Geometrical interpretation of the photon position operator with commuting components

Uniqueness of the photon position operator with commuting components

“…However, as it is well known [11,16,18], all these operators reveal a string singularity analogous to the one observed in the case of the Dirac magnetic monopole. Thus, in Sec.…”

“…To proceed further we employ results of our recent work [18]. In that paper we showed that the photon position operator with commuting components introduced by Hawton [10] is determined by a flat connection in an appropriate open dense submanifold of momentum space R 3 such that tangent planes orthogonal to the momentum are propagated parallel with respect to this connection and that a corresponding covariant derivative is an anti -Hermitian operator with respect to the Bia lynicki -Birula scalar product (2.7).…”

“…2 we find a general form of the photon position operator obeying the conditions mentioned above. The main result of that section is that the Hawton photon position operator as well as other photon position operators with commuting components obtained in [11,16,18] are, up to a unitary transformation preserving the transversality condition (2.6), the general photon position operators with commuting components.…”

“…On the other hand in 1999 a very promising photon position operator with commuting components was found by M. Hawton [10]. This idea was then developed in several publications [11][12][13][14][15][16][17][18]. Significant progress in understanding Hawton's photon position operator was made in [11,16], where it was shown that one can construct other Hawton -like operators by choosing a suitable orthonormal basis in a 3-D Euclidean momentum space.…”

“…Significant progress in understanding Hawton's photon position operator was made in [11,16], where it was shown that one can construct other Hawton -like operators by choosing a suitable orthonormal basis in a 3-D Euclidean momentum space. Very recently [18] we developed the idea presented in articles [11,16] and proposed a simple geometrical interpretation of the Hawton -like photon position operators as objects determined by a flat connection (covariant derivative) on an appropriate dense subset of the momentum space R 3 such that 2-planes orthogonal to the momentum vector are propagated parallel, and moreover, the covariant differentiation is an antihermitian operator with respect to the Bia lynicki -Birula scalar product [19,20]. Note that the similar axioms have been assumed in the work by H.Babaei and A.Mostafazedeh [17].…”