Question

I'm aware that we can describe the time evolution of states/operators (choose your favourite picture) of non interacting quantum fields and that perturbation theory is very effective in computing S matrix elements between free states in the remote past and free states in the remote future. Clearly the non-perturbative description of what's going on at finite times, where the interaction is active is intractible, but my question is - are there simplified toy models (scalar electrodynamics ? reduced numbers of dimensions ?) where we can describe what's happening non perturbatively.

Even if nothing like electrodynamics has been treatable like this, any results on the other "textbook" QFTs (like ?4) would be interesting.

Answer 1

A lot is known about QFTs (including QED) at finite time. It is tractable approximately (just like scattering). though in 4D no rigorous treatment is available (neither is there one for scattering).

One can compute - nonrigorously, in renormalized perturbation theory - many time-dependent things, namely via the Schwinger-Keldysh (or closed time path = CTP) formalism.

For example, E. Calzetta and B. L. Hu, Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation, Phys. Rev. D 37 (1988), 2878-2900. derive finite-time Boltzmann-type kinetic equations from quantum field theory using the CTP formalism.

There are also successful nonrelativistic approximations with relativistic corrections, within the framework of NRQED and NRQCD, which are used to compute bound state properties and spectral shifts. See, e.g., hep-ph/9209266, hep-ph/9805424, hep-ph/9707481, and hep-ph/9907240.

There is also an interesting particle-based approximation to QED by Barut, which might well turn out to become the germ of an exact particle interpretation of standard renormalized QED. See A.O. Barut and J.F. Van Huele, Phys. Rev. A 32 (1985), 3187-3195, and the discussion in Phys. Rev. A 34 (1986), 3500-3501,3502-3503.

Approximately renormalized Hamiltonians, and with them an approximate dynamics, can also be constructed via similarity renormalization; see, e.g.,
S.D. Glazek and K.G. Wilson, Phys. Rev. D 48 (1993), 5863-5872. hep-th/9706149

In 2D, the situation is well understood even rigorously:

For all theories where Wightman functions can be constructed rigorously, there is an associated Hilbert space on which corresponding (smeared) Wightman fields and generators of the Poincare group are densely defined. This implies that there is a well-defined Hamiltonian H=cp_0 that provides via the Schroedinger equation the dynamics of wave functions in time.

In particular, if the Wightman functions are constructed via the Osterwalder-Schrader reconstruction theorem, both the Hilbert space and the Hamiltonian are available in terms of the probability measure on the function space of integrable functions of the corresponding Euclidean fields. For details, see, e.g., Section 6.1 of J. Glimm and A Jaffe, Quantum Physics: A Functional Integral Point of View, Springer, Berlin 1987. In particular, (6.1.6), (6.1.11) and Theorem 6.1.3 are relevant.

[Information extracted from the Section ''Relativistic QFT at finite times?'' of Chapter B3: ''Basics on quantum fields'' of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html ]

[Edit October 9, 2012:] On the other hand, a lot is unknown about QFTs (including QED) at finite time. Let me quote from the 1999 article ''Some problems in statistical mechanics that I would like to see solved'' by Elliot Lieb http://www.sciencedirect.com/science/article/pii/S0378437198005172: But there is one huge problem that everyone avoids, because so far it is much too difficult to handle. That problem is Quantum Electrodynamics, and the problem exists whether we are talking about non-relativistic or relativistic quantum mechanics. [...] The physical picture that begs to be understood on some decent level is that the electron is surrounded by a huge cloud of photons with an enormous energy. We are looking for small effects, called 'radiative corrections', and these effects are like a flea on an elephant. Perturbation theory treats the elephant as a perturbation of the flea. [...] After renormalizing the mass so that the 'effective mass' (a concept familiar from solid state physics) equals the measured mass of the electron we are supposed to obtain an 'effective low energy Hamiltonian' (again, a familiar concept) that equals the Schroedinger Hamiltonian plus some tiny corrections, such as the Lamb shift. From there we should go on to verify the levels of hydrogen (which, except for the ground state, have become resonances), stability of matter and thermodynamics and all those other good things. But no one has a clue how to implement this program. [...] On the other hand matter does exist and the sun is shining, so the theory must exist, too. I would like to see it someday