Renormalization of the Semiclassical Hamiltonian Field Theory

TL;DR

This paper explores the renormalization process for the Hamiltonian formulation of quantum field theory, addressing unique challenges like divergences and initial state dependencies, and discusses semiclassical approximation methods.

Contribution

It introduces a renormalization scheme for the Hamiltonian approach considering initial state singularities and compares semiclassical quantization methods beyond WKB.

Findings

Initial states must depend on cutoffs to avoid divergences.

Semiclassical quantization aligns with complex-WKB theory.

Different initial condition prescriptions are analyzed.

Abstract

The Hamiltonian approach to the quantum field theory is considered. Since there are additional difficulties such as the Haag theorem and Stueckelberg divergences, renormalization of the time-dependent dynamical quantum field theory is much more complicated than renormalization of the S-matrix. It is necessary to consider the regularized theory with ultraviolet and infrared cutoffs and impose the conditions not only on the dependence of the Hamiltonian on the cutoffs (as usual) but also on the dependence of the initial states. It happens that one should consider the initial states to be singulary dependent on the cutoffs in order to avoid the Stueckelberg divergences. Different types of semiclassical approximations to quantum theory are discussed. It happens that the method of quantizing classical solutions to field equations corresponds not to the WKB-approach but to the complex-WKB theory. The problem of imposing conditions on the semiclassical initial states is discussed. Different prescriptions for choice of initial conditions are analysed.