Localization and position operators
Position operators are part of the toolkit of relativistic quantum
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,
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
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
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.
Reeh-Schlieder meets Newton-Wigner
This paper gives some relations to quantum field theory
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ