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arXiv:2104.04351 · Physical Review A

Geometrical interpretation of the photon position operator with commuting components

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

Published Oct 7, 2021• 9 citations
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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.

DOI
10.1103/physreva.104.042206

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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(R3\ensuremath∖(0,0,k3)\ensuremath∈R3:k3\ensuremath≥0)T({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0})T(R3\ensuremath∖(0,0,k3​)\ensuremath∈R3:k3​\ensuremath≥0) over R3\ensuremath∖(0,0,k3)\ensuremath∈R3:k3\ensuremath≥0{\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,{k}_{3})\ensuremath{\in}{\mathbb{R}}^{3}:{k}_{3}\ensuremath{\ge}0}R3\ensuremath∖(0,0,k3​)\ensuremath∈R3:k3​\ensuremath≥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⟨\ensuremathΨ∣H\ifmmode^\elseˆ\fi\ensuremath−2s∣\ensuremathΦ\ensuremath⟩\ensuremath{\langle}\mathrm{\ensuremath{\Psi}}|{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}^{\ensuremath{-}2s}|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}\ensuremath⟨\ensuremathΨ∣H\ifmmode^\elseˆ\fi​\ensuremath−2s∣\ensuremathΦ\ensuremath⟩. 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.
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 $({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}$.
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.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
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.