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
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.
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  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 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.