Model of Coupling-Constant Renormalization

The charged scalar theory of $\ensuremath{\pi}$ mesons interacting with a fixed nucleon source is truncated as follows: $\ensuremath{\pi}$ mesons are permitted to exist only in a set of discrete states ${\ensuremath{\psi}}_{m}(k)$ such that $k$ is of order ${\ensuremath{\Lambda}}^{m}$ in the state ${\ensuremath{\psi}}_{m}(k)$; $\ensuremath{\Lambda}$ is an arbitrary constant above 4\ifmmode\times\else\texttimes\fi{}${10}^{6}$. Also, two mesons of the same charge cannot occupy the same state. The resulting Hamiltonian can be solved by a perturbation expansion in ${\ensuremath{\Lambda}}^{\ensuremath{-}1}$ provided there are only a finite number $M$ of states ${\ensuremath{\psi}}_{m}$. When $M\ensuremath{\rightarrow}\ensuremath{\infty}$, the renormalized coupling constant and ground-state energy diverge in perturbation theory (in the coupling constant). If the unrenormalized coupling constant is allowed to go to infinity as $M\ensuremath{\rightarrow}\ensuremath{\infty}$, it is proven that the renormalized theory exists (without ghost states) for any value of the renormalized coupling constant. The proof uses the perturbation analysis in ${\ensuremath{\Lambda}}^{\ensuremath{-}1}$ carried to all orders. This analysis leads to the definition of a transformation $T$ which eliminates one meson degree of freedom from any given Hamiltonian, replacing it by an effective Hamiltonian with one less degree of freedom. The effective Hamiltonian gives exactly all energy levels of the original Hamiltonian except those with mesons in the removed degree of freedom. The renormalizability of the theory is proven using topological properties of $T$. In particular, there is a substransformation ${T}_{A}$ with a nontrivial fixed point ${P}_{c}$ whose properties determine the principal features of the renormalized theory. The idea of the fixed point is a generalziation of the Gell-Mann-Low eigenvalue condition for the bare coupling constant of quantum electrodynamics.