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.Correction: a correspondent reminds me that for the next part to be true this definition needs to be supplemented by an extra condition, since as stated SO(3,1) has two components. One version of the extra condition is to take the connected component of the identity, another is to take the component that preserves time orientation. Many use a different notation for this component to make this explicit, I’ll define SO(3,1) as the connected component.
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.
Is it just me, or LaTeX isn’t rendering? You forgot to include the word “latex” after the dollar sign.
Re the signs and typographic errors: in the definition of SU(2) you wat $\alpha\bar{\alpha} + \beta\bar{\beta} = 1$; so that the determinant is 1. Then you also actually get the 3-sphere.
—,
Thanks, fixed.
Maxis,
I’m doing this using MathJax, works for me. If people are having trouble with this let me know.
I am trying to write up some notes for a course on Lie theory and applications I would like to teach when I retire ( but probably never will ). Your notes are some of my best references.
It’s great that you posted this comment, and it’s a wonder it’s really necessary. Getting the notation right using Latex is so easy these days ….
Peter,
LaTeX isn’t rendering for me either.
Note that MathJax generally does render in my browser (if I visit my own website, or arXiv, or various other places), and everything works fine — except for your website, which just displays LaTeX source instead.
HTH,
Marko
Peter
First, the LaTeX is displaying fine for me, Safari 10.1. Second, physicists really can’t use different notation for the group and the algebra? In math, anything in Fraktur is the algebra. I assume this is still true, it certainly was when I took Lie Algebras in grad school in ’87.
My broswers (Firefox, Edge, and Chrome) does not render the Latex because it views the Latex rendering as insecure. On Chrome, I could easily turn this off.
Looking at your page source, I see at least part of the problem:
http://www.math.columbia.edu/department/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML&ver=4.7.3
is not https, which is how I’m seeing most of this page. Note that
https://www.math.columbia.edu/department/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML&ver=4.7.3
seems like it works, but my https everywhere extension does not automatically translate that link to https – which is why the rendering wasn’t working on my browsers.
Fred P,
I see the potential problem, changed the link to https.
All, please let me know of any continuing problems with the Latex rendering.
Yes, that fixed it, now it renders correctly for me too!
🙂
Marko
Ok, now that I see the equations rendered correctly, shouldn’t the sums be “circled”, \(\oplus\), instead of an ordinary plus, \(+\), when writing sums of algebras at the end?
I don’t want to sound like a nitpick, but an algebra is also a vector space, and its addition of vectors (usually denoted with an ordinary plus sign) should be distinguished from the notion of the direct sum of two algebras. Especially if the two algebras are actually two copies of the same algebra.
HTH, 🙂
Marko
Marko,
You’re probably right, but remember, here I’m trying to convince physicists that the distinctions mathematicians make are important ones….
Aside from remembering to type {-}\sin instead of -\sin when it’s standing alone, among other things. 😉
Peter- Maybe it’s ok to mention the March For Science tomorrow (Saturday):
it’s supposed to be nonpartisan and a celebration of science…
of course barely in the background are worries about climate change, budget cuts,
the destruction of the EPA, NIH…
Here’s TYT:
https://www.youtube.com/watch?v=pCGFqbRC6do
There are supposed to be marches in 500 cities worldwide.
Information here:
https://www.marchforscience.com/
I noticed a very minor pet peeve of mine in the book. It is really minor, so please feel free to not fill up the space under your post by publishing this comment: right after introducing “the Schrödinger equation” you call it “Schrödinger’s equation.” How could I not search the text after spotting an inconsistency! The text calls it “the Schrödinger equation” throughout, except for the aforementioned place, Fig. 19.1 and page 558.
I also noted that on p. 466 you dropped the umlaut in naming the Pauli-Schrödinger equation. I actually only found this latter instance because your old-school LaTeX setup doesn’t actually output the letter “ö,” but instead takes \”o very literally. In other words, a search for “Schrödinger” in the pdf turns up empty, and I had to search for “Schr” instead. Thankfully, you don’t cite Peskin & Schroeder all too often.
Congrats on the book, and thanks for the explanation of the mistake in the decomposition of the Lorentz group that you gave here. That was a magic bit of common textbook lore that had puzzled me for the longest time! Too bad that I no longer do physics, and the only group that I encounter regularly these days is the Quaternion group.
In the two equations identifying R4 with 2×2 self-adjoint matrices, the signs of the x2 entries are opposite those of the R3 case (and of your book).
Why do mathematicians use Fraktur for the Lie algebra so(3) when lowercase works fine?
My suspicion is that they do it to show off their superior knowledge of fonts. Of course, this may confuse physicists into using the same notation for the Lie algebra and the Lie group, because they may not know how to use Fraktur in LaTeX and may be too lazy to bother looking it up.
Dear “To ask a silly question…”: Your guess for the reason that fraktur fonts are used to denote Lie algebras is incorrect. The reason is the same as why fraktur fonts are commonly used to denote ideals in commutative algebra: many concepts in abstract algebra and Lie theory were initially developed by those who worked in Germany or very closely with the German school of mathematics in the late 19th and early 20th century, among whom the fraktur font was quite common to use (unsurprisingly).
Lie spent some early pivotal years of his professional career in Berlin, where he struck up a friendship with the German mathematician Klein that was to become very influential on the development of Lie’s own work (prior to the unfortunate demise of their friendship over some priority disputes). Other early fundamental work in the subject of Lie algebras was done by Killing (also German), to say nothing of the early fundamental work in representation theory by Frobenius, Schur, and Weyl. You can read about all of this and much more in Thomas Hawkins’ great book “Emergence of the Theory of Lie groups” (as well as a bit in the Historical Notes at the end of Chapters 1-3 of Bourbaki’s “Lie groups and Lie algebras”).
It was due to the aftermath of World War II that German became less common as a language for math papers and the fraktur font consequently less widely seen. But the tradition was already firmly established in Lie theory and various parts of ring theory, a tradition that has remained to this day; this was widespread long before modern type-setting.
Moreover, due to the extensive need for notation in mathematical proofs, it is a tremendous convenience that one can use g to denote an element of a Lie group G while considering proofs that involve both the Lie group and its Lie algebra $\mathfrak{g}$ at the same time while not thereby writing “g” to denote two very different things in the same discussion. The presence of many font options in LaTeX is due to the widespread notational needs of mathematicians, and not the other way around. In math papers and books, notational choices and traditions are generally made to promote clarity of discussion, not to show off.
To ask a silly question,
As BCnrd comments, mathematicians often have a need to clearly notationally distinguish unambiguously different kinds of mathematical objects, so it’s not surprising that they use more fonts and yes, are more likely to know the relevant TeX off the top of their heads.
I understand why physicists are reluctant to use fraktur fonts, its use is culturally alien. What I can’t understand is why they won’t use upper vs. lower case. Why not write the Lie algebra of SU(2) as su(2)?
In your book, section 40.2, p. 448, the complexified so(3,1) “splits into a product of two sub-Lie algebras”, and the equation is written with a multiplication sign. I think it should be a direct sum.
Art,
Thanks. Fixed sign, and yes, for Lie algebras should be sum not product.
For completeness, the same equation shows up at the end of section 40.4.
SU(2) and SO(3) simplified with quaternions (H). Let X be in H, and U a unit element in H. Then
X –> UX is an SU(2) action,
X –> UXU* is SO(3).
In the former case all 4 components of X are mixed,
and in the latter only the 3 imaginary components
(U* is quaternion conjugation, and because U is
a unit quaternion, it is also the inverse of U).
There must exist some imaginary element v in H
(no real part) such that U = exp(v). The Lie algebra
of SU(2) and SO(3) is the set of such elements v.
The study of the Lie groups and algebras important
to theoretical physics is greatly simplified by using
division algebras. (Am I biased in this matter?
Hmm, let me check …)
PS: How does one exploit Latex in these comments?
Geoffrey Dixon,
I do think that using quaternions and Sp(1)=Spin(3) is the best way to understand the relationship between Spin(3) and SO(3), and that’s the way I do it in my book. However, this post is aimed at keeping math apparatus to a minimum for an everyday physicist, so I avoid quaternions and use two by two complex matrices instead.
You can use latex in comments, using MathJax. I should add this info to the “leave a reply” text, once I figure out how to escape the delimiters correctly. One way to convey this info is to say I’m using the default delimiters, see
http://docs.mathjax.org/en/latest/tex.html#tex-and-latex-math-delimiters
The (astro)physicists seem to be incorrigible… they write in https://arxiv.org/pdf/1704.05067.pdf , p.42:
“11)We acknowledge the pain of our more mathematically inclined readership at our deliberate, yet well-intentioned ambiguation of a group and its matrix representation. Sorry!”
TG3D,
When people are having trouble figuring out the difference between a Lie group and a Lie algebra, not a good idea to tell them about representations…
Thanks, Peter, for stressing that we should always keep Lie groups and Lie algebras separate!
A perspective that I’ve always found clarifying is that a Lie algebra can be taken to be the tangent space to the Lie group at the identity. Of course for that to make sense you have to remember that Lie groups are manifolds that have a tangent space. But this is worth knowing because it makes obvious and intuitive some of the Lie group isomorphisms you talk about: since the group SU(2) is the double cover of the group SO(3), they are locally isomorphic and therefore have isomorphic tangent spaces. Viola! That’s just another way of saying that su(2) is isomorphic to so(3). You can even give a geometrical explanation of the exponential map as a map from the tangent space (Lie algebra) to the manifold (Lie group) if you feel ambitious.
Sure, that needs more mathematical machinery than you wrote about, but I find that machinery very helpful for explicating the difference between Lie groups and Lie algebras. I’ve even used this to explain the difference to non-math folk using pictures!
People who can’t distinguish between a Lie group and a Lie algebra have no business calling themselves physicists or being employed as researchers. Sticking with 19th century confusions is just unprofessional and uneducated, manifesting a lack of curiosity and an intellectual laziness that shouldn’t be allowed in professional contexts.
pedant,
I think you’re being unfair to the 19th century…