Introduction to theN-Quantum Approximation in Quantum Field Theory

Abstract

The $N$-quantum approximation is designed to find approximate operator solutions of theories characterized by a specific Hamiltonian. The Heisenberg field operators of the theory are approximated by finite-degree normal-ordered expansions in an irreducible set of in-fields. The $c$-number functions which are the coefficients of these expansions are the unknown quantities in the approximation. The approximation assumes that the dominant contributions to the vertex function, scattering function, and other low-order functions come from functions of similar low order. The $c$-number functions correspond to the connected graphs with a given number of external lines. Thus in graphical language the approximation assumes that the connected graphs with few external lines dominate. The $N$-quantum approximation is manifestly covariant, treats positive and negative frequencies in a symmetric way, allows a calculation of several different physical processes simultaneously, allows incorporation of bound states, and requires extrapolation off the mass shell in fewer variables than the usual Green's function approaches. After describing the $N$-quantum approximation, it is shown to be compatible with renormalization theory in first order of the approximation in the model with ${\mathcal{L}}_{I}=g{A}^{3}$. It should be emphasized that all powers of the coupling constant occur in first order of the $N$-quantum approximation in this model. A quadratic integral equation is obtained for the vertex function, and it is shown that the vertex function satisfies the renormalization criteria that the particles in the theory have a given observed mass, and that the vertex function has a given coupling constant as the residue of a pole ${({m}^{2}\ensuremath{-}{k}^{2})}^{\ensuremath{-}1}$ in the unphysical region. It is also shown that the power-series-expansion solution is finite term by term in all orders of the coupling constant.