Model Hamiltonians for Local Quantum Field Theory

We study the problem of renormalization for the interaction of a charged scalar meson field with a fixed two-level point source. Our especial interest is coupling-constant renormalization. We study in particular the problem of obtaining eigenstates and eigenvalues of the Hamiltonian for the fixed-source theory. We propose a model for the fixed-source theory, in which mesons exist only if their momenta ($k$) lie within an infinite set of intervals: $0<k<\mathrm{}{k}_0$, $\frac{1}{2}\Lambda <k<\mathrm{}\Lambda$, $\frac{1}{2}{\Lambda }^2<k<\mathrm{}{\Lambda }^2$, etc., where $k_0$ is of the order of the meson mass, and $\Lambda$ is much larger. We solve this model by treating the mesons in the nth interval (or lower) as a perturbation on mesons in the (n+1)st interval (and higher). This reduces the problem to the solution of two strongly cut-off Hamiltonians, one of which must be solved for an infinite sequence of coupling constants $\left\{g_n\right\}$, one for each momentum interval. We show that even if the low-momentum coupling constants $g_1$, $g_2$, etc., are small, the sequence goes to infinity as $n\to \infty$. We analyze the Lee model similarly; here the sequence is undefined above some finite value of $n$. We show a close analogy between our analysis and the analysis of quantum electrodynamics of Gell-Mann and Low. Then we analyze the full fixed-source Hamiltonian qualitatively. We expland the meson field in terms of a complete set of "wave-packet" states, the coefficients being discrete oscillator variables. The states are so chosen that the self-interactions of oscillators dominate the coupling between oscillators. For each order of magnitude for the meson momentum, there is one pair of oscillators coupled to the source; this coupling can be analyzed analogously to our model. The full fixed-source Hamiltonian is thereby reduced to the solution of a Hamiltonian for two oscillators coupled to a two-level source.