Robustness of least-time dominance for the propagation amplitude of a finite-mass spin-0 particle at an interface between two media

A 2016 analysis computed the detection amplitude for a nonrelativistic, finite-mass, spin-0 (scalar) quantum particle emitted in one medium and detected in another across a planar interface, using the free-particle Green's function convolved with Gaussian impact-region probability densities, and found that the amplitude is maximized when the particle effectively follows the least-time route, a matter-wave analog of Fermat's principle of least time in geometrical optics. That conclusion was obtained in a model explicitly labelled nonrelativistic and spin-0 that disregards the material properties of the two media, and in a single special planar-interface geometry. The open problem is to determine the range of validity of the least-time dominance: recompute the interface Green's function, its convolution with the impact-region weights, and the location of the amplitude maximum once a refractive-index or potential contrast between the media is introduced, and settle whether the maximum then lies on the least-time path, the stationary-action path, or elsewhere.

Background

In geometrical optics, light traveling between two points across the boundary between two transparent media follows the path of least total travel time (Fermat's principle of least time), which at a flat interface implies Snell's law of refraction. Feynman's popular account of quantum electrodynamics (QED, the quantum field theory of light and charged particles) explains this as a stationary-phase effect: the probability amplitude for a photon to propagate is a sum over all routes, each contributing a phase set by its travel time, and contributions from routes far from the least-time one cancel. Maksuwan, Wongkokua and Viriyasirisakul (2016) asked whether an analogous least-time dominance holds for a massive quantum particle. Here spin-0 means a scalar particle carrying no intrinsic angular momentum, treated nonrelativistically with the Schrödinger equation; the Green's function, or propagator, is the kernel $G(x_2,t_2;x_1,t_1)$ giving the probability amplitude for the particle to travel from spacetime point $(x_1,t_1)$ to $(x_2,t_2)$. In their model the particle is emitted by a source in medium 1 and registered by a detector in medium 2, the media meeting at a planar interface; every point of the interface may act as an intermediate scattering site with its own amplitude, and the emission and detection regions are weighted by Gaussian probability density functions (PDFs), the impact regions, with widths $\sigma_j$. Each interface impact point thus carries its own amplitude, a product of propagators convolved with these Gaussian weights, and integrating those impact-point amplitudes over the interface gives the total detection amplitude; the quantity whose maximum is at issue below is the amplitude belonging to a given impact point, in the source's own phrasing the point of most probable detection. Working in the special geometry where emitter and detector sit at equal normal distances from the interface ($z_1=z_2$) and explicitly disregarding the material properties of the two media (no potential or refractive-index contrast, so the particle propagates freely on both sides), the authors found, analytically and numerically, that the detection amplitude is maximized when the particle follows the least-time route, in analogy with Fermat's principle for light. Companion work by the same group built the corresponding amplitude picture for light: a quantum-field-theoretic propagator model of the refractive index that also disregards material properties, a study of radiation from planar secondary sources, and a QED treatment of a massless spin-1 particle (the photon) moving between two media of different dielectric properties through emitter-detector amplitudes. None of these works tests whether the least-time conclusion for the massive particle survives the neglected physical ingredients. Standard semiclassical physics frames the stakes: a matter wave crossing a step in potential energy refracts according to a Snell-like law with an effective refractive index proportional to the local particle speed $v$, which is a stationary-action condition, whereas least time extremizes $\int ds/v$; the two criteria select different broken paths whenever the two media differ, so the contrast between the light and matter cases is exactly where the source paper's idealization bites.

Problem Statement

Determine the range of validity of the least-time-dominance conclusion of Maksuwan, Wongkokua and Viriyasirisakul (2016) for a finite-mass spin-0 quantum particle refracting at a planar interface between two media. The core determination: introduce the material properties of the two media into their Green's-function framework, that is, an effective potential or refractive-index contrast between medium 1 and medium 2; recompute the interface Green's function, its convolution with the Gaussian impact-region PDFs, and the interface point at which the detection amplitude is maximized; and decide whether that maximum coincides with the least-time route (which, once the media differ, is a genuinely broken path obeying Snell's law with medium-dependent speeds) or with a different path such as the stationary-action path. The determination is carried out within the source framework, the nonrelativistic finite-mass spin-0 propagator with Gaussian impact-region weights, now with a medium contrast included, and it asks how the location of the amplitude maximum depends on that contrast and on the remaining parameters of the model (mass, widths $\sigma_j$, emitter-detector geometry): under which physical conditions the detection amplitude for a quantum particle crossing the interface is dominated by the least-time path, and under which the source's Fermat-type conclusion fails. Regimes that call for a propagator model other than the source's, namely relativistic propagation of the finite-mass particle, the massless (photon) case in a full QED treatment, nonzero spin, interference and coherence effects beyond the geometrical-optics description, and non-Gaussian impact-region weights or non-planar interface geometries, lie outside this determination; the Gaussian impact-region widths $\sigma_j$ and the planar emitter-detector placement, including the normal distances $z_1$ and $z_2$, remain within scope.

References

1. Atirat Maksuwan, Wiwat Wongkokua, Pongpich Viriyasirisakul, 'Propagation of finite mass (spin-0) particles in refraction phenomenon', AIP Conference Proceedings 1775, 030047 (2016). DOI: 10.1063/1.4965167, https://doi.org/10.1063/1.4965167
2. Atirat Maksuwan, 'Propagation of Massless Spin-1 Particle between Two Media', Key Engineering Materials 675-676, 687-690 (2016). DOI: 10.4028/www.scientific.net/KEM.675-676.687, https://www.scientific.net/KEM.675-676.687
3. Atirat Maksuwan, 'Quantum Field Theory Viewpoint of Refractive Index', Advanced Materials Research 979, 31-34 (2014). DOI: 10.4028/www.scientific.net/AMR.979.31, https://www.scientific.net/AMR.979.31
4. A. Maksuwan, P. Viriyasrisuwattana, 'Radiation from Secondary Planar Surfaces Sources in Quantum Field Theory', Journal of Physics: Conference Series 622, 012007 (2015). DOI: 10.1088/1742-6596/622/1/012007, https://doi.org/10.1088/1742-6596/622/1/012007
5. Richard P. Feynman, 'QED: The Strange Theory of Light and Matter', Princeton University Press, Princeton, New Jersey, 1985. ISBN 0-691-08388-6.