Geometrical interpretation of the photon position operator with commuting components

Abstract

It is shown that the photon position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ with commuting components can be written in the momentum representation as $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}=i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$, where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is a flat connection in the tangent bundle $T({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0})$ over ${\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0}$ equipped with the Cartesian structure. Moreover, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is such that the tangent 2-planes orthogonal to the momentum are propagated parallel with respect to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ and also $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}$ is an anti-Hermitian (i.e., $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ is Hermitian) operator with respect to the scalar product $\ensuremath{\langle}\mathrm{\ensuremath{\Psi}}|{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}^{\ensuremath{-}2s}|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$. The eigenfunctions ${\mathrm{\ensuremath{\Psi}}}_{\stackrel{P\vec}{X}}(\stackrel{P\vec}{x})$ of the position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}$ are found.