When Is a Quantum Field Theory Exactly Solved?

[Lester]

A question I would like to get an answer is when is a QFT exactly solved? E.g. if I know the solution of the equation for the two-point function I have got all about the theory? This equation is classical in nature being the two-point function defined in the sense of distributions. I have read the original paper of Schwinger about QED2 and he does exactly this.

Jon

 

[strangerep]

A question I would like to get an answer is when is a QFT exactly solved?

One meaning of "solved" is that we have a complete set of eigenstates of the full
(interacting) Hamiltonian. I.e., one has "diagonalized" the interacting Hamiltonian.
With such a complete set, the properties of any scattering scenario or bound state can
be expressed analytically in closed form (exactly).

E.g. if I know the solution of the equation for the two-point function I have got
all about the theory?

That's not enough. But did you mean 2-point or 4-point? (Don't you need 4-point to
describe 2-particle scattering?)

This equation is classical in nature being the two-point function defined in the
sense of distributions. I have read the original paper of Schwinger about QED2 and
he does exactly this.

Could you give a more precise reference, pls?

 

[Count Iblis]

I think that in QED there are only a few exact results, e.g. the exact expression for the pair creation probability per unit volume and time in a constant electric field. You can write this as a functional determinant and exactly evaluate it.

 

[Avodyne]

I think that in QED there are only a few exact results, e.g. the exact expression for the pair creation probability per unit volume and time in a constant electric field. You can write this as a functional determinant and exactly evaluate it.

Nothing is known exactly in QED. The functional determinant only sums one-loop graphs (an electron-positron loop in the external field, but with no virtual photons exchanged).

There are a number of exact results in two spacetime dimensions, however.

 

[Lester]

About reference I mean the Schwinger's paper about QED in 1+1 dimensions (Phys. Rev. 128, 2425 (1962)) but I think that any reliable textbook should give the same information.

For a QFT generally a two-point function is enough to compute scattering amplitudes by LSZ formalism. This formalism is exact and should give also information about states and asymptotic states.

Jon