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

TL;DR

The paper resolves long-standing questions about the photon position operator by presenting three equivalent Cartesian definitions derived from a boost generator, transversality, and helicity projections, and proving their Hermiticity on the physical state domain. It shows that, despite noncommuting Cartesian components, well-defined eigenstates exist in the form of one-dimensional photon strings (electric, magnetic, and helical) along lines and circles, including open and closed configurations. Radial components are defined but are not universally equivalent; two radial operators emerge as twin counterparts under the helical representation, revealing a rich structure of position representations. A helical (frequency/helicity-based) representation is developed, showing invariance of the standard scalar product and revealing dualities between electric and magnetic string states, as well as an extended Hilbert space formalism that cleanly separates physical from longitudinal modes. The findings illuminate photon localization within Maxwellian/QED frameworks and provide a concrete, operational basis for position-like observables beyond commuting-coordinate paradigms.

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.

Figures (7)

• Figure 1: The interference pattern produced by a single-photon source with (a) 30, (b) 300, and (c) 3000 photons registered. In contrast, the decoherent distribution of (d) 30, (e) 300, and (f) 3000 photons lacks the dark fringes. The illustration and description come from the publication Afshar
• Figure 2: Scanning electron micrograph of an amphibian eye retina's rods (and cones) structure. Rods about 1 µ m in diameter and 20 µ m long can absorb single photons of weak light. The photocurrents of similar natural photocells (rods) were studied in the work Pugh. The illustration was purchased from Alamy
• Figure 3: Interaction of a 780 nm laser with 100 µ m thick steel foil under different pulse durations: (a) scheme for long pulse duration, (b) scheme for short pulse duration, (c) electron micrograph of the ablation hole at nanosecond laser pulses of 3.3 ns, (d) electron micrograph of the ablation hole at femtosecond laser pulses of 200 fs (scale bar 30 µ m). Graphics (a), (b) are from Femto 2021, and original micrographs (c), (d) are from Femto 1996
• Figure 4: Graphical visualisation of the $\ket{x_0,y_0}$ photon state as a linear electric string, which is an infinitely thin and infinitely dense bundle (vector Dirac delta). The string is open, but it is infinitely long -- its cross-section is for illustrative purposes only
• Figure 5: Graphical visualization of the $\ket{x_0^{\star},y_0^{\star}}$ photon state as a linear magnetic string, which in the $\hat{\mathrm{p}}\ket{x_0^{\star},y_0^{\star}}(\mathbf{r})$ "energetic" representation is an infinitely tight and dense spiral vector field (a rotation of the vector Dirac delta). The rings are complete but have been cut for illustrative purposes. Basically, this magnetic string is infinitely long, but it can be shortened by separating but not cutting the rings -- without violating the transversality condition of the state