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).
