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

J. Glimm and A Jaffe, Quantum Physics: A Functional Integral Point of View, Springer, Berlin 1987.

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,

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.

E. Calzetta and B. L. Hu, Nonequilibrium Quantum Field Theory, Cambridge Univ. Press, Cambridge 2008.

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, <\UL> 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
It is usually applied in the front form (cf. the FAQ entry on 'Forms of relativistic dynamics'), as the many references to this paper in scholar.google (search for: author:glazek author:wilson) show. See also hep-ph/0009071.
S.D. Glazek and K.G. Wilson, Phys. Rev. D 49 (1994), 4214-4218.
gives a proof that, for renormalizable theories without massless particles, similarity renormalization results in a perturbatively finite Hamiltonian at all orders of perturbation theory.

A different, more explicit renormalized Hamiltonian framework is given for quantum electrodynamics (but with a small photon mass to avoid infrared problems) in the instant form in a book by Stefanovich,

E.V. Stefanovich, Relativistic quantum dynamics, 2015

Apart from the photon mass, his perturbative construction appears to be equivalent with QED on the the renormalized level, and provides on the perturbative level a representation of the Poincare group in the instant form, and therefore a dynamical interpretation. To be equivalent with observation he needs to start with standard quantum field theory (QED). But then he transforms it in perturbation theory - by means of a dressing transformation that breaks manifest Lorentz invariance - into a representation in which particles and antiparticles figure. The result is therefore observer-dependent, violating Lorentz invariance at any finite order of perturbation theory, with the result that there appear artifacts with seeming violation of causality. (Thus many of his foundational views voiced here are far from being justifed.)

Both similarity renormalization and Stefanovich renormalization give infrared-regularized QED a dynamical content at every order of perturbation theory by providing approximate but finite, UV renormalized Hamiltonians to each order that are asymptotic to a formal Hamiltonian that acts formally as the generator of translations in the Poincare group. Convergence questions are not discussed. Also, the infrared divergences are not addressed but must be removed by assuming a tiny photon mass, thus spoiling gauge invariance.

Note that both schemes are related to renormalization in Dyson's intermediate representation, cf.

TS Walhout, Similarity renormalization, Hamiltonian flow equations, and Dyson's intermediate representation, Phys. Rev. D 59, 065009 (1999)
and references there.




While Dyson's argument (see the FAQ entry on 'Summing divergent series') implies that it is not reasonable to demanding a convergent S-matrix expansion, the limit Hamiltonian in these approaches could still be convergent. If this could be shown and the massless limit for the photon performed, it would amount to an existence proof of quantum electrodynamics.

In general, the correct Hamiltonian is

H = lim_{Lambda to inf} H(Lambda,g(Lambda,E,q_phys)),
where H(Lambda,g) is the canonical Hamiltonian with cutoff Lambda and parameter vector g (containing the so-called bare mass, charge or coupling constant, and field renormalization factor), and g(Lambda,E,q_phys) is the cutoff-dependent parameter vector as determined by renormalization conditions at energy scale E, which relate g to a set of physical parameters q_phys (consisting, in case of QED of the physical electron mass m, the physical electron charge e, or, equivalently, the observed value of the fine structure constant alpha=e^2/4pi).

This limit probably exists, at least for renormalizable, asymptotically free theories, at least in 1D and 2D field theory, where this can be proved in certain cases. In 3D and 4D, one probably needs also a Lambda-dependent inner product defining the Hilbert space to ensure that one ends up in the right representation, and Lambda-dependent wave functions to ensure that the limiting renormalized wave functions remain bounded in the limiting renormalized inner product.

The consistency problem in a Hamiltonian approach to quantum field theory is precisely to show that this limit indeed exists.




The missing consistent dynamical theory in 4D relativistic QFT may also have consequences for the foundations of quantum mechanics. Clearly, measurements happen in finite time, hence cannot be described at present in a fundamental way (i.e., beyond the nonrelativistic QM approximation). Thus foundational studies based on nonrelativistic QM are naturally incomplete. This implies that it is quite possible that a solution of the unresolved issues in relativistic QFT are related to the unresolved issues in quantum measurement theory.


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Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ