 Last time we didn't get as far as I wanted to get to because we got a bit digressed into some final points. So let me start today with a very simple quantum mechanic just to get a bit reoriented. Let's consider a two-state system. The goal is to basically introduce some of the essential ideas like entropy, ensembles and so on which will be important for discussing quantum black holes and entropy but in the simplest possible situation. So we said that any physical system should correspond to Hilbert space. In this case the Hilbert space is just two-dimensional. The Hamiltonian is therefore just a 2 by 2 matrix, Hermitian matrix which I can diagonalize and which I will write it as. So in fact so all the discussions that we had about quantum mechanics we can now state everything in a very explicitly. For example I can define the state plus which is just 1 0 and state minus which is 0 1. So plus has eigenvalue plus E of the Hamiltonian. If you make a measurement of the Hamiltonian I will get an answer plus E. Similarly if I measure minus I will get minus E. It's a extremely simple quantum mechanical system but it's actually is very comes up a lot in physics. Some of the context in which it comes is spin systems. So you can have a particle with a spin up or down. That's a two-state system. Fermi oscillator which we will discuss shortly. So there is a analog of a both the oscillator that we consider. There is a generalization of that which is called a Fermi oscillator which basically means whether the particle is present or not present. That's also a two-state system. Then in quantum information theory it's called a qubit and so on. So it's actually a good example to keep in mind. And then we can consider all the general axioms that we had discussed. For example if I have a general state psi then I can write it as a linear combination of this and I'm going to take it to be unit. Okay so we take it to be unit norm. So a square plus b square is equal to 1. This state is clearly not an eigenstate of the Hamiltonian. So as I said you have a Hilbert space. A physical system is represented by Hilbert space. A state in a Hilbert space with a unit norm. And observables are represented by self-adjoint operators. And the probability of obtaining the result e. So if you measure h on the state is given by a square. And similarly the probability of obtaining result minus e was given by b square. And now you can see why it's common in physics literature to normalize things to use an orthonormal basis and to normalize things to 1 because in the orthonormal basis I can just read off the probabilities just by looking at it. I just take the coefficient and take the absolute value square. That's the probability of finding this result. And moreover because a square plus probabilities psi psi equal to 1 guarantees that we are dealing with normalized probabilities. Okay so all if I measured if I take a large number of such identical systems prepared in the same state and I measure repeatedly what I will find is that absolute value a square times I will get result e. Absolute value b square times I'll get result minus e. And a square plus b square is equal to 1. So the total probability is add up to 1. So that's the reason for the way that's the reason for choosing a state with a unit norm and an orthonormal basis. And clearly here these measurements will not depend on the overall phase of the state and therefore you now you can of course restate the whole thing without using an orthonormal basis and relaxing the unit norm. But if you normalize things this way the physical interpretation is more transparent because you can just read off the probabilities. And moreover you get normalized probabilities. So last time we had a 15 minute discussion on this but it was a bit of a digression. Ideas are I think illustrated here and now let's take any of all observables in the system identity and the Pauli matrices. They are proportional to there are some real linear combinations of identity and the Pauli matrices which is sigma 1 is equal to sigma 2 is equal to and sigma 3 is equal to. So our Hamiltonian is nothing but e times sigma 3 but you could choose some other observable a is equal to sigma 1 for example and you might want to look at the eigenstates of that. Of course this as you know is diagonal you can diagonalize it and let me denote those eigenvalues with arrows which have the property that eigenvalue of this is equal to plus 1 and eigenvalue of a on this state is minus 1. This is a state with some eigenvalue minus 1. But now if I major the Hamiltonian on the system then I will get a probability by this axioms if I major and minus e with probability half right because the coefficient of this is 1 over square root of 2. If I square it the probability of obtaining the result plus e as probability half the probability of result obtaining result minus e is probability half and half plus half is equal to 1 and this is true for both these states even though there is a very important minus sign between the two. So here we say that the overall phase has no physical significance we see that the relative phase clearly is important and it's important not of course for the measurement of Hamiltonian because for the measurement of Hamiltonian you cannot distinguish between this state on that state. But if you measured a you will be able to distinguish because the one which has a plus sign here will with unit probability will give you answer 1 and this one will give you unit probability and answer minus 1. So this relative phase is a really of absolutely critical importance in quantum mechanics. There are two really weird things about quantum mechanics which one must appreciate is that a and b are called probability amplitudes probabilities and they are complex numbers and so if you have some what you because the entire Hilbert space structure is linear at the level of the states and not at the level of the norm in quantum mechanics you add probability amplitudes. So if you have two alternative no no I'm not now adding vectors I'm adding probability amplitudes for a process I agree with you is everything clear up to this point or do we still disagree about about this okay now this statement is that it's not about adding probability amplitude is not the same as vectors probability amplitude yeah a and b are probability amplitudes for what for a process that is the probability amplitude of finding the state psi in this case is e after a measurement now let me yes no I don't want to add a so this if what we okay let's see how should I say this in the most if there are two if there are two alternatives to achieving the same result okay so if the system for example if the particle can go this way and come here and that way then you can associate with this process a probability amplitude a and the probability amplitude b if this was a classical particle what you will add is a square plus you will just add the unnormalized sorry a square plus b square the probabilities you will add whereas in quantum mechanics what you will add is a plus b and then you will take the square this is not the same as adding a square plus b square okay okay some of these I think some of these questions I think are more appropriate for a tutorial okay so I think we should probably discuss it because I'm trying to now present a big chunk of the things in a minimal possible way so if this point is not clear let's take make a note of fit and we'll come back to it in a tutorial I mean I'll be happy to take a tutorial on this after afterwards okay but the two important point is that the Hilbert space structure is for these states whereas the probability requires to take your absolute value square so probabilities are defined as I told you by taking this expectation value of a projector and if this state is normalized to one then the probability is normalized is normalized to one so the main point is that the state itself does not have directly it does not have interpretation these coefficients a and b themselves don't have interpretation as probabilities but they're absolutely square is what has interpretation as probability that is one very important difference compared to classical probabilities because here the relative phase okay I should probably not call this a and a prime let me call this a and a prime not to confuse with the other previous discussion so here you will get a a prime star this is star a prime so the fact that you have two complex numbers that you're adding and their phases can be may or may not be aligned is what gives rise to all kinds of interference effects in quantum mechanics and which is the origin of the wave like nature of and these two things are really actually very in a way very surprising because first of all complex number really play a very very crucial role in a description of real world you cannot do this without complex numbers and secondly you have exactly linear structure I mean generically in physics you always encounter some nonlinearities so having an exactly linear structure where complex numbers are having a linear vector space as the fundamental underlying structure of quantum mechanics is really one of the most surprising actually I think Dyson has said that this is one of the two biggest jokes that nature has played is that it has used complex numbers in a very fundamental way and linear structure in a very fundamental way okay let me now introduce one more concept and then we can move on to the field theory and then density matrix to given a state psi I can define a density matrix okay in this Dirac's notation it simply means in the 2 by 2 meter it's a 2 by 2 matrix in this case where you multiply a column vector with a row vector basically all the axioms of quantum mechanics I can state in terms of the density matrix instead of psi for example the probability axiom the probability of obtaining the result I is given by this expectation value of the projector which I can call is as trace of rho times pi and once again it's clear that rho dagger is equal to rho it's a self-adjoint operator and trace rho in our normalization is 1 and one can define now something known as the von Neumann entropy which is as the definition minus trace rho log rho now it's clear that as long as rho is of this form for example here rho can be written as in some diagonal basis 1 0 0 right here because suppose I can choose psi to be one of the basis vectors of an orthonormal basis for example if for example if rho is equal to plus plus the matrix representation it looks like that and therefore the trace rho log rho is 0 you understand that 0 times log 0 is 0 is this clear the trace rho log rho is just eigenvalues pi log pi and so pi is equal to 1 log 1 which is 0 and then it is 0 times log 0 which is 0 and that suggests a generalization because our entire discussion started by assuming that we know in which state the physical system is in but quite often we actually don't know which state the physical system is in you might just have a probabilistic I mean there are two kinds of probabilities that appear in quantum I mean there is very fundamental probability that appears in quantum mechanics which has to do with measurement that exists even if you knew exactly what the state you were talking starting with but there could be an additional source of uncertainty which is more just a source of your lack of your knowledge you may not know whether the state is really this or this you might know that maybe half the time it is this and half the time it is this we are going to come to that you know so that's going to be the main actor of our in the black hole story so entropy is a very fundamental concept and I want to sort of introduce it first in the simplest context and then hopefully but it's a measure it's a so let me try to give some some physical interpretation of it in this two-state system already so one can therefore define a generalized density matrix preserving these two conditions and therefore row generically will take a form p1 p2 in some diagonal basis because since it's Hermitian you can diagonalize it and trace rho is equal to 1 so the probabilities add up so p1 p1 plus p2 is equal to 1 it's clear that if for example you can take rho is equal to half in this basis and what does that say it says that I know that the system is plus in the plus state with probability half and I know that the system is in minus state with probability half I don't know more than that and in that case the form I'm an interface actually not zero because you can now calculate it it is half log half minus plus half half in fact if I used since I'm using a natural logarithm is locked to if I chosen a in a binary log in a log 2 in the two basis if I chosen in base 2 it would have been 1 okay so in some units is locked to and this leads to a notion of pure state on a mixed state that the fundamental entropy here is zero and here it is non-zero mixed it means that I only know the state to be in some probabilistic way that it is in this state or in that state I don't know exactly in which state it is notice that even our state the upstate here also we knew that the probability was half for plus and minus but this is very different from this state because look at the row for this up upstate if you calculate it it was going to be like this 1 1 1 1 it is not the same as this obviously because you should be able to diagonal in the diagonal basis it will go to 1 0 because after all if I choose the orthonormal basis which is the Eigen basis of the operator a which is just the Pauli matrix sigma 1 in that basis in that basis the density metric should look like this and therefore its fundamental entropy should be 0 and therefore it cannot be the same as this state so you see that in general all the weird quantum behavior and the what is called the coherent superposition this this is what I was referring to the interference effects are contained in this off diagonal elements of the density matrix of course this is a basis in dependent statement but the basis independent state yeah so this is a basis dependent statement sometimes it's about quantum coherence but let me give a some kind of a physical interpretation of this this is related to information theory the concept introduced by so this is just again a side remark but it could be important I mean keep it in mind it's not absolutely essential to what we're going to discuss but this is similar to the Shannon entropy and Shannon's idea was very simple let's suppose you have you're trying to communicate with somebody with a bit so this as I said you can think of a classical bit which is like on and off instead of being plus and minus and if your bit is working perfectly then your friend can really tell whether your light is on or off and then you can communicate one bit of information but it quite often can happen that your telephone line is not very good or the it's not so it's foggy and then your friend cannot tell whether the light is on or off and can only tell that the light is on with probability p1 and that it is off with probability p2 in that case the amount of information you can communicate is not log 2 or one bit but it is less than one bit so the amount of information that you can communicate is I max which is the dimension of your state space which is log 2 in this case log of the dimension minus the entropy so in our problem it is just log 2 minus s in particular if your bit was really defective and you cannot really tell whether it is on and off you can only tell with 50% probability that is on and off then basically you cannot communicate any information right because your friend cannot tell whether you're on or off and so your most code will not work so therefore this is a very important connection which was so Shannon was a great electrical engineer actually turns out that in the story of entropy two engineers really made fundamental contributions one is Shannon and other is Karnaugh who we will encounter later on so they really made they were trying to solve some practical problem Karnaugh was trying to design the most efficient steam engine and he hit upon the concept of entropy and Shannon introduced this information theoretic ideas into the notion of entropy so you can see that increase in entropy will mean decrease in amount of information I can transmit so that's kind of one physical interpretation of entropy is loss of information and this will also we will see that it's like increase in disorder and this will relate to the second law of thermodynamics of course a Q bit is very different from Shannon's classical bit because there is no sense in which you can do a linear superposition with complex things often off and on but if you have a spin system it makes sense to have a linear superposition with complex coefficients off and on in both cases they're normalized to one the probabilities are normalized to one but in one case you're allowed to add complex numbers with phases in the other case you're not and this is the difference between quantum information theory and classical information theory and that's a lot of the rage in quantum computing and quantum information really relies on these phases and entropy sort of enters there in an interesting way okay so for us the entropy is going to be very critical for physics is actually I would say is one of the really very deep ideas which are sometimes not very appreciated because you learn it in undergraduate studies and it's some something to do with heat and but it's really very fundamental idea macroscopic window into the micro world how that happens I'm that's what I will try to describe in the next hour and that is how we want to use it the entropy of black holes also to learn about the microscopic world of quantum gravity so this was one of the really most important insights of 19th century physics I would say I mean along with maybe Maxwell's equations in nineteen sixty you know some of the kind of the great masters like Maxwell and Boltzmann I mean of course I have to write Boltzmann first Einstein and so on they really used very deftly this notion of entropy and statistically reasoning to learn about the micro world by doing talking about properties on a very microscopic scale at a time when you could not really look into the for example apparently one of the reasons that contributed to Boltzmann's suicide which happened here was that nobody believed in the atomic hypothesis at that time I mean this was in 19th century 1900 so a lot of the chemists didn't believe that the world is made up of atoms so we are now gleefully saying that okay this room is filled filled with photons that is filled with atoms how do we know if you don't have a microscope how can we ascertain the quantum nature of matter so in 19th century the quantum structure of matter you could look into the use of entropy that's why the entropy is such an important idea so even though you did not really have a microscope to resolve electrons and protons and so on but indirect reasoning a statistical reasoning and using the notion of entropy and the relation between quantum statistical mechanics and thermodynamics we could indirectly gain evidence for the reality of atoms and that's the kind of thing that we want to do you want to study quantum structure of space-time or quantum gravity but it was done through the entropy of gases ideal gases or entropy of harmonic oscillators if you like we will now study harmonic oscillators we have been studying harmonic oscillator the quantum oscillator and one of the very important clues tells us so in some sense that we are in a similar situation like the 20 19th century physicists that the LHC is not good enough to probe the structure of space-time at the length scale that we want to but can we can still learn something about the quantum structure of space-time indirectly by studying the entropy of black holes so that's why entropy of black holes and black holes are so important in the investigations of quantum gravity and I just want you to be able to it's of course not possible to cover this whole thing in two lectures but I want you to be able to appreciate in reasonable detail that how it can be done so the way entropy occurs in physics is through the notion of quantum ensembles so we have been talking about states very often like for the gas in this room if you really seal everything and everything is thermally insulated then only thing that we know about the gas in this room is the total energy we don't know where precisely each molecule is you don't know whether this different molecules are going all over the place so very often we may know so and now from now on actually I'm going to talk when I talk about system I will think of a system of a mesoscopic size I mean a big enough size not one spin or two spins it's possible to actually formulate everything even for us single spin by using what is known as Gibbs ensemble but I won't go into that I'm just think of a system which is large enough so that there are large large number of spins up and down so your Hilbert space is big it's not just two-dimensional it could be a product of many two-dimensional spaces in this case if you know the total energy there are many ways to like we saw for the harmonic I mean there are many ways you can distribute the energy for example I can have energy suppose only the total energy is known then I can think about the Hilbert subspace the Hilbert subspace is just this all states its size equal to E psi so in this two state system of course there was it was that dimension was one but generically if you have lots of them then once when could be up once one one guy could be plus E another guy could be minus E such that the total thing adds up to some big number right then I can look at the dimension of the Hilbert subspace that also is what Boltzmann called the entropy Boltzmann entropy okay Boltzmann wrote it like that very log of this sorry and this is a very famous Boltzmann relation sometimes you'll see there is a factor k but it's ups it depends on some normalizations but I'm trying to give it in the cleanest possible way so yeah so we were all our temperature is measured in units of n energy then this Boltzmann constant you can say to one now what does this have to do with quantum and entropy a priority now h of e is a in most physical applications is finite dimension so d of e are some finite integers then a fundamental postulate of quantum statistical mechanics which which what made all these things possible was that in equilibrium an isolated system which means that at fixed energy has equal a priority probability of being in any of these microstates so what is what is meant by microstate microstate means any of these states let me label them by r there can be a billion states with the same energy for example right if you have a large like in this case in the case of so is this clear for example if it took the our two-state system so the Hamilton and let me call it small epsilon times 1 0 0 minus 1 but if I take n copies of this right then there are many ways given some total energy e I can obtain it in many different ways right suppose I have n copies of this so I have total h is h1 plus h2 hn so this can be plus or minus epsilon plus or minus epsilon so total energy is plus or minus epsilon plus or minus epsilon plus or minus epsilon so there are many possibilities in which I can realize a given energy e and each of those possibilities will represent a unique state in the quantum Hilbert space and that is known as a microstate microstate is simply an element of this Hilbert subspace and this fundamental axiom of quantum statistical mechanics now this is actually is not it depends in many systems this is also known as Argodic hypothesis and so on so there are there is a big there is a huge literature and there is a huge physics behind this which I'm trying to summarize in in the minimal possible way so it's not always true and so on and so on there are caveats but for a very large class of systems this is a good axiom to go with to be very concrete for example if you think of this room as being filled with atoms you can have many different microstates in which one atom is going that way another atom is going that way and so on and so on such that the total energy adds up to e right so you can see that there is very large number of microstates corresponding to this entropy corresponding to this stage in this case one can define a density matrix since it's unit probability it is going to be one along the diagonals divided by d e the Boltzmann entropy just minus trace low roll over on this Hilbert subspace is this clear okay let me take a pause because I think I have thrown a lot of things yeah it's a matrix of size d e by d e each with unit probability so if you just calculate it will be log d times one upon d times d so it will add up to with a minus sign right it will be one upon d x is equal to minus sorry minus of log of one upon d which is log d so this is sometimes known as the micro canonical ensemble okay some historical name actually it's not important which simply says that you have equally a priori probability of the system being in any of the microstates for a fixed energy and actually essentially all of quantum statistical mechanics follows from this all the one of the important thing that follows from it which you can derive from it actually is the canonical ensemble which is kind of essential to make contact with thermodynamics for example what you have is suppose this is a bottle of wine and it's sealed completely so that this energy is fixed but now the canonical ensemble means that suppose you put that bottle of wine in a swimming pool and it's allowed to exchange heat with so we want to now make contact so what is Boltzmann fundamental insight this of course is just a definition you know I can always call log of d e to be S of e it doesn't do anything fundamental insight is that he connected change in entropy I mean in thermodynamics the independent notion of entropy to the engineer Karna that I talked about and his definition is that if you add heat if you add energy to this suppose you put a heater in this room and add energy to the heat heat this room up at room temperature right so the change in energy you increase the energy if you add heat energies if you increase the energies is equal to delta q but the change in entropy he said Karnaugh is dq upon t where t is the temperature so temperature is not a fundamental so so far I think in a mathematician would be completely comfortable since you're talking about Hilbert spaces and so on and so on it's all completely well defined temperature on the other hand is something that we feel but it's a kind of a microscopic concept and it doesn't have any fundamental representation in terms of but it is related to how fast the molecules are moving so temperature temperature is a macroscopic concept dynamics and somehow we have to understand it in the mic microscopic world microscopic description and that's where the canonical ensemble comes in canonical ensemble says that if a system instead of having it at fixed energy like our wine bottle which is completely sealed okay so that it cannot exchange energy is hermitically sealed if a system can exchange heat sorry this temperature is measured in Kelvin yeah that's why this degree Kelvin was introduced you can always shift yeah you can always shift this and change this but the notion of temperature in degree Kelvin appears in a very natural way in in this following following way if a system can exchange heat with a heat reservoir temperature the probability that the system is in is in a microstate energy here is e to the minus beta here the beta is one upon t okay so now look at the difference this is this is called a canonical ensemble and this is called a micro canonical ensemble micro canonical ensemble the total energy is fixed so it's like a wine bottle which is completely sealed wrapped up and therefore it cannot exchange heat and its energy is fixed and in that case all the molecules in the wine bottle could be could be in many of these microstates such that the total energy adds up to what the energy is here that now the wine bottle is in a swimming pool swimming pool could be at very high temperature and the wine bottle is at 12 degrees then at equilibrium the system is a microstate is at equilibrium at thermal equilibrium so now the wine bottle can absorb heat can get heated up if it was at 12 degrees from the refrigerator and you put it in the swimming pool of 25 degrees it will get heated up and then the probability that the system is in a microstate psi r is given by here okay and I won't try to derive it because again it's a kind of a two lecture thing that one would have to do let me just quickly indicate how the canonical ensemble is related to micro canonical ensemble is that you can now consider the total system of the wine bottle with the swimming pool has been totally insulated of fixed energy asked a kind of a probability of probabilistic question that if the total energy is fixed to be E naught what is the probability and they're exchanging heat with each other so that the total energy of the bottle plus the total energy of the swimming pool is fixed then what is the probability of finding the bottle in a state which has energy E bottle okay that's the kind of a statistical question that you can do and you can from that if the system is large is some kind of a saddle point evaluation and you can it's a bit like how how you prove the central limit here I mean it's a it's something that will be that you can derive only the system is large and starting with the axiom of equal a priori probability you can derive the canonical ensemble so the net result is that in the canonical ensemble the density matrix I can in fact think of it as an operator is simply e to the minus beta times the Hamiltonian the density operator or density matrix was the total trace now we are no longer in service or is no longer in the two-state system in the canonical ensemble and temperature is defined simply by del log del of se one upon the temperature beta is defined as in some average so the average energy okay so let me know from here you can define a part so if you now look at just the unnormalized we can define a partition function z of beta which is trace e to the minus beta h then it is equal to sum over all possible states with some density of states so number of states times e to the minus beta i I can do it as a sum over energies number of states with energy e the this trace is equal to the number of states at energy e as times e to the minus beta i no so sup it's a trace right so suppose I have many states with the same energy e that's let omega e be the number of states sorry sorry sorry I use de sorry de sorry sorry de sorry I use the different notation sorry omega is used commonly in physics notation I'm sorry de is the number of states right is that dimension of the against nice easy to see that the average energy is computed as just trace e e to the minus you can show that this is equal to basic rate it's del logarithm of z times del beta so think of log of z beta as some function x of beta and we want to find so this gives us energy average energy is a function of beta average energy is a function of temperature but if you want to invert the relation and we want to get beta as a function of average energy then the way to do it is that you define yeah so e beta is defined as del x minus del x by del beta then one way to invert this is through the use of Legendre transform we different y equal to x plus e bar beta and you have to now keep track of the so del y by del e bar is equal to del x by del beta times del beta by del e bar plus beta plus e bar beta by del e and using the fact that e is equal to minus del x by del beta this term and this term will cancel this is a standard trick in what is Legendre transform is when you have implicit function and you want to invert it when you define a Legendre transform now beta is defined as del y by del e bar and now comes a step which I will not labor to justify here completely because there are five different derivations of this but I will just give you a flavor of it the point is that d of e is a very rapidly growing function energy because as you saw for example when you add to partition the partition number of partitions of an integer you know it very rapidly grows with the number because there are really lots of ways you can split the total energy into its subsystems so it's an extremely rapidly growing function and the exponential beta is an extremely rapidly falling function it has a very sharp peak at e bar which you can evaluate by a saddle point evaluation okay we will come across this calculation in when we talk about the hardier Ramanujan formula also we will do a similar calculation the first first term of that you can get by a saddle point evaluation okay in physics literature what is known as the Cardiff formula but the main point is that so if I write therefore it's e to the s e minus beta e bar evaluated at this bar is a good approximation to z of beta which we said as e to the x so how is this saddle point evaluation determined is determined by demanding that it's a saddle point right so you have to minimize this function minimize this function it's easier to see now x beta is plus beta e bar is s e bar therefore y is equal to s and therefore beta is equal to del s by del e or in other words this is the first law of thermodynamics or ds by de that d is equal to tds which is a 1 upon t so if you use so this really is a very critical point which makes the connection between micro and the micro because now entropy has an independently sorry this is average energy because we can only talk about average energies if you have a wine bottle in a in a swimming pool its energy is not fixed because it is exchanging energy with the swimming pool but its average energy is fixed and it's pretty sharply peaked so it doesn't really make a difference whether it's we talk about the average energy or total energy because it's extremely sharply peaked so you don't the fluctuations the statistical fluctuations are so small that you can there are situations when the statistical fluctuations become big and that's actually whole field of statistical physics but in in a situation like this the fluctuations are so small that whether you talk about the total energy or average energy it doesn't make any difference so that's the key point now we have so I'll summarize now we have connected s which is equal to log log de and I'm henceforth going to blur the distinction between e and e bar average energy is more or less the same as the total is actual energy s e is precisely what appears in the first law of thermodynamics what is the first law of thermodynamics first law of thermodynamics simply says that energy is conserved so if I heat this room the energy must increase by the same moment as the heat that I'm adding you cannot all that heat must go into some energy otherwise energy is not conserved and then by Carnot's relation this is equal to t delta s which is the same as this relation so this is how why entropy is so fundamental in our discussions because now by doing something very accounting problem this is what number theory is like to do or this is how it makes connections with our modular forms and so on story because there we're doing a counting problem but the counting problem by this long chain of reasoning is related to very physical quantity namely the amount of heat you're adding to the system and some thermodynamic property of the system and so now even if you cannot have a microscope you can determine how whether or not your microscopic model of the world is correct or not and for that I will now take one example going back to quantum field theory so this actually summarizes my discussion of quantum mechanics and quantum statistical mechanics and entropy so which is a long which requires usually three or four courses but okay I think I have I hope I have conveyed all the essential concepts right I mean everything so far is well-defined I hope apart from okay we have some issues with the finer way to put things properly exactly that's what I'm going to do next and now you will see how we connect to quantum fields and we will derive Plunk's distribution law one of the most famous celebrated results in quantum statistical mechanics by doing quantum set quantum harmonic oscillators so I said last time that for most practical purposes essentially so let me qualify my statement so that I don't make a wrong statement in perturbative QFT what is known as perturbative quantum field theory which actually is good enough to describe essentially everything except for the nuclear force so all of quantum electrodynamics all of weak interactions electric theory falls in this class of perturbative quantum field theory which means that essentially uncoupled which means that a quantum field can be thought of as an essentially uncoupled or very weakly coupled harmonic oscillators quantum oscillators and to a good approximation you can just treat them to be just exactly harmonic oscillators and the interactions correspond to unharmonicities so for example if you recall our harmonic oscillator it had the equation of motion dA by dt is equal to omega a and similarly similar equation for a dagger sorry maybe a and dA dagger by dt is equal to minus omega a and if you wrote a is equal to let's say q plus ip divided by square root of 2 I had put some factors of omega there to be consistent with the physics conventions but let's just do it like that you think of a as a complex number the two-dimensional plane right and what this is telling you that is a rotating that's the harmonic motion this number is rotating because what is the solution of this is a is equal to a 0 e to the minus i omega t and if I write this in terms of real and imaginary coordinates like that then you can see that it satisfies the harmonic equation the famous harmonic oscillator that we have seen in school you can check that d d square q by dt square equal to minus omega square q if you just plug it in here because of this I you can work this out right I mean it will have basically dq by dt plus i dp by dt equal to omega dq by d sorry omega q plus ip and then it depends I want to eliminate this so I take one more derivative and I use this equation and you can convince yourself that you get this equation if I wrote it on this side unharmonious city see this is a linear equation in q unharmonious it is which is like lambda q q or something like that all my discussion is valid when lambda is much more than one in some perturbative sense so basically you can drop this term solve this equation and treat this lambda q cube as a perturbation and that's basically all of quantum field theory that is required for quantum electrodynamics you can do in this language just in terms of harmonic oscillators okay you have to be a little bit it's not the best way to do it if you most efficient way of doing it but conceptually is the simplest way to think about it conceptually it's completely correct to think about the electromagnetic field are just a collection of harmonic oscillators and the iraq electron field is a collection of harmonic oscillators of a different kind and they are interacting with so you can have a more than one harmonic oscillators and they're interacting with each other and the interaction is small that's what we meant by perturbative quantum field theory so now coming back to sharia's question let's look at the photons electromagnetic field so the electromagnetic field last time I wrote down the electromagnetic field actually in in four dimensions has two polarizations okay so the electromagnetic field is actually a mu as we saw it's a connection one form if you drop this term then what you get is very similar to a wave equation if you have not k harmonic oscillators some large number of harmonic oscillators where k could be a momentum vector for each of them I will get a harmonic oscillator equation and where does this momentum vector come from if you had a field phi of t and x if we describe fields right so scalar field in d plus one dimension then I can define ddk okay this is some normalization which is not important okay it's normalized in this way okay to the power half okay some normalization the important point is a k e to the i k dot x minus i k dot x plus a k dagger e to the minus i k dot x so there is a there are different ways to think about it you can start with a scalar field which satisfies the Klein Gordon equation as we saw the Klein Gordon equation would be something like minus del by del t square plus del square of x phi of x t is equal to 0 and if you now do a Fourier decomposition of phi then you see this will turn into you can bring down this derivative and it will look exactly like this equation each of the phi k's satisfy exactly a harmonic oscillator equation omega k is just the absolute value of k of the vector k yeah so all that the point that I want to make is that a field which satisfies a wave wave equation a quantum field a classical field which satisfies a wave equation so there are two ways to do it I think last time it was also raised why don't we do the canonical quantization the point is that a field in d plus one dimensions can be thought of as a collection of harmonic oscillators quantum oscillators I mean if you have a quantum field it's a quantum oscillator if it's a classical field it's a classical oscillator of frequencies so k is a d-dimensional vector d-dimensional spatial vector so I'm not doing something fancy I'm just doing Fourier analysis because I want to solve a wave equation in flat space but there is a generalization of these two curved space time which okay I may not get time to do it but it's also pretty straightforward and which also you can state in terms of harmonic oscillators that I want to solve an equation like that minus del square by del t square plus grad square acting on phi is equal to zero and the natural thing to do is to do a Fourier analysis and then if phi goes as e to the minus i omega t let's say and e to the i k dot x then one of the modes of this is will satisfy the equation that minus omega square something is wrong plus omega square minus k square this implies that omega is equal to omega k from the equation of motion so this is the intuitive idea that I had discussed and you are familiar with that I think of the electromagnetic wave as this is a collection of light going in this direction going in that direction so that is a wave vector and the magnitude of the wave vector determines the frequency of light so if it is a red light the frequency is different if this blue light in the frequency is higher and so as you span the whole spectrum you can get all kinds of frequencies so now this room if you think of it as now you want to follow the statistical mechanics and want to derive the average energy of the gas of photons using exactly the calculation that we did last time and last time we computed and these harmonic oscillators are completely independent of each other so you remember we did a single harmonic oscillator and then we did a multiple harmonic oscillator but that was trivial because you just take the tensor product of the Hilbert spaces the Hamiltonian just adds so calculations are completely trivial and if you remember the partition function z of beta for a single harmonic oscillator z of q I had called it for a single harmonic oscillator labeled by k so my total Hamiltonian now is omega k hk summed over k of course I'm writing it as a discrete sum but you should really understand it as some d cube k I'm just summing over the energies of all and here is this very interesting thing beautiful thing about the harmonic oscillator the quantum oscillator which now is what makes the particle interpretation correct possible you recall that for each mode each harmonic oscillator I had a fork vacuum right so I had an oscillator let's say ak and ak dagger whose product was one sorry whose commutator was one ak commutes with itself in fact I can put a prime here and I can put a delta function this is true if I have multiple nice thing about the harmonic oscillator was the Hamiltonian was just omega k times a dagger k e k plus half and this we recognize as a number operator in fact if you remember the Hilbert space representation of this algebra is based on it's a fork representation it's called the fork representation based on a fork vacuum and then we had all these states I could act with ak dagger and zero that was my state one in fact I had a representation of the state nk as ak dagger to the power nk and the important point was that n acting on nk was an integer for Hamiltonian therefore it is can be identified as a state with n particles so somehow this very simple looking system undergraduate level system is what makes it possible to view a quantum field as a collection of particles because particles if they are not interacting I should be able to just add them then energy should add right so if I have each this overall this is just some shift in the origin of the energy but every time I go up in nk the energy goes up by omega k so that means if nk goes up by one the energy goes up by omega k see so this is very important that the energy levels eigen levels are integers based energy eigen values of the harmonic oscillator and therefore the quantum field which is just a collection of harmonic oscillators is Fox Space representation can be equivalently interpreted as so for example I can now take the most general state ak1 dagger to the power nk1 ak2 dagger to the power nk2 to some ak whatever your number you want akn to the power nkn on the Fox vacuum it's a normalization then this represents nk1 photons with wave vector k1 and so on so therefore any state of the electromagnetic field in this room I can represent in this way it's just five green photons going that way and six blue photons going that way so what do I have to do I just choose k to be that direction and says that the k square is the frequency of the green light and another k to be of that direction and I take nk1 to be five and nk2 to be six moreover it's a system of independent harmonic oscillators so I can just complete the partition function very trivially and which we already did last time we called it q to the power range k if I I'm assuming of course that q is less than one and so on and then I get a geometric series it is an overall e to the minus beta omega k over 2 which as we saw in some cases is important but for our current discussion is not so important and then it's one minus q to the n right this was so it's one minus e to the minus beta omega k for a single harmonic oscillator and what is the average energy you can calculate it's minus the log z the small calculation which you can do which turns out to be h omega k sorry omega k e to the beta omega k minus one and you probably if you remember Planck's distribution law this is already Planck's distribution law if you remember from your if you have vague memories of the undergraduate course you did no so I'm sure you will get confused because quantum field theory appears in many many contexts and that was a one plus one dimensional system now I'm talking about a three plus one dimensional system also that one plus one dimensional can be the space time in which you live or it yeah so that context was 24 bosons in one plus one dimensions beta is one upon t in our calculation there beta was e to the minus beta was q but then of course we wanted to make q complex if you want to think about modular forms but if you want to think it in terms of the physical system of a one so that system corresponded to instead of living in three dimensions suppose you lived in a one dimension our system there was one dimension it was on a circle though which is why k was not continuous see here you have an integral d cube k there we had a sum and we had time so you could think of them as photons traveling along this direction and their frequencies happen to be so it sure happened that because k in one dimension is n over l if l is the length if this is the 2 pi length of 2 pi l your Fourier integral in that case became a Fourier sum and omega k happened to be just proportional to k which is just n if we put l is equal to 1 so we had in that case an infinite set of oscillators whose frequency were themselves integers spaced so there were two integer spacing right these excitations are integers spaced but also their frequencies were integer space so you had one oscillator with frequency like that the other oscillator what frequency twice this and the third oscillator was frequency three times this so you had an infinite collection of such oscillators and 24 varieties of them and that's what gave rise to one upon delta could be identified could be identified with the temperature one upon the temperature and in fact one can derive the Cardiff formula or the the exponential growth of the Hardy Raman Rogen formula from this point of view why it goes the square root of n because it's you can think of it as one plus one dimensional thermodynamics okay I'm sure this is going a bit too fast but yeah so the thing is that notice that so therefore the total energy average energy which is going to be some of our the average energies of each of these guys summed over k all possible wave vectors and the all possible wave vectors is it's easier to think of it put it in a finite volume and then you take volume to go to infinity limit so basically k is going to be divided by 2 pi say l a vector like that 2 pi n1 sorry n1 divided by l and 2 divided by l my length of the box is l i called it l that let me call it r and the so basically it's a set of points three dimensional space it's a separation between them as 2 pi upon r if r is the size of this room and then as standard you can sort of convert this integral into us in the large r limit you can convert this sum into an integral using the fact that delta k 2 pi r right so you can basically write it as delta k that's so basically you can write e bar is equal to d cube k r cube divided by 2 pi cube times omega k e to the beta omega k minus 1 so this is just the volume of the room so the energy density therefore is proportional to this quantity which is the famous Planck distribution law now what do we notice that beta was 1 upon k 1 upon t and omega k was absolute value of k so this is nothing but k square dk times k divided by e to the k over t minus 1 so we immediately see that the energy density I can divide by t here and take out to t to the 4 outside as the form eta square eta cube d eta e to the eta minus 1 the eta is equal to k upon t and this is the famous Stefan Boltzmann law that now we have derived that if I hit this room the energy density will go as the fourth power of the temperature so this is the promised window into the microstructure now this you can check and this is this is a all the experiments in blackbody radiation going back to Kirchhoff's and so on so on they were checking the frequency dependence on the temperature dependence and now and it is true that the average energy density scales as the fourth power of the temperature and in fact you can see that if you are in d plus one dimensions it will scale as the t to the power d plus one in two dimensions it will one one plus one dimensions it will scale as t square average energy density and average entropy density will scale as the total volume of the room because if the volume is infinite the energy is infinite right so so you can only talk about energy density in this room but this is a really remarkable thing that we had a microscope this implies that the microscopic model of the electromagnetic radiation as a collection of harmonic oscillators makes a very sharp prediction through Boltzmann's and you can similarly check that the entropy scales as t cube implies that energy density feels as t to the power d plus one in the in d plus one dimensions and entropy density scales as t to the power d so in this sense I can measure this now right I put a heater and if I increase the this is the amount of heat that I have to add if I want to increase the temperature in this room by some factor the amount of say if I need to double the temperature I need to add eight times so the scaling is you can verify very easily and in fact now one can very precisely verify also the frequency dependence so this is my kind of my today's conclusion I think okay I succeeded in doing something today it's a micro window into the microscopic world okay and in our you remember we had a one plus one dimension we had computed z of cube for a collection of harmonic oscillators which was some let's say one upon delta q but you can have many other things possible it is well known that d of n which you can expand as d of n q to the n and d of n has a scaling is e to the square root of n with some coefficient that is depends on the details of the modular form but what is this this is the scaling of this is e to the entropy at large n and you can see that there is a simple physical understanding of this namely entropy score scales as the temperature in one one plus one dimensions and energy scales as t square therefore entropy scales the square root of the energy if you remember n was the eigenvalue of the Hamiltonian so that's exactly the so this calculation will actually come up exactly identical calculations are very similar calculations will come up in the so let me give you a preview of what we will do next time so just to summarize what we saw is that if I didn't know anything about the so we started with the atomic hypothesis that the world is made up of atoms which means world is made up of quantum fields okay it's all big words but therefore I have a microscopic model of the light in this room or the photon gas in this room and if that atomic hypothesis is correct then that microscopic model implies that I have these infinite number of harmonic oscillators with frequencies given in this particular way and then just following the quantum statistical reasoning you conclude that the average energy in this room should scale as t to the 4 with this very precise Planck distribution law as a scaling with frequencies and this you can verify quantum black holes so there were two things that are required for this to be a useful enterprise that we had an independent way of determining the entropy as temperature to the cube right we had some way of measuring the average energy in this room by putting the thermometer right and we can check whether it goes up as a which power of temperature as it goes up as I hit the room right so we required a macro side and a micro side in the case of the room the macro side was a heater and the micro side was a model of photons we verified that average energy density goes as t to the 4 in the case of black holes the macro side is basically general relativity micro side is always quantum mechanics or quantum field theory it's going to be a black hole solution of general relativity and on this side is going to be some brain configuration which I will explain next time today also I can explain the remaining minutes and here there is an independent calculation just from the geometry of the black hole you have a notion of Beckenstein Hawking entropy and in this side it's a counting problem of some Hilbert subspace of the brain system and I will look at this brain system might sound a bit strange but I will try to demystify it in a moment so this is some problem in enumerative geometry okay and this turns out to be related to some Euler character of some modular space and it's related to various modular forms Fourier coefficients of modular forms so just like our counting problem for the photons was this very simple problem of right this problem right this problem is actually very generic in fact this problem this is the kind of thing that Goetsche did for example he when he was computing the modulus some piece of Euler character of the elliptic genus sorry Euler character of symmetric products of k3 he got this kind of a object because of the Euler character of a single k3 is 24 and if you have symmetric product they basically behave like bosons because they symmetrize symmetric they commute with each other and therefore that counting problem is actually related to bosons because they all have even cohomology if you had odd cohomology then they have to come up with some minus signs and they are related to what we will call fermions which I'll do next time so basically all this technology that we used is very directly related to this counting problems of various modular spaces and let me now very quickly say what is a brain system and then I will stop so what for our purpose is you don't need to know anything about various D brains and this and that so we said that there is a space time manifold which was say m1 comma d in string theory the typical m1 comma d is for example you encounter is m1 comma 9 which can be some m1 comma 4 times some kalabia 3 some 6 manifold which is a kalabia 3 4 something like that m1 3 sorry now inside this m1 d you can have a sigma 1 comma p sitting inside an m1 d okay just some you can think of it as some homology cycle some complicated homology cycle inside this or just some plane a p dimensional plane in a d dimensional space now we have been discussing quantum fields on m1 d so let me say that the coordinates of this are sigma mu or sigma alpha and the coordinates of this are x mu so x mu is equal to your time x 0 and x so mu goes from 0 to d whereas sigma alpha goes from sigma 0 that means sigma vector which is so alpha goes from 0 to p such an object is called a p brain is the generalization of if you had a string if you had a point particle it's a zero brain if you had a one dimensional sigma then it's a one brain and there is a more general general notion of a p brain but of course it's not just the homology cycle the point is that there is a quantum field theory which is living on that and I think this is what so it's it's like a you have to think in terms of sheeps or something like that because for example you can have a connection one form which is localized on this quantum field theory on sigma 1 comma p belonging inside an m1 comma d such as a young mills theory I mean this theory of connection one forms so by theory I mean there was a Hamiltonian that we wrote down some equations of motion that we wrote down one forms of group g of say u1 let's say to be very simple then the by theory I mean the equations of motion so a now lives a mu d sigma mu is localized onto this brain and the equations of motion the Hamiltonian equations of motion are just I write f is equal to df this is the Maxwell equation in p plus one dimension the Bianchi identity is this and the Maxwell equation is d star f is equal to zero so this system describes a theory on a p brain and you can have different brains with different quantum use a different quantum field theories and so on so there is a whole variety of things so the simplest quantum field theory that we considered was a one brain with 24 scalar scalar fields phi of sigma and tau some label I I going from 1 to 24 and that's what gives rise to this one upon delta q so this is how this is what I will try to describe next time that somehow in the same way that we were able to understand get some insight about the microscopic world about of matter microscopic structure of matter by just you know heating the room and measuring the temperature without really having a microscope in the same way one can actually learn about quantum gravity by studying black holes and their entropy because then we have to be able to explain that entropy is logarithm of something else logarithm of some dimension of some Hilbert subspace and that counting problem of dimension of Hilbert subspace is very interesting mathematically because it is related to all kinds of interesting enumerative geometry problems and very often they assemble themselves into modular forms and that again is not an accident it's related to the fact that we are dealing with a conformal field theory so we saw last time that we got this weird one to the 24 I mean it was really important that that ground state energy somehow could be renormalized or regularized using the zeta function and the fact that the minus 1 upon 24 came out the way it came out was because if you want to maintain conformal invariance then that's though that that answer is uniquely determined so I think I'll stop here yeah so the typically the black the brain will be some cycle inside this case this six dimensional manifold so think of this four dimensional the ten dimensional manifold in string theory is say some m one three and at each point there is a k six and if this brain is some complicated cycle wrapping something some brains cycle wrapping in this k six then in four dimensions it looked like a point but there is a Newton's there is a coupling constant which I have been setting to taking it to be small right we have been treating free field theories there is a way to increase the coupling constant in which case the Newton's constant becomes large and this point sort of swells up into a black hole but because you can go from here to there by continuously varying that coupling constant you expect that you should be able to complete the degeneracies yeah and then then you have a nice metric on it which is what I will describe next time so once the coupling constant becomes large this brain configuration which is localized at a point swells up into a black hole and that geometry I can work out by solving some differential equations and that entropy I can calculate and by some continuity arguments which are actually subtle because of wall crossing and so on and that's where the whole mock modular story comes in but that's roughly the intuition that you should be able to discover okay I think I'll stop here