Are quasi-particles really particles?

There are basically two types:

1. Quasiparticles, if they are related to fermions

2. collective excitations if they are related to bosons

Now there are four main differences between real elementary particles (or real particles and composite particles) and quasiparticles:

1. real particles are or are made up of the elementary particles in the standard model, our currently accepted theory. On the other hand, quasiparticles are non of these (usually) but an emergent phenomenon that occurs inside a solid (usually we use quasiparticles in solids)

2. it is possible to have an real elementary or composite particle in free space. On the other hand, quasiparticles need to exist inside an interacting many body system (usually solid)

3. real particles in a solid have a very complicated interacting way, whereas quasiparticles are exactly there to make the modeling of these solids easier, since they act like non-interacting

4. quasiparticles are a math tool for describing solids

A very good example for quasiparticles (fermion related) is to understand the difference between the speed of electricity itself, almost c, and the drift velocity of real electrons inside the conductor, very slow. The difference is because the real electrons are interacting with the hole system of the conductor lattice, whereas the electricity itself (this is where we use the word electron holes) is a phenomenon that is emerging because the conductor is packed so densely with electrons that electricity itself will travel almost at c. This is easiest to understand if you imagine the wire with electrons in it fully packed (this is not correct but makes a good example) and if you push the first electron with an external field, it will push the others and the electrons at the other end of the wire will move almost at speed c.

Now a good example for collective excitations (boson related) is phonons. Phonons are movements inside a solid lattice molecular structure. You can imagine that phonons are not real particles, but are an emergent phenomenon based on a wave that travels in the lattice structure of a crystal. It is the relative changes in the positions of the molecules that make up the lattice that create a phenomenon that looks like a wave in a solid. This acts like a quasiparticle, a phonon.

They have kinetic energy and momentum, some have rest energy. They only live in a medium but some would argue that space-time itself is a medium. Let me reverse the question: why not ?

Quasiparticles are not "real" in the sense that the trial wavefunction represented by an n-quasiparticle state is not a state of good particle number, and therefore can't represent a physical state. I think this fact tends to get forgotten in condensed matter physics, where I suppose the particle number fluctuations are on the order of the square root of Avogadro's number, which is huge but negligible compared to Avogadro's number. But when we use quasiparticles in nuclear physics, this is a serious issue. You're trying to study a nucleus with 37 protons and 42 neutrons, but the model effectively mixes in nuclei with other numbers of protons and neutrons.

This applies not just to excited states but also to the 0-quasiparticle state, i.e., the vacuum, which would be the ground state of an even-even nucleus. So for instance in nuclear physics we try to predict the shapes and vibrational states of nuclei by calculating their potential energy surfaces as a function of shape. Say we do this for the vacuum state in an even-even nucleus. Experiments show that although the observables connected to shape are usually slowly varying functions of particle number, it can happen, especially for light nuclei, that you get sudden changes. A quasiparticle model will never be able to correctly reproduce this type of sudden transition. The failure of the model in these cases arises from the fact that it's effectively averaging properties over different nuclei.