The $\hslash$ expansion in quantum field theory

We show how expansions in powers of Planck’s constant $\hslash =h/2\pi$ can give new insights into perturbative and nonperturbative properties of quantum field theories. Since $\hslash$ is a fundamental parameter, exact Lorentz invariance and gauge invariance are maintained at each order of the expansion. The physics of the $\hslash$ expansion depends on the scheme; i.e., different expansions are obtained depending on which quantities (momenta, couplings, and masses) are assumed to be independent of $\hslash$. We show that if the coupling and mass parameters appearing in the Lagrangian density are taken to be independent of $\hslash$, then each loop in perturbation theory brings a factor of $\hslash$. In the case of quantum electrodynamics, this scheme implies that the classical charge $e$, as well as the fine structure constant are linear in $\hslash$. The connection between the number of loops and factors of $\hslash$ is more subtle for bound states since the binding energies and bound-state momenta themselves scale with $\hslash$. The $\hslash$ expansion allows one to identify equal-time relativistic bound states in QED and QCD which are of lowest order in $\hslash$ and transform dynamically under Lorentz boosts. The possibility to use retarded propagators at the Born level gives valencelike wave functions which implicitly describe the sea constituents of the bound states normally present in its Fock state representation.

Figure 1

The first two ladder diagrams contributing to nonrelativistic atoms in the limit of small coupling $\alpha$ are shown.