Quark confinement in the infinite-momentum frame

We formulate the problem of quark confinement in the infinite-momentum frame. In this frame the dynamics is naturally described as a many-body problem: Quarks and gluons can be thought of as nonrelativistic particles moving in the two-dimensional transverse space with mechanical mass related to the longitudinal momentum $P^{+}$. In this language, a natural quark-confining mechanism is the condensation of gluons along a tube joining a separated quark and antiquark. This condensation is favored by two circumstances: (1) The fact that bare gluons are massless reduces the minimum energy for gluon pair production to zero, and (2) the octet color structure allows gluons to form into chains with long-range attractive nearest-neighbor interactions. We investigate the viability of this mechanism first in the limit $N_c\to \infty$, $N_c{g}^2$ fixed, where $N_c$ is the number of colors in the theory. We analyze in detail a simplified version of the $N_c\to \infty$ dynamics which preserves the essential features of the full problem. This simplified model exhibits quark confinement and describes mesons as relativistic open strings. It also yields a relationship between the Regge slope $\alpha$' and the scale ${\mu }_0$, measured in deep-inelastic leptoproduction: ${\mu }_0^2\approx \frac{2}{\left(\pi \sqrt{3}{\alpha }^{\prime }\right)}\approx 0.4$ Ge${\mathrm{V}}^2$, which is not too far from the experimental number 0.25 Ge${\mathrm{V}}^2$. We discuss next the problem of finite $N_c$. We argue that the $\frac{1}{N_c}$ expansion is likely to have a vacuum instability which must be handled nonperturbatively before $\frac{1}{N_c}$ corrections can be calculated. We suggest that the true vacuum is a condensate of closed strings which have a finite density by virtue of repulsive interactions inherent in the fourgluon term in the Hamiltonian. A crude estimate of the condensate energy density yields an order-of-magnitude relation of the form $\epsilon \sim \frac{{N_c}^2}{{\alpha }^2}$. ${\epsilon }^{-\frac{1}{4}}$ should be a rough estimate of the thickness of the string.