# Can interacting quantum field theory describe more than just scattering?

From my understanding we do not yet know how to make much out of interacting QFT other than scattering amplitude at asymptotic infinity. (Correct me if I misunderstand.) But path integral, in principle should allow us to compute different sorts of transition probabilities. So is lack of understanding more about constructing observables such that we can test with measurements? That is, is this lack of understanding about our current lack of how to interpret states at times not sent to asymptotic infinity?

• We can compute the mass of the proton by numerically simulating the path integral of QCD. We can compute the critical exponents of thermodynamic phase transitions by using the epsilon expansion and the renormalization group for statistical field theories. We can compute the anomalous magnetic moment of the muon. We can compute the equation of state and the conductivity of QED and QCD plasmas. None of these things are scattering amplitudes. – Thomas Nov 7 '18 at 16:16

Conceptually, a QFT is a net of algebras of local observables (as explained in Haag's book Local Quantum Physics), and this can be realized explicitly in many interacting QFTs if we don't mind treating spacetime as a discrete lattice (as explained in Montvay and Munster's book Quantum Fields on a Lattice). In those cases where we know how to construct the model on a discrete spatial lattice (which excludes chiral non-abelian gauge theories like the Standard Model, but includes both QED and QCD), we can even write down the Schrodinger equation for a general state-vector like this: $$i\frac{\partial}{\partial t}\Psi[t,\phi] = H\Psi[t,\phi] \tag{1}$$ where $$\Psi[t,\phi]$$ is the wavefunctional as a function of time $$t$$ and all of the field variables $$\phi$$, which may include scalar fields, spinor fields, and gauge fields. The Hamiltonian $$H$$ is expressed in terms of field operators, which are in turn expressed in terms of things like derivatives with respect to the field variables $$\phi$$. So, in principle, we can treat interacting QFT just like any other quantum theory, in terms of time-evolving states, without relying on the asymptotic past/future.