GAMBIT

The LHCP 2017 conference was held this past week in Shanghai, and among the results announced there were new negative results about SUSY from ATLAS with both ATLAS and CMS now reporting for instance limits on gluino masses of around 2 TeV. The LHC has now ruled out the existence of SUSY particles in the bulk of the mass range that will be accessible to it (recall for instance that pre-LHC, gluino mass limits were about 300 GeV or so).

Over the years there has been an ongoing effort to produce “predictions” of SUSY particle masses, based on various sorts of assumptions and various experimental data that might be sensitive to the existence of SUSY particles. One of the main efforts of this kind has been the MasterCode collaboration. Back in 2008 before the LHC started up, they were finding that the “best fit” for SUSY models implied a gluino at something like 600-750 GeV. As data has come in from the LHC (and from other experiments, such as dark matter searches), they have periodically released new “best fits”, with the gluino mass moving up to stay above the increasing LHC limits.

I’ve been wondering how efforts like this would evolve as stronger and stronger negative results came in. The news this evening is that they seem to be evolving into something I can’t comprehend. I haven’t kept track of the latest MasterCode claims, but back when I was following them I had some idea what they were up to. Tonight a large collaboration called GAMBIT released a series of papers on the arXiv, which appear to be in the same tradition of the old MasterCode fits, but with a new level of complexity. The overall paper is 67 pages long and has 30 authors, and there are eight other papers of length totaling over 300 pages. The collaboration has a website with lots of other material available on it. I’ve tried poking around there, and for instance reading a Physics World article about GAMBIT, but I have to confess I remain baffled.

So, the SUSY phenomenology story seems to have evolved into something very large that I can’t quite grasp anymore, perhaps a kind reader expert in this area can explain what is going on.

Posted in Uncategorized | 20 Comments

This Month’s Hype

It seems that a couple of the authors of the recent Cosmic Controversy letter (discussed here) are going on a campaign to embarrass the 29 physicists who were convinced to sign their letter. Andrei Linde has gone to Lubos Motl’s blog to thank him for his blog entry which lauded Linde as having eaten from the biblical tree of knowledge and which denounced his critics as imbeciles. To deal with Linde and his claims, Ijjas, Steinhardt and Loeb have added a new webpage to their website called Fact Checking. It lists the four “predictions” of inflation claimed to agree with experiment by Linde et al. and gives four references to papers published by Linde touting different “predictions” for the same quantities, predictions not agreeing with experiment.

This month’s Scientific American has a remarkable cover story, The Quantum Multiverse from one of the other four letter authors, Yasunori Nomura. I’ve seen some fairly bizarre stories about fundamental physics in Scientific American over the years, but this one sets a new standard for outrageous nonsense, and I’m wondering whether it too may cause some of the 29 co-signers of the letter co-authored by Nomura to question the wisdom of joining with him and Linde. Nomura is well known for a definite prediction based on the multiverse: in 2009 he co-authored a paper claiming that the multiverse predicted the Higgs mass would be 141 GeV +/- 2 GeV. This played a major role in the film Particle Fever. That three years later the Higgs was discovered at 125 GeV seems to have had no effect on his multiverse enthusiasm.

The new SciAm cover story is not about anything new, but is based on a 6 year old paper by Nomura discussed here. At the time I wrote about this “I’m having trouble making sense of any of these papers” and quoted Lubos’s evaluation: “They’re on crack”. Nothing I’ve seen about this over the past six years seems to me to make any sense at all, including the new SciAm cover story, which just seems even more content-free and meaningless than previous efforts to explain this “multiverse interpretation of quantum mechanics”. On the obvious question: how would you test this, Nomura just has this to say:

Evidence so far indicates that the cosmos is flat, but experiments studying how distant light bends as it travels through the cosmos are likely to improve measures of the curvature of our universe by about two orders of magnitude in the next few decades. If these experiments find any amount of negative curvature, they will support the multiverse concept because, al­­though such curvature is technically possible in a single universe, it is implausible there. Specifically, a discovery supports the quantum multiverse picture described here because it can naturally lead to curvature large enough to be detected, whereas the traditional inflationary picture of the multiverse tends to produce negative curvature many orders of magnitude smaller than we can hope to measure.

This paragraph manages to put together three different misleading and unsupported claims:

  • “If these experiments find any amount of negative curvature, they will support the multiverse concept because, al­­though such curvature is technically possible in a single universe, it is implausible there.” This is just nonsense.
  • “the traditional inflationary picture of the multiverse tends to produce negative curvature many orders of magnitude smaller than we can hope to measure”. What is the inflationary multiverse “prediction” for negative curvature? As far as I can tell it’s compatible with pretty much any level we might observe.
  • “the quantum multiverse picture described here because it can naturally lead to curvature large enough to be detected.” I can’t find anywhere a calculation of the negative curvature expected by the “quantum multiverse picture”, and I don’t believe any such calculation is possible.

Given some of the outrageous hype I’ve seen in recent years in respectable publications, it’s gotten rather hard to shock me with this sort of thing, but I do find this Scientific American cover story shocking.

Update: For some reason this was not mentioned in the SciAm article, but the paper justifying Nomura’s claims about negative curvature is here.

Update: A modest proposal: Given the situation, I think someone needs to write a letter to SciAm complaining about the Nomura article and get leaders of the community to sign in support of it. They could start gathering signatures by writing to the 29 signers of the earlier letter. If these people were willing to object to the Steinhardt et al. article, they should be willing to object to the far worse Nomura article.

Posted in Multiverse Mania, This Week's Hype | 49 Comments

A Cosmic Controversy

A couple months ago Scientific American published an article by Ijjas, Steinhardt and Loeb (also available here), which I discussed a bit here. One aspect of the article was its strong challenge to multiverse mania, calling it the “multimess” and accusing multiverse explanations of being untestable and unscientific.

Yesterday Scientific American published, under the title A Cosmic Controversy, a rebuttal signed by 33 physicists, together with a response from the authors, who have also set up a webpage giving further details of their response. Undark has an article covering this: A Debate Over Cosmic Inflation (and Editing at Scientific American) Gets Heated.

As Ijjas, Steinhardt and Loeb point out on their webpage, the story of this letter is rather unusual. It was written by David Kaiser and three physicists well-known for their outspoken promotion of the multiverse (Guth, Linde and Nomura). Evidently these authors decided they needed reputational support on their side, and sought backing from other prominent names in the field (I’m curious to know who may have refused to sign if asked…). Their letter starts out with a claim to represent the “dominant paradigm in cosmology” and notes the large number of papers and researchers involved in studying inflation.

If you read carefully both sides (IS&L and GKL&N) of this, I think you’ll find that they are to a large degree speaking past each other, with a major problem that of imprecision in what one means by “inflation”. To the extent that there is a specific identifiable scientific disagreement, it’s about whether Planck data confirms predictions of the “simplest inflationary models.” IS&L write:

The Planck satellite results—a combination of an unexpectedly small (few percent) deviation from perfect scale invariance in the pattern of hot and colds spots in the CMB and the failure to detect cosmic gravitational waves—are stunning. For the first time in more than 30 years, the simplest inflationary models, including those described in standard textbooks, are strongly disfavored by observations.

whereas GKL&N respond:

there is a very simple class of inflationary models (technically, “single-field slow-roll” models) that all give very similar predictions for most observable quantities—predictions that were clearly enunciated decades ago. These “standard” inflationary models form a well-defined class that has been studied extensively. (IS&L have expressed strong opinions about what they consider to be the simplest models within this class, but simplicity is subjective, and we see no reason to restrict attention to such a narrow subclass.) Some of the standard inflationary models have now been ruled out by precise empirical data, and this is part of the desirable process of using observation to thin out the set of viable models. But many models in this class continue to be very successful empirically.

I take this as admission that IS&L are right that some predictions of widely advertised inflationary models have been falsified. Of course, if these had worked they would have been heavily promoted as “smoking gun” proof of inflation, as was demonstrated by the BICEP2 B-mode fiasco. After BICEP2 announced (incorrectly) evidence for B-modes, Linde claimed this was a “smoking gun” for inflation (see here) and the New York Times had a front page story about the “smoking gun” confirmation of inflation vindicating the ideas of Guth and Linde. A couple months later, before the BICEP2 result was shown to be mistaken, Guth, Linde and Starobinsky were awarded the $1 million Kavli Prize in Astrophysics.

GKL&N don’t mention the sorry story of the BICEP2 B-modes, what they have to say about this is

the levels of B-modes, which are a measure of gravitational radiation in the early universe, vary significantly within the class of standard models…

The B-modes of polarization have not yet been seen, which is consistent with many, though not all, of the standard models.

About the IS&L “unexpectedly small (few percent) deviation from perfect scale invariance” all GKL&N have to say is

The standard inflationary models… predict the statistical properties of the faint ripples that we detect in the cosmic microwave background (CMB). First, the ripples should be nearly “scale-invariant”

This doesn’t seem to address at all the IS&L claims, which they make in more detail as

The latest Planck data show that the deviation from perfect scale invariance is tiny, only a few percent, and that the average temperature variation across all spots is roughly 0.01 percent. Proponents of inflation often emphasize that it is possible to produce a pattern with these properties. Yet such statements leave out a key point: inflation allows many other patterns of hot and cold spots that are not nearly scale-invariant and that typically have a temperature variation much greater than the observed value. In other words, scale invariance is possible but so is a large deviation from scale invariance and everything in between, depending on the details of the inflationary energy density one assumes. Thus, the arrangement Planck saw cannot be taken as confirmation of inflation.

GKL&N argue for three other confirmed predictions of inflationary models:

Second, the ripples should be “adiabatic,” meaning that the perturbations are the same in all components: the ordinary matter, radiation and dark matter all fluctuate together. Third, they should be “Gaussian,” which is a statement about the statistical patterns of relatively bright and dark regions. Fourth and finally, the models also make predictions for the patterns of polarization in the CMB, which can be divided into two classes, called E-modes and B-modes. The predictions for the E-modes are very similar for all standard inflationary models

On these issues I don’t see anything from IS&L and would love to hear from an expert.

The main issue here comes down to the question of the flexibility vs. rigidity of inflationary models. Is the inflationary paradigm rigid enough to make solid predictions, or so flexible that it can accommodate any experimental result? GKL&N are making the case for the former, IS&L for the latter, and they point out the following quote from Guth himself:

when asked via email if they could name any pro-inflation scientists who believe that the theory is nonetheless untestable, the trio pointed to a video of a 2014 panel during which Loeb asks Guth directly whether it’s possible to do an experiment that would falsify inflation.

“Well, I think inflation is a little too flexible an idea for that to make sense,” Guth replied.

A fair take on all this would be to note that it’s a complicated situation, and I doubt I’m the only one who would like to see an even-handed technical discussion of exactly what the “simplest” models are and a comparison of their predictions with the data. Claims to the public from one group of experts that Planck data says one thing, from others claiming it says the opposite are generating confusion here rather than clarity about the science.

I’m strongly on the side of IS&L on one issue, that of the danger of theories that invoke the multiverse as untestable explanation. I don’t think though that they make a central issue clear. The simple inflationary models whose “predictions” for Planck data are being discussed involve a single inflaton field, with no understanding of how this is supposed to couple to the rest of physics. One is told that eternal inflation implies a multiverse with different physics in different universes, but in a single inflaton model this physics should just depend on a single parameter, and such a theory should be highly predictive (once you know one mass, all others are determined). What’s really going on is that there is no connection at all between the simple single field models that GKL&N and IS&L are arguing about, and the widely promoted completely unpredictive string theory landscape models (involving large numbers of inflaton-type fields with dynamics that is not understood).

I think IS&L made a mistake by not pointing this out, and that Guth, Linde, Nomura and some of the signers of their letter (e.g. Carroll, Hawking, Susskind, Vilenkin) have long been guilty of promoting the defeatist pseudo-scientific idea that “evidence for inflation is evidence for a multiverse with different physics in each universe, explaining why we can’t ever calculate SM parameters”. By defending the predictivity of “inflation” while ignoring the “different physics in different parts of the multiverse” question, I think many signers of the GKL&N letter were missing a good opportunity to make common cause with IS&L on defending their science against an ongoing attack from some of their fellow signatories.

Update: There are sources with technical details of the arguments being made by both parties:

I’ll try to find time to read these carefully and try and understand exactly what claims are being made. Would love to hear from experts who may have looked at these and are better placed to evaluate what the arguments are.

This controversy continues to involve an unusual level of PR rather than science. The Stanford press office has just put out this, where Linde makes it clear that he sees this as a political and PR fight:

Linde added that he worries about the younger generation of scientists getting the wrong impression from this story. “I don’t want them to read this article and think that they are spending their time on inflationary theory in vain. But the enthusiastic support that we are receiving makes us optimistic that this is not going to happen,” he said.

There’s no mention in this press office story of their last press office story about Linde and inflation, which promoted the BICEP2 “smoking gun” vindication of Linde and inflation.

Some more takes on this story can be found here, here and here.

Update: Another article about this is at the Atlantic. A crucial issue here is whether inflation has now entered the realm of unfalsifiability. Given any likely new data that could appear, is there any way it could falsify inflation, or can one just come up with some version of inflation that will match it? Guth and Linde seem at times to be taking the attitude that this is fine, I take Steinhardt et al. as pointing out that this is no longer conventional science. Replacing falsifiability by arguments about how many prominent people have signed your letter is a worrisome development. From the article:

In 2014, for example, Loeb asked Guth during a panel discussion if inflation was falsifiable—whether you could design an experiment to disprove it. Guth called that a “silly question,” suggesting that inflation was an umbrella over many theories, making it very hard to knock them all out at once. The hope right now, he says, is to use observation and further theory to winnow inflation down to just one specific version.

“Our point is that this kind of reasoning is inconsistent with normal science and cannot be resolved by invoking authority,” Ijjas, Steinhardt, and Loeb wrote to The Atlantic.


Update
: More about this from John Horgan.

Update: It seems that Andrei Linde is a Lubos Motl fan. This is getting very weird. It’s not normal to respond to a scientific argument by enlisting letter writers on your behalf, even less normal to put your university press office to work on a response, and truly far out there to think that it’s helpful to have Lubos announce that you have eaten from the tree of knowledge and that your opponents are imbeciles.

Update: For a sensible take on this that I think likely reflects well the views of most mainstream cosmologists, see Peter Coles.

Update: IJS have put together a Fact-Checking page. It lists the four “predictions” of inflation claimed to agree with experiment by Linde et al. and gives four references to papers published by Linde touting different “predictions” for the same quantities, predictions not agreeing with experiment.

This month’s Scientific American has a bizarre cover story by Nomura on “The Quantum Multiverse”, see here.

All in all, I don’t know what other people’s reactions to this have been, but before this started I was a lot more sympathetic to the argument that Steinhardt was treating the case for inflation unfairly.

Update: Yet more coverage trying to make sense of this, from Nick Stockton at Wired.

Posted in Multiverse Mania | 53 Comments

Some Quick Items

A few quick items, I may use this posting to add a couple more later, the next posting will discuss today’s letter to Scientific American about inflation.

  • Today’s LHCC meeting at CERN had reports from the LHC machine and experiments. About two weeks to go before collisions and data-taking start again.
  • Physics Today has a report this month on the LHeC proposal, something that has not gotten as much attention as it deserves. This is a proposal to collide protons and electrons, by building a new electron machine and a detector at a collision point with the LHC beam. Unlike proposals for a 100 TeV proton-proton machine that are getting a lot of attention, this would not push the energy frontier, but it would cost a great deal less (estimate is half a billion to a billion, vs. multiple tens of billions for the 100 TeV machine). In a few years when the question of a follow-on machine to the LHC starts to get very pressing, this idea and the HE-LHC idea (higher field magnets in the LHC tunnel, maybe doubling the energy) may get a lot more attention as the only financially viable ways forward.
  • The Université de Montpellier today has started to make accessible about 18,000 pages of its archive of Grothendieck’s mathematical writings. For anyone interested in Grothendieck’s work, this should keep you busy for a while…

Update: A few more.

  • I was sorry to hear of the recent death of Cecile DeWitt-Morette, a mathematical physicist responsible for the Les Houches physics summer school. Her books on geometry and physics (with Yvonne Choquet-Bruhat) were influential, and her more recent book with Pierre Cartier on Functional Integration contains a lot of interesting material.
  • Every so often I’d wondered what the Chudnovsky brothers have been up to, some information about this emerged today.
  • For a story about problems at the science magazine, Nautilus, which is having trouble transitioning from its original Templeton Foundation funding, see here.
Posted in Uncategorized | 12 Comments

Theories of Everything

I’ve written a review for the latest issue of Physics World of a short new book by Frank Close, entitled Theories of Everything. You can read the review here.

As I discuss in the review, Close explains a lot of history, and asks the question of whether we’re in an analogous situation to that of the beginning of the 20th century, just before the modern physics revolutions of relativity and quantum theory. Are the cosmological constant and the lack of an accepted quantum theory of gravity indications that another revolution is to come? I hope to live long enough to find out…

Posted in Book Reviews | 26 Comments

Why String Theory is Still Not Even Wrong

John Horgan recently sent me some questions, and has put them and my answers up at his Scientific American site, under the title Why String Theory is Still Not Even Wrong. My thanks to him for the questions and for the opportunity to summarize my take on various issues.

Posted in Fake Physics, Multiverse Mania | 37 Comments

Two Pet Peeves

I was reminded of two of my pet peeves while taking a look at the appendix A of this paper. As a public service to physicists I thought I’d go on about them here, and provide some advice to the possibly confused (and use some LaTeX for a change).

Don’t use the same notation for a Lie group and a Lie algebra

I noticed that Zee does this in his “Group Theory in a Nutshell for Physicists”, but thought it was unusual. It seems other physicists do this too (same problem with Ramond’s “Group Theory: a physicist’s survey”, the next book I checked). The argument seems to be that this won’t confuse people, but, personally, I remember being very confused about this when I first started studying the subject, in a course with Howard Georgi. Taking a look at Georgi’s book for that course (first edition) I see that what he does is basically only talk about Lie algebras. So, the fact that I was confused about Lie groups vs. Lie algebras wasn’t really his fault, since he was not talking about the groups.

The general theory of Lie groups and Lie algebras is rather complicated, but (besides the trivial cases of translation and U(1)=SO(2) groups) many physicists only need to know about two Lie groups and one Lie algebra, and to keep straight the following facts about them. The groups are

  • SU(2): the group of two by two unitary matrices with determinant one. These can be written in the form
    $$\begin{pmatrix}
    \alpha & \beta\\
    -\overline{\beta}& \overline{\alpha}
    \end{pmatrix}$$
    where \(\alpha\) and \(\beta\) are complex numbers satisfying \(|\alpha|^2+|\beta|^2=1\), and thus parametrizing the three-sphere: unit vectors in four real dimensional space.
  • SO(3): the group of three by three orthogonal matrices with determinant one. There’s no point in trying to remember some parametrization of these. Better to remember that a rotation by a counter-clockwise angle \(\theta\) in the plane is given by
    $$\begin{pmatrix}
    \cos\theta & -\sin\theta\\
    \sin\theta & \cos\theta
    \end{pmatrix}$$
    and then produce your rotations in three dimensions as a product of rotations about coordinate axes, which are easy to write down. For instance a rotation about the 1-axis will be given by
    $$\begin{pmatrix}
    1&0&0\\
    0&\cos\theta & -\sin\theta\\
    0&\sin\theta & \cos\theta
    \end{pmatrix}$$

The relation between these two groups is subtle. Every element of SO(3) corresponds to two elements of SU(2). As a space, SO(3) is the three-sphere with opposite points identified. Given elements of SO(3), there is no continuous way to choose one of the corresponding elements of SU(2). Given an element of SU(2), there is an unenlightening impossible to remember formula for the corresponding element of SO(3) in terms of \(\alpha\) and \(\beta\). To really understand what’s going on, you need to do something like the following: identify points in \(\mathbf R^3\) with traceless two by two self-adjoint matrices by
$$(x_1,x_2,x_3)\leftrightarrow x_1\sigma_1 +x_2\sigma_2+x_3\sigma_3=\begin{pmatrix} x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{pmatrix}$$
Then the SO(3) rotation corresponding to an element of SU(2) is given by
$$\begin{pmatrix} x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{pmatrix}\rightarrow \begin{pmatrix}
\alpha & \beta\\
-\overline{\beta}& \overline{\alpha}
\end{pmatrix}\begin{pmatrix} x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{pmatrix} \begin{pmatrix}
\alpha & \beta\\
-\overline{\beta}& \overline{\alpha}
\end{pmatrix}^{-1}$$

Since most of the time you only care about two Lie groups, you mostly only need to think about two possible Lie algebras, and luckily they are actually the same, both isomorphic to something you know well: \(\mathbf R^3\) with the cross product. In more detail:

  • su(2) or \(\mathfrak{su}(2)\): Please don’t use the same notation as for the Lie group SU(2). These are traceless skew-adjoint (\(M=-M^\dagger\)) two by two complex matrices, identified with \(\mathbf R^3\) as above except for a factor of \(-\frac{i}{2}\).
    $$(x_1,x_2,x_3)\leftrightarrow -\frac{i}{2}\begin{pmatrix} x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{pmatrix}$$
    Under this identification, the cross-product corresponds to the commutator of matrices.

    You get elements of the group SU(2) by exponentiating elements of its Lie algebra.

  • so(3) or \(\mathfrak{so}(3)\): Please don’t use the same notation as for the Lie group SO(3). These are antisymmetric three by three real matrices, identified with \(\mathbf R^3\) by

    $$(x_1,x_2,x_3)\leftrightarrow \begin{pmatrix}
    0&-x_3&x_2\\
    x_3&0 & -x_1\\
    -x_2&x_1&0
    \end{pmatrix}$$
    Under this identification, the cross-product corresponds to the commutator of matrices.

    You get elements of the group SO(3) by exponentiating elements of its Lie algebra.

If you stick to non-relativistic velocities in your physics, this is all you’ll need most of the time. If you work with relativistic velocities, you’ll need two more groups (either of which you can call the Lorentz group) and one more Lie algebra, these are:

  • \(SL(2,\mathbf C)\): This is the group of complex two by two matrices with determinant one, i.e. complex matrices
    $$\begin{pmatrix}
    \alpha & \beta\\
    \gamma& \delta
    \end{pmatrix}$$
    satisfying \(\alpha\delta-\beta\gamma=1\). That’s one complex condition on four complex numbers, so this is a space of 6 real dimensions. Best to not try and visualize this; besides being six-dimensional, unlike SU(2) it goes off to infinity in many directions.
  • SO(3,1): This is the group of real four by four matrices M of determinant one such that
    $$M^T\begin{pmatrix}-1&0&0&0\\
    0&1&0&0\\
    0&0&1&0\\
    0&0&0&1\end{pmatrix}M=\begin{pmatrix}-1&0&0&0\\
    0&1&0&0\\
    0&0&1&0\\
    0&0&0&1\end{pmatrix}$$
    This just means they are linear transformations of \(\mathbf R^4\) preserving the Lorentz inner product.

    Correction: a correspondent reminds me that for the next part to be true this definition needs to be supplemented by an extra condition, since as stated SO(3,1) has two components. One version of the extra condition is to take the connected component of the identity, another is to take the component that preserves time orientation. Many use a different notation for this component to make this explicit, I’ll define SO(3,1) as the connected component.

The relation between SO(3,1) and \(SL(2,\mathbf C)\) is much the same as the relation between SO(3) and SU(2). Each element of SO(3,1) corresponds to two elements of \(SL(2,\mathbf C)\). To find the SO(3,1) group element corresponding to an \(SL(2,\mathbf C)\) group element, proceed as above, removing the “traceless” condition, so identifying \(\mathbf R^4\) with self-adjoint two by two matrices as follows
$$(x_0,x_1,x_2,x_3)\leftrightarrow\begin{pmatrix} x_0+x_3&x_1-ix_2\\x_1+ix_2&x_0-x_3\end{pmatrix}$$
The SO(3,1) action on \(\mathbf R^4\) corresponding to an element of \(SL(2,\mathbf C)\) is given by
$$\begin{pmatrix} x_0+x_3&x_1-ix_2\\x_1+ix_2&x_0-x_3\end{pmatrix}\rightarrow \begin{pmatrix}
\alpha & \beta\\
\gamma & \delta
\end{pmatrix}\begin{pmatrix} x_0+x_3&x_1-ix_2\\x_1+ix_2&x_0-x_3\end{pmatrix} \begin{pmatrix}
\alpha & \beta\\
\gamma& \delta
\end{pmatrix}^{-1}$$

As in the three-dimensional case, the Lie algebras of these two Lie groups are isomorphic. The Lie algebra of \(SL(2,\mathbf C)\) is easiest to understand (please don’t use the same notation as for the Lie group, instead consider \(sl(2,\mathbf C\)) or \(\mathfrak{sl}(2,\mathbf C)\)), it is all complex traceless two by two matrices, i.e. matrices of the form
$$\begin{pmatrix}a&b\\
c&-a\end{pmatrix}$$

For the isomorphism with the Lie algebra of SO(3,1), go on to pet peeve number two and then consult a relativistic QFT book to find some form of the details.

Keep track of the difference between a Lie algebra and its complexification

This is a much subtler pet peeve than pet peeve number one. It really only comes up in one place, when physicists discuss the Lie algebra of the Lorentz group. They typically put basis elements \(J_j\) (infinitesimal rotations) and \(K_j\) (infinitesimal boosts) together by taking complex linear combinations
$$A_j=J_j+iK_j,\ \ B_j=J_j-iK_j$$
and then note that the commutation relations of the Lie algebra simplify into commutation relations for the \(A_j\) that look like the \(\mathfrak{su}(2)\) commutation relations and the same ones for the \(B_j\). They then announce that
$$SO(3,1)=SU(2) \times SU(2)$$
Besides my pet peeve number one, even if you interpret this as a statement about Lie algebras, it’s not true at all. The problem is that the Lie algebras under discussion are real Lie algebras, you’re just supposed to be taking real linear combinations of their elements. When you wrote down the equations for \(A_j\) and \(B_j\), you “complexified”, getting elements not of \(\mathfrak{so}(3,1)\), but what a mathematician would call the complexification \(\mathfrak{so}(3,1)\otimes \mathbf C\). Really what has been shown is that
$$ \mathfrak{so}(3,1)\otimes \mathbf C = \mathfrak{sl}(2,\mathbf C) + \mathfrak{sl}(2,\mathbf C)$$

It turns out that when you complexify the Lie algebra of an orthogonal group, you get the same thing no matter what signature you start with, i.e.
$$ \mathfrak{so}(3,1)\otimes \mathbf C =\mathfrak{so}(4)\otimes \mathbf C =\mathfrak{so}(2,2)\otimes \mathbf C$$
all of which are two copies of \(\mathfrak{sl}(2,\mathbf C)\). The Lie algebras you care about are what mathematicians call different “real forms” of this and they are different for different signature. What is really true is
$$\mathfrak{so}(3,1)=\mathfrak{sl}(2,\mathbf C)$$
$$\mathfrak{so}(4)=\mathfrak {su}(2) + \mathfrak {su}(2)$$
$$\mathfrak{so}(2,2)=\mathfrak{sl}(2,R) +\mathfrak{sl}(2,R)$$

For details of all this, see my book.

Posted in Uncategorized | 28 Comments

Quick Links

A few quick items:

  • I was very sorry to hear recently of the death of David Goss (obituary here), a mathematician specialist in function fields who was at Ohio State. David had a side interest in physics and was a frequent e-mail correspondent. From what I recall I first heard from him in 2004 soon after the blog started, with my first reaction when I saw the subject and From line that of wondering why David Gross wanted to discuss that particular article about physics with me.

    Over the years he often sent me links to things I hadn’t heard about, with always sensible comments about them and other topics. I had the pleasure of meeting him a couple years ago, when he came to Columbia to drop off his son, who is now a student here. My condolences to his family and friends.

  • The AMS has a wonderful relatively new repository of mostly expository documents called Open Math Notes. The quality of these seems to uniformly be high, and this is a great new service to the community. I hope it will grow and thrive with more contributions.
  • Peter Scholze has now finished his series of talks at the IHES about his ongoing work on local Langlands, the talks are available here.
  • Jean-Francois Dars and Ann Papillault have a web-site called Histoire Courtes, with short pieces in French, many of which are about math and physics research.
  • The LHC is starting to come to life again after a long technical stop. Machine checkout next week, recommissioning with beam during May, physics starts again in June.
  • There’s a new book out with string theory predictions from Gordon Kane, called String Theory and the Real World. Kane has been writing popular pieces about string theory predictions for at least 20 years, with a 1997 piece in Physics Today telling us that string theory was “supertestable”, with a gluino at 200-300 GeV. Over the years, his gluino mass predictions have moved up many times, as the older predictions get falsified. I don’t have a copy of the new book, but at Google Books you can read some of it. From the pages available there I see that

    the compactified M-theory example we will examine below predicts that gluinos will have masses of about 1.5 TeV…
    The bottom line is that with about 40 inverse fb of data the limits on gluinos are just at the lower range of expected masses at the end of 2016.

    Right around the time the book was published, results released at Moriond (see here) claimed exclusion of gluinos up to about 2 TeV. Assumptions may be somewhat different than Kane’s, but I suspect his 1.5 TeV gluino is now excluded.

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The Social Bubble of Physics

Sabine Hossenfelder is on a tear this week, with two excellent and highly provocative pieces about research practice in theoretical physics, a topic on which she has become the field’s most perceptive critic.

The first is in this month’s Nature Physics, entitled Science needs reason to be trusted. I’ll quote fairly extensively so that you get the gist of her argument:

But we have a crisis of an entirely different sort: we produce a huge amount of new theories and yet none of them is ever empirically confirmed. Let’s call it the overproduction crisis. We use the approved methods of our field, see they don’t work, but don’t draw consequences. Like a fly hitting the window pane, we repeat ourselves over and over again, expecting different results.

Some of my colleagues will disagree we have a crisis. They’ll tell you that we have made great progress in the past few decades (despite nothing coming out of it), and that it’s normal for progress to slow down as a field matures — this isn’t the eighteenth century, and finding fundamentally new physics today isn’t as simple as it used to be. Fair enough. But my issue isn’t the snail’s pace of progress per se, it’s that the current practices in theory development signal a failure of the scientific method…

If scientists are selectively exposed to information from likeminded peers, if they are punished for not attracting enough attention, if they face hurdles to leave a research area when its promise declines, they can’t be counted on to be objective. That’s the situation we’re in today — and we have accepted it.

To me, our inability — or maybe even unwillingness — to limit the influence of social and cognitive biases in scientific communities is a serious systemic failure. We don’t protect the values of our discipline. The only response I see are attempts to blame others: funding agencies, higher education administrators or policy makers. But none of these parties is interested in wasting money on useless research. They rely on us, the scientists, to tell them how science works.

I offered examples for the missing self-correction from my own discipline. It seems reasonable that social dynamics is more influential in areas starved of data, so the foundations of physics are probably an extreme case. But at its root, the problem affects all scientific communities. Last year, the Brexit campaign and the US presidential campaign showed us what post-factual politics looks like — a development that must be utterly disturbing for anyone with a background in science. Ignoring facts is futile. But we too are ignoring the facts: there’s no evidence that intelligence provides immunity against social and cognitive biases, so their presence must be our default assumption…

Scientific communities have changed dramatically in the past few decades. There are more of us, we collaborate more, and we share more information than ever before. All this amplifies social feedback, and it’s naive to believe that when our communities change we don’t have to update our methods too.

How can we blame the public for being misinformed because they live in social bubbles if we’re guilty of it too?

There’s a lot of food for thought in the whole article, and it raises the important question of why the now long-standing dysfunctional situation in the field is not being widely acknowledged or addressed.

For some commentary on one aspect of the article by Chad Orzel, see here.

On top of this, yesterday’s blog entry at Backreaction was a good explanation of the black hole information paradox, coupled with an excellent sociological discussion of why this has become a topic occupying a large number of researchers. That a large number of people are working on something and they show no signs of finding anything that looks interesting has seemed to me a good reason to not pay much attention, so that’s why I’m not that well-informed about exactly what has been going on in this subject. When I have thought about it, it seemed to me that there was no way to make the problem well-defined as long as one lacks a good theory of quantized space-time degrees of freedom that would tell one what was going on at the singularity and at the end-point of black hole evaporation.

Hossenfelder describes the idea that what happens at the singularity is the answer to the “paradox” as the “obvious solution”. Her take on why it’s not conventional wisdom is provocative:

What happened, to make a long story short, is that Lenny Susskind wrote a dismissive paper about the idea that information is kept in black holes until late. This dismissal gave everybody else the opportunity to claim that the obvious solution doesn’t work and to henceforth produce endless amounts of papers on other speculations.

Excuse the cynicism, but that’s my take on the situation. I’ll even admit having contributed to the paper pile because that’s how academia works. I too have to make a living somehow.

So that’s the other reason why physicists worry so much about the black hole information loss problem: Because it’s speculation unconstrained by data, it’s easy to write papers about it, and there are so many people working on it that citations aren’t hard to come by either.

I hope this second piece too will generate some interesting debate within the field.

Note: It took about 5 minutes for this posting to attract people who want to argue about Brexit or the political situation in the US. Please don’t do this, any attempts to turn the discussion to those topics will be ruthlessly deleted.

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Some Math and Physics Interactions

Quanta magazine has a new article about physicists “attacking” the Riemann Hypothesis, based on the publication in PRL of this paper. The only comment from a mathematician evaluating relevance of this to a proof of the Riemann Hypothesis basically says that he hasn’t had time to look into the question.

The paper is one of various attempts to address the Riemann Hypothesis by looking at properties of a Hamiltonian quantizing the classical Hamiltonian xp. To me, the obvious problem with an attempt like this is that I don’t see any use of deep ideas about either number theory or physics. The set-up involves no number theory, and a simple but non-physical Hamiltonian, with no use of significant input from physics. Without going into the details of the paper, it appears that essentially a claim is being made that the solution to the Riemann Hypothesis involves no deep ideas, just some basic facts about the analysis of some simple differential operators. Given the history of this problem, this seems like an extraordinary claim, backed by no extraordinary evidence.

I suspect that the author of the Quanta article found no experts in mathematics willing to comment publicly on this, because none found it worth the time to look carefully at the article, since it showed no engagement with the relevant mathematical issues. A huge amount of effort in mathematics over the years has gone into the study of the sort of problems that arise if you try and do the kind of thing the authors of this article want to do. Why are they not talking to experts, formulating their work in terms of well-defined mathematics of a proven sort, and referencing known results?

Maybe I’m being overly harsh here, this is not my field of expertise. Comments from experts on this definitely welcome (and those from non-experts strongly discouraged).

While these claims about the Riemann Hypothesis at Quanta look like a bad example of a math-physics interaction, a few days ago the magazine published something much more sensible, a piece by IAS director Robbert Dijkgraaf entitled Quantum Questions Inspire New Math. Dijkgraaf emphasizes the role ideas coming out of string theory and quantum field theory have had in mathematics, with two high points mirror symmetry and Seiberg-Witten duality. His choice of mirror symmetry undoubtedly has to do with the year-long program about this being held by the mathematicians at the IAS. He characterizes this subject as follows:

It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofs. Mathematicians are close to applying this exactitude to homological mirror symmetry, a program that vastly extends string theory’s original idea of mirror symmetry. In a sense, they’re writing a full dictionary of the objects that appear in the two separate mathematical worlds, including all the relations they satisfy. Remarkably, these proofs often do not follow the path that physical arguments had suggested. It is apparently not the role of mathematicians to clean up after physicists! On the contrary, in many cases completely new lines of thought had to be developed in order to find the proofs. This is further evidence of the deep and as yet undiscovered logic that underlies quantum theory and, ultimately, reality.

I very much agree with him that there’s an underlying logic and mathematics of quantum theory which we have not fully understood (my book is one take on what we do understand). I hope many physicists will take the search for new discoveries along these lines to heart, with progress perhaps flowing from mathematics to physics, which could sorely use some new ideas about unification.

Update: Some comments sent to me from a mathematician that I think give a good idea of what this looks like to experts in number theory:

The “boundary condition” is imposing an identification with zeta zeros by fiat, so the linkage of any of this to RH is basically circular. The paper at best just redefines the problem, without providing any genuine new insight. More specifically, as the experience of more than 100 years has shown, there are a zillion ways to recast RH without providing any real progress; this is yet another (if it makes any rigorous sense, which it does not yet do, yet the absence of rigor is not the reason for skepticism about the value of this paper, whatever the pedigree of the authors may be).

One has to find a way of encoding the zeta function that is not tautological (unlike the case here), and that is where deep input from number theory would have to come in. This is really the essential point that all papers of this sort fail to recognize.

Real insight into the structures surrounding RH have arisen over the past decades, such as the work of Grothendieck and Deligne in the function field analogue that provided a spectral interpretation through the development of striking new tools inspired by novel insights of Weil. In particular, the appearance of the appropriate zeta functions in such settings is not imposed by fiat, but is the outcome of a massive amount of highly non-trivial constructions and arguments. In another direction, compelling evidence and insight has come from the “random matrix theory” of the past couple of decades (work of Katz-Sarnak et al.) was inspired by observations originating with Dyson merged with work of the number theorist Montgomery.

Number theorists making a major advance on the puzzles of quantum gravity without providing anAbdelmalek Abdesselam identifiable new physical insight is about as likely as physicists making a real advance towards RH without providing an identifiable new number-theoretic insight. There is no doubt that physical insights have led to important progress in mathematics. But there is nothing in this paper to suggest it is doing anything more than providing (at best) yet another ultimately tautological reformulation by means of which no progress or insight should be expected.

Update: Another way to state the problem with this kind of approach to the RH is that without number theoretic input, it is likely to give a much too strong result (proving analogs of the RH for functions that don’t satisfy the RH). For example, see the comment here (I don’t know if this correct, but it explains the potential problem).

Update: Nature Physics highlights the Bender et al. paper with “Carl Bender and colleagues have paved the way to a possible solution [of the RH] by exploiting a connection with physics. Some wag there has categorized this work as work with subject term “interstellar medium”.

Update: There’s an article about the Bender et al. paper here, with extensive commentary from oneAbdelmalek Abdesselam of the authors, Dorje Brody, who addresses some of the questions raised here (for example, why PRL if it’s not a physics topic?).

Update: Belissard has put up a short paper on the arXiv explaining the idea of the Bender et al. paper, as well as the analytical problems one runs into if one tries to get a proof of the RH in this way.

Update: One of the authors has posted on the arXiv a note with more precise details of the construction of a version of the operator discussed in the PRL paper.

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