Localization and position operators

Position operators are part of the toolkit of relativistic quantum mechanics.

In a relativistic setting, one always has a representation of the Poincare algebra. From the generators of the Poincare algebra (namely the 4-momentum p, the angular momentum \J, and the boost generators \K) one can make up (in massive representations) a nonlinear expression for a 3-dimensional \x (the position operator) that together with the space part \p of the 4-momentum has canonical commutation rules and hence gives a Heisenberg algebra. (The backslash is a convenient ascii notation to indicate bold face letters, corresponding to 3-vectors.)

The position operator so constructed is unique, once the time coordinate is fixed, and is usually called the Newton-Wigner position operator, although it appears already in earlier work of Pryce. Relevant applications are related to the names Foldy and Wuythousen (for their transform of the Dirac equation, widely used in relativistic quantum chemistry) and Bakamjian and Thomas (for their relativistic multi-particle theories); both groups rediscovered the Newton-Wigner results independently, not being aware of their work.

That the time coordinate has to be fixed means that the position operator is observer-dependent. Each observer splits space-time into its personal time (in direction of its total 4-momentum) and personal 3-space (orthogonal to it), and the position operator relates to this 3-space. By a Lorentz transformation, one can transform the 4-momentum to the vector (E_obs 0 0 0), which makes time the 0-component. Most papers on the subject work in the latter setting.

For massless representations of spin >1/2, the construction breaks down. This is related to the fact that massless particles with spin >1/2 don't have modes of all helicities allowed by the spin (e.g., photons have spin 1 but no longitudinal modes), which makes them being always spread out, and hence not completely localizable. For details, see the FAQ entry ''Particle positions and the position operator''

Here are a few references:

  • J.P. Costella and B.H.J. McKellar, The Foldy-Wouthuysen transformation, arXiv:hep-ph/9503416
    This paper discusses the physical relevance of the Newton-Wigner representation, and its relation to the Foldy-Wouthuysen transformation

  • T. D. Newton, E. P. Wigner, Localized States for Elementary Systems, Rev. Mod. Phys. 21 (1949) 400-406
    The original paper on localization

  • L. L. Foldy and S. A. Wouthuysen, On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit, Phys. Rev. 78 (1950), 29-36.
    On the transform of the Dirac equation now carrying the author's name

  • B. Bakamjian and L. H. Thomas Relativistic Particle Dynamics. II Phys. Rev. 92 (1953), 1300-1310.
    and related papers in
  • Phys. Rev. 85 (1952), 868-872. Phys. Rev. 121 (1961), 1849-1851.
    First constructive papers on relativistic multiparticle dynamics, based on a 3D position operator

  • L. L. Foldy, Synthesis of Covariant Particle Equations, Phys. Rev. 102 (1956), 568-581
    A lucid exposition of Poincare representations which start with a 3D position operator, and a discussion of electron localization Before eq. (189), he notes that an observer-independent localization of a Dirac electron (which generally is considered to be a pointlike particle since it can be exactly localized in a given frame) necessarily leaves a fuzziness of the order of the Compton wavelength of the particle. (This is also related to the so-called Zitterbewegung, see, e.g., the discussion in Chapter 7 of Paul Strange's "Relativistic Quantum Mechanics".)

  • A. S. Wightman, On the Localizability of Quantum Mechanical Systems, Rev. Mod. Phys. 34 (1962) 845-872
    A group theoretic view in terms of systems of imprimitiviy

  • T. O. Philips, Lorentz invariant localized states, Phys. Rev. 136 (1964), B893-B896.
    A covariant coherent state alternative which does not require to single out a time coordinate

  • V.S. Varadarajan, Geometry of Quantum Theory (second edition), Springer, 1985
    A book discussing some of this stuff

  • L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.
    The bible on quantum optics, a thick but very useful book. Relevant here since it contains a good discussion of the localizability of photons (which can be done only approximately, in view of the above) from a reasonably practical point of view.

  • G.N. Fleming, Reeh-Schlieder meets Newton-Wigner http://philsci-archive.pitt.edu/archive/00000649/
    This paper gives some relations to quantum field theory

    Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
    A theoretical physics FAQ