# Introduction to 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 counterintuitive. But it doesn’t have to be this way…

Table of Contents

**Traditional approaches**

The starting point of any quantum field theory is free fields that serve to define irreducible representations of the Poincare group with physically correct mass and spin. 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 the 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 of making time ordering mathematically rigorous by distribution splitting techniques borrowed from the microlocal analysis.)

In contrast to the more standard approaches mentioned before, causal perturbation theory works throughout with free asymptotic 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 consistency, 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.

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

**Axioms for causal quantum field theory**

By design, the causal approach achieves UV renormalization without any regularization, since it uses at every stage the correct covariant singular distributions. But to be able to work with free fields it regularizes the physical S-matrix in the IR by means of test functions with compact support (rather than arbitrary smooth ones), which amounts to switching off the interaction at large distances. The adiabatic limit restores long-distance interactions.

This is fully analogous to truncating short-range potentials in quantum mechanics in order to be able to use free particles at large negative and positive times to obtain an S-matrix without any limit. In quantum mechanics, the adiabatic limit restores the original potential. The mathematically proper treatment has to introduce a Hilbert space of asymptotic states and a Möller operator that transforms from infinite time to finite time. This makes the whole procedure less intuitive and requires more machinery from functional analysis, described rigorously in the 4 mathematical physics volumes of Reed and Simon.

In causal quantum field 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 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 **adiabatic limit**, i.e., the weak limit where ##g## tends to the constant function ##1## (Scharf 1995, Section 3.11) defines the physical S-matrix. The physical fields satisfying **microcausality** (i.e., commutation or anticommutation of smeared field operators with causally disjoint support) can be found from ##S(g)## by functional differentiation (Scharf 1995, Section 4.9), a procedure going back to Bogoliubov and Shirkov 1959. It is important that the adiabatic limit ##g\to 1## is not needed for the construction of the local field operators and hence for the perturbative construction of the quantum field theory in terms of formally local operators in a Hilbert space. The adiabatic limit, on the perturbative level the only limit appearing in the causal approach, is needed only for the recovery of the IR regime, including the physical S-matrix.

The operator-valued function ##S(g)## of ##g## should satisfy the following 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). (Scharf 1995, (3.1.6) and (3.1.9))

(C) **Causality**: ##S(g+h)=S(g)S(h)## if ##g## and ##h## have causally disjoint support. (Scharf 1995, (3.1.23))

(U) **Unitarity**: ##S(g)## acts unitarily on the physical subspace of gauge-invariant states. (Scharf 1995, (3.1.4) and (4.7.1))

(V) **Vacuum stability**: In the adiabatic limit ##g\to 1##, the S-matrix ##S(g)## maps the vacuum state to itself. (Scharf 1989, (3.3.2) and (3.6.31) assumed this for all ##g##, which is too strong. Scharf 1995 (4.1.32) has the limit)

(S) **Single-particle stability**: In the adiabatic limit ##g\to 1##, the S-matrix ##S(g)## maps each ##1##-particle state-space unitarily into itself. (Scharf 1989, (3.6.26) and (3.7.36) assumed this for all ##g##, which is too strong. Scharf 1995 does not mention the limit explicitly but gets the old results in the adiabatic limit in Section 3.11.)

These axioms (not clearly emphasized in Scharf’s books, hence the detailed references) define non perturbatively what it means to have constructed a local covariant quantum field theory. To define particular interacting local quantum field theories such as QED, one just has to require a particular form for the first-order approximation of ##S(g)##. In cases where no bound states exist, which includes QED, this form (Scharf 1995, (3.3.1)) happens to be identical to that of the traditional nonquadratic term in the action, but it has a completely different meaning. Nothing in causal quantum field theory ever makes any use of Lagrangian formalism or Lagrangian intuition. No action principle is visible in causal quantum field theory; it is not even clear how one should formulate the notion of an action! (The widely used action-based functional integration approach has not been made rigorous.)

Instead, causal quantum field theory starts with a collection of well-informed axioms for the parameterized S-matrix (something not at all figuring in the Lagrangian approach) and exploits in causal perturbation theory the relations that follow from a formal expansion of the solution of these equations around a free quantum field theory. The latter can be constructed directly from irreducible representations of the Poincare group, as in Weinberg”s book (where Lagrangians are introduced much later than free fields).

Models for interacting local quantum field theory in 2 or 3 spacetime dimensions) are known for a long time. Unfortunately, however, models proving that QED (or another interacting local quantum field theory in 4 spacetime dimensions) exists have not yet been constructed. On the other hand, the above axioms are consistent with everything known nonrigorously about local quantum field theories. No arguments are known proving rigorously that such models cannot exist in 4 dimensions. Finding a fully rigorous construction for an interacting 4-dimensional local quantum field theory or proving that it cannot exist is therefore a widely open problem. My bet is that a rigorous construction of QED will be found one day.

**Haag’s theorem** states that the representation of the Poincare group of an interacting local quantum field theory cannot be unitarily equivalent to the representation on Fock spaces constructed in all textbooks on quantum field theory for free (or quasifree) fields. It is usually loosely expressed by saying that ”the interaction picture does not exist”. One may also express it by saying that a pure Fock space approach to interacting local quantum field theory is doomed to run into divergences.

**Perturbative construction**

Causal perturbation theory is the name under which a perturbative solution of models of interacting local quantum field theories, including QED, is constructed. In place of an operator-valued function ##S(g)##, only a formal series in ##g## satisfying the five axioms is constructed. This means that truncation at any order produces results that satisfy the axioms (and hence their consequences, such as microcausality) up to terms of the first neglected order in ##g##. In particular, microcausality is only approximate at each order (and only in the sense of formal power series if all orders are considered). Therefore Haag’s theorem does not apply and the whole construction works in a Fock space.

For QED, order 6 is sufficient to get errors smaller than the current experimental resolution. Thus perturbative QED, constructed to order ##p## with ##p=6## or slightly larger is a nearly local QFT sufficient for practical applications. On the other hand, all these formal series expansions are asymptotic only, diverging as the order increases beyond some threshold. For QED with the physical parameters, the threshold is expected to be roughly at the inverse ##\approx 137## of the fine structure constant, far beyond what will ever be experimentally relevant.

The axioms and the first-order approximation assumed 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 like a formal series by causal perturbation theory.

In causal perturbation theory, all UV problems are handled by mathematically safe distribution splitting. In particular, there is no UV cutoff or other regulator; the construction proceeds manifestly covariant from the start. That ##g(x)## must-have compact support allows one to restrict the construction to a finite but arbitrarily large spacetime 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.

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 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 analog 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)##.

Not a single argument is known that would indicate that interactive causal QFTs in 4 spacetime dimensions do not exist. To construct them in the causal approach, a promising way might be to find a summation scheme for which one can prove that the result satisfies the axioms non perturbatively. This is a nontrivial and unsolved step but not something that looks completely hopeless. Borel summation is not sufficient because of the appearance of renormalon contributions. The most promising approach is via resurgent transseries, an approach much more powerful than Borel summation.

The recent book

- Michael Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory, Birkhäuser 2019

treats causal perturbation theory in a different way than Scharf, using off-shell deformation quantization rather than Fock space as the starting point, which makes it closer to a functional integration point of view. In the preface, the author writes among others:

- The aim of this book is to give a logically satisfactory route from the fundamental principles to the concrete applications of pQFT, which is well intelligible for students in mathematical physics on the master or Ph.D. level. This book is mainly written for the latter; it is made to be used as the basis for an introduction to pQFT in a graduate-level course.
- This formalism is also well suited for practical computations, as is explained in Sect. 3.5 (“Techniques to renormalize in practice”) and by many examples and exercises.
- The observables are constructed as formal power series in the coupling constant and in ##\hbar##.
- This book yields a perturbative construction of the net of algebras of observables (“perturbative algebraic QFT”, Sect. 3.7), this net satisfies the Haag–Kastler axioms [93] of algebraic QFT, except that there is no suitable norm available on these formal power series.

**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 traditional settings. This is the ultimate reason why the causal approach gives – though through a 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.]

## The renormalization group in causal perturbation theory

The parameterization of the S-matrix of QED in terms of the physical mass and charge fixes the first-order term in and hence everything, so there is nothing to be renormalized.

But (Scharf 1995, p.260, p.271) there is some freedom in the construction. It can be used to introduce a redundant parameter at the cost of introducing running coupling constants and more complex formulas. Since the physical electron charge corresponds to a running charge at zero energy, the parameterization of the S-matrix in terms of the physical mass and charge corresponds to a conventional renormalization at zero photon mass.

The redundant parameter would have no effect in the nonperturbative solution. But since the expansion point is different, it leads to different results at each order of perturbation theory. These perturbative results are then related by finite renormalizations in terms of a Stückelberg-Petermann renormalization group. It expresses the charge appearing in the coupling constant – now no longer the experimental charge but running with the energy scale – in terms of the physical mass and charge.

Thus renormalization is always finite. In QED, where the free physical parameters have direct physical meaning and the perturbative series is very accurate, it is optional and not really useful.

Note that there are two very different renormalization groups that should not be mixed up. The first one by Wilson is important in nonequilibrium thermodynamics and for condensed but approximate descriptions in terms of composite fields. The second, older one by Stückelberg is the most important one in local quantum field theory and is not related to effective fields but to overparameterization.

- The Wilson renormalization group (actually only a semigroup, but the name has stuck) removes high energy degrees of freedom by repeated infinitesimal coarse-graining, expressed through the Wetterich renormalization group equation. It loses information, hence is not invertible and leads to approximate effective field theories.
- The Stückelberg-Petermann renormalization group (a true group) expresses the running coupling constant through the Callan-Symanzik renormalization group equation. This group is due to the existence of a redundant mass/energy parameter and has nothing to do with effective fields, as it does not change the contents of the theory, only the perturbative expansion.

The Stückelberg-Petermann renormalization group already appears in the quantum mechanics of an anharmonic oscillator when one wants to relate the perturbation series obtained by perturbing around Hamiltonians describing harmonic oscillators with different frequency. The frequency chosen is arbitrary and hence nonphysical; it is the analog of the renormalization scale in QFT.

## The Landau pole in causal perturbation theory

There is a widespread view that QED cannot rigorously exist because of problems with the Landau pole, a pole at an (extremely huge) energy in the partially resummed renormalized propagator computed in low order perturbation theory with explicit cutoff. This means that letting the cutoff move to infinity – a renormalization requirement to restore Poincare covariance – is impossible without going through a singularity where the physical content is lost. Similarly, there is a seemingly universal agreement (and numerical evidence even at very low energy) that lattice QED is trivial, i.e., letting the inverse spacing (the energy cutoff in a lattice theory) go to infinity in the interactive lattice version produces not an interacting continuum limit but a free one. This triviality of lattice QED means that the latter is not a suitable starting point for approximating QED.

In causal perturbation theory, the triviality problem disappears. The causal perturbation approach to QED is not susceptible to the usual triviality arguments, as it is manifestly covariant and works throughout without a cutoff or regulator. It works directly from the nonperturbative axioms without regularizing anything. Instead, it pays detailed attention to the singularity structure to avoid any potentially faulty operation. Partial resummation gives nontrivial partially nonperturbative results. Since there is no cutoff the standard triviality argument – namely that a Landau pole must be traversed by the cutoff – breaks down.

In Scharf’s treatment of QED in causal perturbation theory, the renormalization point is at zero mass, so there is no free parameter in the theory. However, by changing the renormalization recipe, one gets a family of reparameterizations depending on a mass scale. This mass scale has nothing to do with the cutoff; unlike in renormalization procedures with explicit cutoff, any y choice of the mass scale leads to a valid covariant perturbation series. The only effect is that experimental quantities at the desired energy scale are best approximated by choosing the mass scale at the desired energy. In this reparameterization, a Landau pole appears perturbatively at (extremely huge) energies. This renders the Landau pole experimentally irrelevant. Moreover, the very existence of the Landau pole seems to be a perturbative artifact. Indeed, Kallen-Lehmann-based resummation (which – unlike most other resummation methods – resums in a way respecting causality) eliminates the Landau pole.

Full Professor (Chair for Computational Mathematics) at the University of Vienna, Austria

[QUOTE=”philosophus, post: 5311954, member: 579984″]I do not understand the start of induction, unless some renormalization is hidden here to define the normal ordered power of the free fields[/QUOTE]

The whole construction happens in the asymptotic space, which even for an interacting theory is a Fock space, since asymptotic particles are free by definition. In Fock space, normal ordering is all that is required to render a polynomial, local operator meaningful. Thus you can start the induction with an arbitrary local polynomial in c/a operators.

Two restrictions come in to get the most desirable properties:

1. one wants the interaction to be covariant; then the interaction must be Lorentz invariant.

2. One wants to have only finitely many renormalization conditions. This requires that the degree of the interaction is small enough to the usual renormalizable terms.

In particular, for a scalar field theory you can take the interaction to be a linear combination of the normally ordered ##Phi^3## and ##Phi^4## term. If you want to preserve the discrete symmetry of the free theory, only the ##Phi^4## term qualifies.

However, the whole procedure makes perturbative sense also without these requirements.

In particular, for quantum field theory in curved space-time one sacrifices condition 1, with success; see work by [URL=’http://arxiv.org/find/math-ph/1/au:+Hollands_S/0/1/0/all/0/1′]Stefan Hollands.[/URL]

For quantum gravity one sacrifices condition 1, also with success; see, e.g.,, the living Review article

[URL=’http://relativity.livingreviews.org/About/authors.html#burgess.cliff’]Cliff P. Burgess[/URL][URL=’http://relativity.livingreviews.org/Articles/lrr-2004-5/’], Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory.[/URL]

“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).”

Can you, please, expand on this a little bit? How is e.g. the T[SUB]1[/SUB] operator-valued distribution in causal perturbation theory constructed for e.g. φ[SUP]3[/SUP] theory? I think I understand the inductive step from T[SUB]n-1[/SUB] to T[SUB]n[/SUB], but I do not understand the start of induction, unless some renormalization is hidden here to define the normal ordered power of the free fields?

[QUOTE=”Feeble Wonk, post: 5139513, member: 178236″]So, if I understand correctly, you are saying that the (bare) electron itself is a meaningless concept. According to the theory, the EM field is what is “real”, and the electron is just a localized description of the field. Is that accurate?[/QUOTE]

You don’t understand correctly. Both the electromagnetic field and the electron currents are real (measurable), photons are elementary excitations of the electromagnetic field, and electrons are elementary excitation of the electron current field, hence are as real. But they are not bare – the bare photons and electrons are meaningless auxiliary constructs that do not survive the renormalization limit.

[QUOTE=”vanhees71, post: 5140061, member: 260864″]I don’t know, what you mean by “real”. It’s a very confusing word[/QUOTE] Real in theoretical physics is what is gauge invariant, has a well-defined dynamics in time (since reality happens in time), approximates a real world situation (it is always to some extent an idealization).

Real things include renormalized gauge invariant operator products of fields, density matrices, and what is derived from it (e.g., S-matrix elements).

Nonreal things include all bare stuff, virtual particles, and wave functions. The latter since they are not invariant under global phase shifts; the corresponding rank 1 density matrices are invariant and hence are real (though often highly idealized) according to this definition.

I don’t know, what you mean by “real”. It’s a very confusing word, because it’s loaden with unsharply defined philosophical meanings. What’s “real” at the lab are detector responses to what we call “particles” and “fields”. Relativistic QFT is a mathematical framework to predict the corresponding transition probabilities from an initial state (usually two colliding protons, leptons, or heavy ions) to a final state (depending on what you like to meausre, e.g., the Higgs production). These transition probabilities are measured as scattering cross sections. In the theory they are given by the S-matrix elements. In this sense the S-matrix elements are “real”, because they can be checked by observations.

The bare electron is not observable by definition, because it doesn’t interact with anything, and thus also not with the detectors used to measure cross sections. What’s “real” are in some sense the asymptotic free electrons of perturbation theory, and they are quite complicated objects. In a very handwaving way you can say they are “bare electrons with their electromagnetic field around them”, i.e., a “bare electron” together with a “coherent photon state”.

For details, see the above cited paper by Kulish and Faddeev. For more details, see the series of papers by Kibble. A more traditional treatment can be found in Weinberg, Quantum Theory of Fields, vol. I.

[QUOTE=”A. Neumaier, post: 5138264, member: 293806″]The physical electron in QED exists, but it is (like an electron in experiments, but unlike a – nonexistent – bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood.

>>>>>>

(Described in terms of bare stuff – which is frequently done though it is meaningless imagery – the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)[/QUOTE]

So, if I understand correctly, you are saying that the (bare) electron itself is a meaningless concept. According to the theory, the EM field is what is “real”, and the electron is just a localized description of the field. Is that accurate?

[QUOTE=”A. Neumaier, post: 5138264, member: 293806″]The physical electron in QED exists, but it is (like an electron in experiments, but unlike a – nonexistent – bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood. In a nonperturbative construction of QED, the infraparticle structure must be explicitly represented. This means that one would have to do perturbation theory starting in place of the Hilbert space of a free field with a Hilbert space featuring infraparticles instead. People have been trying to build such a Hilbert space but nobody so far has married it with a perturbative construction in the spirit of causal perturbation theory – except in case of an electron in an external electromagnetic field, which is already quite technical.

(Described in terms of bare stuff – which is frequently done though it is meaningless imagery – the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)[/QUOTE]

Indeed, causal perturbation theory is just the rigorous version of BPHZ renormalization.

Just to add another point of view of (non-rigorous) renormalization. A cutoff is not renormalization but regularization, and you have to take the regularization to the “physical point” in order to achieve a Lorentz covariant S-matrix. In the usual framework, cutoff-regularization is not a wise choice, because it complicates the calculations. What’s common to all regularization schemes is that you somehow have to introduce an energy-momentum scale.

The breakthrough in some sense was the discovery of dimensional regularization by ‘t Hooft during his PhD work (adviced by Veltman). There the physics of the scale is a bit obscured, because it comes in very formally by assuming that the coupling parameters in the Lagrangian keep their energy-momentum dimension at arbitrary space-time dimensions. E.g., to keep the em. coupling in QED dimensionless you have to introduce a scale factor ##mu## in the loop-integration measure. The great thing is that you have very easily defined “mass-independent” renormalization schemes (minimal subtraction or modified minimal subtraction, which is the standard use in QCD, leading to the definition of the physical scale parameter ##Lambda_{text{QCD}}## (see the Review of Particle Physics).

Another point of view is not to regularize at all but to do the subtractions directly at the level of the loop-integral’s integrands (BPHZ renormalization). There the introduction of a scale, particularly in the case of theories where massless degrees of freedom are involved, becomes very natural: You cannot subtract at 0 external momenta of the Feynman diagrams, because you hit the cuts of the proper vertex functions in the complex energy plane. So you have to subtract at some space-like momentum, which introduces the energy-momentum scale. Also you should not choose an on-shell renormalization scheme, because this introduces artificial IR problems if massless degrees of freedom are present.

The RG can be formulated (particularly simple for mass-independent renormalization schemes) from the invariance of the physical properties (i.e., S-matrix elements) from the choice of the energy-momentum scale (of course only approximately in the perturbative sense). For an RG treatment of the simple ##phi^4## model in the BPHZ scheme see

[URL]http://fias.uni-frankfurt.de/~hees/publ/lect.pdf[/URL]

The physical electron in QED exists, but it is (like an electron in experiments, but unlike a – nonexistent – bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood. In a nonperturbative construction of QED, the infraparticle structure must be explicitly represented. This means that one would have to do perturbation theory starting in place of the Hilbert space of a free field with a Hilbert space featuring infraparticles instead. People have been trying to build such a Hilbert space but nobody so far has married it with a perturbative construction in the spirit of causal perturbation theory – except in case of an electron in an external electromagnetic field, which is already quite technical.

(Described in terms of bare stuff – which is frequently done though it is meaningless imagery – the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)

Causal perturbation theory can be applied to any quantum field theory for which you can write down a free field theory and a first order term for the S-matrix consistent with causality. Everything else is then determined. This is completely independent of asymptotic safety. The standard model is a theory of which QED is the low energy limit, and yes, you can apply causal perturbation theory. Any other fundamental theory if it satisfies the two criteria just mentioned works as well.

[QUOTE=”A. Neumaier, post: 5138034, member: 293806″]

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.[/QUOTE]

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?

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 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.

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?

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).

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?

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.

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?

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.

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?

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.

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]

[USER=260864]@vanhees71[/USER]: 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.

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 :-).

[USER=123698]@atyy[/USER]: 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.

[USER=35381]@samalkhaiat[/USER]: 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.

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.

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?

Please unblock me somebody.

Help I have been banned!! Apparently for spamming.

Nice first entry @A. Neumaier!