The geometrical interpretation of the photon position operator

Abstract

It is shown that the photon position operator $\hat{\vec{X}}$ with commuting components can be written in the momentum representation as $\hat{\vec{X}}=i\hat{\vec{D}}$, where $\hat{\vec{D}}$ is a flat connection in the tangent bundle $T({\mathbb{R}}^3\setminus \{(0,0,k_3)\in {\mathbb{R}}^3:k_3\ge 0\})$ over ${\mathbb{R}}^3\setminus \{(0,0,k_3)\in {\mathbb{R}}^3:k_3\ge 0\}$ equipped with the Cartesian structure. Moreover, $\hat{\vec{D}}$ is such that the tangent $2$-planes orthogonal to the momentum are parallelly propagated with respect to $\hat{\vec{D}}$ and, also, $\hat{\vec{D}}$ is an anti-Hermitian operator with respect to the scalar product $⟨\mathbf{\Psi }|{\hat{H}}^{-2s}|\mathbf{\Phi }⟩$. The eigenfunctions ${\mathbf{\Psi }}_{\vec{X}}(\vec{x})$ of the position operator $\hat{\vec{X}}$ are found.

23 pages, 1 figure, corrected typos