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arXiv:2205.04791 · Physical Review A

Construction of a photon position operator with commuting components from natural axioms

Michał Dobrski, Maciej Przanowski, Jaromir Tosiek, Francisco J. Turrubiates

Published Apr 11, 2023• 6 citations
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Abstract

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bia\l{}ynicki--Birula scalar product, and is defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over ${\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)}$. This observation enables us to reformulate the quantum mechanics of a single photon on $({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}$.

DOI
10.1103/physreva.107.042208

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Physical Review A•May 1, 1999•55 citations
A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bia\l{}ynicki--Birula scalar product, and is defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over R3\ensuremath∖(0,0,0){\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)}R3\ensuremath∖(0,0,0). This observation enables us to reformulate the quantum mechanics of a single photon on (R3\ensuremath∖(0,0,0))\ifmmode×\else\texttimes\fiC2({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}(R3\ensuremath∖(0,0,0))\ifmmode×\else\texttimes\fiC2.
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.
It is shown that the photon position operator X⃗^\hat{\vec{X}}X^ with commuting components introduced by Margaret Hawton can be written in the momentum representation as X⃗^=iD⃗^\hat{\vec{X}}=i \hat{\vec{D}}X^=iD^, where D⃗^\hat{\vec{D}}D^ is a flat connection in the tangent bundle T(R3∖{(0,0,k3)∈R3:k3≥0})T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})T(R3∖{(0,0,k3​)∈R3:k3​≥0}) over R3∖{(0,0,k3)∈R3:k3≥0}\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}R3∖{(0,0,k3​)∈R3:k3​≥0} equipped with the Cartesian structure. Moreover, D⃗^\hat{\vec{D}}D^ is such that the tangent 222-planes orthogonal to the momentum are parallelly propagated with respect to D⃗^\hat{\vec{D}}D^ and, also, D⃗^\hat{\vec{D}}D^ is an anti-Hermitian operator with respect to the scalar product ⟨Ψ∣H^−2s∣Φ⟩\langle \mathbf{\Psi} | \hat{H}^{-2s} |\mathbf{\Phi} \rangle⟨Ψ∣H^−2s∣Φ⟩. The eigenfunctions ΨX⃗(x⃗)\mathbf{\Psi}_{\vec{X}} (\vec{x})ΨX​(x) of the position operator X⃗^\hat{\vec{X}}X^ are found.
The paper provides three main definitions of the Cartesian photon position operator based on: boost generator, the transversality condition and the helicity operator. In each case, the correctness of the definition and Hermitianness of the operator in the domain of physical states are proven. All considered definitions lead to the same form of the Cartesian position operator in the domain of physical states. Radial photon position operators were also defined, but they turned out to be non-equivalent. Nevertheless, the most useful two radial operators turned out to be twin counterparts in the sense of the transformation to the helical representation, which is an alternative positional representation. The components of the photon position operator do not commute, but in analogy to the problem of eigenangular momentum, its eigenstates do exist. The eigenstates of the two components of the position operator (including the radial component) are called photon strings. Photon strings on a straight line and on a circle have been studied. The usual, previously known photon string states were named electric strings, but new magnetic strings were also introduced. Exact helical photon strings on a straight line, as well as hybrid helical photon strings on a circle, were also considered. On the other hand, infinitely short photon magnetic strings turned out to be flat photon vortices on planes with a normalization formula looks like as point-localized particles.
We extend a procedure for the construction of photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are reestablished by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our position operator, like that proposed by Pryce, is a matrix that does not commute with the spin operator. Unlike the Pryce operator, however, our operator has commuting components, but the commutators of these components with the total angular momentum require an extra term to rotate the matrices for each vector component around the momentum direction. Several proofs of the nonexistence of a photon position operator with commuting components are based on overly restrictive premises that do not apply here.
A photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase. Its eigenkets are transverse and longitudinal vectors, and thus states can be selected that have definite polarization or helicity. For angular momentum and boost operators defined in the usual way, all of the commutation relations of the Poincar\'e group are satisfied. This new position operator is unitarily equivalent to the Newton-Wigner-Pryce position operator for massive particles.