Dirac states from the 't Hooft model

The dynamics of a light fermion bound to a heavy one is expected to be described by the Dirac equation with an external potential. The potential breaks translation invariance, whereas the bound state momentum is well defined. Boosting the bound state determines the frame dependence of the light fermion dynamics. I study the Dirac limit of QCD$_2$ in the limit of $N_c \to \infty$. The light quark wave function turns out to be independent of the frame of the bound state, up to an irrelevant Lorentz contraction. The discrete bound state spectrum determines corresponding discrete energies of the Dirac equation, which for a linear potential allows a continuous spectrum.

💡 Research Summary

The paper investigates the connection between the bound‑state dynamics of a light quark bound to a heavy quark in two‑dimensional QCD (the ’t Hooft model) and the Dirac equation with an external linear potential. Working in the large‑(N_c) limit, where quark‑antiquark pair creation is suppressed, the author derives an exact bound‑state equation (BSE) for the color‑singlet wave function (\Phi(x)) in temporal gauge ((A_0=0)). The BSE contains the Dirac matrices, the quark masses (m_1) (light) and (m_2) (heavy), and a linearly rising potential (V(x)=V’|x|) that follows from the instantaneous gluon exchange in 1+1 dimensions.

First, the rest‑frame BSE is written in terms of four scalar components (\varphi_j(x)). By eliminating three components, the system reduces to two coupled equations for (\varphi_0) and (\varphi_1). Taking the heavy‑quark limit (m_2\to\infty) while keeping the light‑quark mass and the binding energy finite, the bound‑state mass separates as (M=m_2+M_D). In this limit the coupled equations become exactly the Dirac equation for a single fermion of mass (m_1) moving in the linear potential (V’|x|): \