# Action-based quantum field theory and causal perturbation theory

What is the difference between action-based quantum field theory and causal perturbation theory?

The former can be found in any textbook on quantum field theory, the latter is exposed for QEDĀin the bookĀ

• G. Scharf, Finite quantum electrodynamics, Springer, Berlin 1989. (My references are for this edition. There is also a second edition from 1995.)
Other quantum field theories including gauge theories are treated in the book
• G. Scharf, Quantum gauge theories: a true ghost story, Wiley, New York 2001.
More work on on causal perturbation theory can be found at http://scholar.google.com

The main difference is that causal perturbation theory, while producing the same renormalized perturbation series as the traditional action-based approach, is from the start free of divergences, since it only works with free fields (that serve to define irreducible representations of the Poincaré\$ group with physically correct mass and spin), and nowhere introduces nonphysical terminology (such as bare coupling constants, bare or virtual particles). Thus it 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.

In contrast, canonical quantization works with physical, distribution-valued fields satisfying ill-defined nonlinear field equations, and 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.

Scharf's book is 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.

Since Scharf has no action, all physics is introduced axiomatically. With a few later additions, Scharf poses the problem in Section 3.1: To find a formal series S(g) of the form (3.1.1) such that the key properties

• (3.1.13) = unitarity (in the projected form (3.11.25), because of gauge invariance)
• (3.1.16) = translation invariance
• (3.1.19) = Lorentz invariance,
• (3.1.23) = causality,
• (3.3.2), (3.6.31) = stable vacuum state, and
• (3.6.26), (3.7.36) = stable single particle states
hold. (Unfortunately, Scharf clouds the issue a little by not introducing all axioms in one place but postponing the last two to the place where they are needed - to fix the splitting ambiguities. But this is a matter of didactical exposition, not one of mathematical sloppiness.)

These properties axiomatically characterize a successful relativistic quantum field theory, apart from the fact that for a fully rigorous solution, S(g) should be an operator-valued functional of g rather than only a formal series.

The causality requirement (3.1.23) is a relaxed form of the exponential law. it says that the composition of two causally unrelated unitary transformations by S(g_1) and S(g_2) adds canonically, thus expressing that effects in causally unrelated regions are independent.

On a heuristic level, S(g) is the mathematically rigorous version of the time-ordered exponential

S(g)=Texp(integral dx g(x)T_1(x)),
where T_1(x) is the physical interaction given in (3.1.1). It is not too difficult to show that the locality properties of the quantum fields discussed in Weinberg'sĀQFT book imply (3.1.23) at the level of rigor of Weinberg's book. This is the ultimate reason why both approaches give the same final results, though through very different routes. Only the causal perturbation theory route can claim logical coherence, due to its mathematical rigor.

On p.139, Scharf discusses perturbation theory in ordinary quantum mechanics, where working with naive time-ordering is admissible and gives finite results, and explains (in subjunctive language) why the same (i.e., the traditional approach bare theory plus perturbation theory) becomes mathematically meaningless in quantum field theory since it performs operations that are not well-defined. Then he spends Section 3.2 in showing the mathematically correct procedure, and applies it in Section 3.3 to QED. (In his other book about the true ghostĀstory, he successfully extends the approach to other quantum field theories.)

The mathematical correct procedure is determined by microlocal theory, a mathematically well-known technique for the analysis of linear partial differential equations. ĀMicrolocal theory tells when the product of two distributions is well-defined. If one understands these conditions (which in terms of physics is roughly what comes under the heading of dispersion relations, but expressed in precise mathematical terms) then one can tell precisely which splittings are mathematically valid. In case of QED, this gives a 2-dimensional space of solutions, whence there are 2 parameters that are fixed not by the requirement of correct splitting but by the requirement of stability of the vacuum and the single particle states.

The important point is that everything can be uniquely determined from the axioms and the interaction. 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).

Scharf's interaction ej·A in (3.3.1) looks the same as in the traditional formalism, but its interpretation is quite different as (unlike in the action-based approach, where bare quantities figure in all formulas) j and A are the physical (renormalized) current and vector potential to order zero, and e is the physical charge. Scharf explicitly remarks that a bare charge appears nowhere.

Similarly, the other parameter in the theory, the electron mass m (which enters through the free field content of QED), is the physical mass (a zero of the self-energy, Scharf p.176).

Bogolubov & Shirkov show in their book (cited by Scharf on p. 130) how one can construct local operators from S(g). Scharf does not do this, but gives in (2.10.2) a particular example of the B&S recipe - the construction of the local current for a charged particle in an external field. More generally, constructing S(g) for a theory including sources in fact constructs the complete content of QFT. (The second edition of Scharf's book discusses this in Section 4.9.)

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).

More work on causal perturbation theory can be found at http://scholar.google.com.

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