 Can you please slide the board up a bit? Ah, I shouldn't write it. It's okay, it's okay, it was a joke. Okay, okay, okay. Okay, so I shouldn't wait till, no? Okay, so... Our speaker today is Atish Dappelkar, who will speak on black holes, primer for mathematicians. Okay, thank you, Don. It was all planned because the idea of this lecture arose because Don and I, we wrote a long paper together with Sameer Murthy, who also is here visiting from King's College. It started out by a conversation in a café in Paris, and then before we knew it, it became a two-three-year long collaboration and became a 150-page paper. And then Cambridge University asked us whether we would like to turn that into a book. And in a moment of weakness, we agreed because turning it into a book is much more of a work. But I thought that this would be a way, and basically Don wanted us to write the physicist about the subject in the language that the mathematicians can understand. So with that objective, I thought that the best way would be to try it out on an audience of mathematicians. So you are the guinea pigs. And if it is not of any value to this audience, then surely there is no point in writing a book about it. So my goal is to so it's going to be perhaps two elementary for some of the physicists, at least the first lecture. Hopefully by the third lecture it will be of interest to physicists as well. My goal is to explain so there has been quite remarkable progress in understanding the quantum structure of black holes in string theory and actually that progress continues and slowly over the years there have been a number of definitive advances in the subject and some of it has to do with our work. But there is always a problem of and one finds that in dealing with black holes in string theory, one is naturally led to some very interesting topics. Sometimes that really are the forefronts of research in mathematics. So for example modular forms occur very naturally. Then Jacobi forms appear. These are sort of topics from number theory you could say. Siegel modular forms appear. Meromorphic Jacobi forms appear. Our work has to do with mock modular forms and Jacobi forms. I mean this is actually a new object that we introduced in our paper. It's a new mathematical object introduced in this paper and we hope that I will try to convey to you that it's kind of a very generic object which is likely to now appear in mathematics and physics more and more. Then there are these are topics in number theory. Also partitions of integers these kinds of problems. Famous problems going back to Ramanujal for example. Topics in enumerative geometry like Euler characters of modular spaces. Donaldson Thomas invariance things like that. Wall crossing phenomena in the Donaldson Thomas invariance. So these are all a very interesting set of topics clearly of interest to mathematicians. And correspondingly on the physics side they appear in somewhat in different language. For example this is an object of course known also to mathematicians but elliptic genus or partition functions of of conformal field theories or super conformal field theories. Black holes quantum black holes I would say 19th century notions like entropy of black holes in thermodynamics things like that. So clearly there is a language problem I think the left hand side will not understand the right hand side and vice versa. And my goal is to translate in one way namely try to explain these things to mathematicians and try to make a connection with all these objects that they know. So for the physicist I apologize I am not going to be able to explain this that is for Don to do. But in some ways this translation is easier because you know modular form is an object you can define. It is a function defined on the upper half plane which transform in a certain way under SL2Z. To give you that definition everything is clear. But defining what a black hole is and what is holography is much more complicated. Of course this is very ambitious because in order to do this I have to explain start with quantum mechanics or quantum field theory and string theory and clearly it is not something I can do in four lectures. But I thought but since we want to write something like a 40-50 pages introduction to our book which makes the content somewhat intelligible to them I decided that I will give these lectures. So what I have in mind is a smart mathematician like Don let's say who maybe has some vague memories of undergraduate physics that he had or she had done and can I try to explain essentially every term that I use in the language that sort of should make sense and it should not be just some mumbo jumbo. So can I say what is a quantum field theory or can I explain what is black hole or what is a super conformal field theory things like that. Can I try to explain that. So that is I set myself that task. And then Don told me that you have to explain everything. What is gauge theory, what is black hole, what is gravity I said okay it's a lot. But what I've thought that the best way to approach is starting with quantum mechanics because and the simplest quantum mechanical system which is called the harmonic oscillator so that's why physicists can now leave if they want to. Because that's what I'm going to spend half an hour of my time which is something that you do in the first course in quantum mechanics. But surprisingly you will see that's actually a very good way to get so I won't be able to of course do the most general quantum field theory but I will be able to explain what is a quantum field theory a simple quantum field theory or what is a simple conformal field theory or how do you compute the elliptic genus in some simple cases. And basically you can go surprisingly far if you know how to do the harmonic oscillator system. And the reason so I will explain this as we go along. So this is I'm just giving you the background a bit maybe the first half an hour I will give you the background. So I thought that one way to organize this talk would be that today I will just talk about this quantum mechanics. And in that I will explain what is known as a harmonic oscillator what is called a state Hilbert spaces and so on. And again you will see some of it is just a matter of terminology because what physics is called a state mathematician would understand it immediately if I say that it's a vector in a Hilbert space but physicists call it a state or what physics is called observable mathematicians would call a self-adjoint operator on the Hilbert space. So that translation I am going to do and this way I can also introduce so I should probably add quantum ensembles and entropy. And in the second lecture I mean as you can see it's very very ambitious. And so I might feel in which case if there are three people left over at the end of the fourth talk we can have another set of four lectures next semester. So the number has to be greater than or equal to three. Quantum field theory and elementary particles. Once again I will try to get only those very basic concepts that are required for my purposes. Then on the third day I will try to make the connection between say quantum mechanics and quantum field theory let's say which is just called it QFT the super conformal field theory is called SCFT QFT and topics in let's say topology and number theory. And along the way I will have to explain what is super symmetry we are able to do this and maybe fourth lecture I will do general relativity and black holes and clearly that leaves a lot more to be done which perhaps if there is enthusiasm I can do it next semester but I thought that I should not inflict too much upon you in the first go. So let me now give you some physical motivations why this problem is the problem of quantum black holes have come to play a very central role in the recent explorations of quantum gravity so one of the really and it's a fundamental importance because one of the big challenges of 20th century physics is to have a consistent theory of quantum gravity so as I promised you I have to explain to you what is quantum and what is gravity so let me try to say what are the two sides of this so there are two sides of the one side is quantum mechanics and the other side is general relativity so what is general relativity is Einstein's theory of gravity how gravity act at very large distances so gravity at large distances you might have heard about the black holes that were discovered gravitational waves and all kinds of things that you might have heard about in news all that has to do with classical gravity whereas quantum mechanics describes nature at very short distances short distances and it describes basically everything all the way up to starting with the semiconductors and lasers and atoms, molecules all the way to the large headron collider, elementary particles everything is described within quantum mechanics but these two are inconsistent with each other in a very fundamental way and one way to say that classical gravity is non-renormalizable again I will explain this term what it means so next time as a mathematician when you encounter this term you would not be shocked these two contain two completely different set of concepts a priori which have nothing to do with each other and we are going to find so in quantum this so the idea is to find quantum theory of gravity which is called quantum gravity and the most promising candidate for that is what is known as string theory there is basically at the moment no experimental evidence at all for string theory so as a physicist that should make you very uncomfortable why we are even discussing string theory but as I will try to explain to you we have uncovered some really very deep and very important facts about quantum gravity structural facts about quantum gravity in the process which makes me believe and many physicists like me believe that we are on the right track and this has something to do with the real world and that is the thing that I want to explain to you those motivations link between these two at the moment the best link that we have is quantum black holes so that can sort of motivate you the topic of this lecture is that we are interested in quantum black holes because we are trying to unify quantum mechanics with general relativity I mean two amazingly successful paradigms in 20th century physics this is extraordinarily successful at very short distances this is extraordinarily successful you can predict all kinds of things at very large distances but when you try to it just doesn't work and there are some fundamental inconsistencies that arise and the only way that we know in my assessment is basically string theory and it's kind of a very unique very highly constrained structure that we have uncovered more or less by accident and what makes it possible to make any progress at all is because we have some very beautiful results from the physics of black holes which give us some very strong hints about how we should proceed even though we don't have any guidance from experiment so some of the topics that we will relate to each other again you might have learnt thermodynamics in your undergraduate studies in quantum mechanics it goes under the name it's sort of subsumed in what is known as statistical mechanics quantum ensemble so on the left hand side quantum ensembles in quantum mechanics on the right hand side the notion is entropy on the left hand side you have Hilbert space on the right hand side you have space-time geometry like in space and time four-dimensional space-time geometry and here we have a numerative geometry some counting problem in some modellized spaces of some some objects which are called brains in string theory literature and on this side we have black holes on this side we have modular forms of various kinds of the kind that I described so enumerative geometry of course encompasses all these topics like Euler characters of various modular spaces Donaldson-Thomas theory and so on and so on modular forms of all these various kinds they are related on this side somehow they appear in the form of the Hardy-Ramanujan expansion Hardy-Ramanujan Radomaker expansion it's called conformal field theory on this side CFD for conformal field theory and EDS which is called antidecital space-time this again became a very very important development in the last 20 years just recently there was a 20th birthday celebration of the discovery of EDS CFD duality so it was exactly 20 years old so my goal is to explain the various connection in a manner that hopefully is intelligible to the mathematicians and I have in mind a kind of a as I said a smart mathematician who will interrupt often but not too often and ask pesky questions about why this doesn't work so therefore I request you to please ask questions if anything is not clear or if I'm sloppy somewhere it will help me to sharpen my argument if I cannot answer it immediately in the language that you want I will try to give a better answer next time but yeah so since it's a translational kind of it's a procedure of translation if I translate something not in a quite way that you understand please ask me to translate it again okay so that is the plan so this is the motivation and this is the plan and these are the sort of connections that we want to understand so is this point clear and are there any questions at this point so by quantum mechanics I mean it's an enumerative geometry of some gauge theories modellized spaces of instantons and some gauge theories for example yes so it's a kind of a different geometry this is the geometry in space time whereas this is the geometry of the internal Calabria manifold so in some sense both are geometry but the geometry here is kind of in the Lorentzian space time the geometry is that of a black hole that somehow encapsulates information about the enumerative geometry of some six dimensional Calabria manifold so are there any questions at this point now I'm going to give you a quick summary of quantum mechanics just to remind you because I'm sure all of you have seen quantum mechanics so basically quantum mechanics as we said it's the theory at short distances so it goes back to the atomic hypothesis and it's really perhaps the most important principle in physics you could say is that all physical processes can be understood so the atomic hypothesis states that all physical processes can be understood can be understood in terms of motion of atoms and motion you mean dynamics dynamical motion of atoms so just to fix an idea we don't in this room we believe that there is air and we believe that it is made up of molecules how do we know this you just learnt it in school but that is the atomic hypothesis that is really made up of little molecules which are moving around and we can understand the temperature in this room or whether it's hot or cold by how fast these molecules are moving so therefore the notion of for Newton atoms were little particles of dust they were moving around under some gravitational force so the notion of atom has actually evolved starting with a dust particle to atom like a hydrogen atom to more elementary particles like electrons and protons, photons let's say so you all of you have some intuitive notion of what an electron is you know it's supposed to be some very tiny particle which is moving around but if you try to really make that notion precise it's quite subtle and basically quantum field theory is the study of quantum mechanics before quantum field theory quantum mechanics is the framework for describing basically any physical process in quantum mechanics in principle you can describe in terms of motion of atoms for describing this dynamics and so it's an extremely powerful formalism and it seems to work surprisingly well and now the modern version of what an atom is really is the quantum of minimum energy of a quantum field okay so I will explain everything what is quantum, what is energy what is a quantum field but you have a heuristic notion of a field like electromagnetic field so for example in this picture a photon is the quantum of minimum energy of an electromagnetic field similarly electron is a quantum of a new kind of field which physicists have not uncovered until the 20th century namely the Dirac who translated it in a mathematical language electromagnetic field A this is sometimes denoted as psi so A is a U1 gauge connection on spacetime manifold so by spacetime manifold I mean the manifold that we live in it's a four dimensional manifold as in we are talking about Minkowski but in general it can be general manifold so M1,3 spacetime manifold is a four dimensional real manifold M1,3 with a metric whose signature is minus plus plus plus so at least we know what is a spacetime manifold spacetime manifold is a four dimensional manifold if it was plus three four pluses then it would be just a four dimensional Euclidean manifold but it's not a Euclidean manifold this is called a Lorenzian it's a Lorenzian manifold and mathematicians all know what is a, so you have a bundle you have a U1 connection each field the electromagnetic field is basically a connection on the spacetime manifolds you have a spacetime manifold M and at each point there is a U1 fiber and there is a principle so there is U1 fiber you have a connection in that similarly the Dirac field so this is connection U1 connection Dirac field you can think of as a section of a spinner bundle on M so you just like at each point in the manifold you have a tangent space and you have a tangent bundle instead of the tangent bundle is transforms as a vector of okay so let me see more size so Minkowski spacetime is a special case which we will denote as r1,3 it's a special case of M1,3 where the metric g mu nu so mu and nu take values from 0,1,2,3 and we have some coordinate system x mu the physics notation where x0 is to be identified with time and x1,2,3 xi is to be identified with space and g mu nu in this case is eta mu nu g mu nu is a metric tensor so it's a tensor field it's a symmetric 2 tensor that's why it has 2 indices I think it's called covariant covariant 2 tensor so it's in a cotangent bundle it's basically a symmetrized product of cotangent bundles and in this case eta mu nu is particularly simple it's just minus 1 all diagonal so this is the analog of just flat Euclidean space if this minus was turned into a plus your spacetime manifold would be just a flat Euclidean r4 but this minus 1 is very important and all the very interesting physics having to do with molecules and all kinds of things about antiparticles and everything has to do with the fact that there is a minus sign and that we are dealing with a Lorentzian manifold and not a Euclidean manifold so even though mathematicians usually like Euclidean manifolds because then theorems are easier to prove somehow physicists are forced to deal with a Lorentzian manifold because some of the most essential physical things do not happen unless you have a minus sign there is this clear? it follows from the fact very important physical observation that the speed of light is constant in all frames I mean all moving observers even if you have a moving normally if you throw a ball and if you view it in a train then it would look to you to be stationary if you have a ball going at 20 kilometers and a train going at 20 kilometers in the frame of the train the ball is at rest whereas in the frame of the platform the ball is moving but if you replace the ball by ray of light surprisingly you find that whether you are in the train or on the platform it is always moving with a 300,000 kilometers per second and that actually has to do with the if I write that line element corresponding to this metric is actually c squared dT squared minus dxI squared the minus sign and when this vanishes that actually indicates the null ray this is equal to 0 is the so a trajectory of a null ray, a light ray is given by satisfying this equation and its infinitesimal version is this I just told you the speed of light does not change you need some transformation which leaves this metric invariant therefore you need not so eta union is invariant under Lorentz transformations which is a group o1,3 so basically you are used to defining a manifold with a tangent space where o4 acts naturally whereas here because you are talking about because of this relativistic fact you are interested in a manifold whose tangent space has o1,3 acting on them that is what dictates this minus sign any other question so this is the modern version of the atomic hypothesis therefore the modern version of the atomic hypothesis is that all physical processes can be understood in terms of dynamics of quantum fields you will see, you will come to Eisenberg principle so I am going to do really axioms of quantum mechanics from the beginning there will be no hopefully no nothing will be left undefined maybe I cannot explain it fully but everything I will try to define as much as possible so this is a classical field, see electromagnetic field when you think of it as a U1 connection it is just some function or it is some right or a section these are all well defined classical objects mathematics you are using what the important thing is that we have to think of them as a quantum fields and quantum field theory has been developing over the last almost now 100 years starting with or 90 years or so and it is a notoriously difficult subject even for physicists and making sense of it to mathematicians serious people have tried for example I I do not know whether to recommend I recommend Princeton University organized a series of lectures by Witton and Grosse and all the sort of the leaders in the quantum field theory side and I think people in the audience were like the lean who were taking serious notes and they have it published I think it is Princeton lectures on quantum field theory something like that you look it up where they really try to explain the guts of the quantum field theory as far as they could take it and I have to say that okay that experiment has been only partially successful I mean I think there was some communication but I do not think it has really penetrated mathematics community so much and it is I mean of course it cannot be I think all the lecturers were no I mean there were no slouches but it was not it is not an easy subject to explain but I realize that for describing this connection with number theory and counting problems and black holes we do not really need the full machinery of quantum field theory and one can understand a large amount of quantum field theory one can go surprisingly far by understanding a much simpler system can be equivalently understood as a collection of quantum oscillators and that is the point of view that I will try to develop because this is a system that I can really honestly describe to a mathematician without having to fudge or without having to say that okay there are I mean it can be described pretty rigorously where does this connection come from so let me first explain the intuitive idea so you are all used to the fact that if you look at the light in this room which as we know as I told you photon is the particle of light collection of photons quantum mechanically it is a collection of photons but classically I mean this is how Maxwell discovered it these are just Fourier modes of the electromagnetic field we know that light what do you remember Fourier modes basically red whatever various colors green blue these are the frequencies of the Fourier modes and I will explain this in greater detail the point is that you think of that the room is filled with some electromagnetic field and it is just oscillating performing all kinds of oscillations because it is a Fourier analysis it is like a harmonic analysis and that is why the word oscillator is sometimes also called harmonic oscillator so you have an electromagnetic wave in some particular Fourier mode going that way of various frequencies green light blue light and so on I can think of the quantum electromagnetic field simply as a collection of various these frequencies oscillating independently of each other so if you understand a single harmonic oscillator putting it together we will be able to understand the electromagnetic field in this room ok is this clear yes this is the quantum picture as a wave as a collection of waves collection of oscillations but those oscillations if you treat them in a quantum mechanical way quantum mechanically and I will explain what that means then what you will get is you will discover that there is a photon in the room we will shortly discover a photon ok in the room so let me state the axioms of quantum mechanics so as I told you quantum mechanics is the formalism of any possible physical process that you can imagine so a fundamental concept is a physical system so a physical system can be a single elementary particle like an electron flowing in a wire or it could be the air all the air molecules in this room that can be the physical system or the all the atoms inside a star because star so any isolated let me write it more correctly so now I am going to translate for you quantum mechanics is a framework for describing dynamics of which means time evolution of how things develop in time isolated physical systems as I said an isolated physical system could be if you isolate this room you completely thermally seal this all the atoms inside is a physical system or this this is a physical system if it's an outer space but here it is interacting with my hand so if I want to take this interaction to account I have to sort of describe all of us together so given any physical system and the physical system can be in different states for example if you have an elementary particle it could be here or it could be there it could be moving with some velocity it could be moving this way so the state this is where the state of a physical system s is represented by a vector and physicists use this cat notation but okay this is a notation you can just call it psi this is Dirac's notation but it's a vector in a Hilbert space associated with the system so every physical system in whichever state it is is represented by a vector in the Hilbert space of that system so you really have to translate you see it's a you have to translate every physical concept that you want to discuss in some appropriate mathematical language so then we can measure various things of the physical system the energy of the physical system the total energy in this room or the charge of the electron as you know is minus 1 or the momentum of the particle how much velocity of the particle that's called an observable every observable of the physical system so whatever you can measure about that physical system is represented by a self-adjoint operator on s, on h these two are easier right given any physical system I tell you what is a Hilbert space and what the state of the system I tell you what vector it corresponds and if I want to perform any measurement I look at it corresponds to some self-adjoint operator on that Hilbert space so since it's a mass audience I'm going to assume that everybody knows Hilbert space, self-adjoint and so on there is a preferred observable very important observable called the energy which measures the energy called the Hamiltonian of the system yes I'm going to come to that so that's why so the units are just added basically so I will give you an example for example the Hamiltonian of a harmonic oscillator should have units of energy so it is omega which just sets the unit times multiplied by some operator yeah you just take them out of your equation secondly as you said that's why they have to be self-adjoint otherwise their eigenvalues could be complex because their self-adjoint their eigenvalues are real but first of all they have to be real to begin with they have to be real so the self-adjoint property guarantees that they are real and then the units are basically set in by hand I mean you just multiply by the appropriate units or you can remove them and work in your favorite set of units like plump units and then there are no units which enter the story now this is a very important observable because it measures the total energy of the system but I will come to this question what do you mean by measure this will be the fifth axiom the time evolution of an observable so these observables are time dependent some observable A as a function of time I want to figure out A as a function of time is given by this equation this is the Heisenberg equation this is the commutator as usual by A B minus B A so it is completely explicit in fact since this is a first order equation it can be immediately solved and you can write A of t is equal to A of 0 U of t is e to the minus i h t so it is clear that if I now differentiate this I will bring down an h it will work out sorry maybe I put a minus sign wrong U dagger here so there is an i here it will work out so U dagger is so if you plug this into this equation right A of t is just e to the i h t A of 0 e to the minus i h t and therefore i dA by dt will just bring down i h it will give you h times A and when you differentiate this you will get A times minus i so basically if you know the Hamiltonian of the system the time evolution problem which is what you are interested in you want to understand how the planets are moving around the sun right it is a problem of time evolution what will happen tomorrow predicting the future so in any dynamical system what you want to understand is how the system will look like tomorrow so where will the earth be tomorrow that is a dynamical problem and in quantum mechanics that dynamical problem is basically completely solved once you know this unitary operator so U of t is called the time evolution operator and since h is equal to h dagger U dagger U is equal to 1 it is a very fundamental fact of quantum mechanics that time evolution is unitary time evolution of observables unitary that means it is implemented by the adjoint action of a unitary operator and then we come to the fifth axiom and it is like the Euclid's fifth axiom and this is the perhaps the most subtle and most debated axiom of quantum mechanics and I am not going to I do not need all of that so I am going to give a simpler version of the measurement axiom but if the system is in a state which is an eigenstate of an observable E if you perform let us say consider a system a physical measurement it is an eigenstate of the observable I with eigenvalue alpha let us say then the physical measurement of this observable on this system will always yield will always remain with unit probability perform you have many such systems they will always give you the same value alpha so is this clear so if a of psi is alpha of psi that means alpha is the result so this part is straight forward this part is not controversial controversial part is that of course generically psi you could choose an observable which is not for which psi is not an eigenstate you get a probabilistic interpretation that the subtle part of the measurement hypothesis measurement axiom has to do with the fact when the state is not an eigenstate of the observable which you are measuring and I think it is fair to say that because at least in my mind the issue is not really satisfactorily fully understood because essentially let me not go into it so basically there is a kind of the measurement process in this axiom is not described by unitary evolution which is a bit unsatisfactory to have two parts of quantum mechanics which are not mixing this it is a bit like the Euclid's fifth I could say and I am sure there has been a lot of work on this quantum measurement and there has been a lot of progress but I think this remains an open issue but we are not going to require that we just are interested we understand now everything so let me stop here so we are trying to understand the dynamics of every possible physical system such as the electromagnetic field in this room as I told you the electromagnetic field in this room is a bunch of oscillators I want to know which state is that oscillator in it is described by that state corresponds to a vector in some Hilbert space corresponding to those oscillators if I want to measure the energy it corresponds to some observable which is a self-adjoint operator on each if that observable the evolution of that observable is given by this equation and if I perform a measurement and if that system is in the eigenstate of that operator then I will always get the eigenvalue as the result so this actually summarizes all of quantum mechanics so if you have any questions you can ask can we interact with each other? I am not talking about particles it is physical systems which can be a collection of particles that is a physical system so I am talking about a single isolated system you are right that often times what we do is that we consider two isolated systems like two particles and then consider them together and introduce some interaction so I will come to that so that is a good question I will come to that so one comment is what you exactly asked suppose I have two non-interacting systems these are commons two systems S1 and S2 each with the Hilbert space H1 and H2 the Hamiltonian H1 and H2 the total Hamiltonian total Hilbert space of this is just the direct product this makes precise what we mean by two systems interacting with but the Hilbert spaces are a direct product and the Hamiltonian is just H1 cross 1 plus H2 1 cross H2 sometimes we just write it as H1 plus H2 with the understanding that it is acting as an identity on this similarly for all other reservoirs so these are basically product Hilbert spaces if it is non-interacting and we will actually shortly see they are very important and they will hopefully by the end of this lecture they will quickly relate to interesting problems partition of an integer they will come from product Hilbert spaces they will quickly relate to interesting problems in number theory or if it is interacting so like you are imagining even if weak interaction if there is some slight and gravitational interaction between earth and the sun then it can no longer be described I just I mean in that case the Hamiltonian is no longer this in that case we just talk about the full Hilbert space of the total system so that is comment number 1 comment number 2 was about this axiom which I just warn you that it is a very subtle axiom not this version but the full version of the measurement axiom is a subtle one so if you encounter it you can ignore it one important point is for purposes comment 2 is that if you have 2 observables A and B commute then they can be simultaneously diagnosed let me call it A1 and A2 this what is the notation I have used A1 and A2 and you will get some eigenvalues alpha 1, alpha 2 and in some cases they might have a complete of commuting observables I can uniquely specify every state I might have some set of observables A1 to An I can specify any psi belonging to H for every psi belonging to H I can write psi as some linear combination I mean okay so as a basis another way to say that is that alpha 1 i alpha i1, alpha 2, i2 alpha alpha n, iN furnishes a basis okay is this clear in this case things are easy because then you can basically label this can be used to label states so once again this is a translation issue quantum number simply means eigenvalues of a complete set of commuting observables okay in general quantum number means just an eigenvalue also observable so let me write this dictionary here you know so that by the end of the day you will have a dictionary with the system S we have associated a Hilbert space with state we associated a vector with observable we associated an operator self-adjoint operator then quantum number is the eigenvalue of the operator so for example if you want to describe a particle like the electron a state of a particle like electron what do we need to specify it turns out it's enough to specify just three real numbers the mass the spin and the chart the momentum sorry so momentum actually so M let me be more complete it will be E energy, momentum charge and G if the electron is going with some momentum if I climb onto a train which is going with the same velocity as the electron I can bring it to the rest frame so using Lorentz transformation if I want to boost I can bring it to a form M0 qj so it suffices to know this state because from that this can be obtained from this by just a unitary transformation corresponding to this act of getting onto a train with the same speed which is called a boost Lorentz boost this is the way to understand an elementary particle in fact we will come to that so completely abstractly an elementary particle is a unitary irreducible representation of a Poincare group times some internal symmetry so a particle abstract definition of a particle it will satisfy a mathematician hopefully is an unitary irreducible representation of the group Poincare group which is basically the inhomogeneous version of ISO 1.3 so you are used to if you are in R4 the group of rotation is O4 and the group of translation is R4 and then you can take a semi-direct product which is called inhomogeneous rotations I will put an SO4 just to look at the one with determinant ISO4 is basically a semi-direct product SO4 R4 SO1.3 of this group of ISO3 times let's say electron times U1 so does that satisfy you as a definition of a particle for a given electron so I am thinking of an electron as an isolated system a single electron as an isolated system in far out space you have a single electron as an isolated system that corresponds to Hilbert space and that Hilbert space is nothing but a representation of this group and therefore it is completely specified by these three numbers because these are the kartan if you like these are the kartan generators kartan of this so it is completely group-theoretic it is a completely group-theoretic definition of an elementary particle sorry I should not write here in the dictionary that eigenvalue so now here so here this is the charge quantum number this is the energy quantum number this is the momentum quantum number and this is the spin quantum number but you could just think of them as eigenvalues of operators j hat j vector j let's say j as Hamiltonian and some operator q maybe I should denote this by small q or something like that no to distinguish the eigenvalue from the operator so if my electron is in eigenstate of these operators then its eigenvalues are e p q j and those are in the physics language called quantum numbers so in this case it is just minus 1 but if it was a positron it would be plus 1 and in fact j is half or some unit half plus or minus half it can be up or down it's a spin half representation of h2 yeah there are different systems and they are no no no what I am saying is that the electron has a Hilbert space h1 this has a Hilbert space h2 yes but now if you now consider two systems combined like that then you can define total charge operator and in that case here we added h1 plus h2 in that case what will happen is that the q operator will also q1 plus q2 but on the first Hilbert space it will give you eigenvalue 1 and the other one it will be eigenvalue minus 1 I mean you have to it's a relative normalization is fixed yeah so it is true that it for a electron it's just for example spin it's just a two dimensional representation of the rotation group that's the spin of the electron so the total Hilbert space if you like is some direct product of this representation of the Poincare group times representation of the u1 which is just one dimensional what you are saying is that representations of Abelian groups are one dimensional so in this case they are just labeled by something 1 minus 1 2 so if you have a system which has total charge 2 then its q quantum number would be 2 and in general the charge could be some non Abelian charge in which case you have some representation okay any other question I am going to pass in a moment so therefore yeah it's already Lorentzian but that's the reason I I never talked about non-relativistic quantum mechanics because I am going to talk about quantum fields so therefore the quantum fields the representations of the harmonic oscillator will give you actually a representation of the Poincare group the Fock representation of the harmonic oscillator will give you that so that's why I don't want to talk about relativistic non-relativistic at this stage and that's why actually I chose to now that's why I am going to talk about the harmonic oscillator okay so this is just the formalism now can we put some very very concrete meet into it and as I told you please pay attention because this is the system that we will encounter this is the simplest system and if you understand this much you can actually understand a lot of physics in fact as somebody has jokingly said almost all of physics is a most of the 90% of the work of the physicists is to reduce a given physical problem to some form of a quantum oscillator and it's actually it's really true all the calculations in perturbative gauge theories which are required for the understanding the Large Hadron Collider Physics of the Large Hadron Collider is essentially an exercise in understanding quantum oscillators interacting with each other so it's really an important and very simple system and it's a system that basically you can understand with sort of high school okay undergraduate mathematics so we need to specify Hilbert space we're going to represent the specify the Hilbert space somewhat indirectly as a representation of some algebra so so Hilbert space and state we will come to later we will reverse the order a little bit so axioms one and two state on the Hilbert space but let me first specify the Hamiltonian the Hamiltonian is this a dagger a plus a a dagger a and a dagger operators on some Hilbert space and given on a Hilbert space h with the commutation relations a a dagger is equal to 1 a a is equal to 0 and a dagger a dagger equal to 0 so given this one can try to find a representation of this algebra the Hilbert space should furnish a representation of this algebra for that it's useful to define an operator n which is the number operator a dagger a and this implies that the Hamiltonian is simply n plus half because I can now using this commutation relation I can turn a a dagger into a dagger a get rid of this two and because of this one I will get omega over 2 so that's now if I take a state psi I am looking for a unitary representation of this ok so a is equal to a dagger of a is equal to a dagger I want to have a is an operator which is independently defined ok if it is obvious to you then I don't need to write this it's really exactly I could have called it b with this relation and then impose a condition a dagger is equal to b I was just trying to be careful then it follows that n a is equal to minus a n a dagger is equal to plus a dagger so it's like in a group representation theory these are the lowering operators and this is the raising operators like in group representation theory we just use more dramatic terminology they call it annihilation operator and they call it creation operator and we will see why because this actually really creates the quantum of the oscillator that we are interested in psi n psi in a unitary representation this is just psi a dagger a psi which is the norm of a psi square so it has to be positive so basically n psi is equal to 0 must means a psi equal to 0 so like you do in unitary representation theory of groups you start with the lowest highest weight representation or lowest weight representation you define a state 0 which is called Fock vacuum vacuum simply means the ground state or in the maths literature it would be the highest the lowest weight state it has the property that a acting on psi is 0 sorry a acting on the vacuum is 0 but clearly a dagger acting on it does not have to be 0 we just construct infinite representation 0 then a dagger acting on it is 0 then at this level you can do a dagger square 0 you can just keep acting with the raising operator and if I write down the n value it is 0 here 1 here 2 here 3 here 4 here like that ok and in fact you can check just by using the commutation relations this has to be 2 factorial so in general a state n is defined as a dagger to the power n divided by square root of n factorial acting on the Fock vacuum incidentally this is the one way of writing this is the famous Heisenberg algebra on no no this is just to say that you want to furnish a orthonormal basis this guarantees that nm is equal to delta nm non orthogonal basis you could do that ok ok ok ok ok ok like the norm 1 because aha good point good point no so this actually I should have said that the norm of a physical state is 1 because that corresponds to the total probability the norm of vector corresponding to a physical state is always 1 so this is because no no no no no so you are happy to if you prefer that you can write it that way none of this will change if you write it like that ok ok it can be 3 you will just divide it by 3 yeah it is a matter of convention this is totally a matter of convention but you can see that basically if you prefer it like I can also write it in this way a n therefore is equal to square root of n n minus 1 I mean this is the ugliest square root of n that you do not like in the representation of a and a dagger acting on n is square root of n plus 1 so basically this is a lowering operator and this is a raising operator and they have unit multiplicity and this furnishes the representation of the Heisenberg algebra there is also re I mean ok it is a vector up to phase no yes I agree I agree no no this I agree what is the best way to no it is this is fixed but his question about the normalization no no this is the total the real number he wants to have the norm to be 3 for example there is a physical reason for it which I mean ok I think it is a matter of convention I think you can do everything the way you want if you prefer it yeah this is a so you should think of it not as a vector but as a projective only the because the phase is not a physical importance it turns out because we are only talking about the I mean one way to say that is that I talked about all measurements the measurements only and then on the eigenvalues and if I multiply by the phase or the some constant it does not change times a constant yes now I will just apply ok so I have described for you a simplest possible quantum mechanical system but also the most important one in some ways for physics applying the axioms of quantum mechanics as I described them right so now I have Hilbert space which is span by these ends my Hilbert space is basically span by these ends and there are operators the Hamiltonian is defined observe a and a dagger are defined and the time evolution is completely determined so you would declare that the quantum problem of the oscillator is solved in this case it is actually useful to write just to get some intuition but you do not have to but I will just do it nevertheless if I write it is useful to write a is equal to omega over 2 q plus i omega p basically notice that a and a dagger are not observables because they are not Hermitian operators they are not self-adjoint operators right a is the adjoint of a dagger and a dagger is the adjoint of a but you can clearly take the real and imaginary part to define self-adjoint operators and you put this ugly again you might they seem to you ugly but again this is for historical reasons I can write q is equal to 1 upon square root of 2 omega a plus a dagger and p is equal to square root of omega over 2 i so 1 upon i a minus a dagger basically real and imaginary part of a and a dagger such that q p satisfy again this is how Heisenberg described them and in this case the Hamiltonian can be written as z z remember Hamiltonian was like a dagger a star a so in terms of real and imaginary part it will look like the real part square plus imaginary part square and you just insert these omegas so that the omega comes here you could have just defined it without the omegas and this is you can think of as a mechanical oscillator like a ball attached to a spring take a ball attached to a spring and it is oscillating and if the spring constant k of mass equal to 1 the unit mass in some units the mass is 1 and the spring constant is proportional to omega square so this is the problem that you must have done in your undergraduate mechanics course you have a harmonic this is why it's called a harmonic oscillator it's just oscillating so this ball attached to a spring is oscillating and the energy of that is basically the kinetic energy because p is the momentum so if it is oscillating very fast then it costs more energy and if the spring is very stiff it costs more energy and if it is very far from its equilibrium position so q is the position from the equilibrium so if you stretch this string here q is the position from the equilibrium and p is the momentum now if this is too physical for you you don't need to know this just forget about if you want to think abstractly just think in terms of a and a dagger and we have done the problem completely you don't need to know this if you don't like it in fact we are not going to follow this point of view we are not going to worry about mechanical oscillators at all for us e and a dagger are going to be oscillations mode of a quantum field as I told you now let me explain why physicists have this very dramatic terminology and we will see this in our second lecture when we talk about quantum fields but basically as I told you a quantum field, electromagnetic field can be thought of as a collection of various oscillators of various frequencies so e and a dagger of frequency correspond to precisely to a photon of frequency omega right or that your mobile phone has a frequency which is megahertz every photon has a frequency blue has is a higher frequency than the red right so every the entire spectrum of the electromagnetic light x-rays and your mobile phone can be just thought of oscillators of different frequencies and therefore the Fock vacuum is the state now we are talking about therefore an isolated photon of a particular frequency of course the photon has direction also I have suppressed the direction here but I can add it if I want as some additional label which tells me the direction of the photon but the photon actually could be going that way or could be going that way but let me suppress that information for the moment but let's just talk about red light right suppose we have a red light in this room doesn't matter which direction it is going then each with each such oscillation mode in the red frequency of the electromagnetic field I associate a harmonic oscillator and the state when that harmonic oscillator is in the Fock vacuum corresponds to no photons present now you will understand this terminology A annihilates a quantum of photon a quantum of energy of red light if it was present whereas a dagger creates the quantum if it was absent in other words a dagger of 0 is state with one photon a dagger square is state with two photons and so on so N is an N photon state so now you see that with starting with very elementary undergraduate quantum mechanics we are directly doing quantum field theory and this we will develop in the next lecture so there is no reason to be sort of intimidated or sort of mystified by some of this terminology so my goal is to really demystify this terminology so that we can there is better communication between these two communities so just as an application of this now but now you could consider something called a partition function yes I am considering free field theory no even then they are free as long as they are not interacting so in particular yeah so good question so we can consider multi photon but this is a single particle Hilbert space but if you want to no sorry single oscillator Hilbert space exactly yes but I could be interested in for example in this room green light as well as red light right so I could have h1 cross h2 cross hi with frequencies omega 1 omega 2 omega i my state would be a generic state i psi would be ni1 ni2 cross ni3 nii and how would you interpret this it has ni1 photons of frequency omega 1 and nik photons in general of frequency omega k the Hamiltonian is simply omega k omega i ni plus half summed over i going from 1 to i completely trivial thing to do and in fact this is the essentially this completes the quantization of the electromagnetic field we will do it in some more detail next time but this is the quantization of this was done first by Dirac and it has some very important consequences for example it is funny normalization that Don doesn't like you can explain the laser action and what are known as the Einstein coefficients it just follows from the sorry the harmonic oscillator representation I mean it is kind of easy to explain the point is that because of this n plus 1 the probability of creating a photon when n photons are already present is enhanced and that is what is known as the light amplification dimension of radiation which is another for laser so this square root of n plus 1 actually explains how the laser works it is very beautiful actually this is how Dirac explained Dirac basically quantized the electromagnetic field and Einstein had just guessed this made a guess and he proved it using this very simple harmonic oscillator quantum mechanics and this is really the basis of laser physics this very simple fact so I think don't be fooled by the simplicity of this because physically I think this is a very deep there are very deep ideas behind it ok one more thing I want to do is partition function ok I am of course going slower than I had planned but is this piece ok for you or should I go faster or slower faster I thought Don would be the one who says slower so I can really ramp up next time ok sir actually we will go faster next time because once we know this we can actually do so partition functioning you can think of some kind of a character q to the h over the Hilbert space it is called a partition function we will explain its importance for statistical mechanics and entropy and so on later on but so given a system with some Hamiltonian ok so here by the way so the total energy if I diagonalize this Hamiltonian will be basically given to me by n i k omega k k going from or 1 to i right if I n photons of frequency omega I will take that that much energy they need plus if I have another more photons of some other frequency I need to add that energy and the total energy is this that is simply the statement of the total Hamiltonian is this so now if I have a partition function for a single oscillator is very easy to calculate because Hamiltonian was you just trace q to the power so it is nothing but n q to the omega n plus half n sum over n right this is the notation that physicists use for psi phi I think in maths literature we would just write it as psi phi it is the inner product yeah physicists write psi operator phi which translates as psi operator phi this is just the notation so this is equal to q to the omega by 2 1 plus q plus omega q to the 2 omega plus q to the 3 omega this is a simple geometric series which is equal to q to the omega over 2 divided by 1 minus q to the omega now suppose I had more than 4 tons of different frequencies multi harmonic oscillators multi oscillators the Hamiltonian is a sum the trace is a product so z of q in that case is going to be just over i q to the omega i over 2 1 minus q to the omega i now you can already see somehow something looking like a modular form appearing here imagine so is this clear these are very straight forward simple calculations imagine a system with omega i being equal to i so you have infinite number of oscillators 1, 2, 3, 4 like that then the partition function is very simple or let me call it n let's say r since we have used n for something else i 1 upon 1 minus q to the n product over n sorry r times q to the power sum over half r over half right sum over r half but it is famously known that sum over r which is like 1 plus 2 plus 3 is equal to minus 1 upon 24 this is the result known to Euler I think and of course it doesn't make any sense to say this statement but this can be defined by zeta function regularization sorry minus 12 you define a zeta function 1 upon r to the s and you take zeta evaluated minus 1 this actually is this looks a bit of a cheating you know I mean your but there is a very systematic way of understanding this in physics and that goes by the name of renormalization and which I will try to explain next time which says that you get very generically in this kinds of problems you get divergences you get infinities but then you understand those infinities are coming from the fact that you have some under is like if you think of this as some if you truncate this series by some number then you can so there is a way to regularize it this is one way to regularize it first of all you have to regularize the infinities to talk about meaningfully but then you are allowed to throw away this means that in qft one is allowed to add local counter terms this I will not explain now but I will try to explain it with this example in this example I will do it very explicitly that you are allowed to add terms which are local you cannot add what is not local there is a physical principle called locality you can add local counter terms and you can show that it is equal to minus plus local so actually there is a much more physical understanding of why this is one is 1 upon 12 and once you understand that then it does not matter whether you use zeta function regularization or heat kernel regularization or there are several regularizations that people use incidentally these kinds of regularizations were also encountered by if you are trying to prove I do not know athya singha index I mean whenever people use heat kernel methods so anyway now you should be yes so that is the theorem that is called renormalizability it is a kind of a theorem in physics that it is called universality or renormalizability so this actually it is a kind of a limiting procedure and the physics is completely understood I mean I would say you can put it on a computer and really do it explicitly it is not fully understood in a mathematical sense I mean that is the million dollar problem of the clay foundation to show that this limit exists for what is known as quantum chromodynamics I mean another quantum field theory but we have a physically we have a pretty good understanding of this fact that independent of how you regularize it meaning whether you use zeta function or whether you do this or that what will change is this local counter term this minus 1 upon 12 is universal and that is known as universality that the such physical quantities low energy physical quantities do not depend on their independent of these local counter terms so this I will explain but I want to now just make a quick connection with and stop 1 upon 24 and you should be now clapping at this point in fact I could have taken a slightly different system with 24 oscillators of 24 different colors right instead of taking oscillator with frequency 1, 2, 3, 4, 5, 6 I could take 24 oscillators of frequency 1 24 oscillators of frequency 2 24 oscillators of frequency this will give you the famous Ramanujan function and it is related to a famous problem in combinatorics what is the problem suppose I give an integer any integer and you partition it integers of 24 different colors if I give you 2 and I am using 2 colors I can write it as 1 plus 1 or I could write it as 1 plus a red 1 could write it as 1 plus 1 there are 3 ways to partition it using 3 ways to partition using colors right but look at what this is this is the total energy and 2 sorry and you also have 2 so the I am going to identify them my counting problem is not to distinguish between 1 plus no I think they are symmetric right just a second no I don't think so ok sir maybe you are right let me think about it the point is that what is this why how is this problem related to our problem our problem was you have total energy no this problem is therefore related to find is equal to the nth Fourier coefficient or n minus 1 I will have to think about it so basically if I write this as 1 upon q sum over q to the n so the Fourier coefficient of this partition function is related to a famous problem in combinatorics and why is that because after all what does this measure see this I could write it as the number of ways of splitting the total energy e right this is the Hamiltonian right my Hamiltonian is my total Hamiltonian is h1 plus h2 24 times h1 plus 24 right it is the sum over h1 plus h2 plus h3 and so on and so I am given some total energy n my total energy is n and I want to divide it equally I can so if I want total energy say 2 then I can excite a harmonic oscillator of frequency 1 twice so that will be or I can excite the harmonic oscillator of frequency 2 only once and similarly so therefore the harmonic oscillator counting problem is very closely related to this combinatoric problem and moreover I think Lothar might recognize this this is also comes up naturally this 24 for example if you are competing the Euler character of symmetric products of k3's also the same partition function appears because again there the Euler character of a single k3 is 24 and now if you take symmetric products of some manifold and you want to calculate the Euler character if you know the Euler character of a given manifold and if all the if the co-homology is only even then in that case the Euler character can be simply computed by this so bosons in some sense bosonic oscillators seem to correspond to actually are related to even co-homologies of manifolds okay so I think I will stop here but I think it is a good point to stop so I hope that I have starting with some really really basic quantum mechanics starting with the actions of quantum mechanics and a very simple system called the harmonic oscillator you immediately see that by just making a bit of more complications you very are quickly led into some very beautiful mathematics going to say Ramanujan's delta function combinatoric problems of this kind of partitioning an integer or enumerative geometry problems having to do with Euler characters of manifolds so this therefore this is the kind of so if you therefore understand this and some more things that we will do next time most of the terms that we will be able to we will be using in describing quantum black holes should be intelligible to a mathematician okay I think I will stop here