the Planck units)*.

The BFSS model
==============

In this section we present the elementary ingredients of the BFSS theory, to more details the reader can see [@BFSS; @Sochichiu; @Zarembo].\
We consider a stack of $N$ D0-branes in a 3D-space, represented by Hermitian matrices $X_i \in \mathfrak M_{N \times N}(\mathbb C)$ for $i=1,2,3$ (noncommutative coordinates for the D2-brane formed by the stack). Intuitively, $X^i = \left(\begin{array}{cc} x^i_1 & s^i_{12} \\ \overline s^i_{12} & x^i_2 \end{array} \right)$ represents a stack of two D0-branes with coordinates in the target space $\mathbb R^3$ $\{x^i_1\}_i$ and $\{x^i_2\}_i$, linked by a bosonic string of oscillation radii $\{|s^i_{12}|\}_i$ ($\arg s^i_{12}$ is the phase for the oscillation in the $i$-direction).\
We consider also a probe D0-brane of coordinates $x_i \in \mathbb R$ and a fermionic string linking the D2-brane to the probe brane described by a state:\
$|\psi {\rangle \hspace{-0.2em} \rangle}= |\uparrow \rangle \otimes \left(\begin{array}{c} \psi^\uparrow_1 \\ \vdots \\ \psi^\uparrow_N \end{array} \right)+ |\downarrow \rangle \otimes \left(\begin{array}{c} \psi^\downarrow_1 \\ \vdots \\ \psi^\downarrow_N \end{array} \right) \in \mathbb C^2 \otimes \mathbb C^N$. We can interpret $\psi^\alpha_a$ as $\psi^\alpha(x_a)$ ($x_a$ being the pseudo-position of the $a$-th D0-brane of the stack): the spatial delocalisation of quantum particles is replaced by the quantum superposition 