2 Yang-Mills Theory as a Matrix model

In the introduction we referred to the supposed relation between confinement and extended excitations of the Yang-Mills field. Recently, an even deeper and more fundamental relation between branes and Yang-Mills fields has been conjectured in the framework of $M$-theory [13]. Non-perturbative formulations of string theory require the introduction of higher dimensional solitonic objects satisfying Dirichlet boundary conditions [14]. Dirichlet-branes are formally described by non-commuting matrix coordinates. Thus, non-perturbative string theory, or $M$-theory, is conveniently written in terms of matrix dynamical variables. The corresponding low energy, effective, supersymmetric Yang-Mills theory is derived through an appropriate compactification procedure of the original matrix model [11], [15].
In this section we are going to follow a similar path, but in the opposite direction: we shall start from a $SU(N)$ gauge theory in four dimensional spacetime and build a matrix model. Our final purpose is to show that the spectrum of such a QCD matrix model contains dynamical membrane type objects. We thus start from the Yang-Mills action

$$S_{\mathrm{YM}}=\int dt\int_{V_H} d^3x \, {\mathcal{L}}_{\ma... ...ft( \mbox{\textit{\bf {}F}}\ , \mbox{\textit{\bf {}A}} \right)$$ (5)

defined in terms of the Lagrangian density

$${\mathcal{L}}_{\mathrm{YM}}\left( \mbox{\textit{\bf {}F}}\ ,... ...{\textit{\bf {}D}}_{[ \mu} \mbox{\textit{\bf {}A}}_{\nu ]} \ .$$ (6)

The covariant derivative is defined as usual,

$$\mbox{\textit{\bf {}D}}_{[ \mu} \mbox{\textit{\bf {}A}}_{\nu... ...\textit{\bf {}A}}_\mu, \mbox{\textit{\bf {}A}}_\nu \right] \ ,$$ (7)

but the $SU(N)$ Yang-Mills Lagrangian ${\mathcal{L}}_{\mathrm{YM}}\left( \mbox{\textit{\bf {}F}}, \mbox{\textit{\bf {}A}} \right)$ is written in the first order formulation: thus

$$\mbox{\textit{\bf {}A}}_\mu\equiv A_\mu^a\mbox{\textit{\bf {... ...bf {}F}}_{\mu\nu}\equiv F_{\mu\nu}^a\mbox{\textit{\bf {}T}}^a$$

are independent vector and tensor fields respectively⁵ valued in the Lie algebra defined by the commutation relations

$$\left[ \mbox{\textit{\bf {}T}}^a,\mbox{\textit{\bf {}T}}^b \right]= i f^{abc} \mbox{\textit{\bf {}T}}^c.$$

The integration volume $V_H$ will be specified later on. The form (6) is appropriate for an Hamiltonian formulation of the action, as it is required by the quenching approximation that we shall apply in the next section.
Starting from the Lagrangian density we can, as usual, introduce the canonical momentum and Hamiltonian

$$\mbox{\textit{\bf {}E}}^i\equiv {\partial {\mathcal{L}}_{\mathrm{YM}}\over \partial \partial_t \mbox{\textit{\bf {}A}}_i} = - {1\over 2g_0{}^2} \mbox{\textit{\bf {}F}}^{t m}$$ (8)

$$H_0 \equiv \mathrm{Tr} \left( \mbox{\textit{\bf {}E}}^i \partial_t \mbox{\textit{\bf {}A}}_i \right)-{\mathcal{ L}}_0$$ (9)

and rewrite (6), in terms of phase space variables, as:

$${\mathcal{L}}_{\mathrm{YM}} = -{N\over 2 g_0{}^2} \mathrm{Tr} \left( \mbox{\textit{\bf {}F}}^{t... ... -i \left[ \mbox{\textit{\bf {}A}}_t, \mbox{\textit{\bf {}A}}_i \right] \right)$$

$$\qquad +{N\over 4 g_0{}^2} \mathrm{Tr} \left( \mbox{\textit{\bf {... ...tit{\bf {}F}}^{mn} \mbox{\textit{\bf {}D}}_{[ m} \mbox{\textit{\bf {}A}}_{nu ]}$$

$$= -{g_0{}^2\over 2N} \mathrm{Tr} \left( \mbox{\textit{\bf {}E}}^i \... ...\left( \mbox{\textit{\bf {}D}}_{[ m} \mbox{\textit{\bf {}A}}_{n ]} \right)^ 2 .$$ (10)

Accordingly, the phase space action reads

$$S = \int \!\!dt\int_{V_H} \!\!\!\!\!\! d^3x \left[ \mbox{\textit{\bf {}E}}^i \partial_t \mbox{\textit{\bf {}A}}_i - H_0 \right]$$ (11)

$$= \int \!\!dt\int_{V_H} \!\!\!\!\!\! d^3x \left[ {g_0{}^2\over 2} \... ...mbox{\textit{\bf {}D}}_{[ m} \mbox{\textit{\bf {}A}}_{n ]} \right)^ 2 \right] .$$ (12)

Let us remark that $\mbox{\textit{\bf {}A}}_t$ enters linearly in the canonical form of the action (12) and plays the role of Lagrange multiplier enforcing the (non-Abelian) Gauss Law:

$${\delta S\over \delta\mbox{\textit{\bf {}A}}_t }=0\quad\Long... ... \quad \mbox{\textit{\bf {}D}}_i \mbox{\textit{\bf {}E}}^i=0 .$$ (13)

Thus, solving the classical field equation for $\mbox{\textit{\bf {}A}}_t$ is equivalent as requiring $\mbox{\textit{\bf {}E}}^i$ to be covariantly divergence free in vacuum. Thus, inserting the solution of the Gauss constraint (13), the action for $\mbox{\textit{\bf {}E}}^i$ becomes

$$S_{\mathrm{YM}}=\int dt\int_{V_H} d^3x \left[ {g_0{}^2\over ... ...}D}}_{[ m} \mbox{\textit{\bf {}A}}_{n ]} \right)^2 \right] .$$ (14)

We will now go on with the quenching procedure.