Classical to Quantum

34 views Aug 17, 2024

34 views • Aug 17, 2024

Is it possible to introduce Fourier series and harmonic analysis on a circle purely rationally? That is, without any so-called "irrational numbers" (which aren't really properly defined, since our computers can't work with them) or "transcendental functions (which aren't really properly defined either, since they do not give precise outputs when given generic inputs) or "angles" (which aren't really properly defined as infinite processes are required to obtain them, even in the most elementary situations)? Well, yes we can. In this video we introduce the right orientation to this, by regarding the unit circle as the main object of study, and interpreting polynomial functions, like f(x)=x + x², as maps from the circle to the (rational) complex plane. We touch base with some interesting geometry and show how the usual sin / cos story is really just a view of this geometry which results by (wrongly) placing the notion of angle at centre stage. 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 ...*****************

0:02

Hello everyone, thanks for joining me on this adventure from classical to

0:05

quantum. We've been talking about harmonic analysis on a circle,

0:10

a building block towards understanding spheres and hyperboloids, and lots of

0:14

things that are important in physics. Now, we're ready to talk about Fourier series,

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which is sort of the start of harmonic analysis on these various higher-dimensional spaces.

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The theory of Fourier series has a two-fold aspect, perhaps. There's a classical applied

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mathematics orientation, going back to Fourier himself, which is concerned with functions on

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the line that are periodic, and trying to write such functions as linear combinations

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of particularly simple periodic functions, namely the functions cosine(nθ) and sine(nθ), where n is

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a natural number like 0, 1, 2, 3, etc. This is a very powerful and important tool in applied

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mathematics, in engineering, signal processing, and physics, but it does have an applied aspect.

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The basic building blocks—the functions cosine and sine and their various dilations like cosine(3θ)

1:31

or sine(5θ)—are not actually exact functions. That is, they are not functions that we can evaluate

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completely and precisely. So, strictly speaking, they do not fall in the realm of pure mathematics,

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where we aspire to deal with objects that are completely 100% defined and where we can,

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in all cases, exhibit exactly what we are talking about completely, precisely, and unambiguously.

2:04

There is another side of Fourier series, a more pure or theoretical side, which I'm

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particularly interested in explaining to you today. That's a very important side

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of the subject that I think simplifies many aspects, not just of harmonic analysis on a

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circle but also of harmonic analysis on these higher-dimensional spaces,

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which are going to be so important for physics, including harmonic analysis on Lie groups and

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the associated representation theory that play a key role in the standard model.

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Today, I'm going to start by saying a few introductory words. Of course,

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you'll all know that I'm sort of responsible for rational trigonometry,

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which is an alternative trigonometry in the plane where we avoid cosines and sines, and we look at

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spreads and quadrances instead of angles and distances. We thereby aspire to have a purely

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rational description of geometrical things in the plane—that's rational trigonometry. Many people

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have asked me, "Okay, well if you're not going to use cosines and sines, then how do you deal with

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Fourier series?" My answer is what I'm going to be explaining to you today: that there is a pure

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side of Fourier series, which can be created and developed purely rationally, okay, purely

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rationally. We don't need irrational numbers at all in this abstract, pure form of Fourier series.

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So, the starting point is to make a shift in your thinking about a function,

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even a very elementary function. I'm going to talk about one particular function today:

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the function, say, f(x) = x + x². If we have f(x) = x + x², and I ask you to graph that function,

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you're going to say, "Well, it's a quadratic, it has zeros at, say,

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0 and -1," so you're going to draw a parabola in the plane, okay, passing through those points

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on the x-axis. And, you know, say there's the graph of the function f(x) = x + x².

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But today, I want to enlarge your point of view to think, "Well, actually, this

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particular function has other manifestations." In fact, it has a lot of other manifestations,

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and a very important one is the sort of circular manifestation that we're going to talk about

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today. So, instead of x representing a real variable or a variable on the line,

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let's talk about w. Here is w, and w, for us, is going to be a complex number.

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And when I say complex number, I always mean a rational complex number—something like 3 + 7i,

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where the coefficients are rational numbers. So, that's what w is.

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Now, our function is f(w) = w + w². This, in principle, takes a complex number and outputs

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another complex number. Note that it's a purely polynomial function, so it's completely precise

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and explicit. Given the complex number w, we can calculate this thing completely and write

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it down, and everyone can check whether it's correct or not. But what we're going to do,

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because we want to make this transfer from thinking about things on the line

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to things on the circle, is to regard this as a function on the circle. So,

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we're going to restrict w to just lying on the unit circle, where x² + y² = 1.

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Have a look at this little diagram here, okay. It's the only diagram we're going to be talking

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about today. So, prominently in blue, we have the unit circle in the complex plane. There's the real

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axis, there's the imaginary axis, and this is the point (1,0), or 1. This is the point (0,1), or i,

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in this complex framework, and right here is a complex number, w, on the unit circle.

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Where is w²? Well, here is w², right here. Now, a traditional analyst might say, "Oh,

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well, the way we get this w² is we take this angle and we double it." Well, no, not really,

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because we can't typically calculate this angle, right? The angle is a transcendental thing that

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we can only get approximately, in general. So, if we're working in pure mathematics,

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which is what we're aspiring to here, we don't want to express things in terms of angles. This

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is something I'm going to have to repeat a lot of times to my physics friends. Okay,

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I know your training is such that you automatically go to parameterizing things

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with angles, but I want you to step back from that. Very often, that's not the right thing

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to do. There are some situations, particularly harmonic motion or uniform harmonic motion like

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a harmonic oscillator or something going around the circle uniformly, where that's a reasonable

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thing to do. But in many other situations, even though we have circular symmetry,

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if we don't have something going around uniformly, then there's no pressing need to describe the

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situation with angles because that already removes us from the domain of pure mathematics.

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So, how otherwise could we get w² from w? Well, one way is to reflect the point (1,0) in this

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line joining w to the origin—just make that reflection. Another way is to take the tangent

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at w to the circle (this tangent line here) and draw a line through the point (1,0) that's

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parallel to that tangent, and see where the other point of intersection with the circle is. So,

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there are these two geometrical ways of describing w² without any recourse to angles.

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So, there's w, and there's w². Where's w + w²? Well, the way complex number addition works,

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basically, is that we're just adding this vector to this vector, so we're getting this point up

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here. It's called z₁, okay. So this is w + w². Now we can think about this function, this polynomial

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function on the circle, as inputting this point w on the circle and outputting this point z₁.

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Let's have a look to see what happens. Let's start with w over here, right? So, there's w, and we see

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z₁ is over here. Now, let's just have a look at what happens as we move w along the circle. w²,

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of course, goes around twice as fast, and you can see what's happening to our little point z₁,

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right about here. It also goes through the real axis, and then it goes around this inside loop

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here. By the time w has reached -1, the point z₁ has gotten to the origin. Then,

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as we continue, well, it just does the mirror thing to what it did before. It goes around the

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circle and gets back to where it started. So, we end up with this nice curve here.

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Think about this: this is a completely legitimate answer to the question, "What does the function

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f(x) = x + x² look like?" It's not the case where x is a so-called real variable on the line;

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it's the case when x is equal to w, which is a point on the circle, which is, in principle,

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just as amenable and interesting a situation as thinking about something on a line. But what

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we're getting is something very different from a parabola, of course—we're getting this curve.

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A natural question is, well, what is this curve? One way of addressing that is to

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parameterize it—not with cosines and sines, but rather with the

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rational parameterization that I've talked about in many places, in terms of t. So,

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1 - t² / 1 + t² would be the x-coordinate, and 2t / 1 + t² is the y-coordinate. With that,

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it's a little calculation to show that this point, z₁, has this as its parameterization:

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The x-coordinate is 2(1 - 3t²) / (t² + 1)², and the y-coordinate is 2t(3 - t²) / (t² + 1)².

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You might like to check that. So, we have this curve parameterized, which is great. That means it

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has an algebraic equation. What is this algebraic equation? Well, I don't really know. I could,

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I suppose, do some work and use Gröbner bases. It's a basic thing in algebraic geometry to go

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from a parameterization like this to an algebraic form of the curve—that is, a polynomial equation

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in x and y which this thing satisfies. My guess is that it's a quartic equation (degree four),

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but I'm not entirely sure. Those of you with some algebraic facility might like to figure it out.

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You could even do it by brute force, by hand, if you were clever enough—just take this x and

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this y, and find the relationship between x and y. You want an algebraic relation, which will

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then be the equation of this red curve, which is the graph of our function f. There are standard

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algebraic geometry routines involving typically Gröbner bases that will also do this for you.

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If you're interested in that, that's a fun thing to do because, of course,

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what I'm doing here is just a very special case of an entirely new way of thinking about

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functions. You've all learned about how to graph functions on a line, like graphing a polynomial,

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maybe a cubic or quartic polynomial, with its ups and downs, and so on. The corresponding

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question here on the circle is: given a general polynomial function like this,

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what is its image as w goes around the unit circle? We're going to get some kind of curve

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in the plane—also a parameterizable curve—and the form of that curve is pretty interesting.

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How does this connect with ordinary Fourier series? Let me try to explain that. To do that,

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I'm going to parameterize this or think about this curve in another way. Suppose I'm interested in,

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say, just the x-coordinate, or perhaps just the y-coordinate of this point, z₁. Let's

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say the x-coordinate. Suppose I'm a traditional analyst, and I want to know how the x-coordinate

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of z₁ depends on w, or more precisely, how it depends on the angle that w makes with the x-axis.

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So, to do this, let me do a few more things here. Let's draw a vector from the origin to B,

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and let's define an angle—say, from B to A to w. There it is: L = 44.23°. Actually,

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I might just rename that because L is more convenient for me. Let's use L,

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and I want to plot the x-coordinate of z₁. So, over here, I'm going to make a vector

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whose first coordinate is L, the angle we've just measured (approximate), and whose second

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coordinate is the x-coordinate of z₁. So, I'll write X of Z sub 1. That should do it.

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You see this point D here? That's obtained—let me make it a little bit more noticeable. I'll

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color it something more interesting, say blue, and make it a little bit bigger.

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There's that point D. That represents a point in the plane whose x-coordinate is the angle,

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"angle" in quotes, and whose y-coordinate is the x-coordinate of what we're doing here. So,

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if we start right there, then the x-coordinate will be, I guess, two—somewhere up there.

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Now, let's just see what happens to this point D as I go around. Let's grab w again. There's

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D again. I'm going to put a trace on it, and then animate the point w to see what happens.

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Okay, so as w moves around the circle, z₁ essentially traces out a circle,

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more or less. It looks almost like a circle, right? So, what we're really getting here is

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very much a graph of a cosine, just expanded by a factor of two, as it

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turns out. But that's only up to a certain point. Once w gets around to here, the curve is no longer

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following exactly a circular trajectory. It ends up doing something different. So,

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what this is... suppose we were interested in this blue, light blue function to begin with.

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It turns out that the Fourier series of this function is essentially w + w². We're sort of

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reverse-engineering this: starting with the Fourier series, which is w + w², and then

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graphing the corresponding function where we're just looking at the x-coordinate of the image.

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We could do this more generally. If this blue curve was more general, we could look for a

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function like f(w) = w + w² that would generate this curve in the way I've just shown you. I want

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to impress upon you that what I'm doing here—like this red curve—may not look like a usual kind

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of Fourier series story, but if we introduce angles and graph either the x- or y-coordinate,

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then we're getting a more traditional function of one variable on the line. The Fourier series

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of that function is essentially coming from this polynomial that we're starting with.

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This is interesting because we're now potentially in a position to look at a lot of interesting

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curves that engineers are interested in, for example, square waves, triangular waves, sawtooth

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waves, etc., and explain their traditional Fourier series in this more algebraic way. So, we have

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the potential of really thinking about familiar things in a somewhat new and more algebraic light.

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