\lambda\in \bigwedge ^p V
\end{equation}
\begin{equation}
\star\lambda\in\bigwedge^{n-p}V
\end{equation}

- I am wondering is this the same operation as used in the Moyal bracket for functions in phase space?

Namely for two functions of the phase space $f$ and $g$, the Moyal bracket is given by;
\begin{equation}
\{f,g\}:=\frac{1}{i\hbar}(f\star g-g\star f).
\end{equation}
I think I'm wrong and that it is somehow a different operation with the same sign, but would really appreciate some help since I'm really not familiar with the Hodge operator other than what I have written above!

- Also if its not too much trouble, could anyone provide a bit of context to the Hodge star operation in physics? e.g. why should I really be interested in vectors in $\bigwedge ^{n-p}V$ space?

This post imported from StackExchange Physics at 2015-01-08 13:59 (UTC), posted by SE-user Janet the Physicist