Skip to content

Pet Peeve Essay On Lying

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
    \alpha & \beta\\
    -\overline{\beta}& \overline{\alpha}
    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
    \cos\theta & -\sin\theta\\
    \sin\theta & \cos\theta
    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
    0&\cos\theta & -\sin\theta\\
    0&\sin\theta & \cos\theta

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}

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}
    x_3&0 & -x_1\\
    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
    \alpha & \beta\\
    \gamma& \delta
    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
    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

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

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.

This entry was posted in Uncategorized. Bookmark the permalink.

Top 40 Pronunciation Pet Peeves

We all know about President Trump’s mispronunciation of “Ji-nah” and his major oops referring to “Be-YONS-ee.” So as not to be partisan, Hillary Clinton had her problems saying the pop icon’s name as well. She called her BAY-on-say once on the campaign trail in Iowa. Former President George Bush, well known for his pronunciation gaffes, said it best, “I have been known to mangle a syllable or two myself.” He’s not alone; even the best American wordsmiths do mispronounce their fair share of words.

Americans are somewhat tolerant regarding pronunciation errors when the mistakes involve infrequently used foreign phrases, place names, technical terms, dialectical differences, or idiomatic expressions. However, for various reasons, we do demand uniform pronunciation of some words. Following are our Top 40 Pronunciation Pet Peeves in no particular order. Also, make sure to check out the Top 40 Grammar Pet Peeves and the Top 40 Vocabulary Pet Peeves. Find out all of your grammatical mistakes and the words you misuse before “You-Know-Who” points them out to you.

  1. Library is pronounced “lie-brair-ee,” not “lie-bear-ee.” [No, it’s not libarian either]
  2. Nuclear is pronounced “nook-lee-er,” not “nUke-U-ler.” [Ode to Bush]
  3. February is pronounced “Feb-roo-air-ee,” not “Feb-U-aire-ee.” [Frequently misspelled, as well]
  4. Orange is pronounced “or-anj,” not “are-anj.” [Orange you glad you know this?]
  5. Prostate is pronounced “praw-state,” not “praw-straight.” [Unless you are lying down]
  6. Height is pronounced “hite,” not “hite with a ‘th’.” [That “e-i” or “width” must confuse us]
  7. Probably is pronounced “praw-bab-lee,” not “prob-lee.” [Or some say “praw-lee”]
  8. Ask is pronounced “ask,” not ” ax.” [Please tell me before you ax me.]
  9. Pronunciation is pronounced “pro-nun-see-a-tion,” not ” pro-noun-see-a-tion.” [But pronounce]
  10. Athlete is pronounced “ath-lete,” not “ath-ah-leet.” [Despite the ath-ah-leets foot commercials]
  11. Strategy is pronounced “strat-uh-gee,” not “stra-ji-dee.” [Though we never say “stra-ji-jick”]
  12. Aluminum is pronounced “uh-loo-mi-num,” not “al-U-min-um.” [Brits have their own version]
  13. Et cetera (etc.) is pronounced “et-set-er-ah,” not “ek- set-er-ah.” [Not “ek-spe-shul-lee” either]
  14. Supposedly is pronounced “suh-po-zed-lee,” not “su-pose-ub-lee.” [Or “su-pose-eh-blee”]
  15. Difference is pronounced “di-fer-ence,” not “dif-rence.” [Often misspelled due to this error]
  16. Mischievous is pronounced “mis-chuh-vus,” not “mis-chee-vee-us.” [You’ll look this one up]
  17. Mayonnaise is pronounced “may-un-naze,” not “man-aise.” [“Ketchup-catsup” is another matter]
  18. Miniature is pronounced “mi-ne-uh-ture,” not “min-ah-ture.” [Who drives an Austin “min-uh”?]
  19. Definite is pronounced “de-fuh-nit,” not ” def-ah-nut.” [For define, it’s “di-fine” not “dah-fine”]
  20. Often is pronounced “off-ten,” not “off-en.” [Probably just sloppy pronunciation]
  21. Internet is pronounced “In-ter-net,” not “In-nur-net.” [Not “in-ner-rest-ing either]
  22. Groceries is pronounced “grow-sir-ees,” not “grow-sure-ees.” [It’s not “grow-sure” either]
  23. Similar is pronounced “sim-ah-ler,” not “sim-U-lar.” [But Websters says “sim-ler” is fine]
  24. Escape is pronounced “es-cape,” not “ex-cape.” [It’s not “ex-pres-so” either]
  25. Lose is pronounced “luze,” not “loose.” [Think “choose,” not “moose”]
  26. Temperature is pronounced “tem-per-ah-ture,” not “tem-prah-chur.” [Cute when kids say it]
  27. Jewelry is pronounced “jewl-ree” or “jew-ul-ree,” not “jew-ler-ree.” [More syllables won’t get you more carats]
  28. Sandwich is pronounced “sand-which,” not “sam-which.” [Or “sam-mitch” either]
  29. Realtor is pronounced “real-tor,” not “real-ah-tor.” [Similarly, it’s “di-late,” not “di-ah-late”]
  30. Asterisk is pronounced “ass-tur-risk,” not “ass-trik.” [It’s not called a star, by the way]
  31. Federal is pronounced “fed-ur-ul,” not “fed-rul.” [Use all syllables to ensure all federal holidays]
  32. Candidate is pronounced “can-di-date,” not “can-uh-date.” [It’s not “can-nuh-date” or “can-di-dit”]
  33. Hierarchy is pronounced “hi-ur-ar-kee,” not “hi-ar-kee.” [It’s not “arch-type”; it’s “ar-ki-type”]
  34. Niche is pronounced “nich” or “neesh,” not “neech.” [This one drives some people crazy]
  35. Sherbet is pronounced “sher-bet,” not “sher-bert.” [I’m sure, Burt]
  36. Prescription is pronounced “pre-scrip-tion,” not “per-scrip-tion.” [and prerogative, not “per”]
  37. Arctic is pronounced “ark-tik,” not “ar-tik.” [Not “ant-ar-tik-ah either]
  38. Cabinet is pronounced “cab-uh-net,” not “cab-net.” [Likewise, it’s “cor-uh-net,” not “cor-net”]
  39. Triathlon is pronounced “tri-ath-lon,” not “tri-ath-uh-lon.” [Not “bi-ath-uh-lon” either]
  40. Forte is pronounced “fort,” not “for-tay.” [But Porsche does have a slight “uh” at the end]

And for the culinary snobs among us… It’s “bru-chet-tah” or “bru-sket-tah,” but definitely not “bru-shet-tah.” And it’s “hear-row,” not “gear-row” or “ji-roh.” If you’re eager for more of the same, check out the 20 Embarrassing Mispronunciations that I have been guilty of over the years.

Grammar, Mechanics, Spelling, and Vocabulary (Teaching the Language Strand) Grades 4, 5, 6, 7, 8

The author of this article, Mark Pennington, has written the assessment-based Grammar, Usage, Mechanics, Spelling, and Vocabulary (Teaching the Language Strand) Grades 4-8 programs to teach the Common Core Language Standards. Each full-year program provides 56 interactive grammar, usage, and mechanics lessons. (Check out a seventh grade teacher teaching the direct instruction and practice components of these lessons on YouTube.) The complete lessons also include sentence diagrams, error analysis, mentor texts, writing applications, and sentence dictation formative assessments with accompanying worksheets (L.1, 2). Plus, each grade-level program has weekly spelling pattern tests and accompanying spelling sort worksheets (L.2), 56 language application opener worksheets (L.3), and 56 vocabulary worksheets with multiple-meaning words, Greek and Latin word parts, figures of speech, word relationships with context clue practice, connotations, and four square academic language practice (L.4, 5, and 6). Comprehensive biweekly unit tests measure recognition, understanding, and application of all language components.

Grammar, Usage, Mechanics, Spelling, and Vocabulary (Teaching the Language Strand) also has the resources to meet the needs of diverse learners. Diagnostic grammar, usage, mechanics, and spelling assessments provide the data to enable teachers to individualize instruction with targeted worksheets. Each remedial worksheet (over 200 per program) includes independent practice and a brief formative assessment. Students CATCH Up on previous unmastered Standards while they KEEP UP with current grade-level Standards. Check out PREVIEW THE TEACHER’S GUIDE AND STUDENT WORKBOOK  to see samples of these comprehensive instructional components. Check out the entire instructional scope and sequence, aligned to the Grades 4-8 Common Core Standards.

The author also provides these curricular “slices” of the Grammar, Usage, Mechanics, Spelling, and Vocabulary (Teaching the Language Strand) “pie”: the five Common Core Vocabulary Toolkits Grades 4−8; the five Differentiated Spelling Instruction Grades 4−8 programs (digital formats only); and the non-grade-leveled Teaching Grammar and Mechanics with engaging grammar cartoons (available in print and digital formats).

Reading, Spelling/Vocabularyalliteration, enunciation, Mark Pennington, pet peeves, pronunciation, pronunciation pet peeves, reading fluency, speech techniques, syllable rules, syllables