Furthermore, TODO confirm it is possible that a solution does not exist at all if and aren't sufficiently small.
This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.
Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.
Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) Updated 2025-01-10 +Created 1970-01-01
This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.
But because it is more basic, it does explain some things quite well.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Updated 2025-01-10 +Created 1970-01-01
The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html
And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.
Overview:
- Chapter 3: gives a bunch of examples of important matrix Lie groups. These are done by imposing certain types of constraints on the general linear group, to obtain subgroups of the general linear group. Feels like the start of a classification
- Chapter 4: defines Lie algebra. Does some basic examples with them, but not much of deep interest, that is mostl left for Chapter 7
- Chapter 5: calculates the Lie algebra for all examples from chapter 3
- Chapter 6: don't know
- Chapter 7: describes how the exponential map links Lie algebras to Lie groups
App-only as of 2023, i.e. for children.
Humans make the table of contents, and then AI fills it. Ciro was thinking about doint the exact same thing at some point, maybe starting from Wikipedia categories.
Funding:
The approach of this channel of exposing recent research papers is a "honking good idea" that should be taken to other areas beyond just machine learning. It takes a very direct stab at the missing link between basic and advanced!
Like the rest of the Standard Model Lagrangian, this can be split into two parts:
- spacetime symmetry: reaches the derivation of the Dirac equation, but has no interactions
- add the internal symmetry to add interactions, which reaches the full equation
Deriving the qED Lagrangian by Dietterich Labs (2018)
Source. As mentioned at the start of the video, he starts with the Dirac equation Lagrangian derived in a previous video. It has nothing to do with electromagnetism specifically.
He notes that that Dirac Lagrangian, besides being globally Lorentz invariant, it also also has a global invariance.
However, it does not have a local invariance if the transformation depends on the point in spacetime.
He doesn't mention it, but I think this is highly desirable, because in general local symmetries of the Lagrangian imply conserved currents, and in this case we want conservation of charges.
To fix that, he adds an extra gauge field (a field of matrices) to the regular derivative, and the resulting derivative has a fancy name: the covariant derivative.
Then finally he notes that this gauge field he had to add has to transform exactly like the electromagnetic four-potential!
So he uses that as the gauge, and also adds in the Maxwell Lagrangian in the same go. It is kind of a guess, but it is a natural guess, and it turns out to be correct.
Why Ciro Santilli removed Disqus comments from his website in 2019-05-04 Updated 2025-01-10 +Created 1970-01-01
Commit: github.com/cirosantilli/cirosantilli.github.io/commit/794705a201a79b5128934e69df85e3511655c03f
As Ciro started getting a lot of comments on his home page about China, he decided that Disqus does not scale, and that it would be more productive long term to remove it and point people to GitHub issues instead.
Upsides of removal:
- Disqus discoverability is bad:
- there is no decent way to search existing issues, you have to do JavaScript infinite loading + Ctrl + F. So every reply that he wrote is a waste of time, as it will never be seen again.
- comments don't have: decent URLs, titles, metadata like tags or open / close
- Disqus archival is bad: web.archive.org/ does not work, and no one knows how to export the issues: www.archiveteam.org/index.php?title=Disqus
- before, there were two places where people could comment, Disqus and GitHub issues. Now there is just one.
- Disqus has ads if you ever reach enough traffic, which unacceptable, especially if the website owner don't get paid for them! It also makes page loads slower, although that likely does not matter much.
Downsides:
- people are more likely to comment on Disqus than to create an issue on GitHub, especially because most people use GitHub professionally. But this has the upside that there will be less shitposts as well.
- with Disqus you can see all issues attached to a page automatically, which is nice. But for as long as Ciro is alive, he intends to just solve the issues, cross link between content and issues and tag things appropriately.
Ciro's stance towards China hasn't changed, and China comments and corrections about his website are still welcome as always.
Related issue: github.com/cirosantilli/cirosantilli.github.io/issues/37
There are unlisted articles, also show them or only show them.