Strong-coupling expansion in quantum field theory

Abstract

We derive a simple and general diagrammatic procedure for obtaining the strong-coupling expansion of a $d$-dimensional quantum field theory, starting from its Euclidean path-integral representation. At intermediate stages we are required to evaluate diagrams on a lattice; the lattice spacing provides a cutoff for the theory. We formulate a simple Pad\'e-type prescription for extrapolating to zero lattice spacing and thereby obtain a series of approximants to the true strong-coupling expansion of the theory. No infinite quantities appear at any stage of the calculation. Moreover, all diagrams are simple to evaluate (unlike the diagrams of the ordinary weak-coupling expansion) because nothing more than algebra is required, and no diagram, no matter how complex, generates any transcendental quantities. We explain our approach in the context of a $g{\ensuremath{\varphi}}^{4}$ field theory and calculate the two-point and four-point Green's functions. Then we specialize to $d=1$ (the anharmonic oscillator) and compare the locations of the poles of the Green's functions with the tabulated numerical values of the energy levels. The agreement is excellent. Finally, we discuss the application of these techniques to other models such as $g{\ensuremath{\varphi}}^{2N}$, $g{(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$, and quantum electrodynamics.