$L^2(\mathbb{R}^4)$ exists an orthonormal basis $\psi_s ,\ s\in\mathbb{R}^4$ of eigenfunctions of $H_m$ with eigenvalues $$\lambda_s(m)=\left(\frac{4\Omega}{\theta} \left(s+2+ \frac{\theta m^2}{2\Omega}\right)\right), \ s\in\mathbb{N}$$ The inverse $H^{-1}_m$ extends to a selfadjoint compact operator on $L^2(\mathbb{R}^4)$ with eigenvalues $\lambda^{-1}_s(m)$. If we look at the trace the operator $H^{-4}_m$ we find: $$\operatorname{Tr}(H^{-s}_m)=\sum_{n=0}^\infty(n + 3)(n + 2)(n + 1)(\lambda_n (m))^s$$ Which is derived simply from the number of possibilities to express $s$ as a sum of four ordered natural numbers. This means that $H^{-4}$ belongs to the Dixmier trace ideal $L^{(1,\infty)}(L^2(\mathbb{R}^4))$ of compact operators and the relation $\mathcal{D}^2=H$ implies that the 4-dimensional Moyal space has spectral dimension 8. From the previous section, we can define a proper Dirac operator just considering the 4-dimensional case obtaining a Dirac operator built from a 8-dimensional Cifford algebra: $$\mathcal{D}_4 = i\Gamma_\mu \partial_\mu + \Omega \Gamma_{\mu+4} \tilde{x}_\mu$$ Here, the $\Gamma_k \in M_{16}(\mathbb{C}), k = 1, . . . , 8$ are the generators of the 8-dimensional real Clifford algebra, satisfying $$\Gamma_k \Gamma_l + \Gamma_l\Gamma_k = 2\delta_{kl} \textbf{I}$$ We take the Hilbert space $\mathcal{H}_4 = S^2(\mathbb{R}^4,\mathcal{S})=L^2(\mathbb{R}^4)\otimes\mathbb{C}^{16} $ of Schwartz functions of spinors over 4-dimensional euclidean space. Accordingly with for $\psi \in \mathcal{H}_4$ we obtain: $$\mathcal{D}^2_4 \psi= \left((-\Delta + \Omega^2 \tilde{x}_\mu \tilde{x}^\mu )\textbf{I} + 