-----------------------------
S4j. Do free particles exist?
-----------------------------
Free particles are a convenient mathematical abstraction.
In Nature, there are - strictly speaking - no free particles,
only interacting ones. This holds both for photons and for other
more tangible particles like electrons. However, in sufficiently
localized (and nearly empty) regions of space, particles can be
approximately free. Again, this holds for both photons and other
particles.
It is very convenient to approximate such states by free states.
For example, this allows to explain much of quantum mechanics
in terms of particle scattering. The S-matrix interpretation
depends crucially on the fact that the ingoing and outgoing
asymptotic states of photons, electrons, quarks, etc. are free.
Thus, in this sense, free photons exist just as much (or just as
little) as free electrons.
In matter, particles are never free since they interact with the
surrounding material. However, one often can interpret what happens
in matter in terms of quasiparticles that behave in a good
approximation freely over sufficiently short time spans, the mean free
path time. These quasiparticles may be the original particles equipped
with an extra dressing due to the environment, or composites like
Cooper pairs of electrons in superconducting materials, or entirely
new species of field excitations such as phonons in crystals.
In quantum field theory, (quasi)particles only exists as excitation
modes of quantum fields. They behave particle-like only on length scales
larger than the Compton wavelength of the particles/fields. For the
eelctromagnetic field, this corresponds to length scales on which the
geometric optics approximation is valid, in which optical phenomena
can be treated by means of light rays (the travelling paths of photons).
Mathematically, a free particle (in a pure state) is a bound state,
i.e., a normalizable eigenstate psi of the 4-momentum operator p
(in case of a relativistic particle; otherwise of 3-momentum and energy.
Here we use the relativistic terminology). Thus it is a state psi
satisfying p psi = k hbar psi and hence
psi(x)= e^{-i x dot p} psi(0).
Every free particle in a pure state can be describe in this way since
this is what it means to be free. This holds independent of any
particular theory. The latter just specifies the form of psi(0).
Free particles should not be mixed up with bare particles.
The latter are an abstract mathematical tool for approximating
processes inviolving real particles. The free particles that can be
found in Nature are all dressed (renormalized).