Relativistic QFT at finite times

Although many time-dependent observable consequences of QED can be deduced in a nonrigorous way in the Schwinger-Keldysh (= closed time path, CPT) formalism, there is at present no rigorous relativistic quantum field theory at finite times in 4 dimensions.

In lower dimensions, 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

In particular, (6.1.6), (6.1.11) and Theorem 6.1.3 are relevant.

Unfortunately, no Wightman functions have been constructed so far for interacting 4D quantum field theorys; see the FAQ entry on 'Is there a rigorous interacting QFT in 4 dimensions?'.

However, the functional integration measure of Euclidean QED is known to exist perturbatively at all orders (Tomonaga, Schwinger and Feynman got the Nobel prize for this), though a nonperturbative construction is still missing. By analytic continuation as in the Osterwalder-Schrader reconstruction theorem , one should be able to obtain a perturbatively valid Hamiltonian for QED (cf. Theorem 6.1.3 in Glimm and Jaffe).


Current 4D QFT in its usual textbook form is based on perturbation theory for free (i.e., asymptotic in- and out-) states; therefore it gives only predictions that relate the in- and out-states. (But see below for the CTP techniques, which are not of the standard perturbative form and give far mor information.) This information is contained in the S-matrix elements. From the S-matrix, one can the derive further information, e.g., about bound state energies as poles.

In nonrelativistic QM, one has a well-defined dynamics at finite times, given by the Schroedinger equation. This dynamics can be recast in terms of Feynman path integrals. Unfortunately, this does not extend to the relativistic case.

The problem with relativistic path integrals is that they are formal objects without a clear numerical meaning: whatever one tries to compute with them turns out to be infinite.

Only selected objects derived from path integrals can be given meaning by means of the renormalization procedure. The books show how to give meaning to S-matrix elements between asymptotic in and out states.

The (Minkowski space) path integral is ill-defined as a number, but, after regularization, well-defined as a formal power series in hbar (the latter is often set to 1 to simplify typography, but this make things more difficult to grasp). The Legendre transform of the logarithm is then also defined as a formal power series, and by letting the coupling constants depend on the regularization parameter eps (or Lambda), one can take the limit eps to 0 (or Lambda to infty) to get the effective action, again as a formal power series.

From there, one can get the S-matrix, again as a formal power series. FOR QED, the first few terms give highly accurate approximations; for other QFTs, partial resumming of these series give acceptable results in agreement with experiment.

Expanding objects of interest as power series is the hallmark of the so-called perturbative approach. In contrast, nonperturbative methods try to give meaning to the actual sums, though no one succeeded so far. Indeed, convergence questions are open, although it is generally believed that (as most series coming from a saddle point expansion of an integral) the series is only asymptotic. See the section on 'Summing divergent series' in this FAQ.

But it is unknown how to give rigorous meaning (i.e., infrared and ultraviolet finite, renormalization scheme independent properties) to, say, quantum electrodynamics states at finite t and their propagation in time.

People don't even know what an initial state should be in a 4D relativistic QFT (i.e., from which space to take the states at finite t); so how can they know how to propagate it...

Thus the standard textbook theory gives an S-matrix (or rather an asymptotic series for it) but not a dynamics at finite times.


This does not mean that there is no dynamical reality underlying 4D relativistic QFT. It only means that no one has been able to find a working, rogorous and logically consistent framework for it.

Probably people working in QFT imagine something like a state evolution in some unspecified Hilbert space underlying their formalism. After all, this is how one justifies that the functional integral works.

Indeed, one can compute - nonrigorously, in renormalized perturbation theory - many time-dependent things, namely via the Schwinger-Keldysh (or closed time path = CTP) formalism; see, e.g., here. For example,

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