General interpretation of the poles of the propagator

Question

I am somewhat familiar with the fact that the poles of the Feynman propagator in QFT give the momentum of particle states. I'm also familiar with the KL spectral representation in that context (See Physical interpretation of propagator pole). What I'd like to know is if there is a general result like this that works for all physical situations with Green's functions. Are the poles of the Green's function of a theory always its free solutions? Is this a mathematical fact? If I calculate the Green's function of linearized gravity in general relativity, will the poles be the wavenumbers of gravitational waves? Or is the pole structure of Green's functions and their relation to wave solutions (particle states in QFT) a property of quantum theories?

I'm looking for a general result connecting the poles of Green's functions to solutions of the theory in a way that applies to other areas of physics which are not quantum mechanical in nature.

Answer 1

Take an arbitrary QFT, possibly strongly-coupled, even a non-Lagrangian one.

If the theory is Poincaré invariant, then there are conserved operators $J^{\mu\nu},P^\mu$ that obey the Poincaré algebra, by definition. So in particular, $[P^\mu,P^\nu]=0,[J^{\mu\nu},P^\sigma]\propto P^\rho$, etc.

As all $P^\mu$ commute with each other, we can diagonalize simultaneously. So we have a basis of states $|p^\mu,\lambda\rangle$ where $p^\mu$ are the eigenvalues of $P^\mu$, and $\lambda$ are other quantum numbers (so in particular, spin, which comes from $J^2$, which also commutes with $P^\mu$, etc.)

Let us also assume that the Hamiltonian $P^0$ is bounded from below, and denote $|0\rangle$ its ground state (it may be degenerate).

Let $\phi(x)$ be an arbitrary operator (not necessarily one appearing in your Lagrangian). Assume that the matrix element $\langle 0|\phi(x)|p^\mu,\lambda\rangle$ is non-zero. Then, it is a theorem that the correlation functions of $\phi(x)$ will have poles in momentum space when any Fourier variable approaches $p^2$, with $p^\mu$ the momentum of $|p^\mu,\lambda\rangle$. See ref.1 for the proof.

The residue at the poles carries information about $\lambda$ (in particular, it is entirely fixed by symmetries, up to multiplicative constants, often called the "wavefunction renormalization" of $\phi$).

Comments:

• The theorem is true for any theory, not necessarily one that is close to a free theory. So no notion of "free solution" is required.

• The theorem is true for any operator, not necessarily a "fundamental" one.

• The theorem is true for any correlation function, not necessarily the two-point function (but of course, it is true in particular for such function).

• As for linearized gravity: by definition this describes massless particles with helicity $2$, so the states $|p^2,\lambda\rangle$ include, among others, a state with $p^2=0$ and $\lambda=\pm$ (the two polarization states of a helicity $2$ state). Such states have non-zero matrix elements with respect to the metric, so the two-point function of the latter indeed has a pole at zero-momentum (and the residue is the usual tensor projection onto the helicity-2 subspace).

References.

1. Weinberg S. - Quantum theory of fields, Vol.1. Foundations, §10.2.