Quantum Hadrodynamics in Two Dimensions

Abstract

A nonlocal and nonlinear theory of hadrons, equivalent to the color singlet sector two dimensional QCD, is constructed. The phase space space of this theory is an infinite dimensional Grassmannian. The baryon number of QCD corresponds to a topological invariant ('virtual rank') of the Grassmannian. It is shown that the hadron theory has topological solitons corresponding to the baryons of QCD.plays the role ofh in this theory; N c must be an integer for topological reasons. We also describe the quantization of a toy model with a finite dimensional Grassmannian as the phase space. In an appendix, we show that the usual Hartree-Fock theory of atomic and condensed matter physics has a natural formulation in terms of infinite dimensional Grassmannians.

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

“…Using exactly the same methods as in [7,18], we can show that it is closed and nondegenerate. In fact the above form is also a homogeneous two-form invariant under the group action, as can be verified in a simple manner.…”

“…If one attempts a geometric quantization of this model, the symplectic form should be an integer multiple of the Chern character of the line bundle, the sympectic form we have in the next section is in fact the basic two form. The other possibility is to write the symplectic form in the action and use single valuedness of the path integral as is done in [7]. (We note in passing that there is a factor of 2 missing in the reference [18], due to an error in the conventions, but we scale the Hamiltonian with the same parameter so the large-N c results are the same.…”

“…In this article we study the large-N c limit of certain two dimensional theories following a general approach developed by S. Rajeev [6,7] (see also [8] and [9] for similar approaches). In the large N c limit of various quantum field theories (e.g., Quantum Chromodynamics or QCD) the quantum fluctuations become small and the theories are well described by a classical limit.…”

“…This classical limit however is different from the conventional one in that many of the essential non-perturbative features of the quantum theory survive the large N c limit [10,1,11]. In the formulation of [7] the classical theory corresponding to large N c limit of 2D QCD is described by a Hamiltonian system defined on an an infinite dimensional Grassmannian. The points of this infinite dimensional manifold can be identified with subspaces in infinite-dimensional Hilbert space (see the main text for precise definitions).…”

“…The approach of [7] was extended to the bosonic case in [18] (see also [19] for a similar approach to the problem).…”