by momenta, we have, from exchanging $p$ and $q$ in (\[tbasis\]), $$|p, q\rangle_{S_\theta,A_\theta} =\ \pm\,e^{ i \theta^{\mu\nu}p_\mu
 q_\nu}\,|q,p \rangle_{S_\theta,A_\theta}.$$ This is the origin of the commutation relations $$\begin{aligned}
a_p^\dagger\,a_q^\dagger\,&=& \pm e^{ i \theta^{\mu\nu}p_\mu
  q_\nu}\,a_q^\dagger\,a_p^\dagger \, , \\ 
a_p a_q &=&  \pm e^{ i \theta^{\mu \nu} p_\mu q_\nu} a_q a_p \, .\end{aligned}$$

Gauge Theories
--------------

The algebra ${\cal A}_\theta({\mathbb R}^N)$, regarded as a vector space, is a module for ${\cal A}_0({\mathbb R}^N)$. This observation is of central importance to us, as it allows us to write gauge theories based on arbitrary gauge groups (as opposed to just $U(N)$). We can show this as follows.

For any $\alpha \in {\cal A}_\theta({\mathbb R}^N)$, we can define two representations $\hat{\alpha}^{L,R}$ acting on ${\cal
A}_\theta({\mathbb R}^N)$: $$\hat{\alpha}^L \xi = \alpha * \xi, \quad \hat{\alpha}^R \xi = \xi
* \alpha \quad {\rm for} \quad \xi \in {\cal A}_\theta({\mathbb
R}^N) \ ,$$ where $*$ is the GM product defined by Eq.(\[starmult\]) (or, equivalently, by Eq.(\[starmult1\])). The maps $ \alpha \rightarrow
\hat{\alpha}^{L,R}$ have the properties $$\begin{aligned}
\hat{\alpha}^L \hat{\beta}^L &=& (\hat{\alpha}\hat{\beta})^L, \\
\hat{\alpha}^R \hat{\beta}^R &=& (\hat{\beta}\hat{\alpha})^R, \label{right}\\
{[}\hat{\alpha}^L, \hat{\beta}^R] &=& 0. \label{LRcommute}\end{aligned}$$ The reversal of $\hat{\alpha}, \hat{\beta}$ on the right-hand side of (\[right\]) means that for position operators, $${[}\hat{x}^{\mu 