Localization of scalar quantum fields on Minkowski space-time

TL;DR

This paper introduces a novel localization scheme for Klein-Gordon particles using a second quantum field in momentum space, enabling the definition of position and time operators in relativistic quantum field theory.

Contribution

It proposes a new method to define space and time operators in Klein-Gordon theory by constructing a second field in momentum space, allowing for localized particle states.

Findings

Particle states are eigenstates of the new space and time operators.

The resulting space-time is flat and consistent with special relativity.

Provides a new interpretation of relativistic notions like inertial observers.

Abstract

A new localization scheme for Klein-Gordon particle states is introduced in the form of general space and time operators. The definition of these operators is achieved by establishing a second quantum field in the momentum space of the standard field we want to localize (here Klein-Gordon field). The motivation for defining a new field in momentum space is as follows. In standard field theories one can define a momentum (and energy) operator for a field excitation but not a general position (and time) operator because the field satisfies a differential equation in position space and, through its Fourier transform, an algebraic equation in momentum space. Thus, in a field theory which does the opposite, namely it satisfies a differential equation in momentum space and an algebraic equation in position space, we will be able to define a position and time operator. Since the new field lives in the momentum space of the Klein-Gordon field, the creation/annihilation operators of the former, which build the new space and time operators, reduce to the field operators of the latter. As a result, particle states of Klein-Gordon field are eigenstates of the new space and time operators and therefore localized on a space-time described by their spectrum. Finally, we show that this space-time is flat because it accommodates the two postulates of special relativity. Interpretation of special relativistic notions as inertial observers and proper acceleration in terms of the new field is also provided.