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.
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.
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.