Newton–Wigner localization

Newton–Wigner localization (named after Theodore Duddell Newton and Eugene Wigner) is a scheme for obtaining a position operator for massive relativistic quantum particles.^[1]

The Newton-Wigner concept was developed for "elementary systems", an abstraction similar to elementary particles but without the requirement of non-decomposability. For example, a hydrogen atom is an elementary system but not an elementary particle, while an electron is both.^[2]

In the relativistic quantum mechanics of a single particle, the Newton–Wigner position operators x₁, x₂, x₃ have the same commutation relations with the 3 space momentum operators and transform under rotations in the same way as the x, y, z in ordinary QM. Though formally they have the same properties with respect to p₁, p₂, p₃, as the position in ordinary QM, they have additional properties: One of these is that

${\displaystyle [x_{i}\,,p_{0}]=p_{i}/p_{0}~.}$

This ensures that the free particle moves at the expected velocity with the given momentum/energy.

Apparently these notions were discovered when attempting to define a self adjoint operator in the relativistic setting that resembled the position operator in basic quantum mechanics in the sense that at low momenta it approximately agreed with that operator. It also has several famous strange behaviors (see the Hegerfeldt theorem in particular), one of which is seen as the motivation for having to introduce quantum field theory.