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 | 15 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.

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 | 26 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.

Posted in Uncategorized | 22 Comments

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.

Posted in Uncategorized | 71 Comments

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.

Posted in Uncategorized | 21 Comments

New LHC Results

This week results are being presented by the LHC experiments at the Moriond (twitter here) and Aspen conferences. While these so far have not been getting much publicity from CERN or in the media, they are quite significant, as first results from an analysis of the full dataset from the 2015+2016 run at 13 TeV, This is nearly the design energy (14 TeV) and a significant amount of data (36 inverse fb/experiment). The target for this year’s run (physics to start in June) is another 45 inverse fb and we’ll not start to hear about results from that until a year or so from now. For 14 TeV and significantly larger amounts of data, the wait will be until 2021 or so.

The results on searches for supersymmetry reported this week have all been negative, further pushing up the limits on possible masses of conjectured superparticles. Typical limits on gluino masses are now about 2.0 TeV (see here for the latest), up from about 1.8 TeV last summer (see here). ATLAS results are being posted here, and I believe CMS results will appear here.

This is now enough data near the design energy that some of the bets SUSY enthusiasts made years ago will now have to be paid off, in particular Lubos Motl’s bet with Adam Falkowski, and David Gross’s with Ken Lane (see here). A major question now facing those who have spent decades promoting SUSY extensions of the Standard Model is whether they will accept the verdict of experiment or choose a path of denialism, something that I think will be very damaging for the field. The situation last summer (see here) was not encouraging, maybe we’ll soon see if more conclusive data has any effect.

If the negative news from the LHC is getting you down, for something rather different and maybe more promising, I recommend the coverage of the latest developments in neutrino physics here.

Update: Lubos has paid off his bet with Jester. Losing the bet hasn’t dimmed his enthusiasm for SUSY. No news on whether David Gross has conceded his bet.

Posted in Experimental HEP News | 60 Comments

This Time It’s For Real

Several months ago I was advertising a “Final draft version” of the book I’ve been working on forever. A month or two after that though, I realized that I could do a more careful job with some of the quantum field theory material, bringing it in line with some standard rigorous treatments (this is all free quantum fields). So, I’ve been working on that for the past few months, today finally got to the end of the process of revising and improving things. My spring break starts today, and I’ll be spending most of it in LA and Death Valley on vacation, blogging should be light to non-existent.

Another big improvement is that there are now some very well executed illustrations, the product of work in TikZ by Ben Dribus.

I’m quite happy with how much of the book has turned out, and would like to think that it contains a significant amount of material not readily available elsewhere, as well as a more coherent picture of the subject and its relationship to mathematics than usual. By the way, while finishing work on the chapter about quantization of relativistic scalar fields, I noticed that Jacques Distler has a very nice new discussion on his blog of the single-particle theory.

There’s a chance I might still make some more last-minute changes/additions, but the current version has no mistakes I’m aware of. Any suggestions for improvements/corrections are very welcome. Springer will be publishing the book at some point, but something like the current version available now will always remain available on my website.

Update: After writing to someone to answer a question and what is and isn’t in the book, and other things to read, I thought maybe I’d post here part of that answer:

For the main topics about QM and representation theory that I cover in the book, I don’t know of a better reference, even assuming an excellent math background. That’s one of the main reasons I wrote the thing… The problem with other books on QM for mathematicians (e.g. Hall, which is very good on the analysis point of view) is that they don’t do much from the representation theory point of view. Weyl’s book was written very early, when a lot of what was going on was still unclear, I don’t think it’s a very good place to try and learn this material from. One topic that is in there that I don’t cover at all is basically Schur-Weyl duality, but even for that arguably Weyl is not a good place to learn that theory.

One thing to keep in mind about my book is that the early chapters are deceptive. I wanted to start out with very simple things, make the simplest examples accessible to as many people as possible, mathematicians or physicists. If you know basic facts about Lie groups, Lie algebras, finite dim unitary representations, Fourier analysis and how to use it to solve e.g. the heat equation, then the first quarter of the book is only going to be of interest in telling you about some applications of math you know. Mathematicians generally should be learning the basic rep theory elsewhere (lots of good books on these topics, and the main reason I’m doing many things in a mathematically sketchy way is that doing them in full would take too long, and has been done better elsewhere). In early chapters, all I’m really doing is working out very special cases of Lie groups/algebras that are rank one or products of rank one, and the irreps of sl(2,C). I never touch higher rank or general semisimple theory (would argue this actually doesn’t get used much in physics, other than some simple SU(3) examples).

Around chapter 12 though, things get much more non-trivial. From a high mathematical level, a lot of what’s going on in the middle part of the book is the representation theory of the Heisenberg group (over R and C) and the implications of the action by the symplectic group by automorphisms (e.g. the metaplectic representation). This is done in a very detailed and concrete way, together with the relation to QM, although for some of the trickier parts of the mathematics (especially the analysis, e.g. the proof of the Stone-von Neumann theorem) I just give references. This is followed by discussing Clifford algebras, the orthogonal group and spinors (over R and C), in a very parallel way (interchanging symmetric and antisymmetric). I wish I knew of a good pure mathematics source for this material aimed at students, stripped of the quantum mechanics apparatus, but I don’t. It (as well as material about reps of the Euclidean groups) is not covered in any conventional rep theory textbook I’m aware of.

Much of the last third of the book, on quantum field theory, I think is just inherently quite challenging, for either mathematicians or physicists. From the representation theory point of view, the basic framework is that of an infinite dimensional Heisenberg group or Clifford algebra, but this is a difficult mathematical subject, and I think the physics point of view helps make clear why. For this stuff the rigorous treatments are quite specialized, I try and do some justice to what the main issues are and give references that provide the details.

Posted in Quantum Theory: The Book | 39 Comments

Can the Laws of Physics be Unified?

There’s a new book out this week from Princeton University Press, Paul Langacker’s Can the Laws of Physics Be Unified? (surely this is a mistake, but there’s also an ISBN number for a 2020 volume with the same name by Tony Zee). It’s part of a Princeton Frontiers in Physics series, in which all the books have titles that are questions. The other volumes all ask “How…” or “What…” questions, but the question of this volume is of a different nature, and unfortunately the book unintentionally gives the answer you would expect from Hinchliffe’s rule or Betteridge’s law.

This is not really a popular book, rather is accurately described by the author as “colloquium-level”. Lots of equations, but not much detail explaining exactly what they mean, for that some background is needed. The first two-thirds of the book is a very good summary of the Standard Model. For more details, Langacker has a textbook, The Standard Model and Beyond, which will have a second edition coming out later this year.

The last third of the book consists of two chapters addressing the question of the title, beginning with “What Don’t We Know?”. Here the questions are pretty much the usual suspects:

  • Why the SM spectrum, with its masses and mixing angles?
  • The hierarchy problem.
  • The strong CP problem.
  • Quantum gravity.
  • Problems rooted in the cosmological model: Baryogenesis, dark matter, dark energy and the CC,

In addition, there are problems listed that are only problems if you philosophically think that a good unified model should be more generic than the SM, leading you to ask: why no FCNC? why no EDM?, why no proton decay?

The last chapter “How will we find out?” lists the usual suspects for ideas about BSM physics: SUSY, compositeness, extra dimensions, hidden sectors, GUTS, string theory. We are told that this is a list of “many promising ideas”. While in general I wouldn’t argue with most of the claims of the book, here I think the author is spouting utter nonsense. The ideas he describes are ancient, many going back 40 years. In many cases they weren’t promising to begin with, introducing a large and complex set of new degrees of freedom without explaining much at all about the SM. Decades of hard work by theorists and experimentalists have not been kind to these ideas. No compelling theoretical models have emerged, and experimental results have been strongly negative, with the LHC putting a large number of nails into the coffins of these ideas. They’re not “promising”, they’re dead.

Langacker does repeatedly point out the problems such ideas have run into, but instead of leaving it at “we don’t know”, he unfortunately keeps bringing up as answer “the multiverse did it”. On page 151 we’re told the most plausible explanation for the CC is “the multiverse did it”, on page 160-163 we’re given “multiverse did it” anthropic explanations for interaction strengths, fermion masses, the Higgs VEV, and the CC. Pages 167-173 are a long argument for “the multiverse did it”. The problem that this isn’t science because it is untestable is dismissed with the argument that it “may well be correct”, and maybe somebody someday will figure out a test. On page 203 we’re told that string theory provides the landscape of vacua necessary to show that “the multiverse did it”.

The treatment of string theory has all of the usual problems: we’re assured that string theory is “conceptually simple”, despite no one knowing what the theory really is. The only problem is that of the “technical details” of constructing realistic vacua. I won’t go on about this, I once wrote a whole book…

In the end, while Langacker expresses the hope that “sometime in the next 10, 50, or 100 years” we will see a successful fully unified theory, there’s nothing in the book that provides any reason for such a hope. There is a lot that argues against such a hope, in particular a lot of argument in favor of giving up and signing up for a multiverse pseudo-scientific endpoint for the field. I suspect the author himself doesn’t realize how much the argument of the book is stacked against his expressed hope and in favor of a negative answer to the title’s question.

Update: If you just can’t get enough multiverse mania, you can watch Joseph Silk’s talk Should We Trust a Theory? (more talk materials here). I’m not quite sure, but I think we agree that the multiverse is not currently science (he writes “The multiverse might or might not exist, but no physicist should waste his or her time chasing the unchaseable”), but not about about string theory. I have no idea what is behind his claim that “String theory has been very successful”, and, since he’s not a string theorist, I suspect that neither does he.

Posted in Book Reviews, Multiverse Mania | 25 Comments

Reality is Not What It Seems

This Sunday’s New York Times has a rather hostile review by Lisa Randall of Carlo Rovelli’s popular book Reality is Not What It Seems, which has recently come out in English in the US. Rovelli responds with a Facebook post. Another similar recent book by Rovelli got a much more positive review in the NYT, his Seven Short Lessons on Physics.

I haven’t written about these two books mainly because I don’t think I have anything interesting to say about either of them (although if someone had asked me to review one of them I might have tried to come up with something). They’re aimed very much not at physicists but at a popular audience that doesn’t know much about physics. From the parts I’ve read they seem to do a good job of writing for such an audience, and I noticed nothing that seemed to me either objectionable or particularly unusual. Rovelli’s two slightly different angles on this topic are an interest in the ancient history of speculation about physics and a background in loop quantum gravity rather than HEP theory/string theory. Instead of wading into the controversy over string theory, he just ignores it and writes about what he finds interesting.

I’m not so sure why, since to me this seems harmless if not particularly compelling, but Randall strongly objects to Rovelli’s attempts to draw connections between modern physics and classical philosophy:

Wedging old ideas into new thinking is analogous to equating thousand-dollar couture adorned with beads and feathers and then marketed as “tribal fashion” to homespun clothing with true cultural and historical relevance. Ideas about relativity or gravity in ancient times weren’t the same as Einstein’s theory. Art (and science) are in the details. Either elementary matter is extended or it is not. The universe existed forever, or it had a beginning. Atoms of old aren’t the atoms of today. Egg and flour are not a soufflé. Without the appropriate care, it all just collapses.

She’s also quite critical of the way Rovelli handles the unavoidable problem of writing about a complicated technical subject for the public:

The beauty of physics lies in its precise statements, and that is what is essential to convey. Many readers won’t have the background required to distinguish fact from speculation. Words can turn equations into poetry, but elegant language shouldn’t come at the expense of understanding. Rovelli isn’t the first author guilty of such romanticizing, and I don’t want to take him alone to task. But when deceptively fluid science writing permits misleading interpretations to seep in, I fear that the floodgates open to more dangerous misinformation.

Here I’m a bit mystified as to why she finds Rovelli any more objectionable than any other similar author (or maybe she doesn’t, and he just happened to be the lucky one to have the first such book she was asked to review in the New York Times). As should be clear from this blog and book reviews that I’ve written, I agree with Randall about a problem that she leads off the review with:

Compounding the author’s challenge is the need to distinguish between speculation, ideas that might be verified in the future, and what is just fanciful thinking.

However, to me it seemed that Rovelli met this challenge better than many, far better than any of the huge popular literature about supersymmetry, string theory and the multiverse. She may be right that someone not paying careful attention could get the wrong idea from Rovelli about cosmological loop quantum gravity models. It’s equally true though that readers of her own books about extra dimensions, dark matter and the dinosaurs might come away not understanding exactly what the strength of evidence was for those speculations.

Note added for clarification 3/6/2017: the following is not part of the commentary on Randall’s review, it’s another related topic I thought readers would find interesting. The relation between the two parts is that they both have to do with the question of distinguishing solid argument/speculation, but it’s not about Randall, and the context is different (communicating with other physicists versus communicating with the general public}.

On this question of how/whether physicists (here mathematicians are very, very different) make clear what is a solid argument and what is just speculation, another interesting case is that of Nima Arkani-Hamed, who came to prominence in particle theory with Randall, both of them working on extra dimensional models. Both of them got a huge amount of attention for this, from the public and from within physics, although these ideas were always highly speculative and unlikely to work out.

There’s a wonderful new “Storygram” by George Musser of a great profile by Natalie Wolchover of Arkani-Hamed. It’s all well worth reading, but related to the topic at hand I was struck by the following:

Arkani-Hamed considers his tendency to speculate a personal weakness. “This is not false modesty, it’s really a personal weakness, but it’s true, so there’s nothing I can do about it,” he said. “It’s important for me while I’m working on something to be very ideological about it. And then, of course, it’s also important after you are done to forget the ideology and move on to another one.”

Arkani-Hamed is an incredibly compelling speaker, but his talks often have struck me as putting forward very strongly some particular speculative point of view, while ignoring some of the obvious serious problems. If you’re not pretty well-informed on the subject, you might get misled… From the quote above he seems to have a fair amount of self-awareness about this. Also interesting in this context is his talk last year at Cornell on The Morality of Fundamental Physics. He gives an inspiring account of the intellectual value system of theoretical physics at its best. On the other hand, he pays no attention to the very real tension between that value system and the way people actually pursue their work, often very “ideologically”. For particle theory in particular and the current situation it finds itself in, this seems to me an important issue for practitioners to be thinking about.

Posted in Uncategorized | 19 Comments

Bertram Kostant 1928-2017

I was sorry to just hear via a comment here about the recent death of Bert Kostant, at the age of 88. MIT has a story about him here.

Kostant was a major figure in the field of representation theory, and perhaps the leading one during the second half of the twentieth century among those with a serious interest in the relations between representation theory and quantum theory. These relations have for a long time now been a deep source of fascination to me, and Kostant’s work has had a great impact on how I think about the subject.

I’ll just list here some of his major papers that I’ve spent significant amounts of time with, characterized by a few major themes:

Borel-Weil-Bott, Lie algebra cohomology, BRST and Dirac cohomology

Quantization of the dual of a Lie algebra, W-algebras

The dual of a Lie algebra is a Poisson manifold, and you can ask what happens when you quantize this. For semisimple Lie algebra, reduction with respect to the nilradical is an idea that Kostant pursued, with two examples the following two papers. Applied to loop groups, this is a central idea of the geometric Langlands program. The theory of W-algebras is also an outgrowth of this.

Geometric quantization theory and co-adjoint orbits

Starting around 1970 Kostant did a great deal of work developing the theory of “geometric quantization” and the idea of quantizing co-adjoint orbits to get representations (other figures to mention in this context are Kirillov and Souriau). Some of his papers on this are:

All of the three general themes above are closely intertwined, and the relations between them indicate that there is still a lot more to be understood about how quantum theory and representation theory are related, with Kostant’s work undoubtedly playing a large role in developments to come.

Posted in Uncategorized | 4 Comments