David Tong has produced a series of very high quality lectures on theoretical physics over the years, available at his website here. Recently a new set of lectures has appeared, on the topic of the Standard Model. Skimming through these, they look quite good, with explanations that are significantly more clear than found elsewhere.

Besides recommending these for their clarity, I can’t help pointing out that there is one place early on where the discussion is confusing, at exactly the same point as in most textbooks, and exactly at the point that I’ve been arguing that something interesting is going on. On page 7 of the notes we’re told

We can, however, ﬁnd two mutually commuting $\mathfrak{su}(2)$ algebras sitting inside $\mathfrak{so}(1, 3)$.

but this is true only if you complexify these real Lie algebras. What’s really true is

$$\mathfrak{so}(1, 3)\otimes \mathbf C = (\mathfrak{su}(2)\otimes \mathbf C) + (\mathfrak{su}(2)\otimes \mathbf C)$$

Note that

$$\mathfrak{su}(2)\otimes \mathbf C=\mathfrak{sl}(2,\mathbf C)$$

Tong is aware of this, writing on page 8:

The Lie algebra $\mathfrak{so}(1, 3)$ does not contain two, mutually commuting copies of the real Lie algebra $\mathfrak{su}(2)$, but only after a suitable complexiﬁcation. This means that certain complex linear combinations of the Lie algebra $su(2)\times su(2)$ are isomorphic to $so(1, 3)$. To highlight this, the relationship between the two is sometimes written as

$$\mathfrak{so}(1, 3) \equiv \mathfrak{su}(2) \times \mathfrak{su}(2)^*$$

This is a rather confusing formula. What it is trying to say is that the real Lie algebra $\mathfrak{so}(3,1)$ is the conjugation invariant subspace of its complexification

$$(\mathfrak{su}(2)\otimes \mathbf C) + (\mathfrak{su}(2)\otimes \mathbf C)$$

where the conjugation interchanges the two factors. Tong goes on to use this to identify conjugating an $\mathfrak{so}(3,1)$ representation with interchanging its properties as representations of the two $\mathfrak{su}(2)\otimes \mathbf C=\mathfrak{sl}(2,\mathbf C)$ factors.

For a very detailed explanation of the general story here, involving not just the Lorentz real form of the complexification of $\mathfrak{so}(3,1)$, but also the other (Euclidean and split signature) real forms, see chapter 10 of the notes here. My “spacetime is right-handed” proposal is that instead of identifying the physical Lorentz Lie algebra in the above manner as the “anti-diagonal” sub-algebra of the complexification, one should identify it instead with one of the two $\mathfrak{sl}(2,\mathbf C)$ factors (calling it the “right-handed” one). Conjugation on representations is then just the usual conjugation of representations of the right-handed $\mathfrak{sl}(2,\mathbf C)$ factor.

Forgive my asking what I know you’ve almost surely addressed before on the blog, but am I right in remembering that this is also related to the highly confusing fact that the Spin groups in signature (1,3) and (3,1) are not isomorphic?

S.

(First, should note that Spin(1,3)=Spin(3,1), the problem is not there, but with the Clifford algebras, or pinors. Pin(1,3) is not equal to Pin(3,1) and one of the Clifford algebras is 4 by 4 real matrices, the other is 2 by 2 quaternionic matrices).

No obvious relation to that (at least not that I understand).

But in the picture I’m proposing there’s a fundamental right-left asymmetry, unlike the usual Clifford algebra/space-time story. So, the issue you are mentioning, which shows up for pinors, not spinors, may show up somehow.

Update: Moderating out further discussion of this issue as off-topic.Hi Peter,

Thanks for having brough forward this link. I read some pages here and there, and I agree that it really look as a very clear and good tutorial on the Standard model.

I learned the Poincaré group from the ancestor of Tong’s lecture notes at Cambridge DAMTP in 1980 and they were similarly careless about combining generators of rotations and boosts with complex coefficients, this being an undefined operation in the context of the group theory used up to that point. Interestingly, our scratchy lecture notes on Lie groups were all we had as there did not seem to be any text books worth the mention on the subject at the time. Maybe Peter’s book fills this void. When I have time, when I retire, if I am ever able to, I will check it out properly!

I’ve never found anyone who can explain this careless mixing of complex combinations of rotations and boosts convincingly. It happens all the time, and is usually mathematical nonsense. I assume they must be trying to describe some real physics, but I can’t see the connection between the physics and the mathematics. Tong, for example, follows convention in putting a scalar i into the definition of the Lie bracket, which unfortunately makes it impossible to talk about symplectic Lie algebras in any reasonable way, and makes it almost impossible to distinguish different real forms of the same complex Lie algebra. If you think, as I do, that the distinction between so(3,1) and so(4) is of utmost importance for quantum mechanics, then this is a rather serious issue.

Robert A. Wilson,

I think there is a straightforward mathematical version of what Tong and everyone else is trying to say in their textbooks, but to state it you need to distinguish between a real Lie algebra and its complexification, and distinguish between different real forms. I don’t know of any textbook that does this (would be curious to hear if anyone else does know of such a thing). Some say as little as possible to avoid writing down nonsense, others do write down something that makes no sense. Behind this I think are various combinations of the author not understanding what is really going on, and the author understanding what is going, but deciding that keeping track of the necessary distinctions is too much trouble for this kind of textbook.

On page 26 of Tong’s notes, he basically says that the physicists’ version with Hermitian matrices generating unitary Lie algebras is equivalent to the mathematicians’ version with anti-Hermitian matrices, but this isn’t true for real (orthogonal) Lie algebras or quaternionic (symplectic) Lie algebras, only for complex (unitary) Lie algebras. In the real case there are more Hermitian than anti-Hermitian matrices, and in the quaternionic case it is the other way round. The quaternionic case does arise even in classical physics, for example in the symplectic gauge group of phase space, and it is quite tricky to handle this example using complex Hermitian matrices, rather than the natural anti-Hermitian (split) quaternionic matrices. This can lead to error, for example to using the 21 independent symmetric 6×6 real matrices instead of the 21 independent anti-Hermitian 3×3 quaternionic matrices, and thereby generating the wrong Lie algebra.

Not sure if this is directly on point, but Richard Behiel just a few days ago put out an amazing youtube titled “the mystery of spinors”. For a layperson, it is helping me get the most tenuous foothold on some of the notation and concepts that folks here know in their bones. https://www.youtube.com/watch?v=b7OIbMCIfs4&t=1374s

Peter,

”you need to distinguish between a real Lie algebra and its complexification, and distinguish between different real forms. I don’t know of any textbook that does this (would be curious to hear if anyone else does know of such a thing). ”

Try Section 2.7 and Section 11.6 of v2 of the Lecture Notes

‘Classical and Quantum Mechanics via Lie algebras’

by A. Neumaier and D. Westra,

https://arxiv.org/abs/0810.1019

Arnold Neumaier,

What I had more specifically in mind was the case of the Lorentz group and its finite dimensional representations, where I don’t know of a physics textbook that distinguishes between the real Lie algebra and its complexification, as well as between the other real forms of that complexification.

Woit, perhaps you do not consider these to be physics textbooks; however, they are quite accessible for hep-th:

Miller, Symmetry Groups and Their Applications

Lounesto, Clifford Algebras and Spinors

Fadde’ev-Popov ghosts,

I don’t have a copy of the Miller book, but from what I remember it was a general group theory book aimed at physical applications, unlikely it had the specific facts about spinors and real forms that I’m referrring to.

The Lounesto book is not the kind of book explaining relativistic quantum mechanics or quantum field theory that I had in mind. It does have a great deal of detail about the Clifford algebra in Minkowski signature, but what I’m referring to is not that, but the complexified and real form stories of the Lorentz and rotation groups in 4d, which I don’t see there (although I didn’t look closely, it possibly is there in some form).

There may be something in one of the following books, but I don’t own them and cannot check:

A. Barut and R. Raczka, Theory of group representations and applications. World Scientific 1986.

B.C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer 2003,

A.W. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton University Press 2001.

B.G. Wybourne, Classical groups for physicists, Wiley 1974.