Nonlocality of the energy density for all single-photon states

The nonlocality of single-photon states has been analyzed from several different but interrelated perspectives. In this article we propose a demonstration based on the electromagnetic energy density observable and on the antilocal property of the frequency operator Omega = c(-Delta)(1/2). The present proof is based on the standard quantization of the electromagnetic field, which can be formulated equivalently in the momentum representations or in the position representations of Landau and Peierls [Z. Phys. 62, 188 (1930)] and of Bialynicki-Birula [edited by E. Wolf (Elsevier, Amsterdam, 1996)]. Our proof extends to all single-photon states the results of Bialynicki-Birula, which were formulated for two particular classes of states, those involving a uniform localization [Phys. Rev. Lett. 80, 5247 (1998)] or alternatively states that are electrically or magnetically localized [Phys. Rev. A 79, 032112 (2009)]. Our approach is formulated in terms of Knight's definition of strict localization [J. Math. Phys. 2, 459 (1961)], based on the comparison of expectation values of single-photon states of local observables with those of the vacuum.