Steven Weinberg has a new preprint out entitled Living with Infinities, which is the written version of a recent talk given in memory of Gunnar Källén. Källén was a Swedish mathematical physicist, who died in a plane accident in 1968 at the age of 42. For more about him, see this by Ray Streater.
Weinberg begins by recalling his first first trip to the Bohr Institute in 1954, where he met Källén, who suggested a research problem involving an exactly solvable QFT model invented by TD Lee. The solvability of this model made it possible to use it for investigating renormalization outside of perturbation theory. Källén and Pauli showed that the model was non-unitary, Weinberg showed that it had states with complex energies.
In his talk, Weinberg describes Källén’s work during the 1950s investigating the question of how QED gets renormalized, outside the context of perturbation theory. Källén found an argument showing that at least one of the renormalization constants must be infinite but Weinberg notes that Källén never claimed the argument was rigorous, and concludes: “As far as I know, this issue has never been settled.” He goes on to give the now conventional Wilsonian description of the non-perturbative situation and the possibility that no non-trivial continuum limit exists. This question is now considered somewhat academic, since it is assumed that QED gets unified with other interactions and ultimately with gravity at energies below those at which the behavior of the coupling becomes problematic.
Weinberg states in his abstract that he will present his personal view on how the problem of infinities may ultimately be resolved. Here’s what he has to say about this:
My own view is that all of the successful field theories of which we are so proud — electrodynamics, the electroweak theory, quantum chromodynamics, and even General Relativity — are in truth effective field theories, only with a much larger characteristic energy, something like the Planck energy….
None of the renormalizable versions of these theories really describes nature at very high energy, where the non-renormalizable terms in the theory are not suppressed. From this point of view, the fact that General Relativity is not renormalizable in the Dyson sense is no more (or less) of a fundamental problem than the fact that non-renormalizable terms are present along with the usual renormalizable terms of the Standard Model. All of these theories lose their predictive power at a sufficiently high energy. The challenge for the future is to find the final underlying theory, to which the effective field theories of the standard model and General Relativity are low-energy approximations.
It is possible and perhaps likely that the ingredients of the underlying theory are not the quark and lepton and gauge boson fields of the Standard Model, but something quite different, such as a string theory. After all, as it has turned out, the ingredients of our modern theory of strong interactions are not the nucleon and pion fields of Källén’s time, but quark and gluon fields, with an effective field theory of nucleon and pion fields useful only as a low-energy approximation.
But there is another possibility. The underlying theory may be an ordinary quantum field theory, including fields for gravitation and the ingredients of the Standard Model…
He then goes on to describe the “asymptotic safety” scenario where the renormalized couplings approach a non-trivial fixed point as the energy cut-off is taken to infinity, a fixed point which presumably cannot be studied in perturbation theory, writing:
Other techniques such as dimensional continuation, 1/N expansions, and lattice quantization have provided increasing evidence that gravitation may be part of an asymptotically safe theory.
and referring to papers by Reuter/Saueressig, Percacci, and Litim (Percacci has a web-page about asymptotic safety here). He ends with the conclusion that, since string theory might not have any role in a fundamental theory, with only QFT needed to understand quantum gravity:
So it is just possible that we may be closer to the final underlying theory than is usually thought.
In these days of string landscape ideology, this possibility is an important one to keep in mind.