4 Performing the quenching

Quenching Approximation amounts to take into account the contributions of the slow modes, described by the eigenvalues of the momentum matrix $\mathbf{q }$ and ``integrate out'' the non-diagonal fast modes. The final result is to turn the original gauge theory is transformed into a quantum mechanical model where the physical degrees of freedom are carried by large coordinate independent matrices. Finally, the spacetime volume integration is regularized by enclosing the system in a ``box'' of four-volume $V$

$$\int d^4x\quad\longrightarrow \quad V$$

In a Yang-Mills framework one can relate $V$ to the $QCD$ scale, i.e. $V=\left(\, 2\pi/\Lambda_{\mathrm{QCD}}\,\right)^4$ [11]. Here, we shall determined $V$ by matching the large-$N$ limit of our matrix model with the Nambu-Goto action.

The resulting quenched action is

$$S^{(\mathrm{q})}_{\mathrm{YM/\Phi }}= {NV\over 4g_0^2 } \ma... ...thrm{q})} , \mathbf{A}_\nu^{(\mathrm{q})} \right]^2 \right)\ .$$ (16)

The large-$N$ quantum properties of the matrix model (16) can be effectively investigated by means of the Wigner-Weyl-Moyal correspondence between matrices and functions, i.e. Symbols, defined over a noncommutative phase space. The resulting theory is a deformation of an ordinary field theory, where the ordinary product between functions is replaced by a non-commutative $\ast$-product. The deformation parameter, measuring the amount of non-commutativity, results to be $1/N$, and the classical limit corresponds exactly to the large-$N$ limit. The final result is a string action of the Schild type, which is invariant under area-preserving reparametrization of the world-sheet.
More recently, we have also shown that bag-like objects fit the large-$N$ spectrum of Yang-Mills type theories as well, both in four [9] and higher dimensions.
Here, we would like to explore a different route leading in a straightforward way to a Nambu-Goto string action. Having discussed the role of the auxiliary field $\Phi$ in bridging the gap between Schild and Nambu-Goto actions at the classical level, we propose the following improved gauge matrix model

$$S^{(\mathrm{q})}_{\mathrm{YM/\Phi\, }}= {N\, V\over 4g_0^2 ... ...]^2 \right)\,\right] +{1\over 4}\,\mathrm{Tr}\mathbf{\Phi} \ .$$ (17)

where, $\mathbf{\Phi}$ is a $N\times N$ diagonal matrix. In a different framework, a similar matrix variable have been introduced to build a consistent path integral for the matrix version of type IIB superstring model [10].
We see, as things stand up to now there seems to be a qualitative difference between equation (17) and equation (9). This is the absence of the analogous to the last term in eq. (17), the one that contains only $\mathrm{Tr}(\Phi)$, which is absent in eq.(9). Nevertheless the corresponding demand that the "measure" $\Phi$ be a ``total derivative'' can be implemented in simple ways also in the matrix model.