Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model

Details Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model Equivalence of Two Dimensional QCD and the $c=1$ Matrix Model

1993-03-27

arxiv.org/abs/hep-th/9303153v1

Phys.Lett. B312 (1993) 155-165

Physics

High Energy Physics

High Energy Physics - Theory

15 pages, CERN-TH 6843/93, UVA-HET-93-02

Scientific paper

10.1016/0370-2693(93)90504-B

We consider two dimensional QCD with the spatial dimension compactified to a circle. We show that the states in the theory consist of interacting strings that wind around the circle and derive the Hamiltonian for this theory in the large $N$ limit, complete with interactions. Mapping the winding states into momentum states, we express this Hamiltonian in terms of a continuous field. For a $U(N)$ gauge group with a background source of Wilson loops, we recover the collective field Hamiltonian found by Das and Jevicki for the $c=1$ matrix model, except the spatial coordinate is on a circle. We then proceed to show that two dimensional QCD with a $U(N)$ gauge group can be reduced to a one- dimensional unitary matrix model and is hence equivalent to a theory of $N$ free nonrelativistic fermions on a circle. A similar result is true for the group $SU(N)$, but the fermions must be modded out by the center of mass coordinate.

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