Hodge dual and the Moyal bracket? Any link?

Question

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;

$$\begin{equation} \lambda\in \bigwedge ^p V \end{equation}$$ $$\begin{equation} \star\lambda\in\bigwedge^{n-p}V \end{equation}$$

1. 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!

1. 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

Comments

Moyal product is a product on a certain space of functions, while the Hodge operator is a map between exterior forms

This post imported from StackExchange Physics at 2015-01-08 13:59 (UTC), posted by SE-user Phoenix87

---

Why should it be the same operation? They act on totally different spaces! Also, the exterior product arises naturally in the course of looking at differential forms.

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

---

I just wondered if the Moyal bracket was some form of "natural relation" between functions on phase space, just like we have these natural mappings between $\bigwedge^p V$ and $\bigwedge ^{n-p}V$.

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

---

Crossposted from math.stackexchange.com/q/1095512/11127

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