Classical to Quantum | Harmonic Analysis and the Oscillator Representation of sl(2)
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Classical to Quantum | Harmonic Analysis and the Oscillator Representation of sl(2) | Wild Egg Maths
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The complex polynumber basis of the affine plane dramatically simplifies aspects of Fourier series or harmonic analysis on a circle, as we have seen. It also allows us to consider in an elementary way an interesting advanced topic called the Oscillator Representation of the Lie algebra sl(2), a favourite topic of my Yale PhD thesis advisor, Roger Howe.
So this is an excellent opportunity to introduce some important notions from Lie theory and representation theory, centered around the Laplace operator on polynomials and a multiplicative operator associated to the basic quadratic function q=alpha2+beta2 = uv which defines the unit circle. We compute some commutators generated by these two operations, connect with the basic three dimensional Lie algebra sl(2) which plays a central role in quantum mechanics.
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hello everyone thanks for joining me on this adventure from classical to Quantum we've been looking at harmonic analysis
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on a circle and the basic structure that we've outlined there in terms of complex
0:15
basis for the space of polinomial or polum functions on the circle uh that
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lends itself to the introduction of some important ideas having to do with Le
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algebras and representations which turn out to be of central importance to
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quantum mechanics as we move forward in that direction so this is actually a a nice opportunity for me to introduce
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some really fundamental ideas and some fundamental objects in a very natural
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situation so uh the motivation is already clear because we're just making observations about some Elementary
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things that we've already been working with okay so there's some additional structure that's visible that comes out
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from the analysis that we've carried on so far in terms of this um 4A series
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harmonic analysis on the circle so what we've started with is poly numbers in Alpha and beta representing X and Y
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coordinates so these can be interpreted as functions on the plane which can then be restricted to the circle and we saw
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that it was very useful to extend our coefficients from rational numbers to complex rational numbers and in
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particular introduce U = Alpha plus I beta and V = Alpha minus I beta as sort
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of alternate generating um constituents for this algebra of polinomial we saw
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the usefulness of introducing Q which is U * V or Alpha squ plus beta Square
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that's a particular quadratic U polinomial which of course is invariant on the circle and then here
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is the basis of um of CP obtained by just taking powers of u and v so
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starting with degree 0 degree 1 u and v degree 2 u2 UV V ^2 degree 3 Etc degree
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four going down and then we had these um these two operators one was the laian
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okay which takes an arbitrary monomial like this U to the m v to the n and sends it to 4 * m * n time U the m
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minus1 v the nus1 so it lowers the degree of both of these exponents by one
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and introduces this multiplicative factor out front another operator is just the
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multiplication by Q which obviously just raises the exponents by one so what we
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can imagine here is that these two operators are sort of working in opposite directions so if we take some
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element uh like uv^ s for example if we apply the lass in that's going to
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decrease the power so it's going to send us to a multiple of this this thing here while on the other hand if we start with
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this one and we apply multiplication by Q it's going to send us to a multiple of this one actually to to this one
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exactly so for example we have some arrows going up so for example this Arrow here represents the fact that if
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we apply the lassan to this monomial here we're going to get a multiple of this one and what's the multiple well we
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just compute from here four * m * n so there's a one and a three exponents so
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the multiple is 12 and we can represent that by putting a little 12 there beside the arrow telling us that
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if we apply the lassan to this term here we're going to get 12 times this thing
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here all right so we at every one of these nodes we have arrows going up and arrows going down and it's important to
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remark that in terms of the llin the arrows going up from the top boundary send us into the void which is to say
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that we are getting zero when we apply the lassan to these harmonic guys on the
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boundary here okay so what we're going to do now is we're going to look at these two
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operators uh not individually but together and sort of think about the algebraic structure that they have as
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operators acting on poly numbers and we're going to meet the oscillator
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representation of SL2 SL2 is arguably the simplest and
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most important Lee alge albra and plays a very very big role in quantum
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mechanics and in fact in the standard model also all right so we're going to regard
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both llan and this Q as linear operators on the space of bipoles the complex
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space of bipoles so in particular Q although it's actually just a bipol number we're going
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to think of it as representing a multiplication operator so it's something that takes a poly number and
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multiplies it by Q okay so it's sort of playing this dual role or another way of saying is that
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we're elevating it from just being a poly number to actually an operator on bipoles so that's an important idea in
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quantum mechanics so now that we have these two operators well we can talk about composition of them what happens
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if we do one of them followed by another so let's uh look at the lassan composed
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with Q by this we mean we do Q first and then the llin so acting on a typical
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element U to the m v to the N this is llin of what happens when you apply Q to
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this well when you apply Q to this it's just multiplying by Q so you get the Pline acting on U to the n+1 V to the
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n+1 and then the way theasian works is you get four times the product of these exponents times what you get when you
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lower the indices by one U to the m v to the N so now let's look at the same
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thing but with the operators applied in the opposite order so Q Circle lassan
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acting on the same thing so we're going to apply we're also going to state that just without the composition that's just
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as Q lassan so this means we're first going to apply the lassan natural order
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and then we're going to apply Q so when we apply the Lan well we get 4 MN U the n minus1 v v to the nus1 and then when
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we apply Q well we have to multiply by Q okay the effect of multiplying by Q is just to raise the exponents here so
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altogether we get 4 MN U to the m v to the N so we see that there's a difference okay there's a difference
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between applying these two operators in this order we get this while applying it
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in this order we get this there's a difference and one way of sort of
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describing how much um non non-commutativity there is in this
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situation is to look at the difference between these two compositions okay the
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bigger that difference is then the more non-commutative these two operators
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are so if we take this Delta Q minus
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this Q Delta acting on this thing so we have to take this expression subtract this expression well there's a common U
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to the m v to the N there's a common four if we take this thing here and expand it and subtract MN we just get
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the other terms which are 1 plus M plus n so uh this commutator in the
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literature is often represented by a bracket okay so the bracket lassan Q means llan Q minus Q
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lassan that's a commutator or a bracket and what we're seeing here is that we have an explicit expression for how this
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commutator operator looks so we're getting a third operator we started with these two operators and then looking at
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the commutator of them we get a third operator which is this particularly sort of pleasant operator because it takes
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one of these guys here and doesn't substantially change it in fact it just
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multiplies it by a a factor that means that this in the sense of linear algebra
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is like an igen Vector for this operator it's sent to a multiple of itself and
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this number here is the associate igen value okay so that's a important and uh
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interesting situation in linear algebra so now we have three operators floating
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around so it turns out that these three operators that we've introduced are actually a reflection of a much simpler
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algebraic structure called this Lee algebra SL2 and um maybe the simplest way of
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introducing that is just to introduce three 2x two matrices X which is this
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Matrix 0 1 0 0 Y which is this Matrix 0 0 1 0 and H which is this diagonal matx
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Matrix 1 0 0 -1 now these are matrices so we can multiply them but we can also
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take commutators of them and if we take the commutator of X and Y which is x *
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Yus y * X well what do we get if we take x * Y and do the multiplication the only
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place where we're going to get an entry is in the one one entry so we're going to get a one in the top entry and then
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when we take the uh the negative of the product in the other order we it in the
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other order then the only place where we get something is when we take this row here times this column we get a minus
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one so we get that thing there so x * Yus y * X is
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H and now I want you please to check that we have also other commutators that
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are keeping things closed that these
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three matrices spanning this sort of threedimensional Subspace of the space of 2x2 matrices is
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closed under brackets because if we take the bracket of H and X we get 2 * X and if we take the
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bracket of H * y we get min - 2 * y so
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these are the commutation relations in this Lee algebra so what do we mean by that well
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it means that we have a three-dimensional space with a bracket okay and the bracket is is a um a
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bilinear operation okay it's it's linear in the first if you take the linear combination of x's and expand out you're
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getting a linear combination on the right hand side so it's linear in both the first and second
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positions and it has some additional features first of all it's anti-symmetric or skew symmetric if you
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take the bracket of A and B you get minus the bracket of ba it's kind of
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obvious much less obvious is this more complicated identity called the jacobe
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identity which is some kind of analog of associativity so we're not we're not
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thinking about this space as having a a multiplication in the usual sense right the product of these two things here
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okay the product of these two things is not um in this span it's outside the
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span of these three things so this is not closed under ordinary multiplication it's only closed under this bracket
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operation and the bracket operation instead of having an associ BL has this jacobe identity which is that the
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bracket of a with the bracket of b c plus the bracket of B with the bracket of C A plus the bracket of C with the
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bracket of a b so kind of a cyclic rotation as we go from here to here to here the sum of those three is zero so
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another great exercise just check that this uh this identity is going to be
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true whenever you have um uh commutators coming from um basically multiplying
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matrices right so it's these properties here that make this three-dimensional
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space into what's called a Lee algebra and the name of this Le algebra we say is
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SL2 and what does it look like specifically well we're just talking about linear combinations of these things so that means 2x two matrices of
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the form a b c minus a so this top entry is arbitrary this bottom entry down here
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is arbitrary but the diagonal entries have to be negatives of each other because we just allow a multiple of this
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one okay so it's a three-dimensional Subspace of the space of all 2x two
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matrices it's a Le algebra and it is the most important Le algebra okay it's the
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most important Le algebra it's sort of at the heart of so many things including
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a lot of quantum mechanics okay so now let's introduce the important idea of a representation
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of a Le algebra and I'm going to do it just in the context of this one example
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SL2 okay but this is a very big subject representation Theory um is a major
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Topic in 20th century pure mathematics and part of the huge interest in that
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subject is exactly because of the connections with physics both in the direction of quantum mechanics but also
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in the direction of the standard model so it turns out this is like of central importance
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to physics and there have been I many physics textbooks which have been devoted to trying to understand
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representation Theory so in the case of SL2 what do we mean by a representation
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well remember we had these three matrices x y and H and I just remind you
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here are the commutation relations that they satisfy so this means for example if you
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take the product of H * X and you subtract the product of x * H then the
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Matrix that you're getting is 2 * The Matrix X okay so what we're interested in is
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finding perhaps other matrices x y and H that satisfy exactly the same
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commutation relations if we find such a thing we'll see that it's a
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representation of SL2 it's like another Incarnation or another model for this
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algebraic structure well so what are some examples well here's a pretty trivial example uh if we
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just look at the onedimensional representation where each one of these is zero um well yeah um if you take the
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product of this times this and subtract the product of this times this yes you're going to get this one so it's a representation but it's kind of trivial
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but it still plays a role here's another example um it's a
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way of cooking up bigger representations from smaller ones okay what we're doing
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is we're looking at 4x4 matrices but we're thinking about them as a bunch of 2 by two blocks and in the top left
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corner and in the bottom right corner we're just sticking a copy of the familiar X or Y or H and then if you
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think a little bit about how you multiply matrices you realize that when you take the product of these things you're really just doing the the 2x two
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matrix multiplication so these commutation relations are sort of automatically more that's going to be
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satisfied by these matrices as well so this is called a direct sum of
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representations and that's a way of always cooking up you know bigger representations is to take direct sum
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exactly in this fashion of representations that you already have now that's not without interest that
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actually plays an important role but somehow we would like to have some examples which are not quite as trivial
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as that and it turns out that there's a whole bunch of really interesting
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non-trivial examples starting with this one okay so here are three 3x3
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matrices X is this one is an upper triangular Matrix with one and two along
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that sort of directly uh you know upper diagonal kind of thing and here's y with
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two and one on the sort of lower sub diagonal and H now instead of having one
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and minus one adjacent it has two zero and minus two or two
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bar so very important exercise please check that these three matrices also
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satisfy exactly these commutation relations so you have to make three separate calculations so please do that
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and now another really important exercise okay very important is to find
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the analogous four-dimensional representation so there's a kind of a an extension of this
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going from the 2x two to the 3X3 that step can be extended to go to a 4x4 and
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in fact to the 5x5 and six six so there's a whole sort of family of representations which are all sort of in
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this direction and I want you to write down the four-dimensional representation once you've done that you'll be easily
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be able to see what the pattern is and write down the general one okay so we get an entire sort of family of
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representations starting maybe with the trivial one then this 2x two representation this 3x3 representation a
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4x4 a 5x5 Etc okay and that's um a really rich
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story a really rich story all these different incarnations of this one
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algebraic structure so if someone says to you well what's the what's the Le algebra SL2 well one answer is it's this
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2x2 matrix this 2x2 matrix this 2 X2 Matrix but another sort of almost equally valid answer is it's this 3x3
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Matrix this 3x3 Matrix and this 3x3 Matrix they have exactly the same
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algebraic properties when viewed as Le algebras they're isomorphic Le algebras
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and now we come to the oscillator representation of SL2 which is a different kind of
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representation but it's bigger and better in some sense it's um so-called
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infinite dimensional representation me meaning that the matrices involved are sort of ongoing they're not finite end
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byend matrices like the ones I've just shown you so this oscillator representation has uh a very rich
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history and connection with all kinds of things so originally introduced in the 1960s by seagull Shale and Andre v um
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it's also been uh extensively studied by my thesis adviser Roger how at Yale
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University who made uh important sort of new uh disc iies that the these these
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oscillated representations for for more General situations allow you to extract all kinds of really rich information
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between the relationships between various League groups okay so um it's a
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very rich subject I'm only going to sort of give you a a simplified introduction
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to this okay so I want to introduce that via our two operators q and the laian
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that we've already just been looking at okay but I'm going to modify things just a
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little bit um because the commutation relations that if you look at that we had you know before um had some factors
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of four and uh to make that a little bit simpler I'm going to just renormalize things okay so we're going to introduce
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operators x y and H now acting on our space of Pol numbers complex space of
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Pol numbers in particular we have powers of u and v and I'm going to Define these
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operators by what they do on these basis elements these monomials so X is basically multiplying
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by q but with an additional factor of a half just for technical reasons to make it a little bit simpler okay that things
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work out a little bit nicer so it sends this thing here to2 U to the n + 1 V to
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the n+ 1 so that's a very simple operator okay X that's the operator X
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the operator Y is again more or less the lassan but again there's a little bit of
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an adjustment we're going to put a minus a half here so the effect of that instead of having a four here as we did
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previously we're going to have a minus 2 * m * n * U to the nus1 V to the N
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minus1 so that's the operator Y and then there's an operator H and the
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operator H in some sense is the simplest one because it sends any one of these
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monomials to just a multiple of itself okay that's the same monomial being repeated here not raising or lowering
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just the same one but there's a factor and the factor is 1 plus M plus n so 1 plus the the degree of this thing here
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okay so just sort of dilates in all these various basis directions so we
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have x y and H and you can guess what
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the claim is the claim is that these three satisfy the commutation relations
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of SL2 okay so let's check this let's have
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a look at the bracket of X and Y acting on this basis element so that means we
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act by X Y so that means y first and then X and we subtract YX times the same
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thing okay so what is this first thing so we're going to apply y first and then we're going to apply X so we're going to
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apply x to the result of applying y to this so remember Y is the the laan
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normalized so there's a - 2 MN okay and then we lower the exponents okay over here we have Min - y
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the result of applying X and what x does it just raises the exponents and
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tosses it a half okay so now when we apply X to this what happens well we
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raise the exponents and throw in a half okay so we get the half cancels with the
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two but we still have a minus sign so we get minus MN * U the m v the n and over
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here we have to take -2 times the product of these uh things
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-2 um times the llin of this thing okay but the minus two cancels with a half
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there so we get a plus sign with a one so we just get m + 1 n + 1 U to the m v the
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n and uh that's just a multiple of U to the m v to the n and what's the multiple
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well it's this thing here minus MN and that's just 1 + m plus n which is indeed
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h of U the m v the N okay and then here are corresponding
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computation relation for H and X so we apply HX minus xh and that means h of X
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of this thing minus X of H of this thing and then you do the same thing calculate
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what it is and please check that you end up getting 2 * X of U the m v again so
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the bracket of H and X is indeed 2 * X please check the
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algebra and here is the corresponding calculation for the bracket of h and y do H Yus YH applied
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to the same thing please check that get this and It ultimately reduces to minus
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2 * y of U the m v the N so the
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commutation relations are satisfied that means we have a representation of SL2
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here involving the line up to a multiple this multiplication by Q operator up to
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a multiple and the commutator which is essentially this um operator that takes
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you the MV the n and just multiplies it by this 1 plus the degree
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number so can we write down corresponding matrices for these operators well yes we can but we have to
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go beyond the usual Matrix formulations because uh there's an unbounded aspect
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to these poly numbers right there's not a finite number of them they they keep on going they keep on increasing in
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degree so we have to be prepared to sort of think in terms of U ongoing maybe
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sometimes called infinite matrices in fact this is a a big story of uh sort of the Revolution that
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Heisenberg introduced in the 1920s uh sort of Matrix mechanics approach to
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Quantum Mechanics what we're doing here is not exactly like that but it's pretty similar to uh what was going on there
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okay so let's have a look at this oscillator representation and and what
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actually is happening to these various monomials as we apply these various operators so here's our triangle going
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down unbounded triangle going down of our various monomial basis elements in our complex basis and now what happens
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when we apply Q well we've seen what happens when we apply Q is we just sort of go down down vertically down and what
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happens when we apply L plasan well we go up we go sort of vertically up except that there's some multiples involved but
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basically that means that any one of these columns okay any one of these columns is going to be maintained by
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both The Operators X and Y and therefore also by the the bracket of them H so
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what we really have in fact is uh sort of a a sum actually of of representations one corresponding to
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each column and one natural way of indexing that is sort of to think of this one as the central one so maybe we
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could call let's say an index value of L so maybe this is L equals z and this one here the to the left
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where we increase the power of U by one that might be Lal 1 here's Lal 2 Lal 3
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Etc and correspondingly over here maybe min-1 - 2 - 3 so if we identify say one
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of the column say the one starting with u to the L then the one low it would be U to the L + 1 V we just raising the
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exponents both of them by one as we go down or if we started with v to the L well again we just raise the exponents
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going down so this is the column that we would sort of index by L this is the column that we would index by minus L
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okay so we have columns which are indexed by integers so we have actually a um an integer parameterized family of
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representations which are all sort of part of this big illat representation of
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these x y and H operators acting on the entire space of poly
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numbers so let's just have a look at this Central one okay the L equals zero representation so here are the basis
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elements 1 UV U2 v s u cubed V cubed Etc okay so we have we have operators so the
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X operator takes one of these guys and raises the exponents and there's a half
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out front the Y operator that's leoan so it lowers the exponents here multiplies
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by MN and there's a minus two out front and our H operator is this diagonal thing that just leaves the monomial
27:35
where it is but multiplies it by 1 + m+ n so we can visualize these things now
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um yes in terms of a matrix but there's in some sense a kind of a simpler way of visualizing things which is going to be
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very important for us as we carry on with quantum mechanics okay so this is a kind of a graphical way of visualizing
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this repres entation is thinking in terms of arrows that take basis elements
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to other basis elements with numbers sort of attached to denote the coefficients
28:10
involved okay so let's look at U the X operators which are the simplest ones so
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the X operator is going to take this to 1/2 UV so we put an arrow from here to here say a Black Arrow corresponding to
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X we put a half here telling us we take X of this we're going to get half of
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this and then there's another Black Arrow with a half if we take X of this we're going to get half of that Etc so
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there's a whole sequence of arrows black ones going down with halves beside them
28:38
representing the action of that operator the action of the Y okay it's going in
28:44
this direction but it's represented going up okay it takes this thing here to this thing here so it takes um one
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and it actually kills it okay so I hope we all appreciate that when m is uh zero
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this this expression is supposed to be zero okay so if we apply y to this thing it kills it so we don't have an arrow
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going there but if we apply y to UV okay UV um with one and one exponents then
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we're going to getting a minus 2 * 1 * 1 we're getting a minus two coefficient so y acting on this thing gives us min-2 *
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this one and what is y acting on this one well we're going to get - 8 because
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it's - 2 * 2 * 2 so - 8 * this one and what's it acting on this thing we're
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going to get -2 * 3 * 3 that's - 18 * this one okay so we get these these
29:37
arrows these red arrows going up and they have Little Numbers Associated to them that tell us what the coefficients
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are that describe the action of this operator corresponding to these
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arrows and what about H which is kind of a diagonal operator okay well it doesn't
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change any of these basis elements just sends it to a multiple so all would we really need to do is record the multiple
30:04
okay if M and N are both zero then we're just getting one if M and N are both uh
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one then we're getting a multiple of three so we put a three beside that and then a five beside this seven beside
30:15
this Etc so the action of all three operators is pleasantly encoded by this
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kind of graph with suitable arrows going in different directions telling us what the various operators are doing to the
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various basis elements it's all very explicit and we can you know check with
30:35
the commutation relations so you know for example if we're doing X or Y the
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bracket of X and Y so we do y let's say we start here okay we do y first okay
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get minus 8 and then we do this one here we get minus four minus four of those
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and then if we uh do this one first we get a half followed by Min - 18 that's u - 9
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okay so what did we say uh we get U minus uh minus 4 and then when we do
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this and this and we subtract we get plus 9 so -4 + 9 is 5 well that's exactly the value of the H acting on
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this thing here right so we can we can check the commutation rules in terms of doing you know going up and back and
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then subtracting down and and up okay and what we should get is the uh this
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thing here and you can check the other commutation rules as well now if we actually want to get
31:29
matrices or familiar linear algebra system well that's okay too because we have a natural ordering of these basis
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elements that's the first one second one third one maybe we should say this is the zeroth one first one second one
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third one Etc then with respect to this basis think of them as being it columns well X is then the The Matrix given by
31:49
here's diagonal of zeros and below the subdiagonal just has a lot of
31:54
halves so that's telling us for example if we apply X to 1 0 0 0 that column
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Vector representing this thing here then we're going to get Z 1/2 so it means
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we're going to get 1 half the next one which is this one here Y is then represented by this upper triangular
32:11
Matrix with entries -2 - 8 - 18 and H is represented by this diagonal matrix zero
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is off the diagonal entri is 135 and if you're happy sort of doing um
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matrix multiplication with these things just doing it trating piece at a time
32:29
then you can check that x * y - y * x = h and so on this is another
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representation of SL2 but it's U of A considerably more subtle nature um than
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the finite dimensional ones that we started off talking about okay so there's a lot um to
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investigate here uh for prospective physics students or people just interested in in um the relationship
32:56
between physics and harmonic analysis here so I've described what happens when we look at the L equals z representation
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right but you you could also look at L equals 1 or L equals minus one or any of the other ones and write down
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corresponding matrices and they're going to look different and get a different kind of representation for each one of
33:14
these different columns another really great thing to do is to start go right back to the
33:20
beginning of the lecture okay and say well what happens if we consider this
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not from the point of view of this particular um so basis set but in terms
33:32
of these factorial basis elements where there's an underscores right so we can
33:37
do the same game we can describe how the llin acts and how the multiplication by Q acts Etc we're going to get slightly
33:44
different formulas and uh we're going to get an equally valid kind of uh
33:50
incarnation of the SL2 representations but they're going to have sort of different numerical values here and it's
33:57
a interesting exercise to go through that um I recommend that to you sort of
34:03
suggesting there's a kind of a symmetry maybe one of these is not really preferable to the other um we could use
34:08
either of them okay so lots of important material here uh please if you're not
34:15
that familiar with this stuff it's a good opportunity to really anchor uh your understanding of some really important Concepts that will become uh
34:22
more and more crucial as we head towards quantum mechanics