The issue is complex, and very much depends on the configuration of the experiment - frequency and coherence length in relation to separation of the slits, the thickness of the slit material, the distance of the focal plane from the slit, etc. One can take a spatial Fourier transform on certain configurations to model interference, but one can also set the configuration so that one doesn't see a pattern at all (e.g., the separation of the slits greater than the coherence length).

The continuous "wave" characterization is that of space-time with massless electromagnetism where Fourier analysis applies, limited by coherence length which "spreads" the samples due to the convolution process w.r.t. the spatial configuration; the "particle" characterization is ultimately quantum mechanical (photons as particles) where the waves are characterized as particle densities interacting with material in the slits.

If one does see a pattern *unchanging", one is looking at a space transform (a photon density distribution at the focal plane). If one watches individual flashes, one is simply seeing how this distribution is built up over time (photons considered as spatial samples, with intensity the relative number of samples at any one position and time - which is interpreted as a probability of any one photon hitting a given position as time progresses).

The actual microscopic mechanisms for this (field interactions at the slits) is very complex, and such a description with infinite precision is impossible because of the uncertainty principle (since it involves mass transfers between photons ("kinetic" mass) and the electrons/atoms/fields in the slit material. These actions are what Einstein called "hidden variables" and by adding the concept of mass (as continuous functions, rather than discrete particles), he was attempting to apply GTR at the local level - a photon as a spherical wave modified by interactions with external field media. But only god can ignore the uncertainty principle in quantum physics (observe without interacting) ....

(Keep in mind that the prescription E= hv applies to photo electrons ejected from a surface (flat or spherical atom), not to changes of state within the atoms themselves (that just eject photons). Nevertheless, these changes in state can be caused by external field excitations impinging on the atoms. So within the atom, one can think of the electrons as changing its effective mass to account for the energy radiated (plus a neutrino to account for the change in the nucleus for a single atom, as opposed to an averaged response from a change in the Fermi level in a material (solid state) lattice) subject to the impinging photon field.

(That said, observing changes of state caused by electrons within atoms involves comparing them to unobserved states (initial and final conditions) which depend on the relation of electron to nucleus, so in is ultimately a circular description at the Higgs level....

Bottom line: it actually gets very complex indeed when one tries to characterize the "hidden variables", and there is no one model that covers it all. Different models apply under different configurations of the experiment, and even the "Big Bang" model (impulse-response spatial Fourier transform) model fails if the configuration is greater than the coherence length of the signal.

(and, like all physics, one eventually needs a blackboard to explain things properly...

One can only apply models in their "scope of validity"....

IMO, YMMV, etc, etc.....

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