Why does light travel slower in water relative to vacuum? First of all read till the end. Do not mark as duplicate before finishing. The answer to the question above is that photons interact with molecules in the water which takes some time causing the net speed to drop. But lets remember something. For an object to be transparent the incoming light shouldn't be able to ionize its electrons. And indeed water is transparent so there is actually no interaction between the photons and the water molecules. So I guess the speed of light in water or any transparent medium should not change.  But it does. So where am I wrong at?
 A: You say:

And indeed water is transparent so there is actually no interaction between the photons and the water molecules

but this is not true. Water is transparent because no energy is dissipated in the interaction between the light and the water. Objects that absorb light take energy from th light and convert it into other forms such as molecular vibrational energy (i.e. heat).
At the risk of over simplifying, in a transparent medium the light does interact with electrons in the medium because it makes them oscillate at the same frequency of the light. However the oscillating electrons then return the energy to the light but with a small phase shift. It is this phase shift that changes the speed of the light. But since the net energy of the light is unchanged the medium remains transparent.
A: First of all, let's distinguish that your question is about quantum mechanics. Classically, $\epsilon$ and $\mu$ are functions of the material and Maxwell's equations produce a wave equation with velocity $\frac{1}{\sqrt{\epsilon \mu}}$.
In QED, the easiest way to get the idea of what's going on is using Feynman's picture of quantum mechanics. Feynman says that to understand the motion of the photon, you need to consider all possible trajectories for the particle, including those that end with it being absorbed.
The interference between the outcomes in which the photon is absorbed by the medium (or has some other interaction) are what slows down its trajectory. I am not a competent enough field theorist to show this explicitly. Perhaps we will get lucky and someone else will do so.
A: If an electron interacts with something, it does because of electromagnetic force. A photon has the potential to interact with an electron because of that force. The electron maintains that force no matter what it is in. Meaning, an electron in water exerts the same force on a photon as any other electron. 
