Localization in quantum field theory

We review the issue of localization in quantum field theory and detail the nonrelativistic limit.
Three distinct localization schemes are examined: the Newton–Wigner, the algebraic quantum
field theory, and the modal scheme. Among these, the algebraic quantum field theory provides
a fundamental concept of localization, rooted in its axiomatic formulation. In contrast, the
Newton–Wigner scheme draws inspiration from the Born interpretation, applying mainly to
the nonrelativistic regime. The modal scheme, relying on the representation of single particles
as positive frequency modes of the Klein–Gordon equation, is found to be incompatible with
the algebraic quantum field theory localization.
This review delves into the distinctive features of each scheme, offering a comparative
analysis. A specific focus is placed on the property of independence between state preparations
and observable measurements in spacelike separated regions. Notably, the notion of localization
in algebraic quantum field theory violates this independence due to the Reeh–Schlieder theorem.
Drawing parallels with the quantum teleportation protocol, it is argued that causality remains
unviolated. Additionally, we consider the nonrelativistic limit of quantum field theory, revealing
the emergence of the Born scheme as the fundamental concept of localization. Consequently,
the nonlocality associated with the Reeh–Schlieder theorem is shown to be suppressed under
nonrelativistic conditions.