Living With Infinities

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.

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17 Responses to Living With Infinities

  1. Shantanu says:

    Peter you mentioned
    “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.”
    Does this unification solve the problem?
    Also a related question (re QED). Are you worried about issues such as the
    Landau pole?

  2. Peter Woit says:


    The problem that Weinberg explains is the Landau pole problem, and I agree with his description of it.

    Conventional guts don’t solve the problem, because they require a Higgs sector, and this is not asymptotically free, so has the same problem as QED.

  3. Peter Orland says:


    Yes, unification solves the problem for QED. The Landau pole of QED occurs at a higher energy than the unification scale. Unfortunately the standard model still has a problem at high energies, because of the HIggs and the U(1) sector.

    Peter: I think K\”{a}ll\'{e}n would have called himself a theoretical physicist, not a mathematical physicist. In his day, the latter referred to people who used rigorous methods to prove things about physics. There was much interaction between the two subfields, but there was a clear distinction.

  4. Sulfer Surfer says:

    It is remarkable that Weinberg is also the one who did much to bring the anthropic principle into physics. One of the things that has made him a great scientist, time and again, is that he seeks to understand the universe without preconditions on what form the answer will take.

  5. N. Nakanishi says:

    Peter: Your description concerning the Lee model is not precise. The original Lee model, in which the V particle is of positive norm,
    is unitary if cutoff factor is introduced. What Kallen and Pauli investigated is the indefinite-metric Lee model, in which the bare V particle is of negative norm. In this model, there are three cases for the physical V particles: one or two real eigenvalue(s), one of which is of negative norm, one double real eigenvalue, which is Heisenberg’s dipole ghost, and two mutually complex-conjugate eigenvalues. For details, see my review article, Prog. Theor. Phys Suppl. 51 (1972), Sec.12.

  6. Chris Oakley says:

    SW seems to have a knack of capturing the mood of the moment – lauding String Theory when everyone (except Peter, of course) was working on it and then supporting the sceptics when PW’s & Smolin’s books came out. But being a trendsetter himself, his measurements of the state of the art have a habit of forcing the system into a quantum state, and no doubt now that he is saying that people should be going back to QFT basics, a large number of researchers will now do just that. It is high time.

  7. Peter Woit says:


    Many thanks for the added information about the Lee model, it’s not something I knew about before this, quite interesting to hear more about it.


    Actually I think Kallen is an interesting case on the borderlines of classification. He wrote a well-known rather phenomenological book about particle physics in the mid-sixties which I learned a lot from as a student. The bulk of his work was investigation of fundamental issues in QFT, although not at a mathematician’s standard of rigor. Later on I think he worked on more mathematical issues concerning analytic behavior of amplitudes, some published in mathematical physics journals. So, I think both conventional particle theorists and mathematical physicists can claim him as one of their own.

  8. Peter Woit says:


    I’d love to count Weinberg as an enthusiast for my book and arguments about string theory, but I’m afraid the evidence doesn’t really support that. After a few years working on string theory himself, he did vote with his feet and move into cosmology, something that probably encouraged that trend, which is ongoing. But it’s my impression that his attitude for many years has been one more of agnosticism on the string theory/QFT question, that work on both is justified, with string theory remaining as much a possible TOE as any other idea on the subject. As far as I can tell, pointing out that string theory unification has failed as an idea about finding a TOE is not on his agenda.

  9. dir says:

    the landau pole problem has two aspects, i think. one is that whether is it really there? the pole apears from perturbative calculation, but the pole itself is in non-perturbative region. lattice peole said that the landau pole can exist. the other is that even if it exists (almost for sure), does this imply the theory is meaningless? maybe not. it can be an evidence of some phase transition which has to be described by a totally new theory.

  10. Chris Oakley says:


    Didn’t Weinberg say that “the sceptics are right” when yours and Smolin’s book came out 2 1/2 years ago?

    As for Kallen, IMHO he did much more than average in trying to make QFT rigorous. He did not succeed, of course – but then no-one else has either.

  11. Peter Woit says:


    That doesn’t correspond to anything I remember, but I’ve mercifully forgotten some details of the “string wars”.

  12. dan says:

    The Nova program on string theory, hosted by Kaku and Greene, Weinberg was interviewed and asked and answered with a qualified endorsement of string theory.

  13. anon. says:

    Weinberg 2.5 years ago mentioned that there are critics of string theory who perhaps are motivated because they want some of the money string theory is getting. He then added ‘perhaps they are right’ to want funding for alternatives. He didn’t say string critics are right in dismissing string theory, and he still supports string theory.

    See 15 July 2008 interview of Weinberg by Dawkins at where Weinberg states at 9 mins 12 seconds:

    ‘It’s been a little disappointing that it hasn’t led to any specific breakthrough in understanding what we already know, but it’s still the best game in town.’

  14. Chris W. says:

    Off-topic: Ilya Piatetski-Shapiro, died Feb. 21 in Tel Aviv (New York Times).

  15. mike says:


    Has there been any success in resolving perturbative issues with the two-dimensional models and conformal QFTs?

  16. Peter Woit says:


    There are lots of 2d models you can solve exactly, and infinities are much easier to deal with (no coupling constant renormalization), so there isn’t the same kind of problem as in 4d.

  17. Thomas R Love says:

    Peter, Thanks for the info on Gunnar Kallen. After reading some more about his work, I finally ordered two of his books, QED and Elementary particles.

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