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 $\phi^4$) would be interesting.

  • $\begingroup$ I am confused a little bit--- the off shell correlation functions are not matrix elements, but genuine field correlators, and they are perturbatively well defined in QED, and nonperturbatively well defined in lattice QCD. So would this be satisfactory? If you are asking for an initial value formulation, this is difficult because you need a mechanism to specify initial and final states in terms of physical particles, which are only perfectly well defined in scattering limits. I think there is a way to do time dependent processes reasonably, but I think it has not been fully worked out. $\endgroup$
    – Ron Maimon
    Feb 2, 2012 at 12:06
  • $\begingroup$ @RonMaimon: No, I'm certainly not looking for a particle interpretation of what happens at intermediate times. To put the question another way, can we find some greatly simplified example system for which we can describe the whole evolution exactly using the full Hamiltonian (which is time dependent and reduces to the free Hamiltonian at early and late times) without resorting to a Dyson series/perturbation expansion ? cont $\endgroup$
    – twistor59
    Feb 2, 2012 at 13:29
  • $\begingroup$ ...cont I don't know what the mathematical description would look like - some sort of state evolution on a state space which begins and ends in some "Fock sector" of that space ? It may be a silly question, but it's something I've always wondered about. $\endgroup$
    – twistor59
    Feb 2, 2012 at 13:29
  • $\begingroup$ @twistor: i know what you're looking for but I can't formulate the question in any better way either :) $\endgroup$
    – BjornW
    Feb 2, 2012 at 13:53
  • $\begingroup$ @BjornWesen Yes, the sort of answer I was hoping for was either (1) The question is meaningless and cannot be answered because <.....>, or (2) The full Hamiltonian has been explicitly computed for a scalar theory in (1+1)dimensions with interaction term <...> $\endgroup$
    – twistor59
    Feb 2, 2012 at 16:16

2 Answers 2


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.

[The above information was extracted from the Section ''Relativistic QFT at finite times?'' of Chapter B3: ''Basics on quantum fields'' of my theoretical physics FAQ at http://arnold-neumaier.at/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''*
  • $\begingroup$ Dear Arnold Neumaier, For your information, Physics.SE has a policy that it is OK to cite oneself, but it should be mentioned explicitly in each answer, where one does it. $\endgroup$
    – Qmechanic
    Mar 4, 2012 at 0:34
  • $\begingroup$ @Qmechanic: I read the policy. Does it mean one has to qualify such links as bein ones own? Would it be sufficient if I refer to ''my the theoretical physics FAQ'' instead of ''A theoretical physics FAQ''? I self-quoted myself elsewhere in the last few days but will in the future mark my own links. $\endgroup$ Mar 4, 2012 at 11:09
  • $\begingroup$ Yeah, I think that should be sufficient. $\endgroup$
    – Qmechanic
    Mar 4, 2012 at 11:42

Radiation of a classical current is a QED example what is going on at finite times. The occupation numbers evolve, that's it.

As well, an electron in a known external field is also has a finite time dynamics.

In general, it is still the occupation number evolutions, maybe non complete interference in case of a periodical external source, etc.

  • $\begingroup$ For the radiation example, I guess you mean soft photon production...? I was thinking more along the lines of what happens at finite times in a scattering problem. In this case, it wouldn't be an occupation number evolution, because I don't think "numbers" would exist meaningfully. But then I may have this all wrong! That's why I'm asking. $\endgroup$
    – twistor59
    Feb 2, 2012 at 15:05
  • $\begingroup$ The occupation numbers are well defined at all times; another thing they evolve rather than stay stable. As a matter of fact, the whole QFT is about occupation number evolutions. Unfortunately there is much fog due to renormalizations and the corresponding "bare particle physics", but in the end everything is boiled down to the occupation numbers of physical particles. $\endgroup$ Feb 2, 2012 at 15:13
  • $\begingroup$ I can see how, in your radiation example, we can talk about evolving occupation numbers, but let's say we scatter a pair of electrons - as they come together and interact, again the soft Bremsstrahlung photon production hay have evolving photon occupation numbers, but what about the "interesting" evolution of the system state which will take place (and give rise to a scattering behaviour which we could have computed with perturbation theory) ? Is the number of physical electrons in this example always 2 ? $\endgroup$
    – twistor59
    Feb 2, 2012 at 15:42
  • $\begingroup$ The total charge is always 2, the occupation numbers of pairs may be present if the energy is sufficient, or zero, if insufficient. The time-dependent wave function $\Psi (t)$ still represents the probability to find electrons in some states. $\endgroup$ Feb 2, 2012 at 15:48

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