Classical to Quantum

704 views Aug 24, 2024

This is an overview video where I briefly sketch one of the general aims and direction of this series (Classical to Quantum for Members of the Channel) to investigate mathematical aspects of quantum mechanics (QM), special relativity (SR) and related aspects of the standard model (SM). In particular I want to delve into the role of representation theory of Lie groups in modern physics, and the remarkable parallels between the rise of QM in the 1930's and developments in harmonic analysis on Lie groups around the same time, leading to a curious role for notions of quantization which was clarified in the 1960's by Kirillov and became the topic called "geometric quantization", pushed along also by Kostant and Souriau and others. This theory features the symplectic geometry of coadjoint orbits of Lie groups, which play the role of classical mechanical systems, with the associated representations playing the role of their quantum analogs. So the purely abstract theory of representations of Lie groups, such as S¹, SU(2) and SU(3) which figure prominently in the SM, forms a natural laboratory to investigate the wider meaning and significance of QM, and perhaps sheds some light on which interpretations of that subject are likely to be the most fruitful. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... If you would like to support these new initiatives for mathematics education and research, you could also consider become a Patreon supporter. Thank you. Here are the Wild Egg Maths Playlists (some available only to Members!) • Algebraic Calculus One • Classical to Quantum (Members only)[ • Algebraic Calculus Two](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-) • Advice for mathematics exploration (M... • Solving Polynomial Equations (Members...[ • The Hexagrammum Mysticum (Members only)](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_) • Algebraic Calculus and dCB curves (Me... • Dynamics on Graphs (Members only) • Playlist • Triangle Geometry • Explorations with q-series (Members o... • Six: An elementary course in Pure Mat... • Maxel Inverses and Orthogonal Polynom...Here are the Insights into Mathematics Playlists: • MathHistory: A course in the History ... • WildTrig: Intro to Rational Trigonometry • Math Foundations • Wild Linear Algebra • Famous Math Problems • Box Arithmetic • Elementary Mathematics (K-6) Explained • Year9Maths • Ancient Mathematics • Wild West Banking • Sociology and Pure Mathematics • Sociology and Pure Physics • Old Babylonian Mathematics • Probability and Statistics: an introd... • Boole's Logic and Circuit Analysis • Universal Hyperbolic Geometry • Differential Geometry • Algebraic Topology • Diffusion Symmetry: A bridge between ...*****************

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Hello everyone. Well, we've been talking about integration on the circle, and I've

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suggested to you that, for physics, this is going to be important because it extends to

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integration on spheres, and spheres are very important for quantum mechanics. But it also

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extends to give us integration theories on more general objects, including relativistic circles

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and spheres, which are ultimately groups, and these are very important in physics.

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More generally, I want to talk today a little bit about something of a grand aim that I have—a

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rather ambitious aim of trying to explore to what extent we can help physics along

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or provide some interesting new directions by thinking a little bit more carefully and

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deeply about the associated mathematics. So, I'm a mathematician, not a physicist,

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but some of the mathematics that I've done over my career is actually quite closely connected to

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physics. In particular, quantization and the representation theory that I've been involved

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in are very close to those used in both quantum mechanics, relativity, and the standard model.

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What I'm hoping for is that we'll be able to use pure mathematics as something of an additional

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laboratory. Right now, theoretical physics at the highest level, dealing with really foundational

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issues, is in a bit of a dilemma because it appears as if the prime places where important

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new data can be obtained—namely accelerators like the Large Hadron Collider or telescopes peering

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into the early part of the universe, where issues of particle physics are very important—these are

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both very expensive developments, and it seems challenging to push things further

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into energy realms where physicists might be able to extract important new information.

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What I want to suggest to you—and it's not a new idea, of course, but I have some new

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ideas in this direction—is that modern pure mathematics, in fact, in a surprising way,

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provides something of a theoretical laboratory, a potential laboratory, to understand some

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important issues in modern physics, having to do with quantum mechanics, relativity

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(both special and general), and the standard model. In particular, the areas that I think

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are perhaps most fruitful are those associated with the representation theory of Lie groups.

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Now, this is a topic that developed hugely in the 20th century. The work of Killing, then Cartan,

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and notably Hermann Weyl, and many other people, has become a sort of centerpiece for that area

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intersecting physics and mathematics. It's perhaps not widely appreciated amongst physicists

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that there's an aspect of this representation theory story in which quantum mechanical ideas

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arise very naturally on their own, essentially completely independently of their development

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in physics. Somewhat coincidentally, this was first started to be appreciated around the 1930s,

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which was pretty much also the time when quantum mechanics was being solidified.

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So, what am I talking about here? I'm talking about

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a topic in mathematics called quantization, where the analog between a classical mechanical system

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framed in a Lagrangian or a Hamiltonian fashion (these are variants of ways of describing Newton's

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laws developed by Lagrange and Hamilton) comes in. They have a different flavor,

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so the Hamiltonian formulation rests on symplectic manifolds, Poisson brackets, and things like that,

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while the Lagrangian point of view rests on the principle of least action and has the

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Lagrangian at the center stage, while in the Hamiltonian version, one has a Hamiltonian.

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Now, both of these approaches turn out to be important in quantum mechanics,

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and the smoke perhaps hasn't cleared to ultimately decide which one of these, if any,

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is going to be ultimately more insightful. But the Hamiltonian point of view, in particular,

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turns out to have a very strong connection with Lie group representation theory,

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and I want to talk a little bit about that today, just to give you sort of an overview

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and perhaps some motivation for why it's going to be important for us to think about Lie groups and

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issues of integration and harmonic analysis on Lie groups in a way that will help physicists.

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When we're talking about Lie groups, maybe it's useful to restrict ourselves to relatively simple

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ones. We're talking about groups that have a continuous, or actually differentiable,

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kind of aspect. The circle S1 is a prime example. The three-sphere S3, commonly called SU(2), is,

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in some sense, the next one up. It's the first non-commutative Lie group, and it's crucially

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important and very interesting. Basically, the entire quantum theory of the hydrogen

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atom and these spdf orbitals that are crucial in quantum chemistry all arise

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essentially from the representation theory of SU(2) and associated spherical harmonics.

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SU(2) also figures in the standard model along with S1. With S1, SU(2), and SU(3), S1 roughly

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describes the gauge theory for electromagnetism, SU(2) for the weak interaction, and SU(3) for the

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strong interaction. These three Lie groups, sort of coincidentally, are arguably the simplest Lie

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groups that one can study—starting from the very simple S1, which is commutative, then SU(2),

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and then SU(3). But there's a big step from each one of those to the next one—a big step from S1 to

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SU(2), and another huge step from SU(2) to SU(3). One shouldn't think that once one has understood

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SU(3), one understands the general situation, you know, going out to exotic groups, notably E8, just

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sort of looming on the horizon there as an object that one wants to understand and which is becoming

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very important in modern physics, particularly due to the very interesting work of Garrett Lisi.

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So, if we restrict ourselves to S1, SU(2), and SU(3), which are sort of the core Lie

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groups connected with the standard model, we might think of these as local symmetries of

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spacetime—symmetries that have something to do with a structure that's somehow intrinsically

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localized at points, as opposed to large-scale symmetries of spacetime more generally, usually

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associated with the Lorentz group or the Poincaré group. Those groups are also very important.

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When we study harmonic analysis on these groups, especially when we go to the non-commutative

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settings, there's a well-established theory. There's a famous theorem of Peter and Weyl

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which describes the representation theory and construction of representations (let's

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say of a compact simple Lie group like SU(2) and SU(3)) geometrically. It turns out that

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this description has an interesting sort of quantum mechanical aspect,

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and that became clear late in the 1960s when Kirillov was investigating the irreducible

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representations for nilpotent groups (an entirely different class of groups) and found a remarkable

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correspondence between irreducible representations and so-called coadjoint orbits.

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So, what is a representation? A representation is a kind of manifestation of a group in the

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world of linear algebra. You have a group with operations, and you try to make that explicit

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by getting explicit matrices of a particular size that encode the group relations. In other words,

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you're realizing the group as a group of matrices, and that's called a representation. A given group,

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like SU(2), can have lots of different representations,

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and I've talked about this in various places. For SU(2), it's particularly interesting and simple:

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there's one irreducible representation in every dimension, and that's all. For SU(3),

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it's more complicated, and for E8, it's much more complicated. But still,

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there's a reasonably well-understood theory that covers the entire representation theory of compact

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Lie groups, at least up to the point in terms of classification and corresponding characters,

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which are sort of the elementary building blocks for harmonic analysis on these groups.

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What Kirillov discovered is that these coadjoint orbits in the nilpotent case, that were naturally

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corresponding with these irreducible representations in the nilpotent case, were

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like classical mechanical systems. The association of representations to these coadjoint orbits was

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very much in line with, and analogous to, the way physicists constructed quantum mechanical systems

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from classical mechanical systems, going back to Heisenberg, really, and his matrix mechanics.

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Heisenberg initiated this idea that instead of classical observables, like position and momentum

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(say, of a particle moving along a line), in the quantum mechanical framework, one rather wants to

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consider operators. Think of position and momentum rather as operators, now operating in some actual

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linear space, which is, in principle, infinite-dimensional—so much more complicated,

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but still simpler in some fashion because we're still just using linear algebra.

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So, the geometry that you might associate with curved space gets transformed into linear algebra,

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even though you may have to go up in dimensions to capture the full richness. This is a very

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interesting story and the beginning of a program in mathematics called quantization,

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or geometric quantization. A lot of people, Kostant and Souriau, contributed,

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along with Kirillov, to develop a recipe—a very broad kind of recipe for connecting

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the mathematical equivalence of classical mechanical systems (which were coadjoint orbits,

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these orbits in the dual of the Lie algebra) and then the irreducible representations.

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What I would like to do, ultimately, is explore this with the view that, you know,

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what would have happened if, in the 1930s, physicists hadn't discovered quantum mechanics

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and had just kept on going with classical mechanics? What if the mathematicians had

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started to develop the representation theory, and they would have basically discovered at least

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aspects of quantum mechanics directly from the mathematics associated with these purely abstract

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objects—these Lie groups and their associated representations? Now, we don't know how that

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story would have gone, but I think it's a powerful thing that physicists should be aware of: that

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the mathematical universe has encoded within it many aspects of the quantum mechanical universe.

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If we're thinking about quantum mechanics as being some deep and abiding mystery,

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and we're really racking our brains to try to understand what it really means,

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with all these different interpretations of quantum mechanics, one way of perhaps

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shedding light on this is to look at the corresponding mathematics and say, "Well,

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let's understand what's going on there in a better, more profound way." If we've done that,

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then maybe the mathematics itself will reveal ways of thinking or will reveal avenues of exploration

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that will then be translatable into physics, and into physics understanding and progress.

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So, that's more or less the reason why I want to slowly, and somewhat systematically,

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and perhaps in a novel way, develop fundamental aspects of harmonic analysis, starting with the

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circle. But you have to get the circle right. Then we're in a position to go to SU(2),

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and then we'll be in a position to go to SU(3). Down the road, I know it's a long road,

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but down the road, we'll even be able to think about E8, which is sort of like the Holy Grail

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in the subject, to try to understand E8 and whether it really holds the key, or a key,

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to understanding the world in which we live. I'm not saying it does, but I'm not saying it doesn't.

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So, lots to think about, and I'm hoping you're going to be joining me on this adventure,

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contributing, and helping me move into interesting waters.

English

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