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Physicists, smart folks use same symbols for Lie groups, algebras for good reasons

I have always been amazed by the sheer stupidity and tastelessness of the people who aren't ashamed of the likes of Peter Woit. He is obviously a mediocre man with no talents, no achievements, no ethics, and no charisma but because of the existence of many people who have no taste and who want to have a leader in their jihad against modern physics, he was allowed to talk about physics as if his opinions mattered.

Woit is a typical failing-grade student who simply isn't and has never been the right material for college. His inability to learn string theory is a well-known aspect of this fact. But most people in the world – and maybe even most of the physics students – misunderstand string theory. But his low math-related intelligence is often manifested in things that are comprehensible to all average or better students of physics.

Two years ago, Woit argued that

the West Coast metric is the wrong one.
Now, unless you are a complete idiot, you must understand that the choice of the metric tensor – either \(({+}{-}{-}{-})\) or \(({-}{+}{+}{+})\) – is a pure convention. The metric tensor \(g^E_{\mu\nu}\) of the first culture is simply equal to minus the metric tensor of the second culture \(g^W_{\mu\nu}\), i.e. \(g^E_{\mu\nu} = - g^W_{\mu\nu}\), and every statement or formula written with one set of conventions may obviously be translated to a statement written in the other, and vice versa. The equations or statements basically differ just by some signs. The translation from one convention to another is always possible and is no more mysterious than the translation from British to U.S. English or vice versa.

How stupid do you have to be to misunderstand this point, that there can't be any "wrong" convention for the sign? And how many people are willing to believe that someone's inability to get this simple point is compatible with the credibility of his comments about string theory?

Well, this individual has brought us a new ludicrous triviality of the same type,
Two Pet Peeves
We're told that we mustn't use the same notation for a Lie group and a Lie algebra. Why? Because Tony Zee, Pierre Ramond, and partially Howard Georgi were using the unified notation and Woit "remember[s] being very confused about this when I first started studying the subject". Well, Mr Woit, you were confused simply because you have never been college material. But it's easier to look for flaws in Lie groups and Lie algebras than in your own worthless existence, right?

Many physicists use the same symbols for Lie groups and the corresponding Lie algebras for a simple reason: they – or at least their behavior near the identity (or any other point on the group manifold) – is completely equivalent. Except for some global behavior, the information about the Lie group is completely equivalent to the information about the corresponding Lie algebra. They're just two languages to talk about the same thing.

Just to be sure, in my and Dr Zahradník's textbook on linear algebra, we used the separate symbols and I love the fraktur fonts. In Czechia and maybe elsewhere, most people who are familiar with similar fonts at all call them "Schwabacher" but strictly speaking, Textura, Rotunda, Schwabacher, and Fraktur are four different typefaces. Schwabacher is older and was replaced by Fraktura in the 16th century. In 1941, Hitler decided that there were too many typos in the newspapers and that foreigners couldn't decode Fraktura which diminishes the importance of Germany abroad, so he banned Fraktura and replaced it with Antiqua.

When we published our textbook, I was bragging about the extensive index that was automatically created by a \({\rm \LaTeX}\) macro. I told somebody: Tell me any word and you will see that we can find it in the index. In front of several witnesses, the first person wanted to humiliate me so he said: "A broken bone." So I abruptly responded: "The index doesn't include a 'broken bone' literally but there's a fracture in it!" ;-) Yes, I did include a comment about the font in the index. You know, the composition of the index was as simple as placing the command like \placeInTheIndex{fraktura} in a given place of the source. After several compilations, the correct index was automatically created. I remember that in 1993 when I began to type it, one compilation of the book took 15 minutes on the PCs in the computer lab of our hostel! When we received new 90 GHz frequency PCs, the speed was almost doubled. ;-)

OK, I don't want to review elementary things because some readers know them and wouldn't learn anything new, while others don't know these things and a brief introduction wouldn't help them. But there is a simple relationship between a Lie algebra and a Lie group. You may obtain the elements of the group by a simple exponentiation of an element of a Lie algebra. For this reason, all the "structure coefficients" \(f_{ij}{}^k\) that remember the structure of commutators\[

[T_i,T_j] = f_{ij}{}^k T_k

\] contain the same information as all the curvature information about the group manifold near the identity. The Lie algebra simply is the tangent space of the group manifold around the identity (or any element) and all the commutators in the Lie algebra are equivalent to the information about the distortions that a projection of the neighborhood of the identity in the group manifold to a flat space causes.

We often use the same symbols because it's harder to write the gothic fonts. More importantly,
whenever a theory, a solution, or a situation is connected with a particular Lie group, it's also connected with the corresponding Lie algebra, and vice versa!
That's the real reason why it doesn't matter whether you talk about a Lie group or a Lie algebra. We use their labels for "identification purposes" and the identification is the same whether you have a Lie group or a Lie algebra in mind. A very simple example:
There exist two rank-8, dimension-496 heterotic string theories whose gauge groups in the 10-dimensional spacetime are \(SO(32)\) and \(E_8\times E_8\), respectively.

There exist two rank-8, dimension-496 heterotic string theories whose gauge groups in the 10-dimensional spacetime are (or have the Lie algebras) \({\mathfrak so}(32)\) and \({\mathfrak e}_8\oplus {\mathfrak e}_8\), respectively.
I wrote the sentence in two ways. The first one sort of talks about the group manifolds while the second talks about Lie algebras. The information is obviously almost completely equivalent.

Well, except for subtleties – the global choices and identifications in the group manifold that don't affect the behavior of the group manifold in the vicinity of the identity element. If you want to be careful about these subtleties, you need to talk about the group manifolds, not just Lie algebras, because the Lie algebras "forget" the information about these global issues.

So you might want to be accurate and talk about the Lie groups in 10 dimensions – and say that the allowed heterotic gauge groups are \(E_8\times E_8\) and \(SO(32)\). However, this effort of yours would actually make things worse because when you use a language that has the ambition of being correct about the global issues, it's your responsibility to be correct about them, indeed, and chances are that your first guess will be wrong!

In particular, the "\(SO(32)\)" heterotic string also contains spinors. So a somewhat smart person could say that the gauge group of that heterotic string is actually \(Spin(32)\), not \(SO(32)\). However, that would be about as wrong as \(SO(32)\) itself – almost no improvement – because the actual perturbative gauge group of this heterotic theory is isomorphic to\[

Spin(32) / \ZZ_2

\] where the \(\ZZ_2\) is chosen in such a way that the group is not isomorphic to \(SO(32)\). It's another \(\ZZ_2\) from the center isomorphic to \(\ZZ_2\times \ZZ_2\) that allows left-handed spinors but not the right-handed ones! By the way, funnily, the S-dual theory is type I superstring theory whose gauge group – arising from Chan-Paton factors of the open strings – seems to be \(O(32)\). However, the global form of the gauge group gets modified by D-particles, the other half of \(O(32)\) beyond \(SO(32)\) is broken, and spinors of \(Spin(32)\) are allowed by the D-particles so non-perturbatively, the gauge group of type I superstring theory agrees with that of the heterotic S-dual theory including the global subtleties.

(Peter Woit also ludicrously claims that physicists only need three groups, \(U(1),SU(2), SO(3)\). That may have been almost correct in the 1920s but it's surely not true in the 21st century particle physics. If you're an undergraduate with plans to do particle physics and someone offers you to quickly learn about symplectic or exceptional groups, and perhaps a few others, you shouldn't refuse it.)

You don't need to talk about string theory to encounter similar subtleties. Ask a simple question. What is the gauge group of the Standard Model? Well, people will normally answer \(SU(3)\times SU(2)\times U(1)\). But what they actually mean is just the statement that the Lie algebra of the gauge group is\[

{\mathfrak su}(3) \oplus {\mathfrak su}(2) \oplus {\mathfrak u}(1).

\] Note that the simple, Cartesian \(\times\) product of Lie groups gets translated to the direct \(\oplus\) sum of the Lie algebras – the latter are linear vector spaces. OK, so the statement that the Lie algebra of the gauge group of the Standard Model is the displayed expression above is correct.

But if you have the ambition to talk about the precise group manifolds, those know about all the "global subtleties" and it turns out that \(SU(3)\times SU(2)\times U(1)\) is not isomorphic to the Standard Model gauge group. Instead, the Standard Model gauge group is\[

[SU(3)\times SU(2)\times U(1)] / \ZZ_6.

\] The quotient by \(\ZZ_6\) must be present because all the fields of the Standard Model have a correlation between the hypercharge \(Y\) modulo \(1/6\) and the spin under the \(SU(2)\) as well as the representation under the \(SU(3)\). It is therefore impossible to construct states that wouldn't be invariant under this \(\ZZ_6\) even a priori which means that this \(\ZZ_6\) acts trivially even on the original Hilbert space and "it's not there".

The \(\ZZ_6\) must be divided by for the same reasons why we usually say that the Standard Model gauge group doesn't contain an \(E_8\) factor. You could also say that there's also an \(E_8\) factor except that all fields transform as a singlet. ;-) We don't do it – when we say that there is a symmetry or a gauge group, we want at least something to transform nontrivially.

OK, you see that the analysis of the correlations of the discrete charges modulo \(1/6\) may be subtle. We usually don't care about these details when we want to determine much more important things – how many gauge bosons there are and what their couplings are. These important things are given purely by the Lie algebra which is why our statements about the identity of the gauge group should mostly be understood as statements about Lie algebras.

At some level, you may want to be picky and discuss the global properties of the gauge group and correlations. But you usually don't need to know these answers for anything else. The knowledge of these facts is usually only good for its own sake. You can't calculate any couplings from it, and so on. That's why our sentences should be assumed not to talk about these details at all – and/or be sloppy about these details.

(Just to be sure, the global subtleties, centers of the group, differences between \(SO(N)\) and \(O(N)\) and \(Spin(N)\), differences for even and odd \(N\), or dependence on \(N\) modulo 8, may still lead to interesting physical consequences and consistency checks and several papers of mine, especially about the heterotic matrix models, were obsessed with these details, too. But this kind of concerns only represents a minority of physicists' interests, especially in the case of beginners.)

By the way, the second "pet sleeve" by Woit is that one should distinguish real and complexified versions of the same Lie algebras (and groups). Well, I agree you should distinguish them. But at some general analytic or algebraic level, all algebras and other structures should always be understood as the complexified ones – and only afterwards, we may impose some reality conditions on fields (and therefore the allowed symmetries, too). So I would say that to a large extent, even this complaint of Woit reflects his misunderstanding of something important – the fact that the most important information about the Lie groups is hiding in the structure constants of the corresponding Lie algebra, and those are identical for all Lie groups with the same Lie algebra, and they're also identical for real and complex versions of the groups.

(By the way, he pretends to be very careful about the complexification, but he writes the condition for matrix elements of an \(SU(2)\) matrix as \(\alpha^2+\beta^2=1\) instead of \(|\alpha|^2+|\beta|^2 = 1\). Too bad. You just shouldn't insist on people's distinguishing non-essential things about the complexification if you can't even write the essential ones correctly yourself.)

In the futile conversations about the foundations of quantum mechanics, I often hear or read comments like:
Please, don't use the confusing word "observation" which makes it look like quantum mechanics depends on what is an observation and what isn't etc. and it's scary.
Well, the reason why my – and Heisenberg's – statements look like we are saying that quantum mechanics depends on observations is that quantum mechanics depends on observations, indeed. So the dissatisfied laymen or beginners really ask the physicists to use the language that would strengthen the listeners' belief that classical physics is still basically right. Except that it's not! We mostly use this language – including the word "observation" – because it really is essential in the new framework of physics.

In the same way, failing-grade students such as Peter Woit may be constantly asking whether a physicist talks about a Lie group or the corresponding Lie algebra. They are basically complaining:
Georgi, Ramond, Zee, don't use this notation that looks like it suggests that the Lie group and the Lie algebra are basically the same thing even though they are something completely different.
The problem is, of course, that the failing-grade students such as Peter Woit are wrong. Georgi, Ramond, Zee, and others often use the same symbols for the Lie groups and the Lie algebras because they really are basically the same thing. And it's just too bad if you don't understand this tight relationship – basically an equivalence.

I think that there exist many lousy teachers of mathematics and physics that are similar to Peter Woit. Those don't understand the substance – what is really important, what is true. So they focus on what they understand – arbitrarily invented rules what the students are obliged to parrot for the teacher to feel more important. So the poor students who have such teachers are often being punished for using a different metric tensor convention once or for using a wrong font for a Lie algebra. These teachers don't understand the power and beauty of mathematics and physics and they're working hard to make sure that their students won't understand them, either.

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