# Causal Perturbation Theory

Relativistic quantum field theory is notorious for the occurrence of divergent expressions that must be renormalized by recipes that on first sight sound very arbitrary and counter intuitive. But it doesn’t have to be this way….

The basic building blocks of any quantum field theories are free fields that serve to define irreducible representations of the Poincare group with physically correct mass and spin.

**Traditional approaches.** Interactions are then introduced by means of nonquadratic terms in the so-called action (which for a free field is quadratic). These terms are the source of all troubles. Two common approaches start with the action. Canonical quantization works with physical, distribution-valued fields satisfying ill-defined nonlinear field equations derived from the action principle. Path integral quantization uses path integrals, whose definition cannot be made rigorous at present. This lack of mathematical rigor shows in the occurrence of logical difficulties in the derivation of the formulas, although these ultimately lead to good, renormalized formulas whose predictions agree with experiment.

**Avoiding ultraviolet infinities.** Causal perturbation theory is a modern, mathematically rigorous version of old recipes that avoids the worst of these problems. Pioneered in a 1973 paper by Epstein and Glaser and made popular in two books by Scharf (‘*Finite Quantum Electrodynamics, 2nd ed. 1995*‘ and ‘*Quantum Gauge Theories – A True Ghost Story, 2001*‘), causal perturbation theory is an approach to perturbative quantum field theory free of the ultraviolet (UV) infinities that characterize other approaches to quantum field theory. (Epstein and Glaser built on earlier work by Bogoliubov and Shirkov 1955, which is not fully rigorous as it depends on a mathematically ill-defined notion of time ordering. The contribution of Epstein and Glaser consisted in making time ordering mathematically rigorous by distribution splitting techniques borrowed from microlocal analysis.)

In contrast to the more standard approaches mentioned before, causal perturbation theory works throughout with free fields only, and nowhere introduces nonphysical entities (such as cutoffs, bare coupling constants, bare particles or virtual particles) typical of the older, action based approaches.

Divergent expressions are avoided by never multiplying two distributions whose product is not defined. This is a prerequisite for mathematical consistence, that is simply ignored in the other approaches. The mathematical correct procedure is determined by microlocal theory, a mathematically well-known technique for the analysis of linear partial differential equations. Microlocal theory tells when the product of two distributions is well-defined. From an analysis of these conditions (which in terms of physics is roughly what comes under the heading of dispersion relations, but expressed in precise mathematical terms) one can tell precisely which formal manipulations of distributions are mathematically valid.

As a consequence, causal perturbation theory is mathematically well-defined, and falls short of a rigorous construction of quantum field theories only in that the perturbative series obtained is asymptotic only, and that the infrared (IR) limit is not well understood. This means that infrared problems and convergence issues are handled by causal perturbation theory not better (but also not worse) than in the more traditional approaches.

**Axioms for causal perturbation theory.** In causal perturbation theory, all physics is introduced axiomatically. The properties required axiomatically characterize a successful relativistic quantum field theory, and are used as a rigorous starting point for all subsequent deductions. There is neither an ambiguity nor a contradiction – everything is determined by the rules of logic in the same way as for any mathematical construction of unique objects defined by axioms (such as the real numbers).

The axioms give conditions on an operator ##S(g)## characterizing the theory, a version of the S-matrix weighted by a test function ##g## with compact support. The limit where ##g## tends to the constant function ##1## defines the physical S-matrix. There are five axioms:

(P) Poincare covariance: ##TS(g)T^*=S(Tg)## for every Poincare transformation ##T## acting in the standard way (unitarily on states and nonunitarily on test functions).

(C) Causality: ##S(g+h)=S(g)S(h)## if ##g## and ##h## have causally disjoint support.

(U) Unitarity: ##S(g)## acts unitarily on the physical subspace of gauge invariant states.

(V) Vacuum stability: ##S(g)## maps the vacuum state to itself.

(S) Single particle stability: ##S(g)## maps each ##1##-particle state space unitarily into itself.

Together with a particular form assumed for the interaction (which has the same form as the traditional nonquadratic term in the action, but a different meaning), these axioms completely determine – up to a small number of physical constants (in QED the electron mass and charge) – the expansion of ##S(g)## into homogeneous forms of increasing degree in ##g##. This series is constructed as a formal series by causal perturbation theory. For a fully rigorous solution – a problem which for interacting 4-dimensional relativistic quantum field theories is open -, ##S(g)## should be an operator-valued function of ##g## rather than only a formal series. From ##S(g)## one can find the physical fields by functional differentiation, a procedure going back to Bogoliubov and Shirkov 1959.

In causal perturbation theory,there is no UV cutoff; all UV problems are handled by mathematically safe distribution splitting. That ##g(x)## must have compact support restricts the construction to finite volume. Letting ##g(x)## approach ##1## is the infinite volume limit, whose existence or properties are not analyzed in causal perturbation theory. The latter has for gauge theories the usual IR problems that must be handled by coherent state techniques.

**Textbook treatment.** Scharf’s books (which are self-contained, though of course partially based on the work of others) show that the causal perturbation techniques work for realistic quantum field theories including QED, nonabelian gauge theories and theories with broken symmetries.

Scharf’s books are mathematically rigorous throughout. He nowhere uses mathematically ill-defined formulas, but works throughout with mathematically well-defined distributions using the microlocal conditions appropriate to the behavior of the Green’s functions. These enable him to solve recursively mathematically well-defined equations for the S-matrix by a formal power series ansatz, which is sufficient to obtain the traditional results.

Scharf’s construction of QED (as far as it goes) is mathematically impeccable. Indeed, it can be understood as a noncommutative analogue of the construction of the exponential function as a formal power series. The only failure of the analogy is that in the latter case, convergence can be proved, while in the former case, the series can be asymptotic only (by an argument of Dyson), and it is unknown how to modify the construction to obtain an operator-valued functional ##S(g)##.

**The heuristic relation to traditional approaches.** On a heuristic level, ##S(g)## is the mathematically rigorous version of the time-ordered exponential

$$S(g)=Texp\Big(\int dx g(x)V(x)\Big),$$

where ##V(x)## is the physical interaction. With this informal recipe and other heuristic considerations one can also motivate the validity of all axioms in the traditional settings. This is the ultimate reason why the causal approach gives – though through very different route – the same final results as the traditional approaches based on the mathematically somewhat ill-defined time-ordered exponential. Only the causal perturbation theory route can claim logical coherence, due to its mathematical rigor. (For those familiar with different traditional renormalization methods, causal perturbation theory is just the rigorous version of BPHZ renormalization. The ”just” of course makes all the difference in quality.)

[Sources: The above is an expanded summary of material (here, here, and here) from

PhysicsOverflow, where more detailed references can be found. See also the PF thread on Rigorous Quantum Field Theory, though it goes off in a somewhat different direction.]

Here is an open source survey of causal perturbation theory from 2009.

Nice first entry @A. Neumaier!

Help I have been banned!! Apparently for spamming.

Please unblock me somebody.

When you write “As a consequence, causal perturbation theory is mathematically well-defined, and falls short of a rigorous construction of quantum field theories only in that the perturbative series obtained is asymptotic only, and that the infrared limit is not well understood.”, do you mean that this method constructs relativistic QED without a UV cut-off in finite volume?

1) Regarding (P) Poincare’ covariance, I suppose you meant to write [tex]U(L) S(g) U^{dagger} (L) = S(Lg) ,[/tex] where [itex]L[/itex] (an element of Poincare’ group) need not be unitary. [itex]g(x)[/itex] is an ordinary function and transforms according to [tex]g(x) to L g(x) = g(L^{-1}x) .[/tex]

2) Why do you attribute this framework to Epstein and Glaser? I far as I know, this formalism was first introduced by Bogoliubov and Shirkov in their 1955 paper and then included it in their classic text on QFT in the 1957 edition (translated to English in 1959). In my 1976 edition of the book, the formalism is explained in Ch.III, Sec.17&18.

@atyy: There is no UV cutoff; all UV problems are handled by mathematically safe distribution splitting. That g(x) must have compact support restricts the construction to finite volume. Letting g(x) approach 1 is the infinite volume limit, whose existence or properties are not analyzed in causal perturbation theory. The latter has for gauge theories the usual IR problems that must be handled by coherent state techniques.

QED is not constructed in finite volume as S(g) is found only as a formal series and not as an operator.

@samalkhaiat: ad 1) – indeed, my shorthand notation was supposed to mean this.

ad 2) Epstein and Glaser built on earlier work by Bogoliubov and Shirkov 1955, which is not fully rigorous as it depends on a mathematically ill-defined notion of time ordering. The contribution of Epstein and Glaser consisted in making time ordering rigorous by distribution splitting techniques borrowed from microlocal analysis.

Thanks to both of you for your comments. I updated my text to better reflect all this.

Great article!

Aren’t the infrared problems at least FAPP solved by using the correct asymptotic states, which are rather coherent states than plane waves and finally equivalent to the usual soft-photon resummation techniques of the traditional approach (Bloch, Nordsieck, et al)? Shouldn’t smoothed field operators cure both the UV and IR problems within the perturbative approach? I’ve to read Scharf’s books in more detail, but it’s of course simpler to just ask here in the forum :-).

@vanhees71: Scharf’s books contain very little (QED book, Section 3.11: Adiabatic limit) about how to handle the IR problem. He does not use coherent states, although the latter is the right way to proceed. But at present, coherent state arguments in QED, while sufficient FAPP, are not mathematically rigorous. There is some rigorous IR material in the literature (under the heading ”infraparticle”), but it hasn’t been incorporated so far into the causal approach.

Smoothing can be done in different ways, and traditional cutoffs are one way of doing it – the question is always how to undo the smoothing at the end to recover covariant results. The causal approach is throughout manifestly covariant; it smoothes the IR in the most general (and hence covariant) way by introducing g(x), but handles the UV part by microlocal analysis. The latter was developed to rigorously analyze classical PDEs, and it is mathematically natural to expect that one would need these techniques also for a rigorous quantum version.

I see. What I had in mind are the (of course no mathematically rigorous) papers

Kulish, P.P., Faddeev, L.D.: Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4, 745, 1970

[URL]http://dx.doi.org/10.1007/BF01066485[/URL]

Kibble, T. W. B.: Coherent Soft‐Photon States and Infrared Divergences. I. Classical Currents, Jour. Math. Phys. 9, 315, 1968

[URL]http://dx.doi.org/10.1063/1.1664582[/URL]

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. II. Mass-Shell Singularities of Green’s Functions, Phys. Rev. 173, 1527–1535, 1968

[URL]http://dx.doi.org/10.1103/PhysRev.173.1527[/URL]

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. III. Asymptotic States and Reduction Formulas, Phys. Rev. 174, 1882–1901, 1968

[URL]http://dx.doi.org/10.1103/PhysRev.174.1882[/URL]

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. IV. The Scattering Operator, Phys. Rev. 175, 1624, 1968

[URL]http://dx.doi.org/10.1103/PhysRev.175.1624[/URL]

Yes. The Kulish-Faddeev paper you cited counts as the definite paper on the subject. It settles the QED infrared problems on the nonrigorous level. Partial rigorous results along these lines are in

D. Zwanziger, Physical states in quantum electrodynamics, Phys. Rev. D 14 (1976), 2570-2589.

and in papers by [URL=’http://arxiv.org/abs/hep-th/0003087′]Bagan et al.[/URL] and [URL=’http://http://arxiv.org/abs/hep-th/0411095′]Steinmann[/URL], but no synthesis with the causal approach has been tried, as far as I know.

If the causal perturbation construction is only a formal power series, is there any way to transition from that to something with physical meaning?

In the informal viewpoint, we accept that QED is only an effective theory and has a cutoff, so our starting theory is, say, lattice QED. Then we argue that if we take a Wilsonian viewpoint and run the renormalization flow down, we recover the traditionally constructed power series plus corrections suppressed in powers of the cutoff. Here the starting theory rigourously exists, since it is quantum mechanics. It has a UV cutoff and exists only in finite volume, but physically that should be ok since our experiments are low energy and always in finite volume. The part of this framework that is not rigourous is the renormalization flow to low energies. The informal thinking is that since the Wilsonian framework makes physical sense, it should be possible for it to be made rigourous one day.

Can causal perturbation theory fit within the Wilsonian framework of QED as an effective field theory?

Or does causal perturbation theory need to come from a truly Lorentz invariant theory without a UV cutoff?

Causal perturbation theory reproduces the perturbative results of standard renormalized perturbation theory, which also produces only a series expansion of the S-matrix elements.

Unfortunately, in 4D we have no method at all to construct more than a power series (or another uncontrolled approximation like a lattice simulation). One gets approximate numbers with physical meaning by the standard means – neglecting higher order terms, or numerical versions of resummation techniques such as Pade or Borel summation, hoping that these will result in good approximations.

Since causal perturbation theory is manifestly covariant, it cannot be viewed in terms of a cutoff, and hence not in terms of Wilson’s renormalization (semi)group framework. However, as described [URL=’http://www.physicsoverflow.org/20506′]here[/URL], there is still a renormalization group (a true group, unlike Wilson’s). It describes (not the effect of changing the nonexistent cutoff) but the effect of a change of parameters used in the specification of the renormalization conditions. This is just a group of exact reparameterizations of the same family of QFTs.

There are rigourously constructed QFTs in 2 and 3 spacetime dimensions. Does causal perturbation theory construct a theory, or become a mathematically meaningful approximation when applied to such theories?

In 2D and 3D, causal perturbation theory only constructs the series expansion of the then rigorously existing operator S(g), but asserts nothing about the latter’s existence. As such it may serve as an alternative covariant way to approximately compute things that are known to exist by the traditional (noncovariant, lattice-based) existence proofs together with Haag-Ruelle scattering theory.

In the asymptotic series I am familiar with, say, the ones that come up in Stirling’s approximation, the error is on the order of the first term omitted.

Let’s suppose we have rigourously constructed a relativistic QFT (say in 2D or 3D), then, if I understand correctly, the causal perturbation theory is a correct way to construct the power series. Is the error also on the order of the first term omitted?

Generically yes. In truncating any asymptotic series $sum_k a_k phi_k(x)$ in $x$ with nicely behaving $phi_k(x)$ and coefficients of order 1, the error is of the order of $the first neglected $phi_k(x)$. But the error can be much bigger than the first neglected term itself, if the coefficient $a_k$ happens to be tiny. So that the error is of the order of the first term omitted is just a rule of thumb.

With this understanding it is valid for the series calculated by causal perturbation theory (which is identical with the series calculated by other good renormalization methods).

Going back to QED in 3+1D, the match of the traditional renormalization to experimental data suggests that there is a good mathematical theory in which the outcome of traditional renormalization makes sense. It seems there are two heuristics on how to proceed.

1) the Wilsonian view, in which we can use lattice theory at some high energy, and the Lorentz invariance is not necessarily exact, but only a very good low energy approximation.

2) the causal perturbation view – we still have no rigourous theory, but a rigourous formal series that is (like traditional renormalization) nonetheless indicated by experiment to be mathematically completable in a Lorentz invariant way. At present Yang-Mills is the best candidate for constructing a rigourous 3+1D relativistic QFT, but since the experiemental match in QED is so good, what are the potential relativistic UV completions of QED? Would causal perturbation theory be consistent with either (A) asymptotic safety or with (B) the introduction of new degrees of freedom to complete the theory?

I believe that QED exists as a rigorous and mathematically complete theory, and that causal perturbation theory provides its correct perturbative expansion around the free theory. There is no proof that would establish the contrary. The arguments that would suggest its incompleteness are based on logically unjustified inference from uncontrolled approximations without any force.

But the perturbative expansion is not the whole story since there must be modifications in the IR, due to the fact that the physical electron is an infraparticle only.

Ha, ha, brave statement. So that corresponds to the informal notion of asymptotic safety, and from what you have explained, I do see that causal perturbation theory strongly suggests that to be the case.

But just to explore possibilities – if there were a rigourous 4D QFT whose fundamental degrees of freedom are different from QED’s – but from which QED emerged at some lower energy. Would it be from the rigourous point of view justified to apply causal perturbation theory to such a theory? Or is it only strictly applicable in the case of asymptotic safety?

I’ve been trying to follow your discussion, with something (significantly) less than complete understanding. But I’d appreciate it if you could expand on this comment. Are you saying that the physical electron itself does not actually exist other than as a focal point for it’s cloud of virtual photons?