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arXiv:2309.12125

The photon position operator (non-commuting) and its string-like eigenstates

Grzegorz Marcin Koczan

Published Sep 21, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2309.12125

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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 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 $({\mathbb{R}}^{3}\ensuremath{\setminus}{(0,0,0)})\ifmmode\times\else\texttimes\fi{}{\mathbb{C}}^{2}$.
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.
We propose a position operator for the photon. The components commute and commute with helicity, and the two operators form a complete set of commuting observables. However, the simultaneous eigenvectors of this position operator and helicity do not rotate locally. For a normalized wavepacket state vector, the position-helicity amplitudes give a complete characterization of the physical properties of the state vector, yet their moduli-squared have rotation properties that do not have a simple interpretation. Thus we call these pseudo-amplitudes. We show that there is a regime of wavepacket state vectors, those with a small fractional spread around an average momentum, where the position-helicity probability pseudo-densities rotate very nearly locally as scalar functions. This is just the regime that is needed to describe a scattering experiment involving photons. We consider other measures of localization for the photon, including those proposed by other authors. It is important to note that exactly the same problems arise if we try to characterize the state vector of a massive particle with spin by position and helicity. The problems are not the result of the masslessness of the photon but arise because of its limited helicity spectrum.
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.
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.
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.