Perturbation theory in large order

Abstract

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by −″ + ( x 2 4 + ϵx 4 4 −E)y=0, y(±∞)=0 , the perturbation coefficients A n in the expansion for the ground-state energy E( ground state) ∼ ∑ n=0 ∞ A nϵ n simplify dramatically as n → ∞: A n∼−( 6 π 3 ) 1 2 Гn+ 1 2 ) . We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λϕ 4 model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.