# What does a QFT particle state have to do with a classical point particle?

In the question Can one define a “particle” as space-localized object in quantum field theory? it is said that in quantum field theory, a particle state is a state with well defined energy and momentum, related with dispersion relation $E^2=p^2+m^2$. This thing is localized in momentum space, which means it must be delocalized in coordinate space.

On the other hand, in classical mechanics, the most striking feature of a point particle is being localized in coordinate space.

On the first glance, these two objects may seem very different with no obvious link between them. It seems that it is usually just postulated that the above defined QFT states are particles, without any clear justification why should they be related to the point-like classical objects. Satisfying the same dispersion relation isn't good enough justification, as it is not obvious that e.g. some classical field configurations can't satisfy it.

So my question is: Why do we call them both particles, how do we see that they behave similarly (in some appropriate limit)?

• @JohnRennie What do you mean by that? AFAIK, the object created by $a^\dagger(k)$ is not an eigenstate of the field operator. Aug 21, 2016 at 7:52
• Hmm, maybe I'll rethink that statement. Aug 21, 2016 at 7:54
• You should not compare them as they have different definitions. In qft, these particles are defined as irreducible unitary rep. of Poincare group while particles in classical mechanics are defined by a simple energy-momentum relationship. It is like the example of spin which has different meaning in both context. An example of a particle in qft is photon which does not even have a coordinate representation.
– ved
Aug 21, 2016 at 8:31
• @ved It is obvious that they have different definitions, but they should also have enough in common to call them with the same name. Aug 21, 2016 at 8:40

The possible pure states of a particle of spin $s$ are the unit vectors in a Hilbert space $L_2(R^3)^{2s+1}$. It contains neither position nor momentum eigenstates.