{"id":13521,"date":"2023-06-01T18:14:50","date_gmt":"2023-06-01T22:14:50","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13521"},"modified":"2023-07-15T17:39:49","modified_gmt":"2023-07-15T21:39:49","slug":"from-quantum-mechanics-to-number-theory-via-the-oscillator-representation","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13521","title":{"rendered":"From Quantum Mechanics to Number Theory via the Oscillator Representation"},"content":{"rendered":"

This past semester I taught our graduate class on Lie groups and representations, and spent part of the course on the Heisenberg group and the oscillator representation. Since the end of the semester I’ve been trying to clean up and expand this part of my class notes. I’m posting the current version, working title From Quantum Mechanics to Number Theory via the Oscillator Representation<\/a>. This is still a work-in-progress, but I’ve decided today to step away from it a little while, work on other things, and then come back later perhaps with a clearer perspective on what I’d like to do with these notes. In a few days I’m heading off for a ten-day vacation in northern California, and one thing I don’t want to be thinking about then is things like how to get formulas involving modular forms correct.<\/p>\n

There’s nothing really new in these notes, but this is material I’ve always found both fascinating and challenging, so writing it up has clarified things for me, and I hope will be of use to others. The basic relationship between quantum mechanics and representation theory explained here is something that I’ve always felt deserves a lot more attention than it has gotten.<\/p>\n

In the past I’ve often made claims about the deep unity of fundamental physics and mathematics, One goal of this document is to lay out precisely one aspect of what I mean when making these claims. There are other much less well understood aspects of this unity, but the topic here is something well-understood.<\/p>\n

One thing that struck me when thinking about this and teaching the class is that this is a central topic in representation theory, but one that often doesn’t make it into the textbooks or courses. Typically mathematicians develop theories with an eye to classifying all structures of a given kind. This case is a very unusual example where there is effectively a unique structure. The classification theorem here is that there is basically only one representation, but it is one with an unusually rich structure.<\/p>\n

When I get back from vacation, I plan to get back to work on the ideas about twistors and unification that I’m still very excited about, but have set to the side for quite a few months while I was teaching the class and writing these notes. More about that in the next few months…<\/p>\n","protected":false},"excerpt":{"rendered":"

This past semester I taught our graduate class on Lie groups and representations, and spent part of the course on the Heisenberg group and the oscillator representation. Since the end of the semester I’ve been trying to clean up and … 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_post_was_ever_published":false,"_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":""},"categories":[1],"tags":[],"class_list":["post-13521","post","type-post","status-publish","format-standard","hentry","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\/13521","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=13521"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13521\/revisions"}],"predecessor-version":[{"id":13524,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13521\/revisions\/13524"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13521"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13521"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}