We study two continuum methods of regulating the formal strong-coupling expansion of the Green's functions, obtained by expanding the path integral in powers of the kinetic energy (inverse free propagator). Our continuum regulations amount to introducing either a hard ($\theta$ function) or soft (Gaussian) cutoff $\Lambda$ in momentum space. The cutoff takes the place of the usual spatial cutoff, the lattice spacing, which arises when the path integral is defined as the continuum limit of ordinary integrals on a Euclidean space-time lattice. We find, by investigating free field theory and $g{\phi }^4$ field theory in one dimension, that the $\theta$-function regulation is more accurate than the Gaussian and, unlike the Gaussian, preserves certain continuum Green's-function identities. The extension to field theories with fermions is trivial and we give the strong-coupling graphical rules for an arbitrary field theory with fermions and bosons in $d$ dimensions.

• Received 22 June 1981

DOI:https://doi.org/10.1103/PhysRevD.24.2693