{"id":14232,"date":"2024-11-11T19:10:36","date_gmt":"2024-11-12T00:10:36","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=14232"},"modified":"2024-11-12T07:58:20","modified_gmt":"2024-11-12T12:58:20","slug":"why-sabine-hossenfelder-is-just-wrong","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=14232","title":{"rendered":"Why Sabine Hossenfelder is Just Wrong"},"content":{"rendered":"

Sabine Hossenfelder’s latest video<\/a> argues<\/p>\n

    \n
  1. There’s no reason for nature to be pretty (5:00)<\/li>\n
  2. Working on a theory of everything is a mistake because we don’t understand quantum mechanics (8:00).\n<\/li>\n<\/ol>\n

    These are just wrong: nature is both pretty and described by deep mathematics. Furthermore, quantum mechanics can be readily understood in this way.<\/p>\n

    Actually, the title and first paragraph above are basically just clickbait. Inspired by the class I’m teaching, I wanted to write something to advertise a certain point of view about quantum mechanics, but I figured no one would read it. Picking a fight with her and her 1.5 million subscribers seems like a promising way to deal with that problem. After a while, I’ll change the title to something more appropriate like “Representations of Lie algebras and Quantization”.<\/p>\n

    To begin with, it’s not often emphasized how classical mechanics (in its Hamiltonian form) is a story about an infinite dimensional Lie algebra. The functions on a phase space $\\mathbf R^{2n}$ form a Lie algebra, with Lie bracket the Poisson bracket $\\{\\cdot,\\cdot \\}$, which is clearly antisymmetric and satisfies the Jacobi identity. Dirac realized that quantization is just going from the Lie algebra to a unitary representation of it, something that can be done uniquely (Stone-von Neumann) on the nose for the Lie subalgebra of polynomial functions of degree less than or equal to two, but only up to ordering ambiguities for higher degree.<\/p>\n

    This is both beautiful and easy to understand. As Sabine would say “Read my book” (see chapters 13, 14, and 17 here<\/a>).<\/p>\n

    This is canonical quantization, but there’s a beautiful general relation between Lie algebras, phase spaces and quantization. For any Lie algebra $\\mathfrak g$, take as your phase space the dual of the Lie algebra $\\mathfrak g^*$. Functions on this have a Poisson structure, which comes tautologically from defining it on linear functions as just the Lie bracket of the Lie algebra itself (a linear function on $\\mathfrak g^*$ is an element of $\\mathfrak g$). This is “classical”, quantization is given by taking the universal enveloping algebra $U(\\mathfrak g)$. So, this much more general story is also beautiful and easy to understand. Lie algebras are generalizations of classical phase spaces, with a corresponding non-commutative algebra as their quantization.<\/p>\n

    The problem with this is that these have a Poisson structure, but one wants something satisfying a non-degeneracy condition, a symplectic structure. Also, the universal enveloping algebra only becomes an algebra of operators on a complex vector space (the state space) when you choose a representation. The answer to both problems is the orbit method. You pick elements of $\\mathfrak g^*$ and look at their orbits (“co-adjoint orbits”) under the action of a group $G$ with Lie algebra $\\mathfrak g$. On these orbits you have a symplectic structure, so each orbit is a sensible generalized phase space. By the orbit philosophy, these orbits are supposed to each correspond to an irreducible representation under “quantization”. Exactly how this works gets very interesting, and, OK, is not at all a simple story.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Sabine Hossenfelder’s latest video argues There’s no reason for nature to be pretty (5:00) Working on a theory of everything is a mistake because we don’t understand quantum mechanics (8:00). These are just wrong: nature is both pretty and described … 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":[21],"tags":[],"class_list":["post-14232","post","type-post","status-publish","format-standard","hentry","category-quantum-mechanics"],"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\/14232","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=14232"}],"version-history":[{"count":9,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/14232\/revisions"}],"predecessor-version":[{"id":14241,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/14232\/revisions\/14241"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14232"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14232"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14232"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}