So thanks very much. It's a great honour to give this ninth Danny Schama lecture.
So my thanks to Lidia, to the Sharma family, to the organisers, Joe and John, to All Souls College for sponsoring this lecture series.
So as Joe says, my Denis was my dphil supervisor back in the 1970s something.
And yeah, as, as he says for the last gosh, 35 years, I think I've got my maths right, my trajectory has taken a rather different uh, uh, direction.
But what I want to do today is to try to look, uh, retrospectively,
I guess at some of the issues which uh, in the seventies for sure excited Denis and excited me.
Two issues to do with the marriage of general relativity and quantum mechanics, issues to do with the large scale structure of the universe.
Uh, issues to do with the very sort of meaning and interpretation of quantum mechanics.
And I say retrospectively, what I'm going to try to do here is, well, at least ask the question whether some of the mathematical concepts,
which I guess have become pretty much second nature to me in the last, whatever it is,
30 odd years, things to do with, uh, with nonlinear dynamics, with chaos, with fractal attractors and fractal geometry,
things which are very central if one is looking at uh, uh, the climate system or the weather.
But to ask whether these mathematical concepts might actually provide new insights
into these types of problems which which certainly were problems that excited Dennis.
Of course, these are problems still very much alive today.
I think we still don't have any unanimity about how to bring general relativity and quantum mechanics together.
And we still have fundamental problems trying to understand basic quantum mechanics.
So this may be this may seem a tall order,
but I'm going to try to persuade you at least that there are some new ways of
thinking about these old problems that nonlinear dynamics might bring to the table.
Well, at least you can make your own judgement about that by the end of the lecture.
So let me start. The first of my protagonists is Ed Lorenz, protagonist in the title of My Talk.
He's a meteorologist from MIT. Someday, actually, I got to know pretty well.
He died, unfortunately, about three years ago now, I think.
Very famous, of course, outside meteorology, famous around the world for his his 1963 paper on deterministic non periodic flow.
If I had more time, I'd tell you a bit more about it. But I'm sure many of you are familiar with the equations which came out of that paper.
Three coupled ordinary differential equations nonlinear equations which exhibited this concept we now call sensitive dependence on initial conditions.
So here's two animations of two initial conditions, which are almost but not quite identical.
They track each other for a while, and then they they diverge from each other.
And this is often called the butterfly effect, although interestingly, this is actually a bit of a misnomer.
What at Lorenz meant by the butterfly effect is actually somewhat different to this.
But again, that's another title of another lecture, which I won't go into.
Okay. Now, you might well ask if you know, your history was actually Lorenz,
the first person to discover a sensitive dependence of evolution on initial conditions.
And in fact, in some sense, he wasn't. At least 50 years earlier on Poincaré studying the gravitational three-body problem.
It also discovered this phenomenon essentially of what we now call chaos.
So the question you might ask is, well, what's special?
What what's special that Lorenz brought to the table that Poincaré had not already brought to the table?
And in a sense,
this is very much the theme of my talk that what Lorenz brought to the table in the particular equations that he looked at which were not actually
present in the gravitational three-body equations was this type of geometry in the state space of the equations which the system evolved towards.
So take any initial state X, Y and z a time zero evolve it,
and you see eventually it starts no matter where you start in the state space of X, Y and Z, it attracts towards this geometry.
Laurence knew that this geometry had to have zero dimensions.
By looking at the structure of the differential equations, he knew that this Aston toxic attracting set,
as it's called, has to have a zero dimension, 000 volume.
But he didn't. You know, he was struggling for a long time.
It's very interesting reading his papers and his notes and things, how he struggled to try to understand what this was.
He realised it couldn't be a point because the system didn't settle down to a steady state.
It wasn't a circle. The system didn't wasn't periodic.
The title of his paper was non periodic deterministic motion.
It wasn't a surface. So what was it? And he agonised about this.
And in fact, I think this is one of the real pieces of genius of Lawrence to realise that this geometry he was looking at,
which came out of these differential equations, was actually a fractal a fractal structure.
And what I want to talk about is that fractal structures.
If we focus on the fractal structure associated with these equations rather than the differential equations per say,
we can discover some remarkable connections into deep parts of 20th century mathematics.
So I'm going to just highlight very briefly a couple of those things that you would never,
I think, have gleaned just by looking at the differential equations as such.
The reason I'm telling you this is I want to make the point that these fractal geometries, you know,
often one sees fractals on the front covers of books or conference flyers and things just as a kind of a sexy piece of of geometry.
But no, they're much more than this. They have very profound links into into deep areas of mathematics.
So just a couple of examples.
The question is how how might one characterise this fractal attractor if one didn't have the differential equations to use?
And one technique that people mathematicians use to try to characterise the
attractor are looking at periodic orbits that lie in some sense close to almost,
you could say embedded in the attractor. They lie close to within the body of the attractor.
So there are many, in fact periodic but unstable periodic orbits.
These actually repeat each other.
You can characterise these or describe these periodic orbits by a technique called symbolic dynamics or symbolic representation.
So what I'm just there is you just partition the attractor into two lobes called the left hand lobe out on the right hand low bar.
And then these particular periodic or the period in general periodic orbits that one looks at, one can specify a sum symbolic bit string.
So for example, this corresponds to a periodic orbit that starts on the left hand lobe, moves to the right, then goes to the left,
then goes to the right, then goes to the left, then goes stays on the left once more and then keeps repeating itself.
And this turns out to be topologically equivalent to the, say, communal garden if that means something to you.
Trefoil Not. It's one of the basic knots in knot theory and using this kind of jones polynomial type of language,
one can classify these periodic orbits or alternatively these symbolic strings as knots.
The knots of the of the periodic orbits.
One, I think, quite remarkable result which a French mathematician and geese showed in 2000.
And if you have time to write down, if you're interested at all in this, there's a fantastic online, uh,
kind of, uh, discussion of this result with some fantastic movies showing how these knots evolved and so on.
The result is that these Lorenz knots are entirely equivalent to what are called modular knots.
Now, I'm not going to tell you what a modular nut is because don't have time.
But suffice to say that it's related to properties of the modular group and one can think of the modular
group as essentially the group of two by two matrices with integer elements and with unit determinant.
So here's an element of the modular group,
and it turns out it also can be written in as sort of a string of symbols where these symbols now define elemental two by two matrices.
And I guess the work of DS is basically to show an equivalence between these symbolic strings for the modular elements of
the modular group and the symbolic strings of the periodic orbits of the of the Lorenz associated with the Lorenz attractor.
Now, if you go into the mathematics, if you go into this this weblink,
you'll see very quickly that the sort of mathematics guess uses to prove this theorem
is the sort of mathematics that number theorists would feel very at home with.
He talks about lattices on the org and plane.
He talks about via Strauss elliptic functions associated with those lattices.
He talks about the Eisenstein series again associated with those lattices.
And then one sort of moves into the field of elliptic curves and modular forms.
And this is the area that that Andrew Wiles unified in his proof of Fermat's Last Theorem.
So from a set of differential equations, one suddenly finds via the structure of the geometry which these equations produce.
One is moving into areas of quite deep number theory, which you would never have guessed,
I think to just look at the differential equations themselves.
Another problem you might want to think about is this.
If I, uh, if you were to give me, let's say, a point in this three dimensional state space of the Lawrence equations,
and you asked me, is there an algorithm for determining whether that point lies on this, on this Lawrence fractal attractor?
The answer is there is no algorithm. It's actually it's actually a non if you like.
It's a problem that can't be solved by finite algorithms.
And you can imagine there's not sort of a totally unreasonable to imagine why that should be,
because even if one knew one of the points on the attractor, it might take you.
No matter how far you integrated the equations, you might still not have reached the point that you are given.
And so you would never have a procedure for deciding in finite time whether your given point was, was, was belonging to the attractor.
Now, this was proven rigorously in in this book by Blue Mittal, which included the famous Steve Smail, incidentally,
result being basically that's what are called halting sets, must have integral house stored dimension.
And we know that these fractal attractors are characterised by fractional dimension.
And indeed, one can say one can take many of the classic problems in computing science that are known not to be solvable
by algorithm and show that they have an equivalence in terms of of of a of a fractal geometric problem.
If people have heard of the post correspondence problem, this is one of the classic problems that can't be solved by algorithms.
One can show it's equivalent to the question of asking, what does a point lie on the attractor?
But does a line intersect the attractor of a of a of a chaotic dynamical system?
So you can see we're going into the territory here of the girdle incompleteness
theorem and the corresponding sharing non compute ability type of issue.
So just to summarise then, so far, what I'm trying to tell you is that here are these Lorentz differential equations.
I think Newton probably could have understood in principle what these equations were saying.
After all, he did discover the calculus, but he would never have guessed.
I am sure that he would never have guessed that these equations could generate this amazingly rich and deep type of geometry.
And it's through that geometry that we see links into quite into some of the classic problems of 20th century mathematics,
whether it's Wiles, proof of Fermat's Last Theorem or the Girdle Theorem or the Turing Non Compute ability theorems.
So the reason I'm saying this is that it's by focusing on this geometry that one gets these links into these deep areas of maths.
So what I want to do then,
having convinced you that this is these these types of fractal geometries are really serious topics and worthy of serious discussion.
I want to now switch to these three people at the bottom of the of the figure for three more.
These are three, of course, two famous 20th century physicists, Schrodinger, Heisenberg and Dirac.
And ask the question, do this does this type of geometry provide us with any new insights into these deep problems of 20th century physics,
which are characterised by these three people, Schrödinger, Heisenberg and Dirac?
Now, at the level of differential equations,
you would have to say this is a barking mad concept that there might ever be any connection because Schrodinger's equation or
Heisenberg's form of the Schrödinger equation or Dirac's relativistic form of the shooting equation are all linear equations,
and this equation is manifestly nonlinear. It's got these terms X and Y and X, and so it's a nonlinear equation.
So how could there ever be any connection at all? So that's the kind of a many I'm sure you've read books who make this point, you know,
as the unpredictability of quantum theory have anything to do with unpredictability of chaos.
And usually people say, no, they're not, because one is a linear century, a linear problem, the other is nonlinear.
They don't have anything to do with each other. I want to make the point that I don't think this is correct.
I think there are some profound links, but they need to be approached by looking at this intermediate geometry.
This is the key. So this is sort of the basis of the talk to some extent.
I thought I would start with a few quotes before we move on any further.
This is from one of Hawking's popular books where he sets out, I guess, the standard model, if you like, of of quantum interpretation.
The standard interpretation which says there really is no connection between quantum the
unpredictability of quantum physics and the unpredictability of nonlinear dynamical systems.
So according to quantum physics, says Hawking, no matter how much information we obtain or how powerful our computing abilities,
the outcomes of physical processes cannot be predicted with certainty because they're not determined with certainty.
So is this fundamental issue that there is there is no determinism in quantum physics.
That's the standard view. However, I want to contrast that with two other comments from also eminent, very eminent scientist.
So here's Dirac himself, one of the the grandfathers father figures of quantum theory.
One can always hope, he says,
that there will be future developments which will lead to a drastically different theory from the present quantum mechanical theory,
and for which there may be a partial return of determinism.
So this, uh, this from the one of the guys that was the Founding Fathers, uh, at least suggesting that the, the door is not yet closed.
And this is from my third protagonist of my title, Roger Penrose,
whose work will feature more and more as we go through the talk with an even, I think, more positive statement.
Still, he says, it seems quite plausible that the correct theory of quantum gravity might be a deterministic but non computable theory.
Now this non computable action is interesting because we've already seen a maybe is this is this just a spurious link or is there something deep here?
We've already seen these fractal geometries are non computable.
They membership of that set cannot be determined by algorithm.
So is that is that just a coincidence or is there something deep link here?
Those three quotes were sort of deliberately taken from three people that link very closely to Dennis.
Stephen was was one of Dennis's students.
Dirac was Dennis's, uh, supervisor.
And Roger was somebody that Denis inspired to move into mathematical physics from an earlier career career in pure mathematics.
So Denis was very instrumental, pivotal in bringing Roger into the relativity community.
Dennis has had an enormous number of very eminent students, and a few perhaps not quite so eminent students.
Um, but, uh, I find that it's almost impossible not to be sort of proud of my linkage to Dennis because of this,
an ability to talk about my academic brother,
Stephen Hawking, my academic father, uh, Roger Penrose, my academic grandfather, Paul Dirac, you know, my academic uncle Roger Penrose.
So that's I couldn't resist putting that slide up.
Okay. Let's just move back then to the Lorenz system.
And this is a slide, actually, I often show in my climate talks or weather type talks,
because it shows how the predictability of the system is a function of initial conditions.
So some parts of the attractor are extremely stable and predictable.
So if one imagines this as some little error bowl, perhaps a cup through an error ball,
the error actually doesn't grow at all as you propagate as you as you evolve the system forward in time.
Other initial states are more unpredictable and others yet more unpredictable.
Still, the fact that the predictability varies around the attractor is a direct result of the nonlinear arity of the equations.
So if these these are Lorenz equations, we can linear ize them.
So look at the growth of small perturbations and the operator is now the derivative of F with respect to x.
So if F is at least quadratic in x, then df by the x is at least linear in x i.e. this operator depends on the underlying state,
and that's what we're seeing here. So somewhere, you know, some weather forecasts as Michael Fish found to his costs can be very unpredictable.
And in fact, in other cases, we can make very potentially quite long range forecasts with confidence.
That's the sort of basis behind much of weather prediction.
But I'm not going to go down that route. What I want to talk about is what equation is actually describing this evolution of probability.
I mean, for example, one could take this point here and say this supply,
this little ring defines a 90% probability that the true state lies within that ring.
Then as you map this forward in time, you get to some future forecast time and there's a 90% probability that the true state lies in that ring.
The forecast time or this would be a sort of a squashed banana, if you like, if you looked at it properly.
There's a 90% probability it lies in that squashed banana. What's the equation that's describing that evolution of probability?
It's this thing. It's called the Louisville equation. This is written in its general, most general form.
By the way, some people think the Louisville equation only works for Hamiltonian systems.
This is not true. Louisville Equation. Conservation of probability works for Hamiltonian and non Hamiltonian systems.
This is a bit like the mass conservation equation for a compressed, compressible fluid.
If it was incompressible, you could move the Volt. But if it's compressible, the V has to be in here.
Now, this is not this is not mass density and volume and velocity in physical space.
This is probability and velocity and state space, but it's still a conservation equation, nevertheless, conservation of probability.
Now, one thing you can note if you look at this equation is it's linear and in probability it's linear in row.
And this is an important point I want to make. If I gave you that equation.
You would not be able to deduce because it is linear.
It cannot be underpinned by some underlying nonlinear dynamic.
The linearity of this equation says nothing about the weather.
The dynamics that generates that probability is linear or nonlinear.
And in fact, in this system, as I've just pointed out, the system is nonlinear.
So the linearity of the alluvial equation is really just a statement of the fact that the probabilities are conserved as the flow evolves.
Now, if we have a Hamiltonian system, we can write the Louisville equation in this form using Poisson brackets I'm sure familiar to people,
or if indeed if we had a dynamical system which was approximated, approximated a Hamiltonian system, this would be a good approximation.
So I want to contrast this with the direct form, at least of the of the Schrodinger equation.
Which I've written here. And the first point I want to make is how remarkably similar it is formally.
It has a row, a d row by d t, it has a sort of an anti symmetric bracket, it has hamiltonians and so on.
It seems to be some formal close similarity with the classical legal equation.
And I'm sure it's because of that that Dirac perhaps was inspired to make that comment that,
you know, just just as this equation can be underpinned by deterministic dynamics.
So maybe also this equation might be underpinned by deterministic dynamics.
And of course, the point that this this equation is linear says nothing at all about whether
the underpinning dynamics is linear or nonlinear might just be non-linear.
Incidentally, just as an aside, I often find people who who look for nonlinear things that might nonlinear processes that might
come in to quantum mechanics sometimes add non-linear terms to the Schrodinger equation itself.
And I always find this an extremely misguided idea personally, because if this is just reflecting a conservation of probability,
you don't really want to tinker with it by adding non-linear terms.
If you want to look for nonlinearity, try and find the equation that underpins this type of conservation of probability.
Okay. Now of course I recognise that there are differences as well between the classical Louisville equation.
There's a square root of minus one. There's a Planck's constant.
And in fact, these things aren't just well, these aren't functions in state space operators in some Hilbert space.
And this is an operator commentator bracket, not a person bracket.
So there are similarities and there are differences. Do these differences matter?
Does the fact it is operating on a Hilbert space rather than a classical state space, does it matter?
Well, the answer, of course, is yes, it does matter.
And there are a number of no go theorems in quantum mechanics, which, uh, which tell us very much that it does matter.
And the most famous of these, I guess, is, is Bell's theorem, which in a nutshell says no physical theory based on local cores or hidden variables.
So you posit some deterministic system which operates deterministically.
The variables of the of the deterministic system may be hidden to you or hidden certainly hidden to quantum mechanics.
But Bell's Theorem says that one can never replicate the predictions of quantum mechanics.
It's based on this inequality, which I don't proceed.
Don't propose to describe in detail. I don't propose to describe at all.
So you either you either say from this that there is no such thing as deterministic reality,
or you say, well, if there is deterministic reality, it somehow violates principles of relativity.
There's something, to use Einstein's phrase,
some spooky action at a distance going on in if we want to try to underpin quantum mechanics with some underlying deterministic model.
That's the standard interpretation. Now I want to at this stage just talk about an idea which.
Bell himself recognised was a potential way out of the constrictions of his theorem.
I'll just say right from the beginning, it's it's an idea which he himself did not believe in.
And I don't think anybody in the in the community or almost nobody believes in.
So I'm not actually going to advocate this. You'll be pleased to know. But it's an important concept to get over in what I will talk about later on.
And actually at first sight,
it's a rather strange loophole or a escape from the constrictions of bells whom you might think to escape from the constrictions of battle serum.
You might want to somehow relax this notion of determinism and imagine things being more stochastic.
Maybe. But what Bell says actually is the opposite.
There is a way to escape the inference of super luminal speeds and spooky action at a distance, but it involves absolute determinism in the universe.
So going somehow even more deterministic. So what does that mean?
What does this word phrase? Absolute determinism. People now generally use the word super determinism.
So what is super determinism? What's the difference between determinism and super determinism?
Well, this at least, is my understanding of the difference between these words and the normal deterministic system.
You have maybe a set of differential equations describing the evolution of the system,
and you start off with some initial condition and then the equation matches into the future.
Now here the equations are deterministic, the dynamics is fixed,
but you can imagine perhaps you've got a free choice to make in what the initial conditions are,
so you don't have to start from a specific initial condition. In the Lawrence model.
In principle, you can start the equations off from any point in space you like.
So you have a free choice. And to some extent in this idea, the the the initial conditions are independent of the dynamics.
So go anywhere you like.
But when you made that choice, then you run the dynamics and the dynamics are fixed and that provides you with a future state.
The super deterministic idea is where somehow not only is the dynamics fixed, but your initial state is fixed.
Now, how this should be, how your initial state should be fixed?
What determines that your initial state is fixed is another matter.
We'll come on to try to discuss some of these issues.
But just say for the moment, there is some reason, some principle that your initial state, there's no choice at all.
Then absolutely everything is fixed. Your initial state's fixture, deterministic dynamics are fixed.
The reason this is relevant is that in the Bell Theorem, if you posit a so-called hidden variable model,
what that does in general is predict outcomes of of of physical experiments.
You might do so you might measure the spin of a particle and your hidden variable model will tell you for a given particle what the result is,
but only for experiments that you might actually do on that particle.
But for experiments that you might have done but didn't do.
So this is what philosophers would call counterfactual experiments.
So I actually measured the particle in the X direction, but I might have measured it in the x direction.
A standard hidden variable model will give you not only the prediction in the in the Z direction that you actually did.
It would also tell you give you a prediction of that.
Counterfactual experiment that you didn't actually do, but you somehow might have done in the extraction.
Now, if you posit then the world is super deterministic,
then obviously these so called counterfactual measurements become by definition they're not things that correspond to.
There's only that one trajectories. Everything's fixed. So there is no there are no counterfactual worlds to consider.
So it's a kind of almost a trivial way in which the Bell Theorem can be in some sense,
negated by disallowing these types of counterfactual experiments.
But the problem with it is, and this is the reason nobody actually believes this is a viable way in physics,
is that it seems to say something about the world that's just uncomfortably fine tuned, uncomfortably special.
So, for example, I might roll a dice and use that outcome to determine how I'm going to orientate my measuring apparatus.
So if I roll the dice and it's a five, that will determine some angle for the measuring apparatus.
Now, in the super deterministic world, the hidden variable will have to somehow be in harmony with the outcome of that dice in order that the in order
in order to get the correct quantum mechanical probabilities or in order to violate the bell inequalities.
But you might imagine well, just as I was about to roll that dice, I sneezed.
And then the dice actually ended up as a six rather than the five or tiny gust of air.
You know, slightly affected the dice as it was rolling and it ended up as a six.
Then that tiny perturbation that's the sneeze or the or the gust of air would
completely upset that harmony that is required to to to violate the bell inequalities.
So you can take all those little perturbations back to the beginning of time.
And the conundrum you're left with is that any almost infinitesimal perturbation, let's say, to the Big Bang,
would produce a world where this perfect harmony that you'd set up had been somehow destroyed.
So the problem for a super determinist is to say, well,
why should the initial conditions be precisely big bang initial conditions and not some just marginal sort of quasi random perturbation of Big Bang.
And I think all a super determinist could do would say, well, maybe God decided this.
This is on the big bang initial condition and not this other one.
Well, as theorists, we don't like you know, we like our theories to be self-contained.
We don't want external agents to somehow determine important parts of the theory.
So this is this is considered then a sort of an implausible reconciliation of of the Bell Theorem.
And so as things stand, people do indeed believe either.
We live in a world where there isn't any deterministic reality or if there is something,
something spooky action at a distance, something super luminal is happening.
Okay. I'm not satisfied with either of these explanations, so I.
This is why I've been going back and thinking and, as I say, trying to think about the things which are pretty much second nature to me over the years
and ask whether these provide some new ways of thinking about this rather old problem.
So we talked about the Lorenz attractor. At the heart of any fractal attractor is a cantor set.
This is perhaps the simplest of all fractals.
I'm sure most of you are familiar with this idea. You start with the real line between nought and one.
Think of that as your zeroth iterate of the cantor set.
Your first iterate will be throwing away the middle third.
Your second district will be throwing away the middle thirds of the two remaining pieces.
Your next district will be throwing the middle third of all the remaining pieces.
You carry on with those iterates and then you take the intersection of the iterates.
Overall, the all the iterates. And this is the cancel set.
Now you can define from that. A dynamical system which operates on that cancel set.
So this is a very simple dynamical system called an iterated function system.
In this example, there are just two functions which two different things to the state, the current state,
but basically the action of either F one or F two, which is to be considered as a kind of time evolution.
Operator keeps the state on this canter set.
By the way, the council set is another nice and very simple example about how number theory kind of comes into these geometries in this in this case,
in a very simple way, because one can represent a point on the cancel set by just writing a number, a fraction say between zero and one in base three.
And if the base three expansion of the number has no digit one in it but has digits zero and two that it's a point on the cancel set.
By definition, because you've thrown away all the bits where you'd have the digit one.
So now I'm going to consider a perturbation of a point on the on the on the cancel set.
And let's suppose for the sake of argument, my I have no knowledge of this cancel set when I do this perturbation.
So I'm going to take a very small perturbation, but in some sense it'll be random with respect to the real line itself.
So imagine I have no knowledge of the geometry or I choose to ignore the geometry of the cancel set in defining my, my, my small perturbation.
So here we are. I've chosen a what I'm calling a geometrically unconstrained perturbation at some.
It's a number that's very small, but it's been somehow chosen from that, from the real line itself.
And therefore, somewhere there may be lots of zeros to start with, because it's a small perturbation, but somewhere there'll be ones.
And because there are ones, as soon as I in general added on to my point on the capital set, I'll I'll move it off.
So here x prime is being perturbed off the or set.
Now what this is saying is that if you view the cantor set from the outside.
It looks incredibly special. Any tiny perturbation will take you off the capital set,
and indeed that is consistent with a cantor set having zero volume or zero measure relative to the real line.
It looks incredibly special. It looks very high, very finely tuned.
Any tiny perturbation will take you off the cantor set.
So this initial state x nought looks incredibly fine tuned when viewed from the outside.
I'm calling this, by the way, fractal determinism because it it's a type of determinism.
Which either looks deterministic or super deterministic, depending on how you look at it.
And when you look at it from the outside.
It looks a bit super deterministic because any tiny perturbation, uh, uh, destroys this balance associated with the system being on the cancel set.
On the other hand, if I am aware of the of this geometry when I construct these perturbations,
so I'm going to construct now perturbation that what I would call geometrically constrained perturbation,
which ensures that I stay on the cantor set when I add this perturbation.
It stays on the cantor set.
Now at first sight, you might think that these this these perturbations are much less numerous than the more random ones relative to the real line.
But the remarkable property of the Cantor set is that there are as many points in the Cantor set as there are actually on the real line itself.
There's an undeniable, infinite number of points on both.
And a way to see this is if you take a point on the on the rail line between zero and one and express it as a binary number just with zeros and ones.
And then right instead of the one replace each one with a digit two, then that becomes a point in base three on the cancel set.
So as a 1 to 1 correspondence to points on the real line and points on the on the cancel set.
So when you're inside the kantor set, it looks absolutely enormous.
It's it's as big as the real line ever was.
The aficionados of Doctor Who may liken the cancer set to the TARDIS.
If you look at it from the outside, it looks incredibly small.
But open the door and go in. And it's fantastically big and spacious.
This is now, of course, us. Colleagues at the time thought he was completely crazy and I think that drove him a little bit to despair.
But he was absolutely dead right.
And also, these things underpin all of these wonderful geometric attractors, which I talked about at the beginning, such as the Lorenz attractor.
We can do a similar thing for Laurent 63.
Take. Uh, take a point.
Uh, if we start on the attractor and evolve along trajectories of the Lorentz attractor, then we'll stay on the attractor.
There are many perturbations as a non de numeral,
infinite number of perturbations to the initial conditions which keep the initial conditions on the on the attractor.
So there's there's a there's a lot of freedom. So it looks from inside on the attractor, it looks deterministic.
You have a big choice of of degrees of freedom.
But on the other hand, if you do something dynamically unconstrained, you can take yourself off the attractor.
And a possible way of doing of taking yourself off is to is to take a perturbation which keeps,
say, one component unchanged and just perturbs the other two components.
This will take you off the attractor. Now, when I was thinking about this, this struck me as a very.
In my mind, it was kind of reminiscent of those counterfactual experiments where you you think, here's a particle.
I actually measure it in this way. I could have measured it this way.
So when you say that you're perturbing, the sort of conceptual experiment you're thinking about is one where the particle stays fixed.
You do nothing to the particle, but you somehow perturb your measuring system so that instead of the actual direction,
you perturb some degrees of freedom to make the direction slightly different.
And you say, well, what would have happened there? So let's just imagine the toy, the Lorenz model now is a toy universe.
And just think about what this type of perturbation might mean.
It might mean perturbing two degrees of freedom, which correspond to the orientation of your measuring apparatus,
keeping fixed the degree of freedom associated with the hidden variable associated with the particle you're about to measure.
So this would then be a non reasonable, I would say,
physically unreasonable type of perturbation if you believe the universe was constrained to this underlying geometry.
And similarly, the experiment where you say, well, I threw my I threw the dice.
And that determined, again, how I would measure this particle.
I will now consider a hypothetical universe where the particle again was the same particle, but now I sneezed,
which caused the dice to land and on different number and that to to lead to a different measurement orientation.
Again, that corresponds to keeping fixed one of the variables, one of the components of the state vector and varying the other two.
And this would be a perturbation that would be unconstrained.
It would be something that you were just thinking up out of your head as a possible type of perturbation.
But if you had the view that there was some underlying geometric constraint associated with the, uh,
the universe somehow lying on a on or belonging to an invariant set, then this would be a physically unreasonable perturbation.
A physically reasonable perturbation would be one where all three components were perturbed.
So not only that, so there are plenty of universes, perhaps where you did, in fact,
had the orientation of the measuring device was as it was in the counterfactual world.
But in that universe, the hidden variable was also different.
To keep the whole system on the invariant set.
So this is something which I have been, uh, as I say, rather interested in as a kind of this is not my day job, if you like, but it's something that.
I find a compelling idea myself whether I can convince my colleagues of that.
I don't know. But this may actually what I call fractal determinism may actually provide a conspiracy
free loophole for the Bell Theorem and a paper in crisis a few years ago about it.
But there is a fundamental assumption here to make this postulate make any sort of sense,
and that is that one can really think of the universe as a dynamical system in its own right.
But it's a dynamical system to beat all dynamical systems, I suppose.
But more than that, it's a dynamical system which evolves on one of these special types of fractal geometries.
Now. I'm I will. I actually there's I mean one could talk about what is the evidence,
the cosmological evidence that the universe is a dynamical system that evolves on a fractal invariant set.
That's a great topic for discussion, but I'm not going to. I'm going to leave that because we'll we'll run out of time.
So let's just assume that is a viable model for the universe that it actually
evolves on on one of these zero volume invariant sets in its state space.
I want to come back now to then, uh, these three key differences between the Louisville equation and the Schrodinger equation.
Planck's constant. Square root of minus one. And the fact it's a Hilbert space.
And if you do, you want a bit more vivid? Hilbert Space means that cats are both alive and dead.
So the question I want to ask is if this does provide a potential loophole,
can we actually go from just postulating this as a possible way out of the dilemma of underpinning quantum mechanics with something deterministic?
Can we go from that to an actual theory? Can we actually put some meat on the bones, as it were, and construct a real theory based on this this idea?
So can we in particular, can we construct a fractal set bit like I did for the Cantor set,
but something which will obviously be more complex than that from which quantum statistics would emerge naturally?
So I just want to spend the last 10 minutes of the talk going through some slightly more technical stuff,
which is in a paper on the archive which I wrote last year, trying to at least give some outline for how one might do this.
So I want to start by talking about Planck's constant. How might that emerge?
Let's start by thinking about a stern Gerlach experiment.
Now I'm going to be very much motivated by people that have been and Roger is is is certainly well known in the field.
People have been motivated by thinking about gravity as a possible mechanism for collapse, state vector collapse.
Now, I'm not talking about collapse in this model because I'm trying not to view superpositions as something of of fundamental significance.
So but nevertheless, I'm going to use the order of magnitude calculations to motivate this work.
So the idea is we have now a sort of bundle of space times,
a bundle of trajectories on this supposed invariant set, this fractal geometry of the universe as a whole.
And I'm going to use this language of symbolic dynamics. So I want to label each space time with a with a colour which represents its symbolic label.
So these are trajectories corresponding to space times where a particle has gone through the magnet and is on its way up towards the spin up detector.
And here are some trajectories of space times where there's a particle on its way to the spin down.
So this this time t one is after it's left the magnet, but long before it's got to the detector.
I want to ask the question, are these space times gravitationally distinct from one another?
And to do that, I'm going to take pairs of space times and ask whether their interaction,
the gravitational interaction, energy integrated over the trajectory exceeds Planck's constant.
The gravitational interaction. Energy is if you have two, let's say two lumps of matter, the gravitational interaction,
energy and Newtonian gravity anyway is easily defined by merging moving one lump of matter to
the position of the second lump of matter against the gravitational field of the second lump.
So how much energy do you need to move it against the gravitational field of, say, this one?
That's the gravitational interaction energy. Now, at this time, it's this small time here, then for sure.
Although Planck's constant is a small number, this thing will be minuscule. Is small.
And so this criterion will not be satisfied.
And I view that as a statement that each of these will be given the same symbolic label.
They're not distinct enough to be given distinct symbolic labels.
But again, using the work DFC, Penrose, Kebble, Percival, many others,
there are good indications that when the particle is started to interact with atoms
in the detectors and you're starting to see macroscopic differences between the bit,
between whether the particle's excited, the spin up detector or the spin down detector,
then this interaction, gravitational energy can start to exceed Planck's constant.
So this might be a small term, but Planck's concerns are also small and the order of magnitude estimates.
I'm just going to take this on faith now. When when the when you start to get macroscopic effects in the detectors.
So at this point, then you can give these distinct labels.
So Planck's constant in this picture is entering through essentially through gravitational effects.
One can analyse multiple stern Gerlach experiments this way.
So here's here's a particle. Here's a series of trajectories that go through one stern Gerlach apparatus.
So this one is and this one is the top one is blocked or measured somehow.
So these three have the same symbolic label, which is different to that one.
You can take this through a second sequential Stern Gerlach apparatus, and now these three trajectories start to this.
This trajectory starts to become distinguished from these two gravitationally.
And then finally in the third one, these two and one can derive the whole of Stern Gerlach statistics in this in this way, quite straightforwardly.
This, by the way, is how Schwinger introduces his students to quantum mechanics through these sequential Stern Gerlach experiments.
You might. Oh, yeah. So I couldn't resist this, you might say.
Gravitational interaction, energy. It is very Newtonian. How do you define this relativistic li?
Uh, this is actually what I did for my thesis under Dennis defining gravitational energy momentum.
And, uh, we came up with, I think, a rather neat solution to this rather old problem.
How do you define gravitational energy momentum in space times which don't have killing vectors?
And the proposal was a not a tensor field on space time,
but a tensor field on the tangent bundle two space time, which is a bit like a state space for space time.
So this is, this is my, this is in the days when you had typewriters and ink and stuff, uh, it appeared, it appeared eventually in Fitzrovia.
So actually there is a way to do this. I just.
To change this because it amuses me. It links back a bit to my thesis work that there is a way to develop these ideas for
how Planck's constant might be based on relativistic gravitational energy momentum.
So conscious of the time I'm moving on, I want to talk about the second aspect planks, the square root of minus one.
One of the interesting properties of a fractal is itself similarity is an essential part of a fractal, in fact.
So if one was to zoom into a cancel set, you'd see a precise copy of the whole thing.
And you could zoom in again and you'd see another copy. If you need to.
Really, if your theory requires you to keep that, you have that notion of self similarity at the fore.
It turns out that it's very useful to describe the dimension, the house of dimension of the fractal using complex numbers rather than real numbers.
And the, the, the second, if you like, the complex number characterises not only if you like the fractal too,
but also this notion of the self similarity.
As you zoom in, how does it how quickly does it amplify to the back to the original scale?
I guess the mother of all I'm so similar.
I mean, this, this, this, this. I'll just show this.
We people have seen this before, but I feel I haven't done sort of justice to the wonders of fractals.
So I'll just show this movie, which just illustrates this notion of.
If self similarity if you haven't seen it before for the Mandelbrot set, we're focusing on a bit now which where this kind of content,
this scaling symmetry part, which is pretty, pretty manifest, it just keeps repeating and repeating ad infinitum.
Now, if one is I'll just leave this point very briefly.
If one is trying to develop a relativistic theory, a theory that's invariant under Lorentz, Lorentz with a T now transformations.
So you might say suppose you had to spatially extend a fractal set which describes a spatially extended dynamical system,
and you wanted that scaling invariance, that scale invariance to be Lorentz invariant.
Okay. Now, in Lorenz. Lorenz transformation, one man's, one man's time is another man's space and time.
So if you want your scaling symmetry to be Lorenz invariant in a spatially extended dynamical system,
you better have some oscillatory type of structure in the configuration space and the spatial degrees of freedom of spatially extended system.
So in some sense, my I think the picture which I I'm sort of coming to for, you know, why is the wave function wavy?
Is it is actually I mean it's it's needed to make this type of scale invariant a symmetry.
Lorentz invariant under a boost for a system with a spatially extended system.
All right. I'm going to move on. This is the last sort of this is the last bit.
A little bit. Well, no, I didn't start till 25, so I'm okay. But I might have run by a couple of minutes.
I just want a very focussed come back to this last the last bit now the Hilbert space.
How would that fit into this picture of fractal invariant sets?
So I just want to remind a very standard picture about Hilbert space for a simple qubit there.
There's the Hilbert space state for this state, state here on the equator relative to a basis where the north poles up, the south poles down.
But now I want to do a unitary transformation where I rotate the basis.
So now if you like the North Pole points here. And ask, how does this state transform under that rotation of the basis?
And quantum mechanics gives us a very clear answer.
Now one can think of this as this counterfactual experiment.
So the state is well defined that say this is the actual experiment I did, but I ask what would have happened had I done this experiment?
Quantum mechanics gives us a well-defined state, at least in which for giving us probabilities relative to this rotated basis.
In this fractal theory, and I'm jumping over quite a bit now.
The equivalent of the Hilbert space is what I'm calling a symbolic skeleton.
These symbols appear as as long symbolic sequences, a bit like for the lowest Lorenz orbits, the North Pole would have just zeros.
The South Pole have just ones. On the equator you have mixtures of zeros and ones at a particular and antipodal points have opposite,
you know, zero gets flipped to one and a particular co latitude feature.
You'd have a symbolic sequence where the frequency of a zero would be given by cosine squared peter over to.
And these are related to. Certain operators,
which I don't have time to talk to talk about a key point about this symbolic
skeleton is that these and these cosine of co latitude angles are rational.
The cosine is facts are based to rational number.
So these can be as dense as you like these. I've just drawn a rather coarse representation of what is a really dense set of points.
So I want to consider this counterfactual and how, uh, how this, how this, this works in this alternative sort of fractal picture.
So I'm taking this point here. So these dots in some sense of the set of all allowable points, and this is very dense.
This is the set of all the blue points and the red points on the equator, of the set of all the allowable states relative to the rotated basis.
But now I want to know this. If I keep this point, this is this is my original state.
And I'll say, what's that state relative to this rotated basis?
In other words, just this coincides with one of the allowable states. Then another very interesting kind of number theoretic result comes out.
This angle between here and here is pi over eight by construction and cosine of pi over eight is not a rational number.
So it doesn't lie on this set of points which have rational cosine.
And in fact it doesn't matter. This can be any based to rational number between zero and one.
Symbol number. Number. Theoretic calculation tells you this cosine is always irrational.
So this construction captures perfectly this notion of of counterfactual indefinite.
This. This point has no representation relative to this rotated basis.
Uh, Heisenberg.
Usually you will occasionally use this German phrase, thus under-estimate Princip to describe the uncertainty, what we call the uncertainty principle.
But this actually translates better as the indeterminacy principle, rather the uncertainty principle.
And I think this indeterminacy characterises this construction pretty well.
So I view the Hilbert space as a sort of completion onto the continuum of this symbolic skeleton.
A bit like the reals are the completion of the rationals. But you know, and the reels were fantastic.
Of course, for for physics, we would nobody would be without the real numbers.
But you don't want to take the real numbers too seriously, in my view.
You can do some pretty strange things with the real numbers if you take them too seriously.
People might be familiar with the Banach task construction.
You can take a bar of gold, you can chop it up into a finite number of pieces, stick it back together again.
And the bar of gold is twice as big as it was. So here's a way to either get rich very quickly or to devalue the value of gold very quickly.
But of course, the reason it's got twice as big is that you've treated the real numbers too seriously in terms of physics.
My own picture and this is, I suppose, a bit controversial, is that the Hilbert's face is a bit like this.
It's a fantastic tool for doing calculations.
And of course all students should be taught the Hilbert space because you have to use it to do calculations.
But you shouldn't take it too seriously, because if you do take it too seriously, you end up with strange paradoxes like Schrödinger's cat.
And if you. So. So that's my that's my picture of how this.
The Hilbert space comes out. It's a completion of this skeleton.
Uh. But it's and it's and in that sense, it's a useful tool for calculations.
But don't take it too seriously. Right.
I'm just going to finish now. So my claim is that if you if you if the universe evolves on a on a fractal set,
one can overcome the Bell Theorem without any need for conspiracy, no need for God.
And in fact, Bell's implausible conspiracy, like all good conspiracy theories,
is really just all in the mind because the sort of conceptual issues that that lead you to think there
might be some conspiracy are are what I would call geometrically unconstrained types of considerations.
So I have a lot of sympathy with with Dirac and with with Roger.
So I'm less wedded to Stephen's view about quantum physics.
I do want to just finish with the word about quantum gravity,
because it seems to me that if this picture if there's any sense in this picture what one is proposing,
here is some extension of of general relativity theory taking, if you like, Einstein's insight about the importance of geometry.
But now, not only for space time, but for state space itself.
And the claim then is that.
That from this extended type of idea about about geometry and in particular than about gravity, quantum physics, maybe emergent.
Now if that's the case. Then the whole quantum gravity program seems to me to be put at B to be the wrong way round.
It's putting the cart before the horse, the horse being gravity and the cart being quantum mechanics.
So if I had to sort of finish with the prediction,
I would I would say if we ever got to the stage of being able to detect gravitons, we won't ever find them.
Because I think it's personally, I think it's a misguided concept. So my last statement is to say the three.
Great theories of 20th century physics, quantum mechanics, relativity, chaos theory.
They're a little bit just disparate theories as we standard have them currently.
Quantum mechanics and relativity are not combined. Chaos and quantum mechanics are not considered different.
Actually, even chaos. If we think of chaos as defined in terms of of of leaping of exponents,
that actually is not a very relativistic idea because you could scale time logarithmic and then a leaping
of the exponentially diverging trajectories would just look like linearly diverging trajectories,
and then it wouldn't be chaotic. All these things.
My view can come together by thinking about this underlying geometry.
And I suppose if I had to summarise my talk in one sentence,
is that we should think perhaps of the laws of physics in their most primitive expression in terms of state space geometry,
move away from the old paradigm of differential equations and think like Einstein taught us about geometry.
Well. So just to finish by saying Dennis, of course, was an enormously eminent and distinguished scientist.
But he was also somebody who was phenomenally good at inspiring young scientists like myself.
And I hope if you come away with nothing else,
you will understand that it's an education under Dennis Sharma is really something one can never shake off.
Thank you very much.