Are EM waves a state of matter?

From grade school, we learned that matter comes in 3 states: solid, liquid, gas ...& some of us, plasma. Okay there are more, but these are the commons.

If e=mc^2 (energy = mass),

Then what "state" are EM waves in?

ether?

...matter is something that takes up space ...mass takes up space ...mass is energy, so energy takes up space?

I'm not a physicist, but I'll give this a shot. I think most of this is accurate...

States of matter are defined for sets of atoms, usually. Solids, liquids, gasses, plasmas, and all the rest are collections of atoms. E&M waves, though, are at the bridge of the whole particle wave duality theory. It's considered radiant energy, which can be mathematically modeled as oscillating E&M fields or streams of photons. I'll go with the latter, because photons are easier to describe the waves.

Photons are massless, despite having energy. E = mc^2 does not really apply to massless particles; otherwise photons wouldn't have energy, which breaks a lot of physics. They do have momentum, though, which is where it gets really weird; E = mc^2 is only half the equation. The full relativistic equation is E^2 = p^2 c^2 + m^2 c^4; the famous E = m c^2 assumes the mass is at rest. When calculating the energy of a photon, you don't use this equation, though; you'd use E = hc/lambda, where h is plank's constant and lambda is frequency, making h/lambda = momentum, but now I'm getting into territory that I forget from my quantum class...basically, my point is that the energy of a photon = hc/lambda and not mc^2. Since photons are one way of defining E&M waves, you don't really define it as "mass," and so I would argue that it doesn't have a state of matter.

Thus, my first instinct is to say that the classification of an E&M wave as a state of matter doesn't really fit with quantum descriptions.

That said, there's also things called Bose gases, which can be made up of photons. I don't really remember the details of them, but maybe it's possible for them to make an E&M wave? In which case your state would be a "bose gas." That seems a bit fishy to me, but someone with a better knowledge of quantum mechanics should answer that!

Heat is energy, so is light. But they are not matter. Energy and mass are equivalent, but they are not the same thing.