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December 7, 2025

Octonions and the Standard Model (Part 13)

Posted by John Baez

When Lee and Yang suggested that the laws of physics might not be invariant under spatial reflection — that there’s a fundamental difference between left and right — Pauli was skeptical. In a letter to Victor Weisskopf in January 1957, he wrote:

“Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist.”

(I do not believe that the Lord is a weak left-hander.)

But just two days after Pauli wrote this letter, Chien-Shiung Wu’s experiment confirmed that Lee and Yang were correct. There’s an inherent asymmetry in nature.

We can trace this back to how the ‘left-handed’ fermions and antifermions live in a different representation of the Standard Model gauge group than the right-handed ones. And when we try to build grand unified theories that take this into account, we run into the fact that while we can fit the Standard Model gauge group into Spin(10)\text{Spin}(10) in various ways, not all these ways produce the required asymmetry. There’s a way where it fits into Spin(9)\text{Spin}(9), which is too symmetrical to work… and alas, this one has a nice octonionic description!

Continue reading “Octonions and the Standard Model (Part 13)” …
Posted at 4:04 PM UTC | Permalink | Followups (3)

December 3, 2025

log|x| + C revisited

Posted by Mike Shulman

A while ago on this blog, Tom posted a question about teaching calculus: what do you tell students the value of ∫1xdx\displaystyle\int \frac{1}{x}\,dx is? The standard answer is ln|x|+C\ln{|x|}+C, with CC an “arbitrary constant”. But that’s wrong if ∫\displaystyle\int means (as we also usually tell students it does) the “most general antiderivative”, since

F(x)={ln|x|+C − ifx<0 ln|x|+C + ifx>0 F(x) = \begin{cases} \ln{|x|} + C^- &\text{if}\;x\lt 0\\ \ln{|x|} + C^+ &\text{if}\;x\gt 0 \end{cases}

is a more general antiderivative, for two arbitrary constants C −C^- and C +C^+. (I’m writing ln\ln for the natural logarithm function that Tom wrote as log\log, for reasons that will become clear later.)

In the ensuing discussion it was mentioned that other standard indefinite integrals like ∫1x 2dx=−1x+C\displaystyle\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C are just as wrong. This happens whenever the domain of the integrand is disconnected: the “arbitrary constant” CC is really only locally constant. Moreover, Mark Meckes pointed out that believing in such formulas can lead to mistaken calculations such as

∫ −1 11x 2dx=−1x] −1 1=−2 \int_{-1}^1 \frac{1}{x^2}\,dx = \left.-\frac{1}{x}\right]_{-1}^1 = -2

which is “clearly nonsense” since the integrand is everywhere positive.

In this post I want to argue that there’s actually a very natural perspective from which ∫1x 2dx=−1x+C\displaystyle\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C is correct, while ∫1xdx=ln|x|+C\displaystyle\int \frac{1}{x}\,dx = \ln{|x|}+C is wrong for a different reason.

Continue reading “log|x| + C revisited” …
Posted at 2:23 AM UTC | Permalink | Followups (13)