Why are short distances synonymous with high energies? Context: I'm thinking about QFT here and why introducing a momentum cutoff means ignoring short distances.
This has been asked before (Why are high energies equivalent to short distances?) but I didn't find any of the answers satisfying. 
I understand that taking units of $\hbar = c = 1$ gives energy units equal to inverse space units, but that doesn't give me any $physical$ intuition for why short distances is the same as high energy. 
Another answer given on that thread (the accepted answer, in fact) was about how high energy corresponds to short $wavelengths$, which is true for the massive and massless particles, but I still feel like that is missing a real link to the physical intuition:'stuff' still happens at 'long' wavelengths, and I can't see why distance and energy should be considered the same in this context.
Can somebody help me find the physical intuition here?
 A: Think of a one dimensional continuous medium whose behavior we would like to approximate by a collection of linear coupled oscillators. Clearly, the more oscillators we have the better our approximation and the more minute details we can capture. In particular the range of frequencies we can talk about will be very large. But having more oscillators means the distances between each oscillator is really small. So if I decide to reduce the number of oscillators I am reducing the range of frequencies I can consider. 
Picking a momentum cut off therefore is another way of picking how many oscillators will be in your approximation and therefore the distance between each oscillator i.e the lattice spacing.
A: the simplest explanation I have come across is as follows: to probe what's happening on extremely short distance scales requires the use of a wavelength of light that is smaller than that distance scale- otherwise, the scattering of that light will not give you any detailed information about what's going on down there. so for example, the accelerator tube for a benchtop electron microscope might be 3 feet long and the resulting beam energy (~ 1/wavelength) is sufficient to resolve ~10 nanometer-sized features- but to resolve the interior of a proton requires an accelerator tube ~10,000 feet long to get the energy high enough so as to obtain a wavelength short enough. 
