Light and matter are neither particles nor waves. We use the particle and wave analogies to allow us to apply some level of intuition to the effects. The interference effects in light (and matter interference experiments) follow similar equations to waves in water so when we want to talk about the interference, we call light and matter a wave. When we want to talk about effects such as the absorption of a discrete quanta of light in the photoelectric effect or the billiards like collisions of two atoms in a gas, we talk about light and matter as particles.
Reiterating: light and matter are neither particles nor waves... they are something else, something different than a sum of the two catch phrases. That difference generates amazing effects that defy our macro-world based intuition.
As for calculations, quantum field theory (QFT) treats both matter and light in a way that handles both the wave and particle like effects in one theory. One thing to note: light and matter have significantly different properties in QFT. The 'particle' properties of light are not the same as the 'particle' properties of matter.
The concepts "particle" and "wave" started from classical physics and from the everyday use of the terms, to begin with. A particle of dust got into one's eye, and the sea had huge waves.
Physics came into its reign when mathematics was seriously used to model observations.
For classical physics "particle" means an entity with small mass and a center of mass tracked at coordinates (x,y,z) at time t. Solutions of kinematic differential equations described the trajectory with accuracy determined by experimental errors.
For classical physics, waves are modeled by sinusoidal functions, i.e. functions that were the solution of "wave equations", could describe the behavior of sea waves, sound waves, and finally electromagnetic waves. Classically a wave is a variation of a measurable quantity like energy, or electric field, in space at a given time t, and the theoretical models were very successful in describing the observations of periodic energy distributions in bulk matter, and even in empty space ( electromagnetic waves).
Then quantum mechanics became necessary, from the discreteness of atoms, the black body radiation spectrum, the photoelectric effect it was finally understood that there were regions in the variables measured that displayed a quantization of energy.
It so happens that the equations that successfully describe the quantum mechanical state of matter were diferential equations with sinusoidal solutions, i.e. wave equations, like the Schrodinger equation. The solutions for the hydrogen atom were able to explain the spectral series adhoc assigned by the Bohr model, IF the postulate was assumed that the wavefunction squared did not represent the energy of the electron at (x,y,z) at time t, but a probability distribution. i.e. if one accumulated with the same boundary conditions a large number of measurements and plotted the (x,y,z) at time t distributions one would know how probable it would be to find the electron at that location.
As an example, this is similar to taking a census of the population of a city by age, and gauging how probable it would be that the first person you meet will be 8 years old. The wave function's function is just that, to give probabilities mathematically which are checked experimentally, and have been very accurate.
The "wave" part confused and continues to confuse people, because they think that the quantum mechanical entity, the electron for example, is spread out according to the solution of the Schrodinger equation. This is a misunderstanding, as the double slit interference experiments show with incoming single electrons:
Note the top photo, where the electron impinges on the screen, it is one whole electron . The probability pattern accumulated though shows clearly the interference effect that is expected by the sinusoidal form of the wave functions describing the electron when it hits the slits and goes through one or the other.
The "particle" facet of the electron is that it appears as a point at ( x,y,z_0) of the screen, and the "wave" facet is the probability distribution displayed in its trajectories.
If one is becoming a physicist it is simple to accept this fact, that the microcosm behaves differently than the macroscopic world we are used to. Bohm was stuck on classical frameworks and tried to derive the quantum mechanical probabilities from an underlying classical description. He succeeded in reproducing the same results as the usual quantum mechanical solutions, but afaik his model is complicated and limited, and cannot be extended into second quantization where the ball game has gone now.
Best Answer
What they actually measured was not particle behavior. It was just a quantized energy transfer to the probing electrons. That corresponds to the absorption of individual photons, but it doesn't mean the Surface Plasmon Polariton (SPP) field was acting as a particle. It just interacted locally with the electron, as it must. Typically particle-like behavior happens when decoherence happens. In the two slit experiment, a superposition of a photon hitting the screen in two separate places is dynamically unstable and will decohere. Consequently, we only observe a single flash at a single point. Nothing like this happened in their experiment. They set up a standing wave in their SPP, and tossed electrons to image the interference fringes. If any interaction is to happen, it will happen at the electron's position, and if any decoherence is to take place in order to secure particle-like behavior, it will happen when the electron is detected. The SPP itself never acts like a particle -- it just acts like a quantum object, which, of course, it is. This paper would be much improved without mentions of confusing notions of wave particle duality. They have cool results as it is; anything else is spurious.
Here is a very good article explaining what it actually is, thereby clarifying the paper's findings.