Renormalizing the strong-coupling expansion for quantum field theory: present status

Progress made in determining the renormalized strong coupling expansion in quantum field theory is reviewed. Consideration restricted is to lambda phi/sup 4/ field theory in d dimensions. The starting point is the lattice version of the path integral representation for the Green's functions. The unrenormalized strong coupling expansion on the lattice is determined first. This is a series in a dimensionless parameter, x. A simple set of graphical rules for determining the Green's functions of the theory in a power series in x is derived. For finite field theories (d < 2) scheme for extrapolating to zero lattice spacing (a ..-->.. 0) is presented. This scheme determines a sequence of approximants to the unrenormalized strong coupling series in the form of a series. Then the problem of renormalization is considered. For the lattice field theory this consists of eliminating x in favor of y, a renormalized x. For d < 2, M/sub R//sup 2/ = 0 occurs when M/sub 0//sup 2/ ..-->.. - infinity, where M/sub R/ is the renormalized mass and M/sub 0/ is the bare mass, and thus the path integral is dominated by instantons. The renormalized strong coupling expansion is valid for the double well where M/sub R//sup 2/ << lambda/sup 2/(4-d)/. The renormalized strong coupling expansion is obtained by two methods. The extrapolation procedure for obtaining the continuum limit of the renormalized lattice field theory works well in zero, one and two dimensions when compared with the exact instanton results. 2 figures, 1 table. (RWR