inverse operator $\mathcal{D}^{-1}$ which is eight not four. This occurrence connects the spectral dimension to the phase space dimension instead the one of the configuration space [@con-phase]. In order to construct such Harmonic Dirac operator and the spectral triple we will work in the framework of the generalized $n$-dimensional harmonic operators. Will be studied the 4-dimensional case in order to construct the non-commutative spectral triple which is starting point for the field theory we are interested in. Having the 4-dimensional Harmonic Dirac operator with harmonic oscillator spectrum, to implement the Higgs mechanism we will consider the tensor product of the non-commutative triple with a finite Connes-Lott type spectral triple [@con-lot]. We will fluctuate the total Dirac operator following the standard machinery [@S-A; @Connes3] of non-commutative geometry to get “Gauged” Dirac operator. Thus we will proceed to compute the spectral action in which are present two U(1)-Moyal Yang-Mills fields unified with a complex Higgs field.

Harmonic Dirac operators 
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**1**

The Harmonic Dirac operator in $n$-dimensions can be defined using the Clifford algebra of $\mathbb{R}^{2n}$ represented on the Hilbert space $ \mathbb{C}^{2^n}$, it is very useful to consider $n$-dimensional fermionic annihilation and creation operators $b_\mu$, $b_\nu^\dagger$ and $n$-dimensional bosonic annihilation and 