The first-quantized photon wave function (PWF) formalism — together with its second-quantized Green's-function scattering extension — has been shown to reproduce standard quantum electrodynamics (QED) for single-photon processes such as Mach–Zehnder interference and for the photon-number-changing second-order process of sum-frequency generation, but only under three idealizations: a lossless dispersive nonmagnetic medium, an unbounded geometry in which the retarded Green function is truncated to its far-field term, and a purely local electric response. Ritboon, Daengngam and Pengpan (Ann. Phys. 371, 53–66, 2016) explicitly left open whether the formalism can be extended beyond these idealizations. No published work constructs the position-space PWF for absorbing media, for general guided-wave and cavity geometries including near-field terms, or for spatially nonlocal response, nor establishes its observable-level equivalence with macroscopic QED. The problem — lift these three idealizations while preserving the energy-density/probability interpretation and causality — remains open.

Background

The photon wave function (PWF) formalism is a first-quantized description of the single photon: instead of quantizing the electromagnetic field and then restricting to the one-photon sector, one assigns the photon a position-space wave function. In the Białynicki-Birula / Riemann–Silberstein formulation the wave function is the Riemann–Silberstein vector, the complex combination F±=ε0/2 E±iμ0/2 H\mathbf{F}_\pm=\sqrt{\varepsilon_0/2}\,\mathbf{E}\pm i\sqrt{\mu_0/2}\,\mathbf{H} of the electric and magnetic field amplitudes, which obeys a Schrödinger-like equation with the Hamiltonian c ∇×c\,\nabla\times and whose squared norm reproduces the electromagnetic energy density. Sipe (Phys. Rev. A 52, 1875–1883, 1995) systematized this picture and connected it to the one-photon sector of the quantized field. The formalism is attractive for intuitive, position-space analyses of single-photon experiments, but its equivalence with standard second-quantized QED has been established mainly for free or effectively linear, lossless, photon-number-conserving situations.

Ritboon, Daengngam and Pengpan (Ann. Phys. 371, 53–66, 2016) applied the PWF formalism, combined with a second-quantized Green's-function scattering treatment, to single photons in a Mach–Zehnder interferometer and to sum-frequency generation (SFG, a second-order nonlinear process converting two photons into one at the sum of their frequencies), recovering the standard interference probabilities and sinc-type conversion intensities and confirming agreement with QED for these processes. Their derivation, however, rests on three explicit idealizations, each of which they flag as an open issue for future work:

  1. A lossless, dispersive, nonmagnetic medium with real electric susceptibility and negligible free currents, so that absorption and dissipative currents are absent.
  2. An unbounded geometry: the retarded scalar Green function is kept only in its far-field term, proportional to R−1R^{-1}; the mid- and near-field terms (proportional to R−2R^{-2} and R−3R^{-3}), which dominate in optical waveguides, cavities, and microsphere resonators supporting whispering-gallery modes (light trapped by total internal reflection circulating the equator of a small dielectric sphere), are neglected.
  3. A purely local electric response, in which the displacement field at a point depends only on the electric field at that point; spatial nonlocality (spatial dispersion, where the permittivity relates the field at one point to the field in a surrounding neighborhood, as in metals at nanometer scales and near resonances) is excluded — an exclusion that also bears on the long-standing problem of photon localization.

The standard second-quantized treatment of light in exactly these regimes is macroscopic QED, which quantizes the electromagnetic field coupled to a reservoir of matter oscillators and introduces Langevin noise operators so that dissipation and fluctuation remain consistent (the fluctuation–dissipation theorem); its elementary excitations in absorbing media are polaritons dressed by the reservoir. Lossy optical resonators are likewise quantized through quasinormal-mode frameworks (modes of open, leaking cavities with complex eigenfrequencies). What is missing is the bridge: a first-quantized, position-space photon wave function — with the probability/energy-density interpretation intact — valid in absorbing media, in bounded and guided geometries including near-field contributions, and in spatially nonlocal media, together with a demonstration that its observable predictions agree with macroscopic QED.

Problem Statement

Determine whether the photon wave function formalism — the first-quantized, position-space description of the single photon, together with its second-quantized Green's-function scattering extension — can be extended beyond the three idealizations under which its equivalence with standard QED has been established, while preserving its energy-density/probability interpretation and causality; if so, construct it and establish that its observable predictions agree with macroscopic QED. Concretely, this requires constructing the formalism for: (i) absorbing (lossy) media, where the refractive index is complex and dissipative free currents must enter the microscopic current density and the Green's-function treatment; (ii) bounded and guided-wave geometries such as optical waveguides, cavities, and microsphere resonators supporting whispering-gallery modes, where boundary conditions, discrete mode spectra, and the mid- and near-field terms of the Green function (proportional to R−2R^{-2} and R−3R^{-3}) cannot be neglected; and (iii) media with spatially nonlocal electric response, where the permittivity is a spatially nonlocal operator, with the attendant question of a consistent description of photon localization. All three axes must be addressed for the task to be resolved, though they need not be covered by a single unified formalism: separate treatments of the different regimes are acceptable, and for each axis resolving it means either constructing the extension or establishing that it does not exist.

References

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