 Okay, I think I can start now, so are there any questions from last time? Okay, let me try to remind you, basically the plan of my lectures was that, first lecture was quantum mechanics, second lecture was about today is going to be about quantum field theory, quantum fields, third lecture will be about general relativity and classical black holes, and the fourth lecture is going to be about quantum black holes. See, it's a very steep, I don't know how much we will succeed, but actually we more or less succeeded in explaining quantum mechanics. Today I will explain what are quantum fields. And the objective is, as I said for this book, I want to explain the connections between various aspects of quantum black holes and topics in number theory and geometry, in a manner that should be accessible to any mathematician or a student from other areas of physics, with a rudimentary knowledge of the basic undergraduate physics that they may have seen. And with that view, we started to develop a dictionary so that we can communicate with each other. So, we said that a physical system S, with it we associated a Hilbert space, then we said that the state of the physical system, we associated with it a state vector in the Hilbert space. And in fact, as there was already mentioned, physically what is, it's really a ray because the vector and another vector multiplied by a phase have the same physical consequences. So, you can, you have to keep this in mind, so it's really strictly speaking a ray, that in a second. In this notation, it's only a phase and the norm can be anything. No, notation does not imply that it's e to the i theta, where theta is real. No, no, so this can be any real number at this stage. I don't want that, so I will come to that. Yes, I mean, we will come to that, I will mention, I will come to that, yeah. Yeah, we are jumping ahead. A physical observable, yeah, I will try to, I mean, there have been some of these confusions, so I will try to address them so that they are not important confusions, but it can stop you from, so we said that these are self-adjoint operators like A, for example. And the eigenvalues, results of a measurement correspond to the eigenvalues of the operator. Sorry, alpha, let's say, alpha i of the operator. So, if you perform, if you measure some observable given a state, the result that you will obtain will be one of these eigenvalues of the observable alpha. And the probability for if, and let's say P i is a projector. So, this is the measurement axiom which I didn't really explain in detail last time, because that was a bit of a diversion from what I wanted to do, but let me state it anyways. If P i is a projector onto the eigen subspace of alpha i, the probability given a state psi, the probability of obtaining the result alpha i, if you have identically prepared systems, is given by psi P i psi divided by psi psi in measuring. Okay, very good. We introduced, we discussed a very simple system which is the harmonic oscillator system. So, there was a Hamiltonian, another concept was Hamiltonian, which was a self-adjoint operator, which was responsible for the time evolution. The time evolution of the state was given by unitary evolution. And one particularly simple Hamiltonian that we considered, this is called the harmonic, the quantum oscillator, I called it. And the Hilbert space, in this case, was a Fox space representation. Simply given by, it's like a highest weight or lowest weight representation, built upon a Fox vacuum, a set of states a dagger 1, etc. a dagger to the power n, the root of n factorial. So, I'm now going to choose a psi which is normalized to 1, to give an orthonormal basis for this. This is state 1, the state n. The set of vectors n furnish a basis, furnish a basis for the Hilbert space. Now, from here to make a transition to quantum field is actually relatively easy. That's what we are going to do. Intuitive idea is that if you have like an electromagnetic field, a classical field, it's Fourier modes, if you do the harmonic analysis, then the Fourier modes behave like quantum oscillators, oscillators for a classical field. And then by regarding those oscillators as quantum oscillators, you basically get from it a quantum field. And I will explain how that happens for a simple example. Let me erase this now because I want to write the dictionary here. So, before getting into quantum field, let me try to explain what is a classical field and classical fields. So, space time as we will see is going to be a pseudo-Riemannian manifold M1d. Let me explain that a little bit. So, all physical processes take place in space and time. And if you have a d-dimensional space, then locally like in this room, you can select coordinates xi, xm. Let's say m goes from 1 to d. The event, any event in space time like I throw the chop up, this event is specified by or a light flickers is specified by its location. So, some space coordinate and then some time coordinate. For Newton, at Newton's time, time was absolute. What that means is that observers in relative motion record the same time. So, my time and the time of the person who is in a train is the same. So, if I have two observers using two different coordinate frames, then you will have xm prime, but the time will always be the same. If I have an observer O prime and observer O, only the space coordinate was changing and the time was absolute. And the essential insight underlying Einstein's relativity is that time is relative. That's why the name relativity, which means that time also can change. Depending on the time recorded by the person in a train doesn't have time of one particular event or the distance between two. If you have two events and they're separated by some time t1 and t2, here their time would be t1 prime and t2 prime. And they can be complicated functions of this coordinate. So, therefore that the fact that time is relative means that you should really not regard space and time separate with space time as a together as a single entity. And in fact, in spatial relativity space time is just, so in general it's a single entity and it's going to be a Riemannian manifold, a manifold. So locally and it's going to be a pseudo Riemannian manifold. I will just remind you, you might have seen this already some of you. So, what that means is that in the simplest case, it's going to be generalization of R1 plus D, which is a Euclidean flat space. We'll go over to R1, D, flat space and this is called the Minkowski space time. Here the metric, there is a metric tensor you can write down. Here it will be just 1, 1, 1 diagonal for Euclidean space. Whereas here the metric is delta mu nu is the metric. Here the metric has 1 minus sign. The only difference between spatial relativity and Euclidean space. Minkowski space time and Euclidean space time differ only by this very crucial minus sign. Which means that the line element here is just dx mu square dx mu dx nu delta mu nu summed over mu nu. Whereas here it is eta mu nu dx mu dx nu. When in particular dt square, sorry, ds square is going to be minus dt square plus dx vectors. So, x vector is just the space vector. Euclidean Riemannian manifold you define as something which locally looks like flat space. You have a chart and then you locally it's flat. Locally you can have a chart, a map onto. So, basically you have a local chart Minkowski. The pseudo Riemannian manifold is a manifold in the usual sense with a chart. But the metric is not Riemannian, it is not positive definite. It has a signature minus 1 minus and d pluses. The pseudo Riemannian manifold simply means that the metric of signature. Given such a manifold you can consider various things like a tangent bundle on it. Let's say a tangent space at any given point is a tangent space T of m or T star of m. So, given a manifold you can consider various bundles. If you think of it as a, the structure group, I mean if you think of this as a vector here. If you have a metric pseudo Riemannian metric just like, just this is a complete analogy between the Euclidean manifold and the pseudo Riemannian manifold. Riemannian manifold and the pseudo Riemannian manifold. Just that you replace a positive definite metric with a metric with a minus 1 minus sign. This is a metric g mu nu. So, this, the structure group, it's called a structure group right. It's acting here is instead of being SO d plus 1 is SO 1 comma d. And in fact in physics you required, oftentimes you require a spin bundle, which is just a cover of this, which is a spin 1 comma d. Which is again, once again in complete analogy with SO d plus 1 and spin d plus 1. I think everybody, all mathematicians are usually familiar with Riemannian manifolds. And on that you can define tangent space and cotangent space and various bundles right. You can take symmetric product. So, a classical field in general is a section of some bundle. And depending on what bundle you choose, you call it a scalar field or a spinner field or a vector field or a metric field. Okay, so that's the definition of a classical field. Is this clear? Let me take a simplest example of scalar field. That's just a function phi of x. I will use x to denote all the, all these coordinates. So, x will really mean x mu. And when I put a vector on top of that, it really means x m. This is going from 0 to d. This is going from 1 to d. In fact, you can have more general bundles. You can have some other compact group. And you can consider, for example, the principle bundle. Spells principle. And the sections of this are the g, the sections of this bundle also you can consider. And these are called gauge fields. So, you can consider associated bundles in some representation of this group. So, you have basically a, your total group is spin one comma d cross g. And so locally, you have some chart in this manifold m. And the fiber is some representation of spin one comma d and a representation of this. So, let's call this representation r. And this representation r tilde. And if, for example, if you have a explicit vector space representation with some index a and here, here's some index alpha, then a classical field is simply some object with two indices as a function of x, which is a section of this bundle. In fact, the simplest group to consider is g is equal to u1. And if you take a u1 bundle and you define its connection a, then in some local patch, you can write it as a is equal to a mu dx mu. It's a one form. And physicists would call this a mu. It's clearly a vector, a contra variant vector field or a form. It's really a form. They will call, this is known as the electromagnetic potential. No, no. This is a connection of the u1 bundle. So, in other words, you can define a covariant derivative delta is the connection. It's the u1 connection. It's the gauge field of the group. So the electromagnetic potential is basically a gauge field in this definition of u1. This is local. It's in the local. No, so in many situations where it was encountered, the bundle was trivial. So it's a bit pedantic to call it a bundle. For example, if you just take the flat Minkowski space, then the bundle is trivial. But there are situations where the bundle is non-trivial. So, yeah, so therefore, yeah. Sorry, it's a connection. It's a connection. Sorry. I'm sorry. Did I say it wrong? Sorry. Connection. I'm very sorry. Yeah. E is a connection. Other fields, you can have an associated bundle. So you can have other fields can be sections of associated bundles like a scalar field. Suppose you have a scalar field, now what is the connection? Connection. So how do you want to say it? Okay. Sorry. Yeah. Let's say a connection one form corresponds to a gauge field. Does that satisfy you? So for example, here we took a scalar field, but that scalar field could have been, if you had, for example, representation of some group with, let's say, SUN, group like SUN, then Phi could be in the fundamental representation. It could have some additional index. Same thing. I'm just saying that if you encounter these, don't get scared because their physicists use different words. They call it a electromagnetic potential. Some people will call it a U1 gauge field. Or if it's U1, in general, it could be an SU3 gauge field. Surprisingly, in physics, G is equal to SU3 cross, SU2 cross U1 is all that you require to explain essentially all the physics that we have encountered thus far in the, up to the Large Hadron Collider. It's quite a remarkable fact. It's kind of, from a mathematical point of view, it's not clear what is so special about these groups. But from a physical point of view, they are special. And this is related to the strong nuclear force. This is the electromagnetic force. So, these together, in fact, the ICTP were unified by Salaman, which is basically electromagnetism plus weak nuclear force. No, no, no. Not at all. So, in fact, that's, I'm coming to that point. These are classical fields, but they satisfy classical equations of motion. And this is how I'm going to make contact with our previous discussion where I'd considered Heisenberg, Hamiltonian equations of motion. The classical equations of motion are simply some partial differential equations satisfied by the classical fields. The connection is not flat, but it must satisfy a classical partial differential equation. So, it's not any field, but physically relevant fields must satisfy these classical equations of motion. And there is a quantum version of that which we will come to. And that's what we'll make contact with our harmonic oscillators and this discussion, which so far is perhaps a bit abstract. So, let's take the simplest example. If you have a U1 gauge connection, A, it's a connection one form. I can define from it its curvature two form, the curvature two form. It obviously satisfies the Bianca identity. And the equation of motion is D star of F is equal to 0. So, these are some very simple generalizations of basically partial differential equation generalizing Laplace equation. For example, if you have a scalar field phi, it just satisfies D dagger D, I mean D star D plus D star D star, I mean D dagger. You can define a dagger operator which is, ok, but star is the hot star operation. So, in particular for a scalar field, this is nothing but the scalar Laplacian. So, I can call it delta phi equal to 0, delta phi is equal to scalar Laplacian. Delta is equal to scalar Laplacian. It's actually not a Laplacian because it would have been a Laplacian if it was Euclidean with that 1 minus sign, it is called Dalyan version, but ok, you can think of it as a scalar field, a classical scalar field in D dimensions, D plus 1 I mean dimensions because 1 is time satisfies delta Laplacian acting on phi is equal to 0. There can be more generalizations of this. For example, it can be delta phi is equal to m square minus m square phi is equal to 0. Notice that this is still, let me write it like that. My convention maybe there is a plus here. Notice that this is still a linear equation, ok. The sign is important because that actually will determine whether that particle is tachyonic or massive. So, the sign is important. We can figure it out actually. So, this operator is del square delta square. So, it is going to be E square minus P square minus m square. So, this is correct. This sign is correct in my convention, ok. But more generally you can also consider slightly more complicated nonlinear equations. Now, if I choose a coordinate frame and this can be in many dimensions, it can be in four dimensions, one 3 plus 1 dimensions. So, the particularly interesting manifolds of our interest are going to be, of course, we live in four dimensions. So, 1 comma 3 is a very special one. It is going to be what we will be talking about mostly. It can also be R 1 comma 3 if you are in Flatminkowski. Or it can be m 1 comma 9 which is what is required in string theory. Or you could consider m 1 comma 1. For example, I will do it. You can consider any possibility, right? You can consider R 1 comma 1. Or you can consider m 1 comma 9 which is of a special form. R 1 comma 3 cross some 6 manifold. And again in string theory, this 6 manifold is often taken to be a Calabi or 3-fold. In a complex 3-fold. So, with this description, so let me actually take a pause here. So, last time we described quantum mechanics. You know just single harmonic oscillator, Hamiltonian, Hilbert space, a state, etc., etc. We slowly want to make a transition to quantum field theory. We did the part, quantum part last time. Now you want to go to the field part. And eventually we want to go to general relativity. Here is this and make it a bit neater. We had observables, self-adjoint operators. Then we had space time as we said is a manifold m 1 comma d. And fields, they can be either connection forms or they can be curvature two forms. That is also a field because it is defined at every point or sections of various bundles. And now I want to explain what is a quantum field. And it is one way to say that it is kind of operator value distribution. To begin with, you can think of it as a function. But for some other niceties you require it to be distribution to make it more. The notion of a classical field you understand. I mean I hope it is clear. The phi of x is a scalar valued function. A scalar valued map from m. That is the classical field. And we are going to make it an operator valued map. In general, I am just taking one example of a quantum field where it is a scalar valued map to begin with. A classical field which is a scalar valued function to begin with. If you have a connection form, then you will have a vector valued quantum field and so on. For the metric, as we will... Let us not go there. Because we actually don't really know how to associate a quantum field with a metric completely. So I think between classical field and quantum fields for us is going to be through these equations of motion. And to do this, I am just going to illustrate this first in the simplest example. Namely, I will take a scalar field. And this is actually an important example because t and x in r1, 1. So not r1, 1 with a further identification. So the classical field satisfies a simple equation delta phi equal to 0. The plushian in 1 plus 1 dimension is particularly simple because it is just minus del square in r plus del square del x square. And if I write t plus x is equal to x plus and t minus x is equal to x minus, then this becomes minus del plus del minus 2 times maybe. This is exact analog of if you had a plushian with a positive sign. Let us say del x square del y square plus del square del y square del x square. This will be just del z, del z bar on a complex plane. If you had a complex plane and you had a plushian on a complex plane, you can just write that as del z, del z bar. And here it is del plus del minus. So we have an equation del plus del minus phi is equal to 0. Now remember last time we and I want to now get a quantum field. I want to now introduce a quantum field based on what we discussed last time. So last time we discussed consider harmonic oscillators, quantum oscillators with frequencies infinite number of them omega r is equal to just r. Frequences are basically 1, 2, 3, 4 like that. The Hamiltonian is just going to be omega r, a r dagger a r summed over from r is equal to 1 to infinity plus a half. I have an infinite selection of operators a r and a r daggers. I can also consider an identical system with a tilde in the same property a r tilde dagger a r plus half. I have another set of operators a r tilde a r tilde dagger. Recall that a r and a r dagger satisfy the Heisenberg equations. Just a and a dagger if you remember last time we had equations. That I da by dt is equal to minus h a and so on. And more generally I d of any operator a in this theory is defined by particular we last time we defined operators we can write a is equal to some q plus i p with some scaling that we had written and q and p also satisfied. So q also satisfied this equation and the Hamiltonian was q square upon 2 plus sorry omega square q square upon 2 plus p square upon 2. So now I am just going to construct the following operator. Phi of tx left moving I am going to call it. You find this is just a definition as a r e to the minus r t minus x plus a r dagger e to the plus i r. So it is a real field and I can define the right moving field. I will explain the terminology in a moment which is a tilde r e to the minus i r t plus x plus a tilde r dagger e to the plus i r t plus x. I hope I have not lost everybody by this time. Let me recapitulate. So this system we understand very well harmonic oscillators. I will now just take a collection of harmonic oscillators. This is nothing that is as simple. They are completely I just Hilbert space is the product of those Hilbert spaces. Hamiltonian is the sum of the individual Hamiltonian. And I just declare this to be my, this is clearly an operator valued field because it is a map from space to an operator, right? That is what is meant by an operator valued function or distribution. And now I am going to call phi of t x is equal to phi of left t x t minus sorry this really depends on t minus x and this depends on t plus x. Why do I do this? It will become clear to you in a moment. By construction del minus del plus phi is equal to 0 because you see the quantum field satisfies the Laplace equation or the Lahrenberg equation. Moreover, you can define an operator phi which is the derivative of phi d phi by dt which is an exact all of this p. Here if you found, if you look at the equations of motion of p because the Hamiltonian is really just p square upon 2 and the commutation relations of q and p is i. This is the famous Heisenberg commutation relations. p was simply dq by dt. And this is called sometimes the conjugate momentum. Then the Hamiltonian has a very nice property that we wrote down r, ar dagger, ar. The total Hamiltonian is this, right? The left Hamiltonian plus the right Hamiltonian is equal to an integral 0 to 2 pi over x. i is also filled as a function of t and x. It can be written as d phi by dt whole square, I mean del phi by del t plus del phi by del x whole square. And in fact, this is pi, so this is pi square. If I replace pi then it becomes d phi by dt, sorry, partial. Now, this might look a bit complicated but it is not complicated. The point is the following. If I just take this definition of the field, take its derivative and then do Fourier transformations. Then what is going to happen is that the integral over x will basically make sure that the r of this and r, they are coupled together. They will enforce a delta function. And that's how you can go from here to there. So, this is an exercise that take this expression, substitute this into this expression, do the x integral and you will get this. It's a straightforward high school exercise. Now, we come to the important point is that this Hamiltonian is a local density. This implies that the Hamiltonian is an integral of a local density over space. So, let me give you some picture about this. Your manifold is this. Now, this is really r, it's really s1 cross r. This is your time coordinate and your x coordinate is periodic. And you have a field defined on it which is t as a function of x. If you did Fourier analysis along the x direction, what you will get are modes of the field oscillating around with different frequencies. And that explains why the frequencies come out to be integers because they all have periodicity 2 pi, a fixed periodicity. So, to begin with a priori when we just thought about harmonic oscillators, these frequencies had no reason to be integers. They could have been any frequencies. But here in some unit, because this length could have been 2 pi L for example, but the important point is that r is r upon L in that case and they are integers based. We have taken L to be 1, but we'll shortly consider the more general case. So, the important point is that the classical field when you... So, therefore the quantum field, this is the slogan that I had told you, is simply a collection of harmonic oscillators, quantum oscillators. Okay, I think this time I perhaps lost more of the audience. So, maybe I should stop and ask questions, wait for questions. This is a special case of a manifold, right? I'm going to consider more general manifolds later on. Yeah, the point is that the quantum field is well defined even on a curved manifold, but it's much more complicated to describe it. But it can be described, it's done now. It's pretty well understood quantization of quantum fields and curved space then. We will do that because that's what leads to Hawking radiation. I wanted to avoid talking about Lagrangians because it becomes a whole lecture on in itself. And I wanted to take the shortest path to... Another way to do it is using Lagrangian mechanics and actions and that's more naturally suited for path integrals but which are usually harder to explain to a mathematics audience. Yeah, I reversed the order because I... It's again because I think quantum field, I can define it in this manner. You're right that normally one starts with a classical field and does something called quantization. But the quantization is not really... One can always take a classical limit, but going from the classical to quantum is a bit of a... It requires some physics and I wanted to avoid that discussion. That's a good question. Yes, absolutely. Absolutely. If you want to include interactions, basically they are oscillators with unharmonic interactions. So you can also view them as oscillators interacting with each other. Yes, yes, I agree with you. But I will come to interactions shortly because first I want to explain what is the meaning of a particle and what is the meaning of a state and so on. Exactly identical thing exists. It's a lot more complicated to describe it, but quantization of... You can quantize all connections. Quantization of connection, one form has been done. That is known as quantum gauge theory. And that procedure is much more complicated than I want to get into. It's not going to be required for our purposes, but it exists. Yeah, so modular space is a much smaller subspace of the space of all fields. Here we are considering a much bigger space, which is the space of all fields. An equation of motion, but that's much larger than the modular space, generically. In some cases the space can be simple or flat connections. Yeah, but that's actually a good point. In many cases you can quantize that space and that's a simpler version of... So in mathematics literature this appears like a space of flat connections. Those are special cases of this. Okay, let me just recapitulate. The point is the following. A classical field satisfies some equation like Laplacian of phi equal to 0. We constructed an object while the quantum field which satisfies the same equation. So quantum field satisfies the same equation. That equation can be viewed as a Hamiltonian equation of motion because if I write it's basically d phi by dt, you can check it's the same as minus h phi and i d phi by dt that I defined is minus h phi. And if I, using this equation which will turn out to be exactly the same as phi dot, i times phi dot, so if I replace phi in favor of phi dot, I will recover this equation. So the equation of motion satisfied by a quantum field, this was the statement that I wanted to... Sorry, so with the replacement phi is equal to phi d phi by dt. Because there was an i here. Sorry, sorry, what am I saying? With phi, sorry, sorry, sorry. Phi is equal to here. From here I will get i d phi by dt is equal to i phi. So with one of the equations of motion will simply tell you phi is equal to d phi by dt. And if you replace phi everywhere, then you will recover this equation. So see Heisenberg equation of motion are for two objects phi and phi. But that's just a way of writing the second order differential equation in terms of first order differential equations. By introducing phi is equal to d phi by dt. You can always write the second order differential equation in terms of first order differential equation if you double the number of variables. That's basically how the phase space was introduced. If I just define phi is equal to d phi by dt, this equation can be written as a first order differential equation. And that first order differential equation is the same as the Heisenberg equation of motion. The final thing is that the Hamiltonian is local. Meaning it's defined by, depends only on derivatives. It doesn't depend, it's a local meaning that it's an integral of a local density. And that local density can be constructed using just derivatives, local quantities. If you look at it, it involves the derivative with respect to time and it involves derivative with respect to x at the same point. So that is the notion of locality. Now this is really very crucial. This is a very important concept because locality is very closely tied to causality. And it has this, right? Let's see. Yeah, I can, it has this. The Hamiltonian does not include terms like this. Phi x times some kernel. It could have had terms like that, right? This would be non-local because you will require the value of the field here and the value of the field in Andromeda Galaxy. Then you'll have to do something, take an integral of them and this is not local. This is bi-local. You could call it bi-local if you want. But it's really not local. And this is really fundamental to quantum field theory. That the light, the signal doesn't propagate instantly. Causality means that first of all, the cause precedes effect and signals do not propagate faster than the speed of light. You cannot communicate. Okay, let me contrast this with Newton's law of gravity. If you remember, Newton's law of gravity is that the force between the sun and the earth is the mass of the sun times the mass of the earth divided by the distance between them. This is not local in time. It's instantaneous in time. Because suppose the sun disappeared tomorrow, then you will suddenly have to put this force to zero immediately. But you know that the light takes eight minutes to come here. And this actually bothered Newton a lot. I mean, you think that Newton just wrote it down, but Newton, there is a letter by Newton saying that this sounds totally bizarre. This cannot be true. And this is really critical for general theory of relativity. And you probably heard about the gravitational wave detection recently. That has to do with the fact that something happened, the black holes too, black holes merged. But that signal took billions of years to come here, one billion years to come here. And we detected them today. But the event actually took place long time ago. This kind of thing cannot happen in quantum field theory. So Newtonian gravity is not allowed according to our principles of local field theory. And that's why Einstein had to construct his theory of gravity, which is local. Now I'm going to connect it to modular form so that you can appreciate some of these things a little bit better. So you remember we introduced a partition function. I mean, sometimes I call it q, sometimes I call it tau, let's say. But q can be e to the 2 pi i tau. And let's just take the left moving. You remember the Hamiltonian was the left moving Hamiltonian plus the right moving Hamiltonian. And so h left was just with r, e, r dagger, e, r. I'm going to ignore the tildas. There was a similar expression with the tildas. And this we evaluated. Now notice here, it is an infinite constant. And let me call that the energy of the Fock vacuum, the ground state energy. Because now we have a collection of infinite number of oscillators, there is no guarantee that things will be finite. And here is a simple example, that even though this half is a finite, after you sum over infinite number of oscillators, there is a potential for divergences. So let's keep this energy e naught. This is the energy of the, because that's the Hamiltonian, eigenvalue of the Hamiltonian for the Fock vacuum, which by definition is the energy of the Fock vacuum. So h acting on the Fock vacuum is equal to e0 times the Fock. This sum we did last time, you remember. So this is going to be e to the e naught, q to the e naught. That's, we don't know what to do with it. And this leads us, but we said, aha, sum over r, zeta function regularization, that I will regularize e0 s to be half sum over r to the minus s, where s is not equal to, sorry, s is not equal to 1, sorry, s is not equal to minus 1. When s is finite, I can evaluate this. Then I will, so this is, looks like a bit of cheating, but let's do that anyways. That will give us minus 1 upon 25. I agree, zeta of minus 1. Yeah, we have to do an analytic continuation, there is a pole, so it's not, there is no pole at s is equal to minus 1, sorry. Now this looks a bit ad hoc, but the point is that this is one of the examples of what is known as renormalization theory in physics, in quantum field theory. I want to take this as an example so that, of course the renormalization theory is a very vast subject, and part of the Princeton notes were actually trying to, really try to define it in some more complicated way, I mean in the more detailed way, and that is much more complicated than what I'm describing here, but it should, I wanted to give you a flavor of it so that you should understand what it means, and it's not something, I mean after all Euler also did this, so it's not something that only physicists do without any reason. And one of the, let's try to understand it a bit more physically, why we did the zeta function regularization. So one can use a different regularization, and the problem is the following, that we have, our space is like that, think of it as like a violin string, of course the violin string is, has a Dirichlet boundary condition, so the violin string has modes, Fourier modes which are like that, and those are the different integer-valued frequencies. Here we have periodic boundary conditions, so again we have integer-valued frequencies, but of course we know that the violin string cannot oscillate. Once it hits the atomic size, there is no sense in whether, you cannot really take the frequencies of the violin string to be extremely high. Very high octaves of the violin string don't really make sense. So that suggests that we should perhaps, sorry did I bring my notes or not, I forgot my notes. I'll have to do this by memory, which is not a good idea, okay but I hope I don't make a mistake. So you can regularize it differently, so what this is doing is that, it is really cutting off, and this you can do very explicitly, and clearly as epsilon goes to zero, the pole here, as epsilon goes to zero, there is a, in fact when you take a derivative, there will be a double pole, and this again I leave as a homo exercise, because I didn't bring my notes, but we can even do it here. Can I leave this as a homework exercise? You can take this function, and expand it in powers of epsilon. What you will discover is that it has a pole minus one upon half epsilon square, minus one upon 24, plus order epsilon. Notice that there is no term of order epsilon, one upon epsilon, some number times. Now notice that our violin string was, had length 2 pi, but actually it could have had length 2 pi l, and the atomic distance scale was say a, then epsilon which is a dimensionless parameter is really a divided by l. A is another cutoff, so a is the distance, is the atomic distance. A is, if the violin string had length l, then I can define, I'm trying to explain the physical origin of this cutoff. Physical origin of this cutoff is that, very high violin frequencies are getting cutoff, and at what scale does that happen? At the scale where the atomic distance is important. So if the total string, violin string is of length l, you are introducing a cutoff epsilon, which you can take it to be purely mathematical cutoff. But you can imagine it also is coming from this following physical cutoff that the violin string actually has, beyond that point, it doesn't really make sense to think of the violin as a violin string as being a long string, but it's really composed of, because it renders, no, it's a cutoff because basically, R, which is much bigger than one upon epsilon, does not contribute to this sound, right? This is a smooth cutoff. Yeah, it's a smooth cutoff. It's called a cutoff. Okay, let's not argue about it. Okay, but it's a matter of terminology. In physics literature, it is called a cutoff, and it's a smooth cutoff. So you can call it a smooth cutoff, and that will make you happy. But it's a cutoff in the sense that higher frequencies of the violin string are cutoff, meaning they are contributing less and less, exponentially cutoff from there. So it's true that it's not a sharp cutoff, but they're exponentially damped the contribution to this interval. Okay, and that typical, the frequency that you will hear the most, I mean the maximum frequency that you can hear is of the order of one upon epsilon, right? Energy E0, in that case, if you had length L, between one upon L times half R, because the frequencies were one upon L. If I now, if you look at the term which is, this goes as one upon L times epsilon, which is epsilon square, some constant times epsilon square minus one upon 24. So this goes as E square, if you plug it in here, this expression, A square times L minus one upon 24, divided by L. It is going to zero, I mean A upon L is going to zero, right? Now here you see the locality, because you see this quantity can be viewed as an integral over, so the renormalization procedure, no, I have not left the, for now, just give me two minutes to, I will come back to the world of mathematics. Just be patient. If I define epsilon is equal to A upon L, this is a completely mathematical statement, right? What are you not satisfied about? If I substitute epsilon is equal to A upon L, right? That follows? Okay. The point is that this is proportional to the length of the string. And I told you Hamiltonian is a local density. And renormalization is a procedure, says that because we are dealing with infinite number of oscillators, right? And the procedure is twofold. One is called regularization, which is a bit like what Euler did. You know, you can make it a zeta function regularization. So zeta function, you can use the epsilon regularization, et cetera. So the point is that you have infinite number of oscillators, and that's what is causing the divergence in this case. So you cut it off. You basically make sure that not all of them contribute to your problem. Or the ones that contribute, you put a smooth cutoff. You can also put a cutoff. I mean, the kind of sharp cutoff that you want can also be used. And that is another way to do so, yeah. The second step is renormalization. What does renormalization mean? Renormalization means that you're allowed to add local counter terms. Actually, all this is much better stated in the Lagrangian formulation. Since I didn't use it, maybe you should keep it in the back of your mind. The whole thing is much nicer to do it in the Lagrangian formulation. I'm just doing it in Hamiltonian formulation for the sake of economy because otherwise. What does it mean? The point is the following, that if I'm considering the Hamiltonian of a Hawaiian string, how do I know that the energy is really what I wrote down? There could have been an overall constant that I could have added, right? Hamiltonian was some density local density, but surely a number like a square is also local density. It's an appropriate sign with a minus sign. Minus C over, there was some C here, no? If the C is half, then it's half. The point is that that half is unimportant for physics because I can remove it by adding a local counter term. But I cannot remove this. You see, 1 upon L, I cannot remove by adding a local counter term. Looks confusing. No, no, so the point is, let me explain what I mean is that the rules is that by allowed to add local counter interrupts to the Hamiltonian, let me finish the sentence then it will become clear to remove the divergences in the regularize a going to 0. Adjust the coefficient in precisely such a way you can get rid of this divergence. It is local and it... No. So the point is that in this example there is a local counter term allowed. This quantity is local, you agree with me? Yes, this is the first term. Show us the first term. That also actually, in this case it turns out that your question is certainly valid. In more generally, there will be more general local counter terms allowed. There are rules. If you take a more general Hamiltonian, this Hamiltonian plus say, let's we add another term like lambda 5 to the 4. In this case, actually, you are allowed to add more local counter terms which can be proportional to this and also the counter term that you were talking about which is proportional to this. That's what I want to say because this is a low energy physics fix but let's say how I want to state this for a free case. So to remove the divergence and the limit keeping low energy physics the same. So the point is that that's one way to say that. You have some frequencies which were 1 upon 2 pi 1 and 2. We cut out some very high frequency modes which were very high frequency which was the limit in physical situations is taken in such a way that the physics at low energies does not change. Now you can ask does this not change the physics but actually what you measure what you are able to measure are only differences between this is actually a particularly bad example of renormalization because this is the famous cosmological constant problem which you cannot measure without having gravity but it's clear that what is physically important is just differences in energy because the overall origin of where you put the energy actually has no meaning. So therefore this counter term is allowed. If you do what you do of course you will get rid of the whole thing. So you want to keep the physics at low energies fixed. I would have said that except that that's not correct. Because 1 over L is not local if L is the length of the universe you want to have a quantity which does not depend on that. If I change the length of the universe I should not change the local physics based on the length of the universe. So you don't want to add something that is the meaning of locality that if I added something which depends on the length of the universe then it's clearly not local because in this example perhaps what you said is the best way to say that so let me make the correct statement you can actually add terms dependent on 5 square so that's why I didn't want to say it the way you stated it but let me state it more correctly So in a quantum field theory generically you will get divergences because you are dealing with harmonic oscillators of infinite harmonic oscillators but you can always regularize it using the smooth regularization procedure if you don't like a cutoff, smooth regulator let's use a better terminology that will not distract us from the main point so by using a smooth regulator you regularize the theory and what that does is that essentially eliminates the infinite oscillator problem essentially makes it a finite number of oscillators problem and then you are allowed to add local counter terms in the Lagrangian all terms in the Hamiltonian which are local in the situation when the Hamiltonian has terms like lambda 5 to the 4 you are actually allowed to add terms which are proportional to this in that case you make some value of the lambda measured in experiment fixed and then you take the limit so the statement that I would make is that renormalization means that to this regularized answers of quantum field theory we are allowed to add local counter terms to the Hamiltonian so as to remove the divergences in the limit when the regulator is removed the low energy does not change the point is that actually this thing which we are setting to 0 it turns out that if you did the same calculation in 4 dimensions the measured value is actually not 0 so it can be this is to be provided by low energy physics next time I will try to think of a better way of saying this but the main point is that practically you saw that there was a divergence and I could simply drop the divergence by imagining that my Hamiltonian was different that is the main point that is important for this discussion and if I did that then I would obtain z of tau which is exactly the Dedekindeta function now you are absolutely right that a priori this q could have some arbitrary finite constant could have a left over and what is the principle which dictates this one principle is that the physical observables at low energy should not change in this particular problem there is actually a symmetry which tells you which is conformal symmetry at low energies you have conformal symmetry if you demand that then this implies that c is equal to 0 and let me make a short comment about the conformal symmetry because that is what brings us to conformal field theory so the basic principle is that you want to put a regulator and you want to define the theory by regulating it and you are allowed to do renormalization but you want to do renormalization following some principles namely the low energy physics should not change or the symmetries that you think should be there in the low energy physics should not be violated and in this particular case if you look at the equation of motion phi is equal to 0 this is invariant under conformal transformation arbitrary let us consider the real equation sorry in the Euclidean space del del z bar if I take z going to f of z and z bar going to arbitrary f of some other g of z bar then this equation remains invariant let me call this omega and this is equal to omega bar so quantum field theories which have whose equations of motion have conformal symmetry are invariant under conformal transformations they are called conformal field theories it is a conformal quantum field theory sometimes it is written as CFT and in particular in this case conformal symmetry means that if I just scale the only scale in the problem is the length so if I scale the size of the universe the energy should by just dimensional analysis energy should scale as 1 upon length okay so this other examples of conformal field theories example the theory that we wrote down df is equal to 0 d star is equal to 0 in 4 dimensions d is equal to 3 this is for the Maxwell electric U1 U1 it is also conformal invariant what exactly do we mean by conformal transformation you can actually make it more another way to state a conformal symmetry is that the equations of motion are invariant under if you had a curved manifold then they are invariant under while transformations so for example the Laplacian in 2 dimensions that we wrote down you know which is like d star d plus d star you can actually in some coordinate frame you can write it as del by del x mu square root of the determinant of the metric g mu nu del by del x nu let us just take a Euclidean or you have to put a minus sign here now if I do a while transformation then the determinant of g will go as d times 2d times the determinant of g g is the determinant this g is the determinant so square root of g goes as e to the 2 psi g mu nu upper the contra variant goes as e to the minus 2 psi g mu nu so we see that exactly only in 2 dimensions the scalar Laplacian is d psi and therefore this quantity does not depend on psi is invariant only when d is equal to 2 because then this term and this term will cancel what is the conformal field theory more generally if you have a curved manifold or if you have more general forms and so on what do you mean is that the equations of motion should be while invariant I am changing the metric psi is a function of x so it is local so it will not cancel so this will go to yes but there is a derivative here let me do it if I just do this transformation on this Laplacian what will I get I will get this times e to the d minus 2 psi x del by del x nu sorry g mu nu square root of g del by del x nu so this will be the same as before then you will pick up an extra term pick up this derivative term so this term will be 0 only when d is equal to 2 sorry here but the equation of motion will not change the point is that if d is equal to 2 there is minus 2 sorry what do you want this is correct right minus 2 yes sure yes I agree minus 2 but therefore the equation delta phi equal to 0 will be left invariant so this while invariance of the equations of motion in a curved manifold are closely related that is one way to think about the conformal field theories of course there is actually much more abstract definition in terms of purely of conformal field theory purely in terms of conformal group which we could have done but this is another way to introduce conformal field theory so I think I will stop here let me now just very quickly summarize what we did I think today's lecture was perhaps less transparent than last time but it's because we are trying to do more things what I said was that starting with quantum mechanics and harmonic oscillators the quantum field you could think of as just a collection of harmonic oscillators and you could do pretty much everything they satisfy the equations of motion and we I gave one very simple example of renormalization and what you want to take away from it is that by adding a local counter term you could deal with the divergence even though you are naively getting a divergent answer if you imagine that your Hamiltonian was slightly different it was possible to remove that divergence so there was a so the statement of renormalizability of quantum field theories that under certain conditions if the coupling constants are dimensionless they are called renormalizable field theories it is always possible to remove divergences I mean it's a kind of a physics folklore theorem physics theorem you can say really prove it rigorously is this clay million dollar prize for the QCD so renormalization theory is the study renormalization theory in general is the study of this limit epsilon 1 to 0 you have regulated with some regulator epsilon and you want to remove that regulator and you are allowed to add up to local counter terms adding local counter terms and the statement of renormalization is that under certain conditions you can always remove those divergences and all physical answers can be rendered finite so whatever divergence you encounter there are of course a lot more complicated divergences appear but this particular example I think is simple enough to you can really work it out there is a divergence and you can remove it and in this case the physical principle that determines that what constant why we don't keep some finite constant still hanging around is because then we ensure that the theory becomes conformal so your renormalization preserves a certain physical symmetry that you wanted to preserve ok I think I will stop here