Q:

How do photons interact with a very fine edge? 

Suppose you have some material which has a low reflectance and is opaque to an incomming photon when the angle of incidence is small. 
Now take that material and make a narrow (20:1 width:length ratio or narrower) wedge with a fine edge - 1/10th the wavelength of said photon or less, the very edge itself should be transparent - and shoot said photon directly down the edge of the wedge like so. 
How does the photon react?  Does the photon treat the edge as slicing a probability wave, resolving to having been deflected randomly to one side of the wedge or the other when observed... or does it treat the edge as if it was no finer than the wavelength and so act as a perpendicular strike on a blunt edge and be absorbed?  Is there other quantum weridness involved, or other information required to answer fully? 

A:

This situation is really little different from most beamsplitter "which way" thought experiments or the explanation behind almost anything you can cook up on an optics table.  
There's no particular quantum "weirdness" here.  I sense that what may be bothering you is that you may be thinking of the wedge's interaction with the photon as some kind of quantum measurement.  This interaction is not a quantum measurement and excited matter states in the wedge's glass simply exist in quantum superposition with the pure, one photon light states.  Quantum measurement only happens when the photon is detected, say at a screen behind the wedge. 
Unless we're dealing with entangled light states, one simply conceives of the theoretically foretold experimental outcome in the standard way:

Describe and analyze the problem with a standard, linear Maxwellian description;
At our detection plane, calculate the intensity field $\vec{E}\times\vec{H}$ from the Maxwellian solution above.  Or, if you have a resonant situation and you want to ask where in the volume you'll detect the photon, you calculate the energy density field $\vec{E}\cdot\vec{D}+\vec{H}\cdot\vec{B}$. 
The quantities calculated in 2.  above are proportional to the probability density to destructively detect a photon at each point in space when we do the experiment as a repeated, one-photon-present-at-a-time experiment. 

This is because there is a one to one correspondence between one photon states and classical solutions to Maxwell's equations, as I discuss further here and here.