\textbf{I} -2\omega \textbf{I} \otimes \delta^{\mu\nu}(b^\dagger_\mu b_\nu - b_\nu b^\dagger_\mu)=H_n\otimes \textbf{I} +\omega\otimes \Sigma_n$$ where in $H_n$ we can recognize the harmonic oscillator Hamiltonian and the spin operator $\Sigma_n$. The universality property of the Clifford algebra grants the existence of an isomorphism between the 2$n$-dimensional Clifford algebra and the Hilbert space $ \mathcal{H}_n=S(\mathbb{R}^n)\otimes \mathbb{C}^{2^n}$. In this representation the Dirac operator is: $$\mathcal{D}_n=i\Gamma^\mu\partial_\mu +\omega\Gamma^{\mu+n}x_\mu  \label{Diracn}$$ Where $\Gamma^\mu$ turns to be $\Gamma^\mu=(b_\mu+b^\dagger_\mu)$, $ \Gamma^{\mu+n}=i(b^\dagger_\mu-b_\mu)$ which satisfy the relations: $$\Gamma_a\Gamma_b+\Gamma_b\Gamma_a=2\delta_{ab} \ \textrm{with} \ a,b=1,\cdots,2n \label{Gamman}$$ Beside, the grading operator is represented as: $$\chi_n=(-i)^n(-1)^{\frac{n(n-1)}{2}}\otimes \Gamma_1\cdots\Gamma_{2n}$$

An harmonic spectral triple for the Moyal plane
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In the framework of non-commutative field theories on 4-dimensional Moyal plane has been proved [@phi4-non; @matrix-renorm] that the introduction of an harmonic oscillator term makes a $\phi^4$-model on 4-dimensional Moyal plane renormalizable. Such oscillator term can be written as: $$H_m=-\frac{\partial^2}{\partial x_\mu \partial x^\mu}+\Omega^2 \tilde{x}_\mu\tilde{x}^\mu +m^2$$ where $\tilde{x}_\mu:=2(\Theta^{-1})_{\mu\nu} x^\nu$, $\Theta$ can be chosen as two copies of the Pauli matrix $\Theta = i\theta\sigma_2  \otimes \textbf{I}_2$ or explicitly: $$\Theta=\left(
\begin{array}{cccc}
0& \theta &0&0 \\
-\theta &0&0&0 \\
0&0&0&\theta \\
0&0&-\theta&0
\end{array}
\right), \ \theta\in\mathbb{R}$$ With this choice we have $\Theta^{-1} =  -\frac{i\sigma_2  }{\theta}\otimes \textbf{I}_2 $. Quantum mechanics tell us that in the Hilbert space 