4.2 A second implementation

Yet another way of proceeding is to explicitly construct a composite measure in the matrix formulation ( instead of using a property a total derivative must satisfy and then implement this ). The procedure in this case would be:

• i) the world sheet coordinates are replaced by matrices $\xi^m$;
• ii) the derivatives are replaced by commutators between the matrix coordinates and the matrix momentum $q_a$.

The end result is a matrix $\Phi$ : $\Phi = \epsilon^{ab} \epsilon_{mn}\,\left[\, q_a\ ,\xi^m \,\right] \left[\, q_b\ ,\xi^n\,\right]$ which should be used in (17), except that now the second term in (17) and/or a possible lagrange multiplier is now not necessary ( the constraint $\mathrm{Tr}(\Phi) = 0$ is now an identity). Now, going back to the action (17), if $\mathbf{\Phi}$ is taken to be the $N\times N$ Identity matrix, then action (17) takes on the form of the Yang-Mills action. On the other hand, if we let $\mathbf{\Phi}$ to be diagonal, $\Phi_{ab}\equiv \phi_{(a)}\, \delta_{ab}$ ( no summation over the index $a$ ), and determine its eigenvalues by varying (17), we find

$$\phi_{(a)}=\sqrt{{N\, V\over g_0^2}\, \mathbf{F^{(0)}}{}_{cd... ...Tr}\,\mathbf{F^{(0)}}{}_{\mu\nu} \mathbf{F}^{(0)}{}^{\mu\nu} }$$ (18)

All the eigenvalues are degenerate and quadratic in the Yang-Mills field strength. By inserting (18) into (17) one finds:

$$S^{(\mathrm{q})}_{\mathrm{BI-YM}}= \sqrt{\, {N\, V\over 4g_... ...}\, \mathbf{F}_{\mu\nu}^{(0)}\, \mathbf{F}^{(0)}{}^{\mu\nu}\,}$$ (19)

The action (19) is the ``square root'' form of a non-Abelian Born-Infeld type action, where the trace is the standard one. The ambiguity in the definition of the trace over internal indices is removed, in our model, by choosing $\mathbf{\Phi}$ to be diagonal. With hindsight, we know that in the large-$N$ limit the trace over internal $SU(\, N\,)$ indices will turn into an integration over world manifold coordinates. Thus, it is compelling to ``move out'' the trace from the square root, in order to obtain a Nambu-Goto type action integral. What we are going to describe is a procedure such that in the large-$N$ limit:

$$\sqrt{\, \mathbf{Tr}\left(\, \dots\, \right)}\longrightarrow \int_\Sigma d^2\sigma \sqrt{\, \dots\, }$$ (20)

and the non-commuting Yang-Mills matrices are replaced by commuting string coordinates.