Classical to Quantum

59 views May 1, 2024

How does one integrate a function over the unit circle? This is an important question which is at the heart of much of Fourier series and associated harmonic analysis. The analogous question for spheres is also extremely central in quantum mechanical questions related to the hydrogen atom and quantum angular momentum. In this video we will be motivated by the approach of the Algebraic Calculus One course, in which one-dimensional integration is introduced in terms of the properties desired of an integration functional. We review informally the traditional approach to circle integration involving subdivisions and limits of Riemann like sums. We also review classical forms of Folland and Baker involving the Gamma function for the integral of a polynomial over the unit circle. Then we explain an alternative formulation due to Kevin Limanta and myself. Kevin was a Masters student and also a PhD student with me at UNSW, and is currently teaching at UNSW. This alternative formulation is purely rational and algebraic, so provides the possibility of a purely algebraic approach to issues of harmonic analysis and associated quantum mechanics. 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, thanks for joining me on this adventure, from classical to Quantum. We are in

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the process of developing some harmonic analysis on circles and spheres. Today,

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a very important lecture where we look at the measure on the circle, the rotationally invariant

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measure on the circle, and how we integrate functions on the circle, the unit circle. So,

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this is a very interesting point because traditionally this subject is intimately

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associated with pi and certain irrational quantities coming from cosines and sines. So,

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we're going to see an alternate way of thinking about this important subject,

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which ends up extending into spheres. I'm going to be emphasizing a very important

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result due to a former student of mine, Kevin L. Ma, and myself. And I think it's a really

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beautiful new direction, a new way of thinking about harmonic analysis on circles and spheres.

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So, let me just remind you about the usual integration story on a circle, somewhat naively.

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So, here's just the one-dimensional story: if we're integrating a function f(x) from 0 to 1, the

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integral from 0 to 1 of f(x)dx. Well, what do we do? There's a graph of f(x) , there's the interval

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0 to 1. We break the interval up into lots of little pieces. This is the traditional approach,

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Riemann integration. And then, say, taking one of these pieces, calling it Δx_i we

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form a rectangle by choosing some point in that interval, calculating f(x_i) for some

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𝑥 𝑖

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x i

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​in that interval. We get

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a little rectangle which sort of approximates the area under the curve in that little interval. So,

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this product here is the area of that little rectangle. And we're going to sum them all up. And

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then, what we do is we take a limit. What we do, it's just a lovely phrase. What we do is we take a

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limit, that is, we look at an infinite sequence of such computations. We perform a computation, then

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we finish that one, then we reduce the sizes of the intervals, and then we do another computation.

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And we keep doing that to infinity. And once we've done all that infinite amount of work,

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we have the integral. So, there's a corresponding story classically for an integral of the form

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integral over 𝑆¹ of 𝑓(𝑠)𝑑𝑠. So, in this case, f is a function, actually it's convenient to think

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about it perhaps as being a function defined on the entire plane, doesn't have to be, but

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it's the case that we're going to be interested in most, but we're interested in the restriction

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of that function to the unit circle. Okay? And we're interested in integrating that restriction.

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So, what we do is we chop it up into little pieces. There's a little piece. Okay? So,

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the Δs_i here represents, well, it ought to represent like an arc length,

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the length of that little piece, but actually it suffices to just take a line segment between

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here and here. Okay? And calculate the length of that line segment using a square root which

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involves an infinite amount of work. So, we take that little length there and we multiply

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it by the function's value at some point in that little arc, okay, say f(s_i ). So,

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that little product there represents just like over here, a kind of a rectangular

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but arc rectangular area which approximates the area under f over that little subinterval. So,

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we sum that all up as we go all the way around the circle, summing that up. And then, we take

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the limit of that sum as the intervals all become smaller. So, we have more and more intervals,

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each of them is getting smaller and smaller, and we take some kind of limit just as we did

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over here. So, in both cases, infinite procedures are required. This is the classical story. We are

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going to be circumventing this entirely. So, you can actually, if you like, forget about this 100%.

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Now, let me also remind you that in the Algebraic Calculus I course, which is over at Wild Egg

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Maths, there's also a fully-fledged online course at Open Learning with hundreds of work problems

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and homework questions. That'll keep you busy for months, even if you're a pretty capable

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math student. Anyway, in that course, what we do is we have a quite different approach

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to integration. It's purely algebraic, it doesn't require any infinite processes, and it's actually

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more general because we end up integrating over curves U which are given by actually two, say,

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polynomials p and q, so not just a single variable thing. Okay? And geometrically,

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what we're doing is computing some kind of signed area of some triangular wedge like this, where

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there's part of the curve say between parameter T and parameter U, and the signed area of that is

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given by some formula. And how is it defined? So, that's the critical thing. How do we define this

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signed area? Well, it turns out to be defined in a way that really depends on the properties that we

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want this functional to have. We think of this as a mapping from pairs of polynomials with numbers

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to rational numbers. And we want that mapping to have certain properties: a normalization property,

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an additivity property, a linearity property, and then a translation and dilation property,

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both of those as well. And the main thing is that we're able to prove the existence and uniqueness

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of such a functional once we've clarified what all these various criteria are. Okay? So,

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it's very different from the Riemann integral kind of point of view. What we're doing is we're

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asking ourselves what properties do we want this integration functor kind of thing to have? Okay?

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Once we identify that, then we say, okay, can we actually build such a thing and can we show that

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it's unique? That's exactly the approach that we're going to take here in the circular case.

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So, here's a nice picture just to orient you. We can think about the

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XY plane as being horizontal momentarily. So, there's a third dimension, maybe a z

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direction. And then a function f(x,y), say a polynomial function on the plane,

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gives us a surface by looking at z=f(x,y). So, the height above the

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plane is the function's value at that point. So, then when we restrict to the unit circle,

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which is this thing here, we can imagine a cylinder over that circle coming up in the z

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direction. And the function f, when restricted to the circle, can be thought of as a section

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of this cylinder. Okay? So, it's represented by this red line that winds around the

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cylinder and goes up and down depending on the function's values. In this case,

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the function has all positive values, but you can imagine it also dipping down here and having

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negative values as well. So, if s is a point on the circle, then the point directly above it in

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the z direction on the graph of the function is then that height is the function's value f(s).

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Okay? So, that's some physical intuition, and what we're really trying to get at is, in some sense,

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this area of this sort of cylindrical part of the cylinder cut out by that top wavy function. Okay?

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So, our question is, is there a purely algebraic, finite approach to integration on S1 that avoids

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all that subdivision and kind of the Riemann-style integration? And, as I've mentioned, the answer

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very pleasantly is yes. And that will probably be a surprise to many of you because we have

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been taught since an early age that somehow the irrational number Pi is intimately connected with

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the measure of the circle and formulas involving integration. You know, usually they involve cosine

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and sine functions or circular or trigonometric functions, and there's Pi's floating around. So,

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how could we possibly do this purely in the rational world without even any mention of Pi?

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Alright, so here is the classical version of this integration formula in terms of a measure. So,

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let's call it dμ, which is a rotationally invariant measure which has been normalized in

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the Euclidean way so that the total measure of the circle, the unit circle, is 2π,

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the traditional circumference of the circle, however involving this irrational number Pi.

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Then the statement is that the integral over the circle of x^m yⁿ, so we're taking this

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polynomial function which is a polynomial in the entire plane and restricting it to the circle,

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and then we're integrating it over the circle. And the formula, found in Folland and Baker and

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probably goes back considerably earlier than that, at least in some form or another, is that if one

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or both of m and n are odd, then we get zero, some kind of cancellation, like if m is one then

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on this side it'll be positive, this side it'll be sort of equally negative, and they'll cancel

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themselves out. However, if m and n are both even, then we get this expression in terms of the gamma

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function. Here's the gamma function: Γ(t) is the integral from 0 to infinity of s^{t−1}e{−s}ds.

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Now, okay, this is not a formula that I'm happy with, as you can probably guess. I mean,

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it's an important formula, very useful in Applied Mathematics if we think of it, you know,

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as defining something which is approximately, you know, given by this, but the exponential function

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is something that we can only calculate approximately. And in fact, this integral

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itself can only be found approximately. That's sort of the real-life reality. Although pure

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mathematicians like to pretend that it actually has a real value, there's a real number that you

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get when you do this involving an infinite amount of work and some infinite decimal. Okay? The gamma

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function is closely related to the factorial. The factorials, of course, are only defined for

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natural numbers, but if k is a natural number, then Γ(k)=(k−1)!. So, for example, Γ(4) would be

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3!, which is 6. There's also an important formula for a half-integer. So, if k is a natural number

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and we look at Γ(k+1/2), then the formula is that you do the same kind of thing here, it's like a

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kind of a factorial going down one at a time but starting with the fellow which is one less

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than the one you've got. So, we start with k−1/ 2, and then we multiply by what we get when we

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subtract one from that, and then we subtract one again, and so on, until eventually, we get to 1/2.

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But curiously, there's something else. In this case here, there's also a square root of Pi,

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another irrationality that strictly speaking requires an infinite amount of work to obtain, and

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we are not in possession of that infinite amount of work. So actually, this is a nebulous kind

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of thing in an exact sense. So given this about the gamma function, here is the formula in case

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m and n are both even. It's 2⋅ Γ(m+1/2)⋅Γ(n+ 1/2)/Γ(m+n+1). So,

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if m is even, then this will be odd, so odd over two. So, it's really in this case here,

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we're really talking about quantities which are of this kind. However, in the denominator

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here, if the numerator there is even, then that's going to be a natural number. So, in that case,

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it's going to be just a factorial. You can think about the gamma function as somehow

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interpolating between these factorial values given by the ordinary factorial for natural numbers.

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Alright, so this is a classical formula. This is the standard formula for how to integrate a

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polynomial over a circle. Okay? So, here is the formula of Kevin L. Ma and myself

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for how to perform that same kind of integration but purely in a rational form. And Kevin did a

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master's thesis with me and also a PhD thesis, so this is probably already in his master's thesis.

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And he has gone on to study this circular case, it's full of richness and connects with remarkable

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combinatorics having to do with super Catalan numbers. And I'm going to try to get him to give

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us some lectures on that, it's fascinating stuff. But in any case, the whole story extends in a

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remarkable way to integration on spheres too, and Kevin is on top of all of this stuff. So, it'll

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be great to have him here and he can explain more interesting aspects of this in various directions.

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Okay, so here is the basic result in the plane. So, P is our space of polynomial by polynomial

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numbers in the plane. Okay? We like to think of them in terms of alphas and betas, so powers

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of alphas and betas. So, there's a unique linear functional, so it's a linear space. Okay? You can

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take linear combinations of these monomials and there's a linear functional from, called I, from

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the space of polynomials to the rational numbers, so all the coefficients here are rational. That

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has three properties. First of all, it sends the constant function 1 to 1. Okay? Secondly, it has

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a kind of a locality property that if we have some polynomial whose restriction to the circle

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is zero, then I of that polynomial will be zero. So, an example of that would be the polynomial

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𝛼^2+𝛽2−1. If you're on the circle, that's going to have value zero, and so I of that will have

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to be zero. And of course, there are many other polynomials that have that property. You could

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just take that 𝛼^2+𝛽2−1 and multiply it by any other polynomial and you'll get another polynomial

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that has this property. And finally, there's a rotational invariance that for any rotation ρ,

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if you take a polynomial and you rotate it the obvious way, you compose it with a rotation,

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then the I functional has the same value. Okay? And here is the explicit formula for it.

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Okay, so what is 𝐼(𝛼𝑚𝛽𝑛)? m and n are natural numbers. Okay? And it's in terms of these double

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factorials. What's a double factorial? Well, a double factorial is what you get when you

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start with a natural number and you multiply by the next natural number, which is two less than

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the one you started with. So, k⋅(k−2)⋅(k−4)⋅(k−6), it's going down by two each time until you get 2,

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if k started as an even number, and if you started with an odd number, well, you're going to go down

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steps of two till you get to one. And also, 0!! is defined to be one in the usual way. Okay? So,

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the formula is this unique linear functional that has these properties, which is our integration

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functional, is characterized by the following formula: that if one or both m and n are odd,

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then it's zero, just as in the classical case. And if m and n are both even,

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then it's (m−1)!!⋅(n−1)!!/(m+n)!!, and in the case when one of these or both of these are zero,

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then the corresponding double factorial is also deemed to be one. Okay? So,

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if they're both zeros, then we have 1⋅1/1. If say one of them is two, say, let's say we have 𝛼²

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with n equals zero, then this will be one and we'll have a 1!! which is one over 2!! which is

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just two. So, the integral of 𝛼² if you like, will be a half. And what happens if say m is

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four and n is zero, then we have 3!! which is 3⋅1 over 4!! which is 4⋅2, so the answer would be 3/8.

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Okay? So, this is a purely rational thing. There's no pies at all floating around. And crucially,

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however, this normalization condition means that we're assigning a mass of one to the entire

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circle. That's also natural, right? To assign this thing here to have value one. And in that context,

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all these integrals become rational numbers. And we see that there's actually no need of

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introducing the gamma function at all. So, there's a lot more to say about this even the planar case,

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it's full of Rich stuff. But I want to emphasize that in fact, there is a beautiful extension

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of this formula to a higher-dimensional spheres, which is going to be very important,

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very important for physics and especially quantum mechanics, even in three dimensions.

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