Frozen and Broken Color: a Matrix Schroedinger Equation in the Semiclassical Limit.

We study questions of broken color symmetry and/or hidden color degrees of freedom in the context of a semiclassical one-dimensional theory in which a colored spinless quark and anitiquark are bound together by a confining, color -dependent potential. Our purpose is to investigate in more detail the dynamics underlying Lipkin's mechanism of hidden charge, and how his conclusions are modified in the presence of symmetry breaking. We consider the case of "frozen color", i.e., where global color symmetry remains exact, but where colored states have a mass large compared to color-singlet mesons. Using semiclassical WKB formalism, we construct the spectrum of bound states. In order to determine the charge of the constituents, we then consider deep-inelastic scattering of an external probe (e.g., lepton) from our one-dimensional meson. We calculate explicitly the structure function, W, in the WKB limit and show how Lipkin's mechanism is manifested, as well as how scaling behavior comes about. The dominant physical process is one of excitation of a semiclassical state by the hard collision of the probe with the quark or antiquark. We derive the WKB formalism as a special case of a method of obtaining WKB type solutions for generalized Schroedinger equations for which the Hamiltonian is an arbitrary matrix function of any number of pairs of canonical operators. Our solution reduces the problem to that of finding the matrix which diagonalizes the classical Hamiltonian and determining the scalar WKB wave functions for the diagonalized Hamiltonian's entries (presented explicitly in terms of classical quantities). If the classical Hamiltonian has degenerate eigenvalues, the solution contains a vector in the classically degenerate subspace. This vector satisfies a classical equation and is given explicitly in terms of the classical Hamiltonian as a Dyson series. As an example, we obtain, from the Dirac equation for an electron with anomalous magnetic moment, the relativistic spin-precession equation. We generalize these considerations to the case of broken color symmetry--but where the breaking is not so strong as to allow low-lying states to have a large amount of mixing with the colored states. In this case, the degeneracy of excited colored states can be broken. The WKB approximation again suffices to provide a description of the spectra. Again deep-inelastic scattering can be used to measure the charges of the constituents, and there will again be a distinct contribution from each type of "classical" state which can be excited by the external probe. However, in the general case, the charge measured via excitation of a given state can be energy-dependent. We find that local excitation of color guarantees global excitation of color; i.e., if at a given energy colored semiclassical states can be constructed with size comparable to that of the ground state wave function, colored states of that energy will also exist in the spectrum of the full theory and will be observed. However, global existence of color does not guarantee the excitation of colored states via deep-inelastic processes: there may be no overlap of the wave functions of these colored states with the ground state wave function. Finally, even in the absence of a direct physical application, we have examined how to implement the WKB method for bound state problems in the presence of internal degrees of freedom. The methodology we have given may be of use in other semiclassical problems which have internal degrees of freedom.