Weinberg on Symmetry

The latest New York Review of Books has an article by Steven Weinberg entitled Symmetry: A ‘Key to Nature’s Secrets’. It’s a bit unusual for the NYRB, since it is both scientifically more technical than usual for them (coming from a write-up of Weinberg’s talk at this conference), and doesn’t review any books. The printed version tells readers to go to the web version for footnotes, but some of these just note that things are being over-simplified. One of the footnotes is worse than useless: the editors have replaced x^3=x as an example of an equation with solutions that break a symmetry (x goes to -x) by “x 3 equals x”, an equation with the same symmetry but only a symmetric solution (x=0). The idea seems to have been to remove or replace any symbols in the equation that might upset people.

Weinberg tells the conventional story of how the Standard Model emerged during the 60s and early 70s out of the realization that non-abelian gauge symmetries were important and an understanding of what happens when symmetries are spontaneously broken. He tries to do some much more ambitious things, explaining the idea of “accidental symmetries” that are due to the limited number of possible renormalizable terms you can build out of a specified list of fields, but I’m not sure the typical reader of the NYRB is going to get much out of this.

The question of how to explain the notion of “symmetry” is an interesting one, and I thought a lot about it when writing Not Even Wrong, the book. To my mind, most such explanations mix up two conceptually distinct things: the group of symmetries (a group), and the action of the group on some other mathematical object (the representation: mathematically a homomorphism from the group to the group of automorphisms of something). It’s both the group and the representation that are important in the use of symmetries in physics, although often what is important is the trivial representation. From a mathematician’s point of view, the simplest representations to look at are unitary representations on a complex vector space, so the mathematical structure of quantum mechanics is very natural. To each symmetry generator you get a conserved quantity, and it appears in quantum mechanics as the thing you exponentiate (a self-adjoint operator) to get a unitary representation. In Weinberg’s piece, which aims at sophisticated issues in particle theory, the question of the basic relation of symmetries and conservation laws is relegated to a footnote which says only “For reasons that are difficult to explain without mathematics…”.

Weinberg ends with a landscape sort of picture, involving symmetries emerging only when a specific ground state emerges out of an initial chaotic inflation state. Philosophically this is a popular view of the future of the subject these days, but one that has so far led nowhere, and one that I think even in principle can never lead anywhere. Much more interesting would be to try and draw lessons from what has worked well in the past: exactly the gauge symmetries and spontaneous symmetry breaking phenomena that led to the standard model. We may very well soon find out there is no Higgs particle, turning this whole subject into a wide-open one. Future progress may come from exactly the same place as in the past: new ideas about how to exploit the mathematical structures inherent in quantum mechanical symmetries.

Update: The missing exponentiation in the on-line footnote has been fixed.

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22 Responses to Weinberg on Symmetry

  1. peterg says:

    “To my mind, most such explanations mix up two conceptually distinct things: the group of symmetries (a group), and the action of the group on some other mathematical object (…)”

    Very true, but it’s incredibly difficult to explain the difference to a layman.

  2. Peter Woit says:


    I agree. I’ll also admit that I was confused about this myself for an embarrassingly long time as a student, studying the topic from a physicist’s view-point. Part of the problem is that if you think of groups as groups of matrices, your very definition of what a group is comes packaged with a representation.

  3. Pingback: Uncommon Descent | Steven Weinberg: The universe’s symmetry may be an accident

  4. neo says:

    It is not at all unusual that this NYRB article does not review books. Many of their articles do not review books. Kind of strange for a book review periodical.

  5. udi says:

    Can someone explain why Weinberg makes a statement, which seems to me to be wrong:

    “these symmetries, known as lepton and baryon conservation, would dictate that neutrinos (particles that feel only the weak and gravitational forces) have no mass”

    Neutrinos can have a Dirac mass term, which is renormalizable and invariant under all local symmetries.

  6. Not being in physics anymore, I find it a little depressing to see this evolutionary psychology approach to physics being espoused by Weinberg himself. What fraction of physicists ascribe to this?

  7. Peter Woit says:


    I think Weinberg has more sympathy towards this kind of thing than you might expect, since he’s credited with the earliest argument that the anthropic “explanation” for the size of the cosmological constant gives a bound at least the right order of magnitude to correspond to the observed value.

    I’m curious to hear what other’s impressions are of how this kind of empty pre-big bang scenario is thought of among physicists in general. From what I’ve seen, I’d describe physicists who are not theorists as mostly quite hostile to the idea as not being science. Within the theory community there’s a vocal group actively pushing it, but perhaps a silent majority that sees it as kind of an embarassment, best ignored, hoping it will go away.

  8. Giotis says:

    Strictly speaking Weinberg, for the prediction of the (bound of the) value of the CC, would deserve to share this year’s Nobel prize.

    Why not? He is a physicist and he made a prediction based on physics.

  9. Jon says:

    For Higgs stuff, it looks like the last 2011 update will happen in Paris on wedenesday november 16 http://hcp2011.lpnhe.in2p3.fr/hcp-2011_program.pdf (at least I couldn’t find a december conference).

    What puzzles me is that the sections on the Higgs that day consist only on reports on experimental results, except the last one (on “combinations and beyond”) which is planed to end with a talk entitled “Electroweak Symmetry Breaking without SM (or SUSY) Higgs” by C.Grojean: might it be a hint that still nothing has been found? Is that what you are alluding to, Peter?

  10. Peter Woit says:


    If the Nobel Prize committee had done anything that idiotic, I suspect that Weinberg himself would have turned down the prize, and a bunch of other physics medalists would have mailed theirs back…


    As far as I know, the Paris conference will be the next time CMS and ATLAS announce new results about the Higgs (and a combined CMS/ATLAS analysis using this summer’s results also appears). I don’t now how much data they will have analyzed by then, at the moment they’ve collected something like 3 times the amount used in the summer results. This should be very interesting…

    The fact that a theory talk about models without a Higgs is scheduled isn’t surprising, and doesn’t reflect any inside knowledge about the new data. There’s no point to a theory talk about the Higgs, everyone has known for decades exactly what the theory says there. The failure to find the Higgs so far is leading to more and more interest in Higgsless scenarios.

  11. Eric says:

    Lubos Motl is hinting on his blog that he has inside information that a Higgs signal has emerged and he knows its mass….

  12. Chris Austin says:

    An observational test of Weinberg’s anthropic explanation of the small value of the cosmological constant has been proposed by Abraham Loeb. The idea is to search for exo-planets in nearby dwarf galaxies that formed at redshifts up to 10. Discovery of such planets would provide evidence that planets could form even if the cosmological constant was larger than the observed value by a factor of up to 1000. Loeb said that the search could be carried out with 2006 technology, but no results appear to have been reported so far.

  13. Peter Woit says:


    One would expect some news to start leaking out of CMS/ATLAS soon, on the other hand, the idea that Lubos would keep something he knows quiet isn’t all that plausible. Anyway, we’ll see soon….

  14. mo says:

    udi says:
    October 11, 2011 at 12:19 am
    Can someone explain why Weinberg makes a statement, which seems to me to be wrong:

    “these symmetries, known as lepton and baryon conservation, would dictate that neutrinos (particles that feel only the weak and gravitational forces) have no mass”

    Weinberg explains it himself in footnote 9.

  15. DB says:

    @Peter “we’ll see soon…”

    In the valuable link you provided to the ICFA seminar in CERN, Terry Wyatt from Manchester appeals for a 5-sigma standard for exclusion on the Higgs (slide 30), effectively excluding the Tevatron from the exclusion process. He states “The gold standard for SM Higgs discovery or exclusion is the same 5σ” That’s a first for me and was understandably challenged by a questioner after the presentation. He doesn’t indicate how he proposes to rejig the LEP2 results which only reach the 95% confidence level, especially given that LEP2 originally eliminated most of the range originally predicted by precision electroweak data.
    He continues: “Excluding the SM should be regarded (and presented to the outside world – my emphasis) as a discovery of equal or greater significance.”
    Yeah right….

  16. x3 says:

    There may be more typos. Cna’t b too quick to judge.

  17. ZZZ says:

    Re: old planets in dwarf galaxies. There was some report last year regarding finding some very old planet around a star that most likely came from a dwarf galactic neighbor. Another point is that many of them would have been swallowed by their stars by now, so there is a built-in selection bias for lower mass long burning stars.

  18. S. Molnar says:

    I’m a bit late on this, but the missing exponentiation in the footnote was clearly an HTML-type error and not an intentional change to the text. I’m unhappy with the continuing decline of editorial rigor in the New York Review of Books (an article recently referred to a film actor who played Dracula as “Bella Lugosi”, apparently confused by the fact that Béla Lugosi appeared in “Glen or Glenda”), but let’s apply Hanlon’s razor when appropriate.

  19. Chris W. says:

    Footnote 3 of Weinberg’s NYRB piece might catch the eye of a few people. Consider how it would be phrased if one replaces Galilean invariance by Lorentz invariance, and adopts the perspective of general relativity; the Earth’s orbit is locally inertial. There is no need to talk about gravity balancing the effects of the centrifugal force caused by the earth’s curved motion. :)

  20. anonymous says:


    some atlas and cms people seem to believe that there is a higgs signal at 120GeV. others don’t. this is the reason why there is no new update for so long. let’s be patient a bit longer and let the experimenters do their job, then we’ll know.

  21. Lau says:

    I’ve recently come across this discussion between Weinberg and Dawkins and thought of sharing it, in case anyone would like to watch. It is 3 years old, so you may have seen it before.

    Starting approximately at the end of part 2, they discuss the multiverse.


  22. Georges MELKI says:

    I don’t really get the difference between x^3=x and x3 equals x, and how the latter upsets people less than the former! Usually, when the number is at the right of the letter, it’s supposed to be an exponent, not a coefficient…unless the reader is completely illiterate, and he/she wouldn’t be reading an article by S.Weinberg!