Pet Peeves | Not Even Wrong

It’s getting hard to wake up every day, read the latest news of the slaughter of civilians in Gaza and the plans to finish off or exile the rest, then go through the two ID checks at the campus gate designed to make sure that no protests about this happen on campus, and when I get to my office resist the temptation to write a rant. But no one wants to read this, and it would probably violate the new rules we’re now living under here. So, I’ll complain instead about some pet peeves about theoretical particle physics.

This week there is the newest edition of a Pre-SUSY School in Santa Cruz, designed to train graduate students and postdocs. My first pet peeve is the whole concept of the thing. It starts off with an Introduction to Supersymmetry which introduces the MSSM, but why is anyone training graduate students and postdocs to work on supersymmetric extensions of the Standard Model? These were a failed idea pre-LHC (see my book…), and the LHC results conclusively confirm that failure.

The Introduction to Supersymmetry lectures given by Ben Allanach are an updated version of similar lectures given at other summer schools designed to train people in SUSY. These lectures trigger several of my pet peeves even before they get to SUSY. I’ve written about some of this before in detail, see here.

The first pet peeve is about the insistence on using the same notation for a Lie group and its Lie algebra. In both versions of the lecture notes, we’re told that
$$SO(1,3)\cong SL(2,\mathbf C)$$
and
$$SO(1,3) \cong SU(2) \times SU(2)$$
There are lots of problems with this. In the first case this is about the group $SO(1,3)$. In the next it’s about the Lie algebra $SO(1,3)$, but the same symbol is being used for both. One would guess that $\cong$ means two things are isomorphic, but that’s not true in either case.

More completely, in the older version of the notes, we’re told

there is a homeomorphism (not an isomorphism)
$$SO(1,3)\cong SL(2,\mathbf C)$$

“Homeomorphism” is nonsense, which has been fixed in the newer version to

there is a homomorphism (not an isomorphism)
$$SO(1,3)\cong SL(2,\mathbf C)$$

There’s still the problem of why a homomorphism that isn’t a isomorphism is getting written as $\cong$. The text does later explain what is really going on (there’s a 2-1 Lie group homomorphism from $SL(2,\mathbf C)$ to $SO(1,3)$).

The other equation is more completely given as

locally (i.e. in terms of the algebra), we have a correspondence
$$SO(1,3) \cong SU(2) \times SU(2)$$

The “locally (i.e. in terms of the algebra)” does help with the fact that the symbol $SO(1,3)$ means something different here, that it’s the Lie algebra of $SO(1,3)$ not the Lie group $SO(1,3)$ of the other equation. The word “correspondence” gives a hint that $\cong$ doesn’t mean “isomorphism”, but doesn’t tell you what it does mean.

A minor pet peeve here is calling the Lie algebra of a Lie group its “algebra”, dropping the “Lie”. For any group, its “group algebra” is something completely different (the algebra of functions on the group with the convolution product). Mostly when mathematicians talk about “algebras” they mean associative algebras, and a Lie algebra is not associative. Why drop the “Lie”?

What’s really true (as explained here) is that the Lie algebra of $SO(1,3)$ and the Lie algebra of $SU(2)\times SU(2)$ are different real Lie algebras with the same complexification (the Lie algebra of $SL(2,\mathbf C)\times SL(2,\mathbf C)$). In the earlier version of the notes there’s nothing about this. There’s the usual definition of two complex linear combinations
$$A_i=\frac{1}{2}(J_i +iK_i),\ \ B_i=\frac{1}{2}(J_i -iK_i)$$
of basis elements $J_i$ and $K_i$ of the Lie algebra of $SO(1,3)$, giving two separate copies of the Lie algebra of $SU(2)$. All we’re told there is that “these linear combinations are neither hermitian not anti-hermitian”.

In the newer version, this has been changed to describe these linear combinations as “hermitian linear combinations”. We’re told

The matrices representing both $J_i$ and $K_i$ have elements that are pure imaginary. (2.2) then implies that
$$(A_i)^∗ = −B_i$$
which is what discriminates $SO(4)$ from $SO(1, 3)$.

which I don’t really understand. Part of the source of the confusion here is confusion between Lie algebra elements (which don’t have a notion of Hermitian adjoint) and Lie algebra representation matrices for a unitary representation on a complex vector space (which do). Here there are different defining representations involved (spin for $SU(2)$ and vector for $SO(1,3)$).

There’s then a confusing version of the correct “$SO(1,3)$ and the Lie algebra of $SU(2)\times SU(2)$ are different real algebras with the same complexification”

the Lie algebra of $SO(1, 3)$ only contains two mutually commuting copies of the real Lie algebra of $SU(2)$ after a suitable complexification because only certain complex linear combinations of the Lie algebra of $SU(2) \times SU(2)$ are isomorphic to the Lie algebra of SO(1, 3).

Here’s an idea for a summer school for physics theory grad students and postdocs: teach them properly about $SO(3,1)$, $SO(4)$, their spin double covers, Lie algebras, complexifications of their Lie algebras and their representations. About SUSY extensions of the SM, just tell them these are a failure they should ignore (other than as a lesson for what not to do in the future).

Update: I strongly recommend Sabine Hossenfelder’s latest video, Scientific research has big problems, and it’s getting worse. She’s been attacked over the years for this kind of critique, most recently as “a disgusting fraud peddling propaganda for fascist oligarchs”, but it’s a very important one that deserves to be taken seriously. In the video she starts out by pointing to a huge problem with scientific research that is getting much, much worse very fast: paper mills and bogus papers, a problem now being turbocharged by AI.

SUSY research is I think one of the things that has, for good reason, motivated her critique. Why is there a huge still active field of people writing papers about a failed idea? What are the incentives and sociology that create this sort of phenomenon? The topic of this posting explains where I very much disagree with Hossenfelder. She likes to name the problem in fundamental physics as “Mathematical Fiction” (quotes others as describing the problem as “Mathematical Gymnastics” or “Mathematical Cosmology”). But looking at the training of SUSY researchers here, the problem is not too much mathematics, but too little. Too many physicists firmly believe that understanding the basic details of what they are doing is a waste of time, that mathematician’s insistence on clear, unambiguous and precise statements is nothing but pedantry. But if you have only a hazy idea of what the fundamental objects in your theory are and how they behave, absent the discipline of experimental tests, you have no hope of distinguishing what works from what doesn’t. SUSY research is an extreme case, where even failed experimental tests only slow the enterprise down, don’t stop it.

Given what is happening in the US, it is important to make clear the sort of reevaluation of federal support of science that Hossenfelder’s critiques implies is needed. Such a reevaluation would require a strong dedication to distinguishing truth from lies. The current defunding of science at US research universities based on pro-genocide fanaticism and a mountain of lies about “antisemitism” is the opposite of what is needed.

Update: The problem with theorists being totally confused about Lie groups vs. Lie algebras and the symmetry groups of spin in 3 and 4 dimensions is not just in the SUSY subfield. For another example, take a look at Appendix A here which starts off with a definition of the SU(2) group that is half a definition of the Lie algebra (up to i, self-adjoint matrices), half a definition of the group (det=1). Things go downhill from there in the rest of the section. Why would anyone write this “pedagogical” discussion when they didn’t understand this at all? Why did none of the five co-authors or a referee notice that this section was complete nonsense?

Update: For the latest on SUSY claims see here. The same story as at any time for the past forty years: no superpartners = no problem, since SUSY “predicts” superpartners just beyond the reach of current and past searches. Nowadays this is accomplished by invoking the landscape and “string naturalness”.