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Hello again, everybody. Let's hope, but the information technology stands up a second time.

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So I against the explain a little bit about how we tried to develop a theory of quantum gravity and why it's difficult.

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So let's start with particle mechanics, something we're all familiar with.

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So the typical problem in particle mechanics is you have a mass moving in a in a potential and it sorry,

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it starts at some place some time, initial time t zero.

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It starts the place. Xterra has a momentum p0 and then it moves in the potential and sometime follows a path through top.

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Here follows a path and ends up at another place with a different momentum.

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Later, a later time. Now, classically, when H bar is equal to zero.

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We can know both the position and the momentum.

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But quantum mechanically, of course, this is not this is not allowed.

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So the quantum mechanical situation is just a little bit different.

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So. At the initial time, we can't specify the momentum anymore if we specify the position and the final time.

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If we measure the position, then we can't measure the momentum. So if we work in terms of positions,

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what we're left with is an amputee to the quantum mechanical aptitude for a particle two time t zero beer to place X zero.

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And to end up at a later time T one. To place X one.

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And the the path which follows between these two events, if you like, it, is not totally determined.

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In fact, all paths are possible. And this is this is encoded in the final path integral, which is a way of calculating quantum mechanical amplitude,

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which is slightly different from what we first teach undergraduates in terms of solving Schrodinger equations and so on.

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The final particle gives you a sort of direct representation of the amplitude for this

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process in terms of the paths which take you from the first event to to the final event.

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And while the footballer what the pathological says is that I'm sorry about this is that

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you should you should sum over all of the intermediate poles and each and get some weight,

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which is a a face factor. And the fate, the face factor is is computed by by looking at what is actually the action or of of the path.

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So instead of having a single classical path, we have a sum over all possible parts.

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And these these build these this song builds up with all of these face factors to give you the quantum mechanical aptitude for the process.

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Now, out of this, you can actually recover classical physics because if you take a bar to zero,

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then the integral has to be dominated by the stationary point of of the integral and in the exponential.

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So if you have eh eh eh eh classical.

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If you have a classical path and you add a small amount to it, you can compute the variation around the classical part of path of the action.

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And this is something that actually you did when you calculated equations of motion using

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the variation of principle in all the principle of least action in classical mechanics.

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And if you go through this calculation, it's just a very short number of steps.

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We won't do it in detail here, but you can pick it up from the slides afterwards.

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You end up with a classical equation of motion, which is the AirMax double dot.

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Is minder's degrading of the potential. Which is. Which is.

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Which is Newton's law.

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So this Pieman path integral formulation is completely consistent with what you know about classical physics and the ball goes to zero limit.

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Now we can use this formulation now to move on and try to understand something about field theory.

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So now it's the archetypal quantum field theory is quantum electro.

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That's the theory of electrons and positrons interacting with the electromagnetic field.

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And here the situation is slightly different. Now, space and time are our framework within within which the the action takes place.

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And that's the time that the time c0 that there is a configuration of the of the field Soliah electric field takes take some form that sa e0,

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which is a function of X. There are also some fields which describe the matter, the electrons and positrons.

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And then you let this system evolve and some time t one later the field has evolved,

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the electric field is evolved to E one and the matter fields have evolved to something which I'd call Cybele.

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So, so now. And of course the interschool electromagnetism or classical we can mechanics.

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What you would do is you get some bunch of differential equations which you would solve,

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which would tell you how how you how the field's evolved over time.

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But instead in quantum mechanics, what we have to do is calculate this amplitude for starting with a field

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configuration at T zero and ending up with a different field configuration at at T1.

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So that's the basic quantum mechanical object in a can in a quantum field theory now.

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So. So I'm not just. So remind you about it. About.

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About something to do with electromagnets, which is that you'll recall that you write in the magnetic field,

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which is a measurable quantity as the kernel of the facts, a potential which is Vector A.

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And you can write the electric field as mine as the gradient of the potential now.

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Actually, there's another term there, which is the time derivative or of the fact of potential.

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And this can all be encapsulated in one object, which is called field strength tensor,

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which is if you knew and SMU New has an expression in terms of the four vector potential.

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Now, the important thing to realise is Acme New contains E fields and B fields and it something which is directly measurable.

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Whereas actually the full vector potential, you can't measure it, you can't measure it directly.

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All of these things have four components because there's a time and three space components.

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If we're in regular four dimensional, four dimensional space time.

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So that there's a desert and they these things behave well under under Lorentz transformation.

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So they would transforming and they're not what's known as covariance only underlines transformations.

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So it's it that the easy objects to. Right. Lorenz in various theories from.

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They have this property, that unengaging variance, so you can shift the vector potential by a total derivative.

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So that means different Teemu means deep differentiating with respect to each X.

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And what happens is F Munir doesn't change. So the physics doesn't change.

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So so now now we're in a situation where we can we can write down by mimicking the classical case, fine, final path integral.

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And so now we integrate over intermediate field configurations.

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And again, the some face factor, which is now a quite complicated looking action.

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But it's got bits in it which contain the electric and magnetic fields.

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That's the FBI new any new term. And it has other bits in it which contain the matter fields, which is the rest of this.

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This expression, if you if you compute what happens when each ball goes to zero,

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you do this stationary point calculation again is rather more complicated than in in the particle case.

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But actually, it gives you Maxwell's equations. So.

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So the in homogeneous Maxwell's equation is the team you deem you have, if you knew, is equal to the current.

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So the the charge and the and the current are sources of electromagnetic field because that's what happens if H.

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Baatar goes to zero. If they HBO is not zero, then this gives us Quantum at the famous quantum electrodynamics.

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And in principle, you can start from this. This formula here in the further the final path integral,

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you can compute any amplitude you like in terms of the electric charge, the mass or of the of the matter fields.

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The photon, of course, is massless and Planck's constant in practise.

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Of course, it's very hard work and there's a whole industry which has done this so over many years.

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But but the outcome the basic outcome is very interesting. So what you find is that you can if you take a couple of measurable quantities.

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I've just called them and be here and you do a quantum electrodynamics calculation.

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Then there's one extra element that you have to feed into the calculation, which is some ultra, which is some scale capital lambda.

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And the reason for that is that you find that there are ultraviolet divergences in the calculations that you do at your feet.

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Once you've fed this scale in, you can you can extract from your measurements A and B values for the mass or of the electron.

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And for the electromagnetic fine structure, constant output alpha.

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Even an H ball, by the way, is kind of. You defy a modern metrology age bar is defined to be a certain value.

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And then you can take these numbers m an alpha and you feed them into another QED calculation.

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Put the scale in again what you may not. Where did you predict the outcome for another process.

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C. And the interesting thing is that actually, when when you when you've done this,

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you realise that you can relate what your prediction for the measurable speed is entirely back to the measurable for and be using or QED calculations.

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And this scale parameter lambda that you put in and the intermediate steps disappears.

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And that's because QED is a renormalise zable theory.

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So it has this remarkable property that it predicts unambiguous relationships between measure rouble's and an expansion in Alpha Electromagnetic,

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which is roughly 100, well over 137. So it's a fairly small number.

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This works spectacularly well. And probably the best demonstration of this is the anomalous magnetic dipole moment of the electron.

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So according to Dirac's relativistic wave equation, the magnetic dipole ratio of the electron is exactly two.

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So the magnetic dipole moment, the electron is likely to be amongst photons,

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but there's a small deviation from the this which is generated by by quantum mechanical effects.

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And you can measure this small deviation. And this is the result that you get.

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You can see it's spectacularly accurate. The number 28 is the error in the last two digits of the result.

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Seventy three. And you can also calculate you have to work very hard to calculate to this degree of accuracy.

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And these are calculations which have only been developed over many years.

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But you can see that the agreement is spectacularly good.

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So so quantum electrodynamics is a very powerful theory and it's a very accurate theory.

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And it works. It works extremely well. It's one of these are some of the most accurate measurements and calculations of the main.

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So that's field theory in a nutshell. Never can tell about gravity, we better remind ourselves a little bit about general relativity.

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Already heard some of this from from John and in his talk.

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So the basic point is he is here that what one of them was the really important ingredients.

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Well, we have a space M which has some fixed topology for a minute. And on that space, there is a metric.

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And so so the the the degree of freedom which really matters is the distance between points.

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So you you take a point X mbewe and a point X bew plus dxp, which is quite close to it.

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And, and what matters is what's the real distance between these two points or the proper distance.

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If we're talking rigourously and that's determined by the metric which is this this object.

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Jemmy, you and Jamie knew is in general dependent on where you are in the manifold.

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Now, some examples are the Euclidian plane is the easiest example, and that's flat.

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And you know that the DSC ad is just the X squared plus dy Y squared.

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Another example is the surface of the sphere, which is most easily written down in polar coordinates.

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And and then there's a piece which is Defeater squared in a piece, which is Defi Square.

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But you see the component of D5 squared is not constant.

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OK. And K is basically the radius of a sphere. And then a final example is Binkowski Space Time.

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OK. So now we're really talking that now we're really talking about a proper, proper interval.

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And that again, is duty's square by the X squared.

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And the metric for Mankowski space time is often given the name item you knew, which we will come back to in a minute.

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Now, the point about this is that we can we can change coordinate systems.

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Right. For example,

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you can change from Euclidean coordinates to plain polar coordinates on the on the Euclidean plane and the coordinates of the two points and B,

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change. But the proper distance, which is actually just a Judi's existence in this case between them, doesn't.

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So so there is actually a hidden in all of this, a reprivatization of variance, which is that you can change the coordinates.

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And when you change the code, that's a metric function changes. But the distances between points are independent of that reprivatization.

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Now, proper distance is actually not the only independent frame, character frame, independent characteristic of of the space and the match.

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Right. There's another frame, independent characteristic, which is the intrinsic curvature.

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So that's illustrated in the end, the picture on on the right hand side.

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You can you can see that if a space is curved so that there are two.

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Right. Independent principal wages of radii of curvature.

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So just think of a surface. And this is the curvature of the space is it is again, independent of the coldness.

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So, for example, if you are if you are looking on the surface of the O of the sphere.

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Right. It doesn't matter what sorts of coordinates you use to describe that surface, it is still curved.

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So the curvature you're clearly implying is co zero because it flat the curvature of this spherical service is just one of the rageous square.

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It's a product of the inverse principle radii.

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Curvature of Binkowski space time is again zero. Binkowski space time is is flat now.

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So. So in this, G.M. is the dynamical degree of freedom.

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It's sort of like the vector potential intellectual dynamics. But so the whole posture of general relativity is that Jemmy New is a dynamical

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degree of freedom and the curvature of space time is generated by mass and energy.

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And that leads to the Einsteins equations, which which we have written down here.

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And the the important bit is the on the left hand side.

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We have all we have the curvature on the right hand side.

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We have the stuff that generates the curvature, which is the stress energy tense.

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And then we've also written down this other term. This is the cosmological term which Einstein wasn't very happy about.

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But but which there is no no reason in general not to have that.

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And one of one of the puzzles of nature is why the cosmological term is rather small.

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Why is this cosmological constant lambda? Sorry, this is a different quantity from our previous Cut-Off previous scale.

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Why is the cosmological constant so small?

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Is one of one of the mysteries which modern physics still wrestles with.

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A capital G is just Newton's constant.

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So this is very nice, neat way of writing things which are actually non-linear, partial differential equations for the metric.

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Basically, all you have to do is specify the initial conditions and solve.

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Right now, of course, that's easier said than done in practise for complicated situations.

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But nonetheless, you get excellent agreement with observation and with experiment.

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So how we get so then how are we going to quantised gravity?

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What what what must be the basic ingredients? Well,

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the first basic ingredient has to be the physical amplitudes that you talk about must be reprivatization

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invariant if they're not read traumatisation invariant and they just turn out to be zero.

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And so they don't actually tell you anything. Okay, so this is a very easy, reprivatization,

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invariant physical amplitude is basically an amplitude that the exact universe exists for proper time as.

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It starts at some event p0 and ends up at some event P1 after proper time.

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And then the forum appal thing to grow.

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You might speculate, which gives us these Ankit amplitude is basically a sum overall, such overall, such metrics.

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So here I've written down a find important to grow where I sum overall the old metrics which have this characteristic I saw.

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I've got some matter fields in there as well. And the hypothesis is this final part integral would enable you to calculate this amplitude.

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Now, this part this part that we've written down here, this is called the Einstein Hilbert action.

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If you take a bar to zero, the stationary point just this gives you about Eisenstein's equations,

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which we wrote down or on on on the previous previous slide.

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So, again, the pathological gives us as each bar goes to zero, limit the classical field equations, these bets here.

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If I if I just integrate. If I just some over matter configurations and I just use the maths apart in the phase.

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This basically gives me quantum field theory and a fixed background spacetime,

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which which is part of what John was talking about in his lecture, where we have quantum electrons in quantum photons.

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But we have a classical background, spacetime.

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But that's a very easy way of writing down something that could easily be very complicated.

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So what does this integral over the metrics mean?

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What what what do we mean when we are going to sum over all these interpolating metrics between two space time events?

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So we're going to look at the two to two attempts at doing this.

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So the first attempt is to basically copy QED. And you do that with a combined with a perturbation expansion in Newton's constant.

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And you might think that, well, gravity is weak.

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So so perhaps the way to think about this is that basically there is a sort of small wrinkles in in space time.

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And although the metric is a function of position, actually what it is, is it is a sum of two bits.

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It's the sum of something which is completely flat, which is just I mean, calfskin metric.

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And then there's H meunière who which gives this these small wrinkles, the oscillations.

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If you if you if you make that substitution for Jim, you and I start as equations and we just drop the cosmological constant term.

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It goes in practise. It's very small. In this case.

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OK. So you drop.

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So then then what you find is, is that the these small oscillations h they satisfy this equation in the middle, which is basically a wave equation.

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This does tells you that the matter is the source of these small fluctuations.

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And this is essentially the equation which leads to to gravitational waves, for example, being being emitted by by match or in some kind of motion.

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So that's some thought that that's one. That's what that's what happens classically if you try.

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If you do this quantum mechanically.

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And so you're now viewing this as a quantum field theory, then you need an analogue of the fine structure constant.

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So in this little quantum general relativity theory we're talking about.

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We need to have dimensionless fine structure constant. And unfortunately, Newton's constant has dimension.

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So the dimensionless object is actually Newton's constant times ours,

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a scale squared and then divided by by age bonder all the time where you actually can use it in units where the speed of light,

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you said equal of one and all these, these calculations. So it's that scale is the mass of the electron.

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Then this alpha for faja general relativity is about ten to the minus forty six.

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So he's really. The small and the certainly weak.

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But what you should worry because if you now set the scale to be about 10 to the 22 times the mass of the electron or so,

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then Alpha Alpha will will be about one.

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Now, that seems like a very large energy scale, but it's not really a very light Jannuzi scale,

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10 to the 22 times the mass of the electron is only about 20 at 10.

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The mine is eight kilograms. So this looks like there might be a problem.

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And in fact, there is a problem. The problem is that if you start taking measurable.

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And you start doing field theory calculations and you feed in your scale and you try to to deduce what Alpha is and all the other parameters,

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what you find is that the parameters cascade.

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And you can never take the outcomes of a number of measurements and feed them into a calculation, unpredicted other measurement.

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And this is this is basically the statement that this theory we're talking about is not renormalise.

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Now, actually proving it's not renormalise is a tour de force.

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And the famous theoretical physicist Envelopment who died this week was an important figure in that endeavour.

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But she is not totally dead. You can keep your scale and just work in a regime very small.

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And that's called defective field theory.

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And you can learn a lot about the relationship between quantum versions of gravity and classical gravity in that way.

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But it's not it's not a self-contained, fundamental theory of all of or all of everything, because you have to keep this role.

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Small scale in the other wave of other route.

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Is that string theory actually contain this field? H we do this.

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This is in fact as so kind of like a spin to field.

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And it also contains a consistent minimum distance scale. And it is a way of constructing a consistent quantum field theory of gravity.

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And Fernandinho will talk about that soon. But of course you pay a price and the price you pay is that string theory doesn't just contain gravity.

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It contains lots of other degrees of freedom as well. So so the the question is, can we do something else?

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Is there some other way of all of looking at this problem? So so this is the second possibility that I get talked about very briefly.

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So the first word looking at it, we just looked at fluctuations around them and kasky metric.

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But there isn't really any particularly good reason to think that that's the case.

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I mean, why aren't on metrics sort of equal in the eyes of the universe?

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So if we think about metric democracy. OK.

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Then I've hear I've drawn a metric which is very uneven, uneven and higgledy-piggledy, which nonetheless connects these two spacetime points.

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Those are manifold, which is higgledy piggledy worth bearing in mind.

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By the way, the path integral, even in the particle case, includes particle trajectories which are not smooth.

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It doesn't just include smooth trajectories. It includes all possible very, very smart only beauty trajectories.

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So there's one. There's another. There's another.

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And the Parthenon Group will be the sum over all of these potentially very higgledy-piggledy,

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these bases which have the character had the sole common characteristic that

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they have these two events separated by by a fixed a fixed proper distance.

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Okay, but then if you have metric democracy, you should worry about.

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Well, shouldn't one. Why would you have? Why wouldn't you have topology democracy?

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Why should you just consider spaces of one particular topology?

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Surely you should allow the topology to change as well.

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So once you allow the topology to change, make all kinds of complicated things can appear.

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So you can get worm host, so little branches of the universe which split off and then join, read, join later.

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For example, you can get breeks, splits, big joints.

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And so there's a huge number of possibilities.

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And this is really very tough, because if you look at four dimensional manifolds, you're trying to do is Inforum attentions.

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They're only partially tortes charted territory. In fact, this is an extremely hard problem.

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So to finish with what we're going to do is just look at the right toy model

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toy model to give you a very simple flavour of how these things might work.

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So what you're supposed to do is you're supposed to consider a graph of equilateral triangles.

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And you imagine that the minimum distance between two points on your triangulation, that's denoted in red on this on this picture,

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which is just three edges, you imagine you imagine that this is actually the Judi's existence.

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Okay. And then then you look at the configurations of triangles you've ever three-fold vertex and you make all of these triangles equilateral, right?

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Then you can't put this on a plane. It sticks up a tetrahedron.

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And that's that. So that says approximation of a place of positive curvature.

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If you have exactly six of them joined, on the other hand, then you can put them flat on a plane.

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That's a place of zero curvature. If you have seven, then you get a place of negative curvature.

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Because if you put light in the drawing and if you put five of them, you make a flap.

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Then the other two were the last two will stick up like this.

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OK, so the only idea is. Firstly, fixed its apology from an IT case.

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So, for example, you draw these graphs on a sphere,

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then you sum over all possible triangulations and because something over all possible triangulations sums overall possible Judi's distances.

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That's a bit like integrating overall possible metrics. OK.

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So then you sum. And then finally you sum over the Jenice.

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So the end result is that the amplitude in this case is a a sum over triangulations or something, which is a sum over Janice.

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So the simplest one is a sphere. The next most complicated one is a Taurus.

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Then you got a double Taurus.

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And so you just have these two points, which you keep a fixed graph distance fixed Gertie's existence as denoted by the by the red line.

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So this is actually a toy model which has which has caused.

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Which has given great entertainment to theoretical physicists so over a long time.

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And to this day, contained contains interesting stuff which is not possible, not properly understood, but it causes basically two dimensional.

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It's not four dimensional.

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And the reason why it works is because we understand geometry in two dimensions and manifolds in two dimensions and topology in two dimensions.

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Estrela. So what do our universities look like if we do that?

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Well, the most interesting characteristic is that if you if you ask the question, well, how much how much universe,

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how much universe is so it, how much universe is there in a volume which is inside a fixed distance as zero like.

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And the answer is it S02 the fourth. But if you're on a flat space, it would be a zero O squared.

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So actually, the spaces that you get in this model are a long, long way from being a flat, two dimensional space.

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And that's really the message of this. Once you let quantum gravity out of the bag, all right,

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then it is very naive to expect that you will get spaces which are nice and simple and flat and look very friendly.

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And in fact, these extraordinary, non-trivial that the quantum universe is so very four dimensional, both locally and globally.

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And this is absolutely something that we will have to have to understand when we understand properly understand quantum gravity and in a complete way.

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So we'll leave it there. Thank you very much.