{"id":13882,"date":"2024-03-22T16:47:17","date_gmt":"2024-03-22T20:47:17","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13882"},"modified":"2024-03-22T17:22:16","modified_gmt":"2024-03-22T21:22:16","slug":"david-tong-lectures-on-the-standard-model","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13882","title":{"rendered":"David Tong: Lectures on the Standard Model"},"content":{"rendered":"

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

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<\/p>\n

We can, however, \ufb01nd two mutually commuting $\\mathfrak{su}(2)$ algebras sitting inside $\\mathfrak{so}(1, 3)$.<\/p><\/blockquote>\n

but this is true only if you complexify these real Lie algebras. What’s really true is
\n$$\\mathfrak{so}(1, 3)\\otimes \\mathbf C = (\\mathfrak{su}(2)\\otimes \\mathbf C) + (\\mathfrak{su}(2)\\otimes \\mathbf C)$$
\nNote that
\n$$\\mathfrak{su}(2)\\otimes \\mathbf C=\\mathfrak{sl}(2,\\mathbf C)$$<\/p>\n

Tong is aware of this, writing on page 8:<\/p>\n

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 complexi\ufb01cation. 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
\n$$\\mathfrak{so}(1, 3) \\equiv \\mathfrak{su}(2) \\times \\mathfrak{su}(2)^*$$\n<\/p><\/blockquote>\n

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
\n$$(\\mathfrak{su}(2)\\otimes \\mathbf C) + (\\mathfrak{su}(2)\\otimes \\mathbf C)$$
\nwhere 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.<\/p>\n

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<\/a>. My “spacetime is right-handed” proposal<\/a> 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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, … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[31,1],"tags":[],"class_list":["post-13882","post","type-post","status-publish","format-standard","hentry","category-twistor-unification","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13882","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13882"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13882\/revisions"}],"predecessor-version":[{"id":13894,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13882\/revisions\/13894"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13882"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13882"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}