# Hodge dual and the Moyal bracket? Any link? [closed]

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; $$\lambda\in \bigwedge ^p V$$ $$\star\lambda\in\bigwedge^{n-p}V$$

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; $$\{f,g\}:=\frac{1}{i\hbar}(f\star g-g\star f).$$ 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?

## closed as unclear what you're asking by ACuriousMind♦, bobie, Pranav Hosangadi, Danu, NeuneckJan 8 '15 at 13:19

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• Moyal product is a product on a certain space of functions, while the Hodge operator is a map between exterior forms – Phoenix87 Jan 7 '15 at 22:20
• 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. – ACuriousMind Jan 7 '15 at 22:24
• 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 $\bigwedge ^{n-p}V$. – user58536 Jan 7 '15 at 22:41
• Crossposted from math.stackexchange.com/q/1095512/11127 – Qmechanic Jan 7 '15 at 23:21
• Maybe category theory has something to say about this? Also, I don't think this is a question suited to this site: It belongs on Mathematics! – Danu Jan 8 '15 at 11:43