Nonunitary Interaction, Adiabatic Condition, Haag's Theorem and Renormalization

TL;DR

The paper argues that unitarity in quantum field theory transformations is physically unnecessary and impossible, proposing that nonunitary interactions better explain phenomena like renormalization and scattering.

Contribution

It revisits Haag's theorem, showing unitarity is not required or feasible in quantum field transformations, and suggests an alternative approach to scattering.

Findings

Unitarity of field transformations is physically unfeasible.

Interaction cannot be turned on or off adiabatically in electrodynamics.

Renormalization may result from forcing unitarity in scattering matrices.

Abstract

Haag's theorem has shown that the transformation between interacting and free field operators in a reasonable quantum field theory cannot be unitary. Here, the original requirement of unitarity is revisited from a physical point of view to show not only that unitarity is not required but indeed not possible. Electrodynamics is used as an example. In a classical treatment the interaction cannot be turned on or off adiabatically as energy conservation cannot be maintained in a physically meaningful way. In a fully second quantized theory the interaction (or source) term is always present in the equation of motion even if the system is in the vacuum state. So, the interaction cannot be physically turned on or off adiabatically or otherwise. Hence, the transformation V(t) from free fields to interacting fields cannot be interpreted as an actual time evolution. This makes the unitarity of V(t)quite unnecessary as the original reason for unitarity was to conserve probability (or energy) through a time evolution. It is conjectured that the infinite terms in the scattering matrix that need renormalization appear due to the forcing of V(t)to be unitary. An alternate approach to scattering of interacting fields is suggested.