Do all forms of energy have a mass? We know by $E=mc^2$ that mass and energy are directly proportional, but there are massless forms of energy such as electro-magnetic waves. I am also told that there are different forms of mass, such as invariant mass, virtual mass, and relativistic mass. This electro-magnetic wave seems to be a kind of quantum mechanical wave, that collapses into a particle dubbed a photon. When it interacts with matter, it gives that matter energy, which gives it more mass. Particles with an invariant mass also can be described by probabilistic waves, and behave in a similar way as photons. That is, when its wave function is disturbed in some way, it too collapses into a particle of some form, depending on what it is. The wave function must not be able to exceed the speed of light, because it has an invariant mass.
So my question is, what happens to the energy of an electromagnetic wave, when it transforms into an energy that has a form of mass, and where is that mass from, and what is the difference between the wave function of an invariant mass, verses the wave function of a photon without mass? Okay lots of question here, sorry.
Answer
To answer your question let me start with the most basic constituents of the universe, i.e. elementary particles.
Standard model of particle physics contains matter particles (quarks & leptons), force carriers (W/Z, photon, gluon) and the Higgs particle. Photon and gluon are massless, and the rest of the particles have non-zero masses. In a non-interacting situation, all those particles have their masses fixed, independent of their speed (So no relativistic mass which is an old way of thinking about relativistic kinematics anyways).
When there are interactions between particles and enough energy to produce new particles (through $E=mc^2$), weird things happen. At this point we need quantum field theory to fully understand what's going on. Without going much into details, the way we think about the interactions is through force carrying particles I mentioned in the previous paragraph. When two particles interact, they can simply scatter off each other or the interaction can create a complete different set of particles. We understand the process of going from an "initial" set of particles to a "final" set of particles in terms of all the "paths" connecting them with the interactions allowed by the Standard Model. This situation is very similar to what happens in the double slit experiment. During this process virtual particles (from the above set I mentioned) are created. These intermediate particles have the exact same properties as the real ones, except their mass (or invariant mass) can be different than their actual mass. As in the double slit experiment, nature takes all the allowed paths between initial and final states.
At this point we can talk about the energy of our particles. According to special relativity there is a rest frame for any massive particle and in this special frame even though our particle is at rest it will have an energy given by $E_0 = mc^2$. So for a massless particle like photon there is no rest frame and hence no rest energy. But all the particles (massive or massless) have momentum, and a total energy given by $E = \sqrt{(mc^2)^2 + (pc)^2}$. Note that this reduces to $E = pc$ for a massless particle like photon.
If you think in terms quantum physics, an electromagnetic wave contains photons. So the energy of the electromagnetic wave comes from the energy of the photons i.e. the momentum carried by the photons (which is also related to their frequency or wavelength).
So far we talked about mass and energy of fundamental particles. What happens when we have bound states like protons/neutrons or nuclei or atoms or molecules? The mass of these compound objects depend on the bounding energy (or potential energy) that keeps them together. So for example the mass of a proton is not equal to the total mass of the quarks that makes up the proton (two up, one down quark) but is mostly created by the interaction between these quarks explained by quantum chromodynamics (QCD). Since there is more than one fundamental particle making up our bound states, there can also be excited states with slightly different masses. An atom for example can absorb a photon and switch to an excited state for a very brief time which has a different mass. This mass difference is usually very small. This only happens when the energy of the photon matches the difference between the discrete energy levels of the atom. Most of the time they will only scatter like billiard balls.
For an even larger bound system like a solid object, an absorbed electromagnetic wave usually changes the temperature of the object. In this larger system, we can think of atoms organized in a geometric pattern. On the average they sit at fixed points on a 3D lattice but they all individually vibrate around their equilibrium points. So an electromagnetic wave absorbed by the object usually changes the amplitude of these vibrations or the average vibration energy i.e. the temperature of the object.
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