*$n$-particle states* in QFT as irreducible representation of symmetry group?

Question

The Quanta Magazine article titled What is a Particle? has a nice summary of various possible interpretations of particles in theoretical physics, but it appears that there is no universally accepted definition as of now. Let us consider two such description:

• Group theoretic definition of particle:

Following is an excerpt from this article:

Ever since the fundamental paper of Wigner [W39] on the irreducible representations of the Poincare group, it has been a (perhaps implicit) definition in physics that an elementary particle "is" an irreducible representation of the group, $G $, of "symmetries of nature". The search for the elements then has three components:

1. Determine the group $G$ (a problem in physics).
2. Find the irreducible representations of G (a problem in mathematics).
3. Determine which irreducible representations of $G$ occur as elementary particles. (a problem in physics).

• Particle interpretation in QFT:

Although fields are more fundamental, a particle like interpretation can emerge if we look at the interaction b/w quantum field and detector. We can start with a two level quantum system coupled to the field. The field state $|\psi\rangle$ with occupation number $n$ can exchange quanta of energy with the detector and can transition to occupation number $n+1$ or $n-1$ with some non-zero probability. Wald argues that $n$ can be interpreted as no. of particles and that this interpretation works even in curved space-time (See $\S$3.3 of "Quantum field theory in curved spacetime and black hole thermodynamics" by R.M.Wald).

While Wald's argument makes sense intuitively, is it possible to assign a group theoretic description to this process? In other words, does the exchanged quanta b/w the quantum field and the detector can be expressed as an "irreducible representation of some symmetry group"?

In general, all interactions in QFT can be described by processes involving exchange of virtual particles. While the ingoing and outgoing states can be given a group theoretic description, is it also true for virtual particles?

Answer 1

The choice between introducing quantum fields first or particles first is a presentation choice and it doesn't interfere with what the objects really are. Personally, I prefer introducing particles first. The basic reason is that in this approach, one ends up constructing quantum fields to encode particles and this reasoning naturally shows one constraint between the Poincaré representation on particle states and the Lorentz representation on the fields. Obviously, the constraint is also there if you start from fields, but I find then that it is somewhat more obscure.

The so-introduced fields are then the free or in/out ones, linear in creation and annihilation operators, with the interacting fields asymptotically matching to them as $t\to \pm \infty$ according to the scattering assumption $$\Phi(x)\to\sqrt{Z}\Phi_{\rm in/out}(x),\quad \text{as $t\to\pm\infty$}.$$

This is the viewpoint in Weinberg's textbook, exposed in Chapters 2 through 5.

Now, the thing is that no matter how you present the subject to other people (starting from fields and canonically quantizing to get particles, or starting from particles and building fields), something is just a fact and remains true: when you decompose the field $\Phi(x)$ into positive-frequency modes to obtain creation and annihilation operators, the states these operators build out of the vacuum furnish one irreducible representation of the Poincaré group, according to what the Lorentz representation of the field was in the first place.

Wald's description relies on such mode decomposition of the fields, so indeed the connection to the definition in terms of representation is the realization that a quantum field is actually an encoding of an irreducible unitary Poincaré representation into an object transforming in a Lorentz representation.

Finally, virtual particles are not real physical entities. They do not correspond to states at all in the Hilbert space. The thing is that we evaluate field correlators through Feynman diagrams. These are just pictorial representations of terms in a perturbative expansion. It just so happens that the external lines can be connected to real physical entities, after you subject the correlator to the LSZ prescription in order to extract scattering amplitudes.

The internal lines, on the other hand, just represent mathematical objects in that particular term contributing to the perturbative expansion, namely, propagators.