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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?

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    $\begingroup$ 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. $\endgroup$ – Thomas Nov 7 '18 at 16:16
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In principle, we can treat interacting QFT just like any other quantum theory, in terms of time-evolving states in the Schrödinger picture (for example), without relying on the asymptotic past/future.

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.

However, relating those local observables to specific physically-recognizable things like electrons and protons is difficult. The usual approach of looking for poles in time-ordered functions is fine for scattering experiments, but when it comes to studying the time-evolution of state-vectors with clear physical interpretations, we're pretty much stuck — we don't even have an explicit expression for the vacuum state in most interacting QFTs, much less states with specified configurations of particles. Such states exist mathematically (modulo the usual ambiguities in what we mean by "particle"), but actually constructing them in interesting models seems to be beyond our current mathematical abilities. I think this is the single biggest obstacle to teaching QFT in a really satisfying way. The meaning of renormalization is a solved problem ("What is renormalization?", https://arxiv.org/abs/hep-ph/0506330), but conceptually-simple things like writing down an explicit representation for a single-electron state are still beyond our reach, as far as I know.

As an example of what we do and don't know how to do, you might be interested in this classic paper by Feynman, where he studies the Schrodinger equation in SU(2) gauge theory in (continuous) three-dimensional spacetime:

(The Schrodinger equation is written in "solved" form in equation (30) on page 498, using the path-integral trick that helped make Feynman famous.)

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