When is a quantum field theory exactly solved?

16/07/2008

This question is somewhat a headache for someone. The point is that we know how to treat quantum field theory by perturbation techniques but the only typical case we have is a free theory. In this case the generating functional is Gaussian and all functional derivatives are easily computed. So, whenever we are able to reduce to this latter situation we can draw sharp conclusions about the solution of the theory.

Indeed there are two aspects one should manage for a quantum field theory and are exactly the same as for the solution of a quantum mechanics problem where one has to determine both the eigenstates and the spectrum. In a quantum mechanics problem one knows that the poles of the Green function are the points of the spectrum of the quantum system. So, we should expect the same in quantum field theory. But we cannot be sure as the Kaellen-Lehman representation is not always expected to hold and this conclusion is not so straightforward to be drawn. Indeed, formulations of quantum field theories exist that extend the validity of the Kaellen-Lehman representation to the case where positivity is missing (e.g. see the works of Franco Strocchi as here) but currently there is no common agreement about and different points of view are met, mostly due to the fact said above that solvable quantum field theories are quite rare. The other aspect relies on the form of the states. Indeed, using Green functions and asymptotic states one is able to compute scattering sections and decay times through Lehman-Symanzik-Zimmermann reduction formula. The problem is what are these asymptotic states for some theories as confined ones.

So, with our present knowledge, a sharp definition of an exactly solved quantum field theory does not exist and also the way to get the spectrum can be a source of debate. What we are really missing is some example of an exactly solved non-linear quantum field theory. Anyhow, any other point of view about is welcome.


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