Exploring q-series

44 views Apr 3, 2023

We introduce the quantum plane, a "non-commutative" analog of the familiar x-y plane, and show that the Gaussian q-binomial coefficients naturally arise when we consider the binomial theorem in this context. It will turn out that there are several rather different versions of what we might call a "q-binomial theorem", and this one is perhaps the simplest and most accessible. Of course this discussion connects with the conundrum of quantum mechanics, but also with modern developments in pure mathematics around the topic of non-commutative geometry.

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Hello everyone I'm Norman Wildberger thanks for joining me on this adventure we're talking

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about q series and we've been talking about the gaussian binomial coefficients today we want to

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move towards a cube binomial theorem which is a key kind of application of these q binomial

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coefficients and I'm going to give you a version of that which is intimately connected with the

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quantum plane so we're going to talk a little bit about this boundary between mathematics and

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physics and sort of quantization kind of issues in mathematics is a kind of a good simple example

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of that so let me just quickly remind you the usual binomial coefficients M choose R given in

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the usual fashion and now we've generalized those to the gaussian binomial coefficients with a q

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under here so M choose RQ and sometimes in square brackets and it's this expression here which on

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the face of it looks quite different from this one here so 1 minus q to the m one minus q the

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M minus 1 down to 1 minus q to the m minus r plus one and on the bottom 1 minus q 1 minus q squared

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up to 1 minus q to the r now we can Define some auxiliary Concepts which clarifies what's going

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on here so the q number first of all just m q okay is this ratio one minus q to the M over 1 minus q

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which is pretty well just a geometric Series so it's the polynomial one plus q Plus q squared up

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to q to the m minus 1. then having q numbers we can Define the q factorial m q factorial is one

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q two q up to mq and then we can do a little bit of filling around and find that these

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gaussian binomial coefficients can be written in terms of these Q factorials in a way that's then

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completely parallel to the original situation up here so mq M minus 1 q down to n minus r Plus 1q

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over 1q 2q up to RQ in terms of factorials M Cube factorial over R Cube factorial

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M minus r q factorial so this makes the parallel between these things are very

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obvious and furthermore when Q equals one then these numbers here this number here

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just becomes M and this Q binomial coefficient just becomes the usual binomial coefficient and

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it's interesting for us to think about these Q binomial coefficients sort of as an array because

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that's after all the Harriet Pascal Maxwell which I'm always talking about and whose importance is

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in my view Central in mathematics so here we have a maxel or an array what's going on going

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okay and it's indexed by numbers 0 1 2 and the Mr entry so M throw Earth column is just this

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Q binomial coefficient M choose RQ and here's what a few of them look like okay and we've

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seen this before but let me just scroll over a little bit so you can see a little bit more okay

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these various entries here they're polynomials even though actually the original definition

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doesn't make it clear that they're polynomials it just looks like they're rational functions but in

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fact they are polynomials okay so today in today's lecture we want to introduce the quantum plane and

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then show how this leads naturally to a q version of the usual binomial theorem so here's the usual

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binomial theorem that you all know and love X Plus y to the N equals sum from k equals 0 to n of n

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choose k times x to the N minus k y to the K so decreasing powers of X increasing powers of Y and

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the coefficients are given by the usual entries of this table the N choose K binomial coefficients so

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we're going to extend this into the quantum situation so let me talk a little bit about

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some some general issues on the border between mathematics and physics okay so that's the issue

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of classical mathematics or classical physics versus quantum mathematics or quantum physics

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and this is of course a huge development in the 20th century physics um coming of course from

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the uh The crucial work of uh the physicists in the 1920s Heisenberg, Schroedinger, Pauli,

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Born, Jordan ,Dirac mathematicians Weyl, Von Neumann and many others okay but it's

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um a very mysterious business this quantum mechanics uh that even physicists sort of will say

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well we don't really understand what's going on we understand how to how to perform calculations

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but you know with the underlying meaning of the story is still I think a big question and there's

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a particular complication involving in this so-called collapse of the wave function where

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people have quite different points of view as to what that really means now I'm a mathematician

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so I'm interested in the mathematical side of this issue and it turns out that there's all

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kinds of Rich mathematics which is essentially adjacent to this physics of quantum mechanics

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and a lot of that has also been developed in the 20th century in fact at some point I I feel it'd

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be really good idea to have a whole series on this classical Quantum connection so I'm actually

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aspiring to to doing that especially from the mathematical point of view okay so I want to um

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promote various points of view that perhaps shed some light on this really interesting situation

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okay so very quickly now so just as a sort of an overview and I don't expect that you necessarily

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understand what's going on here but roughly In classical physics classical mechanics the

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state space of some system is a simplectic manifold or something like a phase space

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and the observables the things that you can measure like the position or the momentum of

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something they comprise a commutative algebra of continuous functions on this phase space okay so

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you point in face base represents a state of the system and maybe the the X function is telling

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you what the x coordinate is and maybe the P function is telling you what the momentum is

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ETC okay so then the possible values of an observable on this State space consists of

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just those numbers that are actually taken on but another way of saying that is that it's those

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numbers Alpha for which the function Alpha minus f is not invertible because the multiplication is

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just point wise multiplication in terms of the algebra so if you have the function and at some

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point it takes on the value for you're measuring something at some point it takes on the value of

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four then four minus the function has a zero there and so that entire function is not invertible in

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this commutative algebra so that sort of connects with the analysis concepts called the Spectrum

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okay so the usual story as it's presented in most places involves the the so-called real numbers

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but that's actually quite obviously a distortion of what actually happens observationally because

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physicists clearly can make observations only in certain ranges okay you never get an exact value

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if you're measuring a temperature or a momentum or a position or anything like that okay it's not

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possible to get infinite Precision that's absurd okay um so it's really unfortunate that we are

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wedded to this idea that our infinitely divisible number system which we like to believe in which

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actually mathematically is very very dubious that that's actually the appropriate framework physics

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I mean it's almost laughable really but in any way so there's actually considerable work to

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be done even into bringing classical mathematics Newtonian lagrangian mathematics into a coherent

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framework mathematically that Accords with observational physics in other words where you're

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really addressing directly the fact of the the nature of observations in physics that in reality

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you measure a temperature you have a range I'm measuring a temperature and then the the needle

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is somewhere between 43.7 and 43.8 okay so that's a range and I can get more accurate but I'm still

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always only going to have a range okay so now in quantum physics the state space is a linear space

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usually it's called a Hilbert space but it's typically over the complex numbers and that

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turns out to be quite an important um sort of and remarkable feature of quantum physics

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The increased role of complex numbers and these observables now the quantum sort of analogs of the

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classical observables they comprise a non-commutative algebra technically a star algebra

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of so-called self-adjoint operators on H so this is some kind of you know extension of matrices uh

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to a world of unbounded dimensions and there's all kinds of subtle analytic questions involved

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and from my point of view also a lot of subtle logical issues on the mathematical side at least

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and it turns out that the possible values of an observable then are those numbers for which

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again Alpha minus f is not invertible in the sense of operator invertibility

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Okay so this this version of quantum mechanics which clearly involves going from a commutative

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algebra of observables to a non-commutative algebra that was then further promoted in on

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the mathematical side by the introduction of non-commutative geometry by Alan Connes

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so the idea there is that aspects of differential geometry what you study on a differentiable

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manifold M which has a usual commutative algebra say of C Infinity functions or smooth functions

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on on it um that you could get some kind of non-commutative manifold in quotes by taking

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this algebra functions and deforming it in some suitable fashion typically into a non-commutative

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algebra what's really going on here from my point of view is it's really non-commutative algebra

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okay these people are typically C star algebras and such and they are applying the non-commutative

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algebra to geometrical issues and and questions and and finding analogs in the in the algebraic

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world to geometrical Notions so it's a great uh interesting topic but perhaps it should be called

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geometric non-commutative algebra but in any case a main example of such uh a non-commutative

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geometry is going back to Heisenberg and the canonical commutation relations where the

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original observable so the perhaps the simplest ones of position and momentum say X and P end up

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having relations once you've made operators into them of x p minus p x that's like a commutator

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equals H or sometimes I H Okay so where H is some constant typically imaginary in the physics world

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but okay so this kind of algebra is of really big importance and you can see the non-commutativity

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rate in this fundamental equation okay um and it's very important it's a sort of algebraic

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embodiment of the Heisenberg uncertainty principle now a somewhat simpler version of this I think is

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um something that was I think initiated by Y. Manin and now called the quantum plane okay so

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here again we consider a two-dimensional situation an affine plane okay consisting of affine points

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X Y like this thing here and we regard X and Y as coordinate functions on this plane that's fine so

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we get an algebra as before and it's a polynomial algebra okay so you know you could take functions

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of X and functions of Y and multiply them and add them so you're basically talking about polynomials

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in X and Y and maybe various extensions of that algebra okay and certainly a commutative algebra

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because crucially x times y equals y times x now the quantum plane then is obtained by not really

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looking at the plane itself but rather looking at this algebra of polynomials or functions on

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the plane and deforming it okay so the deformation that's considered is one where the two variables

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y and x no longer commute but rather y times x equals Q times x y where Q is what well maybe

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Q is a number like a constant but maybe in some other situations it's a variable okay and

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this is then a a new kind of algebra and we might denote it by P sub Q X Y just a Quantum version of

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the usual polynomial algebra and when Q equals one well then we recover the usual story so e algebra

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if you have some monomial like this right you have to be a little bit careful about rearranging it

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using commutativity because commutativity is not there but we have this replacement for community

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so let's consider how we would be able to convert this into a monomial of what we

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might call standard form where the X's are all at the left and the Y's are all at the

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they're on the right so something would form x to the a y to the B so let's just have a look

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to see uh what what happens here so that'll illustrate you know how you work in such an

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algebra so we're starting with YX squared y cubed x okay and we're going to take this first y x the

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first two terms here and replace the YX with Q X Y so that's the bracket thing there we still

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have another X Y cubed and X okay so you can think about having essentially move that y over one to

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the right by making a swap okay and we're just going to keep doing that so at the next stage

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because we still have associativity okay that's maybe important to say that all these algebras

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are associative so the associative property is always still there so we're able to get rid of

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these brackets and then group it in terms of this y x okay this y x so then we get Q X at

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the front and then the Y X that's right there is replaced with Q x y and the rest is unchanged

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okay so maybe we can bring the Q the Q is just a number or constant so it comes out in front so we

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get Q squared and then there's an x squared and now we have y to the fourth X great so we still

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have to sort of move these four y's to the other side of this remaining X okay so the first step

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we take um you know one of them the Y X the last y x and replace it with Q x y and then at the next

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stage we take another y x and replace it with Q x y and then we keep on doing that until finally

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all the Y's are over on the right hand side oh the X's are on the left-hand side and we have

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a constant and the constant Q is raised to the sixth power because we've done this swap exactly

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six times six because you can sort of see from here that the number of swaps required to get

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all the Y is to the right so this first y will have to go past three X's so this is going to

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be three swaps involved in that and this y cubed these y cubed terms are all going to have to get

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past this single X so that's another Three swap so all together six swaps that's why we

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get a coefficient of Q to the sixth so that's how we do arithmetic or we can do arithmetic

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in this kind of algebra and you see that we can always get to a kind of a canonical form

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okay so now we're in position to introduce our first Q binomial theorem so it turns out that

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there are several different Q binomial theorems okay and this one here is particularly simple

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I think but it does take place in the quantum plane so you have to make this adjustment that

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the variables X and Y that are appearing here are really to be thought of in this Quantum

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plane non-commutative algebra P sub Q X and Y so to get an idea of what's happening let's just

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consider expanding X Plus y squared into standard monomials okay so X Plus y squared that's X Plus y

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times X Plus Y and now we're just going to use the distributive property in the most simple

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fashion okay so there's going to be four terms here so before we start reducing things there's

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just going to be four individual terms because there's two terms here and two terms here so

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there's x times x which is x squared there's x times y there's y times x and there's y squared

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okay these three here are all in standard form because the X's are to the left or the Y's are to

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the right but this one isn't so we can replace this YX with Q X Y when we do that we get x

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squared plus 1 plus Q X Y plus y squared and that in fact that is a form of this binomial theorem in

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this Q World okay it's a q binomial theorem we've expanded X Plus y squared and we're not getting 2

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here ordinarily we would get two but here we're getting 1 plus Q and in terms of the gaussian

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Q binomial coefficients the coefficients here are 1 1 plus q and one and they are indeed

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two to zero Q two choose one q and two choose two Q so two choose one Q is indeed one plus Q

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let's do the same kind of thing with X Plus y cubed so now we have these three things

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multiplying together there's going to be 2 times 2 times 2 or 8 terms when we just expand out before

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we put any things together okay so we have an X cubed there'll be an x squared y when we choose

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the first term in this one the first term and this one and the second term here there will

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be an X Y X this x times that y times this X and there will be um okay a y x squared there'll be

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an X Y squared there'll be a y x y there'll be a y squared x and finally there'll be a y cubed

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now in ordinary algebra we quickly commute things around and then we start lumping these things

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together but here we have to do a bit more work before we can lump things together but

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we have to get everything in sort of standard order so this thing here is in standard order

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um this thing here there's a y x so we could replace that y x with X Y times Q so that'll

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give us Q x squared y this thing here we have to move the Y over twice so we have to make that

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of y x becoming qxy we have to do that twice so we're getting a q squared times x squared

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y this one here is already in standard order this one here we just have to permute once or

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interchange once so get Q X Y squared this one here again we have to have two interchanges to

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get the Y's on the other side so we get Q squared x y squared and this one's as it is

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and now all the monomials that are appearing here are in standard order and we can then group them

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together so we get X cubed plus 1 plus Q Plus Q squared x squared y plus 1 plus Q Plus Q squared

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x y squared plus y cubed and not too hard to see that those are indeed the coefficients are just

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the gaussian Q binomial coefficients three to zero cubed three choose one Q three choose two Q three

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choose three q and it might be worth noting again that if Q equals one in other words if we sort of

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reduce to the commutative situation then these coefficients in here become the familiar three

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three so we get the usual X cubed plus three x squared y plus three X Y squared plus y cubed

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okay then here is our general Q binomial theorem in this Quantum plane situation

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so in the quantum plane with this algebra P sub Q X Y we have for any natural number n

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X Plus y to the N is the sum from k equals zero to n n choose k q times x to the N minus k y to the k

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so um we can think about this in a number of ways so we could derive this result for for Q

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a power of a prime by using the interpretations of these gaussian binomial coefficients which

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we've discussed in previous lectures so let me remind you that these things count subspaces of a

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vector space over a finite field and they're also expressible as expressions in queue that describe

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the number of inversions in a corresponding path okay and it's this second interpretation the

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one that I told you last time where we were talking about young diagrams and paths and

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and Associated permutations and inversions Counting inversions that's really sort of

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um perhaps um directly related to what's going on here and so to illustrate the basic idea let me

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look at an example in fact we'll take the same kind of picture that we've already seen in our

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last example so here is a situation that we looked at when we had a three by four rectangle it was a

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young diagram shape three three one Associated path from the lower left to the upper right

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and there were five of these squares to the to the right and below our path

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Okay so suppose that we're expanding X Plus y to the 7. okay in the quantum plane so this can be

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all kinds of terms that arise but let's consider just the coefficient of x to the fourth y to

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the cube after we've put everything in standard monomial form so after we've got all the X's to

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the lefts and the Y's to the right so what will be the total coefficient of this thing here well um

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so there there's going to be various um you know powers of Q involved here and I want to sort of

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you know illustrate where where this particular one how this one contributes because this is

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um a path that has four right steps and three up steps so this um this particular path corresponds

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to the original term x y x squared y squared x where we interpret X as a right step and Y

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is an up step so we're using X and Y instead of R and up as previously right so when we expand this

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thing out there it's going to be all these these terms okay and before we start rearranging things

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this is going to be one of the terms that appears we choose an X from the first term a y from the

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second term X is from the next two terms y from the next two terms at an X from the last term

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okay so that's going to be one of many actually two to the seven uh different terms that appear

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now this particular one okay this particular one then corresponds to this thing here and the

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there are going to be five inversions required to get all the Y's to the right because we're

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gonna have to pass the Y past this x squared and then we have to pass all three y's that

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are then here we'll pass this x so it'll be two plus three that's five inversions all together

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and that corresponds exactly to the number of um squares underneath this thing here in other

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words to the inversion number of this path those two concepts are really exactly the same right

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so this corresponds exactly to the inversion number of the associated permutation one three

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um the one and then the three four uh seven those are the right steps and then two five six are the

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the left steps that corresponds to this this is just another way of thinking about about what's

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um what's being expressed here in terms of this path so uh this this particular diagram here in

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this particular path contributes a q to the five okay because that's the power of Q that's going

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to be Associated to it a q to the five so it's going to be responsible for one of the four Q

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to the fives that appear okay so the seven choose three Q that's going to be the coefficient of this

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of this particular term in this expansion it's going to be a big polynomial in q

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okay so the coefficient of this particular term here is going to be according to our

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theorem 7 choose three Q This is this rather longish polynomial in Q and The Q to the fifth

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coefficient is going to be coming from paths like this that have exactly five inversions in other

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words that have five squares underneath them or underneath and to the right of them so this is

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one of the contributions to this 4 q to the 5 that is appearing in this big expression okay

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and in fact there's going to be three other ones and you might like to find the three other ones

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um also going to be found in this rectangle so I hope that gives you sort of an idea of how what we

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talked about last time in terms of young diagrams and paths and inversions is really uh manifesting

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itself here in an algebraic form in terms of the um the Q binomial theorem in this Quantum plane

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Now I'm going to just finish with a sort of a speculative perhaps uh

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some food for thought okay so could it be that non-commutative algebra is in some sense simpler

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but crucially not easier than commutative algebra

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okay that's maybe strange to think about because you'll think that oh commutative algebra that's

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what you're used to you're not used to having to be fussing around the order in which x's and

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y's appear however if you think about you know going back to primary school you know

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algebra when you're first learning algebra and you had to say expand out X Plus y cubed right

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arguably this first step is really the the natural one you have these three terms each of them has

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two elements you can have two times two times two different products in fact that's what would

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happen if these things were all different right instead of having repeats if it was

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X Plus y times Z plus W Times U plus v then there certainly would be eight different terms and they

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would all look differently so this is sort of in some sense the the niave the immediate the basic

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calculation where you just expand these things out and you keep things in their proper order and you

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get these eight terms and in some sense this is a more Elementary answer than the subsequent thing

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that you learn okay which is well we can in this particular situation we can commute the X we can

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lump all the uh the x's and y's together with for free okay so we can replace uh this X Y X with x

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squared Y and then we can lump all the x squared y's together to get three of them all together

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okay so there's some sort of additional stuff that's required to go from this step to this step

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even though the final answer might appear simpler so I'm I'm getting you to to think that it might

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be that this is some kind of deep philosophical reason or maybe it's not so deep but sort of you

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know basic philosophical reason or maybe a mathematical computational reason why nature

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or in terms of physics in terms of quantum physics especially appears to want to be operating more

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at the non-commutative level at least when we get to this fundamental kind of quantum scale

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so maybe we should not be so hugely surprised that this quantum physics forces us to move

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more towards a non-commutative algebra maybe that's the the real algebra and we've just

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been sort of looking at some sort of simplified version of it because it's it's easier for us.