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. It is known to largely conflict with the Reeh–Schlieder theorem outside of a very limited scope.

The Newton–Wigner position operators x₁, x₂, x₃, are the premier notion of position in relativistic quantum mechanics of a single particle. They enjoy 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

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

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.

References

• Newton, T.D.; Wigner, E.P. (1949). "Localized States for Elementary Systems". Reviews of Modern Physics. 21 (3): 400–406. Bibcode:1949RvMP...21..400N. doi:10.1103/RevModPhys.21.400.
• M.H.L. Pryce, Proc. Roy. Soc. 195A, 62 (1948)
• V. Bargmann and E. P. Wigner, Proc Natl Acad Sci USA 34, 211-223 (1948). pdf
• V. Moretti, On the relativistic spatial localization for massive real scalar Klein–Gordon quantum particles Lett Math Phys 113, 66 (2023). [1]