The Newton-Wigner and Wightman localization of the photon

Simon N. M. Ruijsenaars

R. M. Mir‐Kasimov

Steven B. Berger

Klaus D. Sattler

You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) œeld. However, Heisenberg’s uncertainty principle tells us that a state of deœnite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indeœnite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with œnite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo˜ of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).

Eugenios Angelopoulos, F. Bayen, Mosh� Flato

Wightman's reformulation of the Wigner-Newton position operators and localizability of a massive particle is generalized so as to include zero mass particles. Different a priori procedures of generalizations are examined: we first prove some negative results based upon a projective induction technique. We then present some positive results which state that in some sense zero-mass particles are localizable and that their corresponding position operators are candidates for observables. Our treatment–which is group theoretical and kinematical in the same sense as that of Wightman–ends with suggestions for further possible developments.

John F. Clauser