Hello again, everybody. Let's hope, but the information technology stands up a second time.
So I against the explain a little bit about how we tried to develop a theory of quantum gravity and why it's difficult.
So let's start with particle mechanics, something we're all familiar with.
So the typical problem in particle mechanics is you have a mass moving in a in a potential and it sorry,
it starts at some place some time, initial time t zero.
It starts the place. Xterra has a momentum p0 and then it moves in the potential and sometime follows a path through top.
Here follows a path and ends up at another place with a different momentum.
Later, a later time. Now, classically, when H bar is equal to zero.
We can know both the position and the momentum.
But quantum mechanically, of course, this is not this is not allowed.
So the quantum mechanical situation is just a little bit different.
So. At the initial time, we can't specify the momentum anymore if we specify the position and the final time.
If we measure the position, then we can't measure the momentum. So if we work in terms of positions,
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.
And to end up at a later time T one. To place X one.
And the the path which follows between these two events, if you like, it, is not totally determined.
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,
which is slightly different from what we first teach undergraduates in terms of solving Schrodinger equations and so on.
The final particle gives you a sort of direct representation of the amplitude for this
process in terms of the paths which take you from the first event to to the final event.
And while the footballer what the pathological says is that I'm sorry about this is that
you should you should sum over all of the intermediate poles and each and get some weight,
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.
So instead of having a single classical path, we have a sum over all possible parts.
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.
Now, out of this, you can actually recover classical physics because if you take a bar to zero,
then the integral has to be dominated by the stationary point of of the integral and in the exponential.
So if you have eh eh eh eh classical.
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.
And this is something that actually you did when you calculated equations of motion using
the variation of principle in all the principle of least action in classical mechanics.
And if you go through this calculation, it's just a very short number of steps.
We won't do it in detail here, but you can pick it up from the slides afterwards.
You end up with a classical equation of motion, which is the AirMax double dot.
Is minder's degrading of the potential. Which is. Which is.
Which is Newton's law.
So this Pieman path integral formulation is completely consistent with what you know about classical physics and the ball goes to zero limit.
Now we can use this formulation now to move on and try to understand something about field theory.
So now it's the archetypal quantum field theory is quantum electro.
That's the theory of electrons and positrons interacting with the electromagnetic field.
And here the situation is slightly different. Now, space and time are our framework within within which the the action takes place.
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,
which is a function of X. There are also some fields which describe the matter, the electrons and positrons.
And then you let this system evolve and some time t one later the field has evolved,
the electric field is evolved to E one and the matter fields have evolved to something which I'd call Cybele.
So, so now. And of course the interschool electromagnetism or classical we can mechanics.
What you would do is you get some bunch of differential equations which you would solve,
which would tell you how how you how the field's evolved over time.
But instead in quantum mechanics, what we have to do is calculate this amplitude for starting with a field
configuration at T zero and ending up with a different field configuration at at T1.
So that's the basic quantum mechanical object in a can in a quantum field theory now.
So. So I'm not just. So remind you about it. About.
About something to do with electromagnets, which is that you'll recall that you write in the magnetic field,
which is a measurable quantity as the kernel of the facts, a potential which is Vector A.
And you can write the electric field as mine as the gradient of the potential now.
Actually, there's another term there, which is the time derivative or of the fact of potential.
And this can all be encapsulated in one object, which is called field strength tensor,
which is if you knew and SMU New has an expression in terms of the four vector potential.
Now, the important thing to realise is Acme New contains E fields and B fields and it something which is directly measurable.
Whereas actually the full vector potential, you can't measure it, you can't measure it directly.
All of these things have four components because there's a time and three space components.
If we're in regular four dimensional, four dimensional space time.
So that there's a desert and they these things behave well under under Lorentz transformation.
So they would transforming and they're not what's known as covariance only underlines transformations.
So it's it that the easy objects to. Right. Lorenz in various theories from.
They have this property, that unengaging variance, so you can shift the vector potential by a total derivative.
So that means different Teemu means deep differentiating with respect to each X.
And what happens is F Munir doesn't change. So the physics doesn't change.
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.
And so now we integrate over intermediate field configurations.
And again, the some face factor, which is now a quite complicated looking action.
But it's got bits in it which contain the electric and magnetic fields.
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.
This expression, if you if you compute what happens when each ball goes to zero,
you do this stationary point calculation again is rather more complicated than in in the particle case.
But actually, it gives you Maxwell's equations. So.
So the in homogeneous Maxwell's equation is the team you deem you have, if you knew, is equal to the current.
So the the charge and the and the current are sources of electromagnetic field because that's what happens if H.
Baatar goes to zero. If they HBO is not zero, then this gives us Quantum at the famous quantum electrodynamics.
And in principle, you can start from this. This formula here in the further the final path integral,
you can compute any amplitude you like in terms of the electric charge, the mass or of the of the matter fields.
The photon, of course, is massless and Planck's constant in practise.
Of course, it's very hard work and there's a whole industry which has done this so over many years.
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.
I've just called them and be here and you do a quantum electrodynamics calculation.
Then there's one extra element that you have to feed into the calculation, which is some ultra, which is some scale capital lambda.
And the reason for that is that you find that there are ultraviolet divergences in the calculations that you do at your feet.
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.
And for the electromagnetic fine structure, constant output alpha.
Even an H ball, by the way, is kind of. You defy a modern metrology age bar is defined to be a certain value.
And then you can take these numbers m an alpha and you feed them into another QED calculation.
Put the scale in again what you may not. Where did you predict the outcome for another process.
C. And the interesting thing is that actually, when when you when you've done this,
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.
And this scale parameter lambda that you put in and the intermediate steps disappears.
And that's because QED is a renormalise zable theory.
So it has this remarkable property that it predicts unambiguous relationships between measure rouble's and an expansion in Alpha Electromagnetic,
which is roughly 100, well over 137. So it's a fairly small number.
This works spectacularly well. And probably the best demonstration of this is the anomalous magnetic dipole moment of the electron.
So according to Dirac's relativistic wave equation, the magnetic dipole ratio of the electron is exactly two.
So the magnetic dipole moment, the electron is likely to be amongst photons,
but there's a small deviation from the this which is generated by by quantum mechanical effects.
And you can measure this small deviation. And this is the result that you get.
You can see it's spectacularly accurate. The number 28 is the error in the last two digits of the result.
Seventy three. And you can also calculate you have to work very hard to calculate to this degree of accuracy.
And these are calculations which have only been developed over many years.
But you can see that the agreement is spectacularly good.
So so quantum electrodynamics is a very powerful theory and it's a very accurate theory.
And it works. It works extremely well. It's one of these are some of the most accurate measurements and calculations of the main.
So that's field theory in a nutshell. Never can tell about gravity, we better remind ourselves a little bit about general relativity.
Already heard some of this from from John and in his talk.
So the basic point is he is here that what one of them was the really important ingredients.
Well, we have a space M which has some fixed topology for a minute. And on that space, there is a metric.
And so so the the the degree of freedom which really matters is the distance between points.
So you you take a point X mbewe and a point X bew plus dxp, which is quite close to it.
And, and what matters is what's the real distance between these two points or the proper distance.
If we're talking rigourously and that's determined by the metric which is this this object.
Jemmy, you and Jamie knew is in general dependent on where you are in the manifold.
Now, some examples are the Euclidian plane is the easiest example, and that's flat.
And you know that the DSC ad is just the X squared plus dy Y squared.
Another example is the surface of the sphere, which is most easily written down in polar coordinates.
And and then there's a piece which is Defeater squared in a piece, which is Defi Square.
But you see the component of D5 squared is not constant.
OK. And K is basically the radius of a sphere. And then a final example is Binkowski Space Time.
OK. So now we're really talking that now we're really talking about a proper, proper interval.
And that again, is duty's square by the X squared.
And the metric for Mankowski space time is often given the name item you knew, which we will come back to in a minute.
Now, the point about this is that we can we can change coordinate systems.
Right. For example,
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,
change. But the proper distance, which is actually just a Judi's existence in this case between them, doesn't.
So so there is actually a hidden in all of this, a reprivatization of variance, which is that you can change the coordinates.
And when you change the code, that's a metric function changes. But the distances between points are independent of that reprivatization.
Now, proper distance is actually not the only independent frame, character frame, independent characteristic of of the space and the match.
Right. There's another frame, independent characteristic, which is the intrinsic curvature.
So that's illustrated in the end, the picture on on the right hand side.
You can you can see that if a space is curved so that there are two.
Right. Independent principal wages of radii of curvature.
So just think of a surface. And this is the curvature of the space is it is again, independent of the coldness.
So, for example, if you are if you are looking on the surface of the O of the sphere.
Right. It doesn't matter what sorts of coordinates you use to describe that surface, it is still curved.
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.
It's a product of the inverse principle radii.
Curvature of Binkowski space time is again zero. Binkowski space time is is flat now.
So. So in this, G.M. is the dynamical degree of freedom.
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
degree of freedom and the curvature of space time is generated by mass and energy.
And that leads to the Einsteins equations, which which we have written down here.
And the the important bit is the on the left hand side.
We have all we have the curvature on the right hand side.
We have the stuff that generates the curvature, which is the stress energy tense.
And then we've also written down this other term. This is the cosmological term which Einstein wasn't very happy about.
But but which there is no no reason in general not to have that.
And one of one of the puzzles of nature is why the cosmological term is rather small.
Why is this cosmological constant lambda? Sorry, this is a different quantity from our previous Cut-Off previous scale.
Why is the cosmological constant so small?
Is one of one of the mysteries which modern physics still wrestles with.
A capital G is just Newton's constant.
So this is very nice, neat way of writing things which are actually non-linear, partial differential equations for the metric.
Basically, all you have to do is specify the initial conditions and solve.
Right now, of course, that's easier said than done in practise for complicated situations.
But nonetheless, you get excellent agreement with observation and with experiment.
So how we get so then how are we going to quantised gravity?
What what what must be the basic ingredients? Well,
the first basic ingredient has to be the physical amplitudes that you talk about must be reprivatization
invariant if they're not read traumatisation invariant and they just turn out to be zero.
And so they don't actually tell you anything. Okay, so this is a very easy, reprivatization,
invariant physical amplitude is basically an amplitude that the exact universe exists for proper time as.
It starts at some event p0 and ends up at some event P1 after proper time.
And then the forum appal thing to grow.
You might speculate, which gives us these Ankit amplitude is basically a sum overall, such overall, such metrics.
So here I've written down a find important to grow where I sum overall the old metrics which have this characteristic I saw.
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.
Now, this part this part that we've written down here, this is called the Einstein Hilbert action.
If you take a bar to zero, the stationary point just this gives you about Eisenstein's equations,
which we wrote down or on on on the previous previous slide.
So, again, the pathological gives us as each bar goes to zero, limit the classical field equations, these bets here.
If I if I just integrate. If I just some over matter configurations and I just use the maths apart in the phase.
This basically gives me quantum field theory and a fixed background spacetime,
which which is part of what John was talking about in his lecture, where we have quantum electrons in quantum photons.
But we have a classical background, spacetime.
But that's a very easy way of writing down something that could easily be very complicated.
So what does this integral over the metrics mean?
What what what do we mean when we are going to sum over all these interpolating metrics between two space time events?
So we're going to look at the two to two attempts at doing this.
So the first attempt is to basically copy QED. And you do that with a combined with a perturbation expansion in Newton's constant.
And you might think that, well, gravity is weak.
So so perhaps the way to think about this is that basically there is a sort of small wrinkles in in space time.
And although the metric is a function of position, actually what it is, is it is a sum of two bits.
It's the sum of something which is completely flat, which is just I mean, calfskin metric.
And then there's H meunière who which gives this these small wrinkles, the oscillations.
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.
It goes in practise. It's very small. In this case.
OK. So you drop.
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.
This does tells you that the matter is the source of these small fluctuations.
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.
So that's some thought that that's one. That's what that's what happens classically if you try.
If you do this quantum mechanically.
And so you're now viewing this as a quantum field theory, then you need an analogue of the fine structure constant.
So in this little quantum general relativity theory we're talking about.
We need to have dimensionless fine structure constant. And unfortunately, Newton's constant has dimension.
So the dimensionless object is actually Newton's constant times ours,
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,
you said equal of one and all these, these calculations. So it's that scale is the mass of the electron.
Then this alpha for faja general relativity is about ten to the minus forty six.
So he's really. The small and the certainly weak.
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,
then Alpha Alpha will will be about one.
Now, that seems like a very large energy scale, but it's not really a very light Jannuzi scale,
10 to the 22 times the mass of the electron is only about 20 at 10.
The mine is eight kilograms. So this looks like there might be a problem.
And in fact, there is a problem. The problem is that if you start taking measurable.
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,
what you find is that the parameters cascade.
And you can never take the outcomes of a number of measurements and feed them into a calculation, unpredicted other measurement.
And this is this is basically the statement that this theory we're talking about is not renormalise.
Now, actually proving it's not renormalise is a tour de force.
And the famous theoretical physicist Envelopment who died this week was an important figure in that endeavour.
But she is not totally dead. You can keep your scale and just work in a regime very small.
And that's called defective field theory.
And you can learn a lot about the relationship between quantum versions of gravity and classical gravity in that way.
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.
Small scale in the other wave of other route.
Is that string theory actually contain this field? H we do this.
This is in fact as so kind of like a spin to field.
And it also contains a consistent minimum distance scale. And it is a way of constructing a consistent quantum field theory of gravity.
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.
It contains lots of other degrees of freedom as well. So so the the question is, can we do something else?
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.
So the first word looking at it, we just looked at fluctuations around them and kasky metric.
But there isn't really any particularly good reason to think that that's the case.
I mean, why aren't on metrics sort of equal in the eyes of the universe?
So if we think about metric democracy. OK.
Then I've hear I've drawn a metric which is very uneven, uneven and higgledy-piggledy, which nonetheless connects these two spacetime points.
Those are manifold, which is higgledy piggledy worth bearing in mind.
By the way, the path integral, even in the particle case, includes particle trajectories which are not smooth.
It doesn't just include smooth trajectories. It includes all possible very, very smart only beauty trajectories.
So there's one. There's another. There's another.
And the Parthenon Group will be the sum over all of these potentially very higgledy-piggledy,
these bases which have the character had the sole common characteristic that
they have these two events separated by by a fixed a fixed proper distance.
Okay, but then if you have metric democracy, you should worry about.
Well, shouldn't one. Why would you have? Why wouldn't you have topology democracy?
Why should you just consider spaces of one particular topology?
Surely you should allow the topology to change as well.
So once you allow the topology to change, make all kinds of complicated things can appear.
So you can get worm host, so little branches of the universe which split off and then join, read, join later.
For example, you can get breeks, splits, big joints.
And so there's a huge number of possibilities.
And this is really very tough, because if you look at four dimensional manifolds, you're trying to do is Inforum attentions.
They're only partially tortes charted territory. In fact, this is an extremely hard problem.
So to finish with what we're going to do is just look at the right toy model
toy model to give you a very simple flavour of how these things might work.
So what you're supposed to do is you're supposed to consider a graph of equilateral triangles.
And you imagine that the minimum distance between two points on your triangulation, that's denoted in red on this on this picture,
which is just three edges, you imagine you imagine that this is actually the Judi's existence.
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?
Then you can't put this on a plane. It sticks up a tetrahedron.
And that's that. So that says approximation of a place of positive curvature.
If you have exactly six of them joined, on the other hand, then you can put them flat on a plane.
That's a place of zero curvature. If you have seven, then you get a place of negative curvature.
Because if you put light in the drawing and if you put five of them, you make a flap.
Then the other two were the last two will stick up like this.
OK, so the only idea is. Firstly, fixed its apology from an IT case.
So, for example, you draw these graphs on a sphere,
then you sum over all possible triangulations and because something over all possible triangulations sums overall possible Judi's distances.
That's a bit like integrating overall possible metrics. OK.
So then you sum. And then finally you sum over the Jenice.
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.
So the simplest one is a sphere. The next most complicated one is a Taurus.
Then you got a double Taurus.
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.
So this is actually a toy model which has which has caused.
Which has given great entertainment to theoretical physicists so over a long time.
And to this day, contained contains interesting stuff which is not possible, not properly understood, but it causes basically two dimensional.
It's not four dimensional.
And the reason why it works is because we understand geometry in two dimensions and manifolds in two dimensions and topology in two dimensions.
Estrela. So what do our universities look like if we do that?
Well, the most interesting characteristic is that if you if you ask the question, well, how much how much universe,
how much universe is so it, how much universe is there in a volume which is inside a fixed distance as zero like.
And the answer is it S02 the fourth. But if you're on a flat space, it would be a zero O squared.
So actually, the spaces that you get in this model are a long, long way from being a flat, two dimensional space.
And that's really the message of this. Once you let quantum gravity out of the bag, all right,
then it is very naive to expect that you will get spaces which are nice and simple and flat and look very friendly.
And in fact, these extraordinary, non-trivial that the quantum universe is so very four dimensional, both locally and globally.
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.
So we'll leave it there. Thank you very much.