Why photons cannot be sharply localized

Abstract

Physicists have pondered over the problem of photon localization and the related problem of the photon wave function for almost 80 years now, beginning with the work of Landau and Peierls 1. An extensive review of the photon localization problem was recently presented by Keller 2. The problem of photon localization is closely related to the widely studied problem of the photon position operator. In this paper we introduce an operational definition of partial localization based on the measurements of correlation functions for electric or magnetic fields. For a want of better names, we shall use the terms electric localization and magnetic localization even though this might erroneously suggest the presence of some electric or magnetic devices that confine the photons. Since a sharp localization of photons according to our operational definition of localization is not possible, a photon position operator compatible with this definition does not exist. Earlier studies of the photon localization emphasized the mathematical aspects see, for example, 3‐5. In this paper we emphasize the physical properties of the electromagnetic field. In particular, we exhibit the role of the photon helicity and the symmetry between the electric and magnetic fields. We proceed in the footsteps of Glauber 6‐9 who was the first to recognize the significance of the space-dependent creation and annihilation operators. This approach was recently summarized and expanded in an extensive paper by Smith and Raymer 10. In our work we concentrate on the analysis of the photon localization in terms of the electric and magnetic field operators. We treat these fields after smearing over space-time regions as bona fide observables. We put emphasis on the field aspect that is complementary to the particle aspect. It might have a weaker connection with experiments usually based on photon counting as highlighted by Glauber, but it is more precise as was explained in detail by Bohr and Rosenfeld 11,12 The standard method of quantization of the free electromagnetic field based on the decomposition into monochromatic modes is not well suited for the discussion of localizability because the monochromatic mode functions are not localized. To overcome this problem we further developed an alternative method of quantization that does not require a mode decomposition. The essential mathematical tools in our analysis are the Riemann-Silberstein RS vector and the helicity operator. This formulation has some merits of its own, and it can also be used to study other general properties of