Quantum chains and regularisation of quantum dynamics

Abstract

We argue that the origin of ultraviolet divergences in quantum field theory (QFT) may be not in the perturbative expansion, but in the fact that, mathematically, the Heisenberg equations of motion are not properly defined. Divergences similar to those in QFT are shown to exist in seemingly simple quantum problems with parametric Hamiltonians. These divergences may be suppressed by formally replacing the system in question by a quantum chain. The latter is a dynamically regularised system specified by two postulates, imposing conditions on its response properties. The kinematical postulate specifies the linear response, the dynamical postulate extends it to nonlinear dynamics. The kinematical postulate uniquely determines the mathematical environment (quantisation with indefinite metric), hence the term. Using the Klein–Gordon, Gupta–Bleuler and Dirac fields as examples we show that the concept of quantum chain generalises seamlessly to relativistic quantum fields. Perspectives of using quantum chains as regularisation devices in QFT are discussed.