Does a photon attract or repel matter? So, I know experimentally that matter and light interact. And I can calculate cross sections in Quantum Electrodynamics. Yet a fundamental intuition about how light and matter interact is completely missing in my understanding.
If a photon were (for a brief instant) next to a proton, would they attract? Repel? How about an electron? Is there some kind of potential $V(x_2-x_1)$ which can approximate the interaction between light in a particle-like state and matter? If we can do it for protons and electrons, which are both particles and waves, why can't we also do it for photons?
Maybe this is going to be categorized as too broad. If that's so, let's just narrow it down to: Does a photon (a gaussian wavepacket of light) attract a proton, or repel it? If the answer depends on the distance, is there must be an optimal distance between them like in the Van der Waals potential?:

 A: I think this question needs to be asked very carefully to get a precise answer. But, the zeroth order answer should be clear: the photon has no electric charge or magnetic moment, so it will neither attract nor repel another electric charge.
One interpretation of this question, given in the comments to another answer, is to ask about the trajectory of a photon passing near an atom.
Well, in fact, even here we need to be careful. When I hear "trajectory" I think of a one dimensional line moving through space; and it makes sense to think of the impact parameter (distance of closest approach) of such a line with an atom. But an EM wave, or a photon wavepacket, is not (generally) of this form.
If the EM wavelength (1 over the photon momentum) is larger than the size of the atom, then the atom will look approximately like a point particle, compared to a traveling plane wave. First, note there is no useful sense in which we can think of the EM wave / photon as having a 1 dimensional trajectory and ask for its distance with respect to an atom in this limit. We should think of a wave extending over space, scattering off of a point source. This is the limit where we expect more or less elastic scattering to apply. There is an interaction, but we describe this interaction in terms of the scattered wave produced at infinity, and not a trajectory.
If the EM wavelength (1 over the photon momentum) is smaller than the size of the atom, then we can think of a trajectory, if we imagine building a wavepacket of high energy radiation. But in this limit, where by definition the photon has a larger energy than the binding energy of the atom, we certainly do not expect an elastic scattering process to occur. Instead, the EM radiation will be absorbed by the atom and re-emitted with a different energy with some probability, or possibly even ionize the atom. If the photon is super-well localized and not on a collision course with the atom, it will have a small probability to interact; the trajectory won't be "dragged toward" or "pushed away from" the atom.
Another possible interpretation of this question is whether one can derive an "effective potential" analogous to the Coulomb potential for photon-charged particle interactions. To address this, one should remember that how Coulomb potential can be derived from quantum field theory: one starts with the full relativistic scattering amplitude, takes the non-relativistic limit, and matches with the scattering amplitude with a potential in non-relativistic quantum mechanics. The problem is that there is no way to take the non-relativistic limit for a photon.
One could ask what happens with a massive, neutral particle -- say a neutron, or imagine the photon really did have a tiny mass. In this case, you can go through the above exercise. You will find that neutral particles will have a zero effective potential in the non-relativistic limit.
A: In photonic crystals, there are bandgaps because there is a difference in energy for standing electromagnetic waves, depending on where the nodes are. The energy is lower when the antinodes (regions with stronger electric fields) are in matter with larger $\varepsilon_r$.
So there is an interaction between light and matter. I am a simple experimentalists and would not dare to say even where the photons are in such a case. Quantum electrodynamics can only treat very simple cases. It cannot really deal with solids or even with larger atoms.
A: You are not specifying matter, but you are asking about a massless photon and a massive proton. And you are not asking about an actual scattering, you are just saying that the massless photon and the massive proton are next to each other (at certain distance).
You are saying that if a massless photon were for an instant next to a massive proton, what you are not specifying is, the instant. Photons are massless particles, travel at speed c when measured locally, in vacuum, and whatever the relative speed of the massive proton is, from its reference frame as per SR, the massless photon always travels at speed c. You cannot specify a reference frame or an "instant" in the temporal dimension where both the massless photon and the massive proton would be at rest relative to each other.
If you are asking about EM attraction or repulsion, then the answer is no. The photon does not EM repel or interact any object in the universe, the photon is EM neutral.
If you asking about gravitational attraction, then the answer is yes. Massless photons do have stress-energy, just like massive protons, they do have their own gravitational (static) field, and both do bend spacetime.
Do photons bend spacetime or not?
