Super-Grassmannian and large N limit of quantum field theory with bosons and fermions

Abstract

We study a large N c limit of a two-dimensional Yang-Mills theory coupled to bosons and fermions in the fundamental representation. Extending an approach due to Rajeev we show that the limiting theory can be described as a classical Hamiltonian system whose phase space is an infinite-dimensional supergrassmannian. The linear approximation to the equations of motion and the constraint yields the 't Hooft equations for the mesonic spectrum. Two other approximation schemes to the exact equations are discussed.

On two dimensional coupled bosons and fermions

“…We note that the lefthanded components of the fermion field are nondynamical, therefore we will remove ψ Lα and its complex conjugate through the equations of motion in the quantum theory (we refer to [18] for our conventions in quantizing this theory, below we summarize the results),…”

“…This is physically meaningful due to charge conjugation invariance, and amounts to the principal value prescription. (see [18,19] for details)…”

“…The corresponding normal orderings are defined in [18]. It is useful to keep in mind the normal ordering rules of the bilinears,…”

“…Following the ideas suggested by Rajeev in [16] we reformulate the problem in terms of color invariant bilinears (see also [17] for some similar ideas). The details of this refromulation are explained in our previous work [18] within the context of gauge theories, therefore in this work we will use the results of the cited article directly. In some sense the present article is a natural continuation.…”

“…In some sense the present article is a natural continuation. The reader should consult [18] for more information on the geometry of the resulting classical phase space. We also recommend the lectures notes of Rajeev on two dimensional QCD [19].…”