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 by deep mathematics. Furthermore, quantum mechanics can be readily understood in this way.
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”.
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.
This is both beautiful and easy to understand. As Sabine would say “Read my book” (see chapters 13, 14, and 17 here).
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.
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.
Can you say exactly what is needed to quantize a given theory? I was told by a physicist that you can never quantize with the Lagrangian alone, which surprised me. I would have assumed that with the Lagrangian alone, one can compute the phase space and then proceed from there.
In other words, what is the smallest list of ingredients from which the rest of the ingredients and then the quantized theory can be cooked up?
Skewered Cigar,
Given a Lagrangian, you can get classical equations of motion (Euler-Lagrange), but you can’t for every Lagrangian construct a quantum theory. If you try to use canonical methods, you first have to construct a phase space, but in general the Legendre transform won’t have the properties you need. In addition, even if you do get that to work, you’ll then have a Hamiltonian function that you don’t know how to quantize (operator-ordering ambiguities, no sensible state space, etc.)
Telling students that all you have to do is pick any Lagrangian and use it to define a quantum theory by a path integral is extremely misleading. That doesn’t actually work except in special cases.
Well played, Peter, certainly got me to click.
I am sure you know this, but just so that no one else gets confused.
My argument is not that nature is not beautiful. My argument is that there is no reason for nature to conform to any preconceived notions of beauty that humans have. This is very different from saying that nature is not beautiful. Indeed the argument in my book is that we learn what beauty even means from nature. (And there are several historical examples for how this happened in physics.) Problems arise if we do it the other way round: try to find natural laws using a certain notion of (mathematical) beauty.
@Skewered Cigar @Peter Woit
You can construct a phase space from the Lagrangian. You start with the Lagrangian and vary it (as usual), but don’t throw away the boundary term (not as usual). This boundary term is a (pre)symplectic potential, and a second variation gives a conserved (pre)symplectic form on the space of solutions of the theory. You can try this with usual quantum mechanics viewed as a field in 1-dimensions
The whole “pre” stuff is about whether this symplectic form is degenerate or not; in any gauge theory it will have degenerate directions. However, you can take functions on the phase space which are constant along the degenerate directions, and define a Poisson bracket on such functions. Quantization then be done is the usual way, along with the usual issues of operator ordering for nonlinear functions, e.g. any Hamiltonian of interest.
Of course, all of this is much easier said than done, but in principle the Lagrangian gives you, at least, all the classical structure needed for quantization.
“have have” is a repeated word in your post.
Since this is extremely pedantic, I’ll also add a relevant question: is “the orbit method” the same thing as the Mackey machine?
All,
Sorry, the reference to Sabine Hossenfelder’s views on beauty was just to get you to click on the post and hope you would read the interesting content here. I’m not about to host a discussion of this topic here now.
S,
Thanks, fixed.
I don’t think there’s much relation between the two. I’ve mainly encountered the “Mackey machine” as a way to understand representations of semi-direct products, don’t know if there’s an analog in terms of behavior of orbits.
The orbit method was originally developed for nilpotent and solvable Lie groups, where it works well, and there are relations to Mackey’s work on induced representations. For the much more challenging issue of semi-simple groups, I don’t know of much relation to Mackey.
Kartik Prabhu,
What you’re describing ends up requiring something like the Dirac method of quantization of constrained Hamiltonian dynamics, with all sorts of trouble in consistently imposing constraints.
In any case, what one is doing when one does this is trying to get to exactly the Hamiltonian context I’m writing about, finding a physical phase space identified with R^2n with the standard symplectic structure and a Hamiltonian function on it. At all stages of this, the fact that you can’t consistently find operators for all functions on phase space is a significant problem.
Kartik Prabhu,
Another point to emphasize is that when you go through the constrained dynamics process you often will end up with a physical phase space which is not R^{2n} and cannot be quantized by canonical methods. This happens for some of the simplest interesting QFTs. A good example is Chern-Simons theory in 3d on a Riemann surface. The Lagrangians is very simple but the degeneracy problem means that your physical state space is going to be the space of gauge equivalence classes of flat connections on the Riemann surface. Quantizing this is a completely different story than canonical quantization and requires very different methods.
Trying to instead make sense of the path integral leads to a host of other issues.
I wonder why, if QM is understood, ‘t Hooft is trying to substitute it with deterministic calculations; why Penrose insists in the unresolved problem of the quantum measurement or why even Steven Weinberg published in his last years some papers introducing non-standard randomness in the wave function. It seems to me that great physicists (I mentioned three Nobel laureates) confess in the end they are not confortable with the current understanding of QM but they dare to speak honestly only when they are old and so respected they are beyond any suspect or reprisals.
Arturo Lopez,
I’ve written extensively about this on the blog in the past, see for instance
https://www.math.columbia.edu/~woit/wordpress/?p=10533
I agree that there is a measurement problem, but I see it as the problem of how the classical description of the world emerges out of the quantum description. The quantum formalism in terms of representation theory is simple, beautiful, deep and very very powerful. The idea that you’re going to replace that with something much more naive and completely different doesn’t seem plausible to me (the Nobel Prize winners who have tried this don’t seem to me to have gotten anywhere at all with it, Weinberg I think admitted as much).
I don’t want to host another discussion of the measurement problem, happy to discuss the ideas about the relation between QM and representation theory in the posting.
Hi Peter,
Some other comments on quantization which I think fit nicely with the orbit method but are not often discussed.
There is a sense in which the “points” of the “symplectic category” are Lagrangian submanifolds. I learned this from an essay by Alan Weinstein. The idea is that one considers a category in which the objects are symplectic manifolds, and the morphisms between a pair of such manifolds, M and N say, are Lagrangian submanifolds of the product M x N. (This is a reasonable notion of morphism because the graph of a symplectomorphim from M to itself is the same thing as a Lagrangian submanifold in M x M. ) In this category, a “point” in M should mean a morphism pt -> M, or equivalently a Lagrangian submanifold in pt x M ~ M.
Now imagine you are a classical physicist and you arrive at this idea that the “points” in a symplectic manifold are really the Lagrangian submanifolds. You might decide that it really isn’t right to consider functions f(p,q) on phase space, but rather that one should choose a foliation by Lagrangian submanifolds and only study things like f(p) or f(q) which are constant on Lagrangian submanifolds. You can almost imagine how following this philosophy could lead a classical physicist to “discover” geometric quantization.
Peter Woit said:
Kartik Prabhu,
What you’re describing ends up requiring something like the Dirac method of quantization of constrained Hamiltonian dynamics, with all sorts of trouble in consistently imposing constraints.
One can get from a general presymplectic form = closed 2-form (no matter what its origin, in particular, the Lagrangian 2-form works) a Poisson algebra on the associalted Hamiltonian vector fields – except that now the latter do not include all vector fields. See Section 13.2 of my recent book
A. Neumaier and D. Westra, Algebraic Quantum Physics, Vol. 1: Quantum mechanics via Lie algebras, de Gruyter, Berlin 2024.
(or Section 18.1 of my free online lecture notes https://arxiv.org/pdf/0810.1019v2 from 2011). Fixing all Casimirs gives a symplectic manifold, and for certain values of the Casimirs these can be extended by geometric quantization.
Dirac’s method is only needed when one wants to have a more explicit description of the reduced symplectic manifold – since one then has to construct a complete set of generating Casimirs.
Hi Anon,
Would you expound on this a bit? I believe that the orbit method uses Kahler-polarizations, which don’t fit into the foliaiton context you described. Also, while states are (sometimes) constant along the leaves of a Lagrangian foliation, observables aren’t, so I find the idea that a classical physicist would discover quantization this way to be a reach. I think quantization is much easier to motivate by making the observation (which is never mentioned) that the quantum mechanics of projective space agrees with the classical mechanics of projective space, ie. the quantum and classical equations of motion agree on quantum observables.
Hi Josh,
I realized after reading your comment that I am bit confused about the point you are making so take what I write now with a grain of salt. Akshay Venkatesh has a nice set of lecture notes on geometric quantization and the orbit method. In the last section, he explains the quantization of coadjoint orbits. It can be described using Lagrangian submanifolds, but they are Lagrangians of the complexified tangent bundle of the orbit, not of the orbit itself, as the story I sketched in my earlier comment would suggest. So as you point out, my earlier comment was too simple. That comment was really only aimed at canonical phase spaces. Why in the orbit method do we need to use Lagrangians in the complexified tangent bundle and not the orbit itself? Here is where I get confused but I think the point is that there is a G-action in the orbit story and a foliation by Lagrangians of the orbit itself would not be G-equivariant. I think my earlier comment needs a disclaimer: “if there is a G-action on the symplectic manifolds, you need to do everything in a G-equivariant way and the constructions are more complicated.” Maybe someone who understands this better than me can chime in.
Josh/anon,
The Venkatesh lecture notes anon refers to are here
https://math.berkeley.edu/~fengt/249C_2017.pdf
This doesn’t address the orbit method in general, for that a place to start is Kirillov’s book
https://bookstore.ams.org/gsm-64/
For the case of semi-simple groups, the story becomes quite intricate, precisely because the representation theory of these groups (in the non-compact case) is similarly intricate, in a way that matches, but not quite. For this, see Vogan’s
https://math.mit.edu/~dav/PCorb.pdf
and a posting here with a link to some of his slides
https://www.math.columbia.edu/~woit/wordpress/?p=631
The relation between the general idea about Lagrangian submanifolds that anon is pointing to and quantization is explained by Segal in lecture 4 of these notes
https://web.math.ucsb.edu/~drm/conferences/ITP99/segal/segal2_n.pdf
This isn’t about the orbit method, it’s about trying to quantize general symplectic manifolds (without the G symmetry of a co-adjoint orbit). It’s not well-known as it should be that there really isn’t a viable way of doing this in general. For the case of cotangent bundles, Segal explains the relation between Lagrangian submanifolds and quantizing by using pseudo-differential operators.
Are there good references on the Poisson geometry approach to classical mechanics? I am aware of pure math references on this topic, but not of references written for physicists. In particular, are there situations, or families of problems, where the Poisson approach makes a mechanical systems easier to study?
Curious Fish,
For some answers to your question, see here
https://mathoverflow.net/questions/28532/classical-mechanics-motivation-for-poisson-manifolds
Curious Fish,,
”are there situations, or families of problems, where the Poisson approach makes a mechanical systems easier to study?”
Hydromechanics can be put into a Hamiltonian framework on Lie-Poisson algebras, but not on symplectic manifolds. See, e.g.,
P.J. Morrison, Hamiltonian description of the ideal fluid. Rev. Modern Phys. 70 (1998), 467.
Curious Fish,
This review article just appeared.
https://arxiv.org/abs/2411.12551