This correspondence between the lines in CP3 and points of a quadric in CP5 is known as the Klein correspondence. http://users.ox.ac.uk/~tweb/00006/index.shtmlHopf Fibration vid
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| Penrose Tribar or Penrose Triangle |
Stereoscopic Klein Bottle vid - the square root of a Klein Bottle is a Torus
Two Moebius Strips makes a Klein Bottle vid
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| Klein Bottle as Hopf Fibration |
Palatial twistor theory and the twistor googly problem Roger Penrose Mathematical Institute, Andrew Wiles Building, Woodstock Road,Oxford OX2 6GG, UK A key obstruction to the twistor programme has been its so-called ‘googly problem’ [from spinning the ball in cricket], unresolved fornerly 40 years, which asks for a twistor description of right-handed interacting massless fields (positive helicity), using the same twistor conventions that give rise to left-handed fields (negative helicity) in the standard ‘nonlinear graviton’ and Ward constructions.An explicit proposal for resolving this obstruction—palatial twistor theory—is put forward (illustrated in the case of gravitation). This incorporates the concept of a non-commutative holomorphic quantized twistor ‘Heisenberg algebra’, extending the sheaves of holomorphic functions of conventional twistor theory to include the operators of twistor differentiation.
Sir Roger Penrose Transcribed:
Why do you believe it's correct? Because you looked at the rules... and you go through all the rules, yes it's true, it's true. So you gotta make sure that these rules are ones you trust. If you follow the rules and it says it's a proof, is the result actually true? Your belief it's true depends upon looking at the rules and understanding them. Now what Goedel shows is that if you have such a system, then you can construct, you can see by the way it's constructed and what it means that it's true but not proveable by the rules that you've been given. It depends on your trust in the rules.
Do you believe that the rules only give you truths? If you believe that the rules only give you truths, then you believe this other statement is also true. I found this absolutely mind blowing. When I saw this, it blew my mind! haha. I thought, "My God!" You can see that this statement is true, it's as good as any proof, but it only depends on your belief in the reliability of the proof procedure, that's all it is. And understanding that the coding it done correctly. And it rules that you transcend that system. So whatever system you have, as long as you can understand what it's doing and why you believe it only gives you truths, then you can see beyond that system.
How do you see beyond it? What is it that enables you to transcend that system? Well it's your understanding of what the system is actually saying. And what the statement that you've constructed is actually saying. So it's this quality of understanding, whatever it is, which is NOT governed by rules. It's NOT a computational procedure.
It's understanding WHY the rules gives you truths that enables you to transcend the rules.
Then I went to this course by Dirac. I think it was the very first lecture that he gave. And he was talking about the superposition principle. And he said if you have a particle, you should think the particle could be over here or over there, but in quantum mechanics it can be over here AND over there, at the same time. And you have these states which involve a superposition in some sense, of different locations for that particle.
And then he got out his piece of chalk, and some people say broke it in two, as a kind of illustration, of how the piece of chalk might be over here and over there at the same time. And he was talking about this, and I, my mind wandered... I don't remember what he said. All I can remember, he just moved onto the next topic. And something about energy, he mentioned, which I had no idea what to do with anything. And so I had been struck by this and worried about this ever since. It's probably just as well I didn't hear his explanation, because it was probably one of these things to calm me down and not worry about it anymore. In my case, I've worried about it ever since.
So I thought maybe that's the catch. There is something in quantum mechanics, where these superpositions become one or the other, and that's not part of quantum mechanics. There's something missing in the theory. The theory is not just incomplete, it's in a certain sense not quite right. Because if you follow the equation, the basic equation of quantum mechanics, is the Schroedinger Equation, you could put that on a computer too. There's a lot of difficulties about how many parameters you have to put in, and so on, it can be very tricky but nevertheless it is a computational process.
Mind you this question about the continuum is before but it's not clear that it makes any difference.
...to make it relate to the world you see you've got to BREAK the Schroedinger equation.
Schroedinger was absolutely appalled by his own equation. That's why he introduced his famous cat. He was really saying, "my equation leads you into this! There is something wrong....
Well it's got to be the Noncomputability comes in there. And I also can't remember when I thought this but it's when gravity is involved in quantum mechanics. It's the combination of those two. And it's that point when you have good reason to believe, this came much later, but I have good reason to believe that the principles of general relativity and those of quantum mechanics, most particularly, is the basic principle of equivalence, which goes back to Galileo: If you fall freely you eliminate the gravitational field.
So you imagine Galileo dropping his big rock and his little rock from the leaning tower (whether he actually did that or not, it's pretty irrelevant) and as the rocks fall to the ground, you have a little insect on one of them looking at the other one. And it seems to think "Oh there's no gravity here!" Of course it hits ground and it realizes something's different going on, but when it's in Free Fall, the Gravity is eliminated. Galileo understood that very beautifully. He used wonderful examples of fireworks and you see the fireworks explode and you see this sphere of sparkling fireworks and it remains a sphere as it falls down, as though there were no gravity. So he understood that principle. But he couldn't make a theory out of it.
Einstein came along, used exactly the same principle. And that's the basis of Einstein's General Theory of Relativity. Now...there is a conflict with something I did much much later. So this wasn't ... much much later. You can see there is a basic conflict between the principle of superposition, the thing that Dirac was talking about, and the principle of General CoVariance, well principle of Equivalence, gravitational field equivalence to acceleration.
In the local neighborhood you can eliminate the gravitation field by falling freely with it. And we know see with astronauts. You know the Earth is right there... you can see the great glow of the Earth right beneath them. But they don't care about it. As far as they are concerned there's no gravity. They fall freely in the gravitational field and that gets rid of the gravitational field. And that's the principle of equivalence.
So what I'm trying to say, trying to put it in a nut-shell but it's not so easy. I'm trying to say that whatever consciousness is, it's not a computation. It's not a physical process which can be described by computation.
And I almost give up. Maybe this is the way around it. And then I thought well maybe this book will at least stimulate young people to do science or something. I got all these letters from old retired people instead. They were the only people who had the time to read my book. Except for Stuart Hameroff. Stuart Hameroff wrote to me and he said, "I think you're missing something. You don't know about microtubules, do you?" He didn't put it quite like that but that was more or less. And he said, "This is what you really need to consider." So I thought, "My God! Yes! That's a much more promising structure."
Apart from red blood cells, they inhabit pretty well all the other cells in the body. But they're not all the same kind, you get different kinds of microtubules. And the ones that excited me the most, this may still not be totally clear, but the ones that excited me most where the ones, where the only ones that I Knew about at the time, because they were; there very very symmetrical structures. And I had reason to believe that these very symmetrical structures would be much better at preserving a quantum state, quantum coherence; preserving the thing without...you just need to preserve certain degrees of freedom, without them leaking into the environment. Once they leak into the environment, you're lost. So you've got to preserve these quantum states at a level which the state reduction process comes in and that's where I think the noncomputability comes in. And it's the measurement process in quantum mechanics, what's going on.
There was a much better chance. It just struck me. Partly it was the symmetry. Because there was a feature of symmetry, you can preserve quantum coherence much better with symmetrical structures. So there's a good reason for that. And that impressed me a lot. I didn't know the difference between the A-lattice and B-lattice at that time, which could be important.
You see it's more outrageous than that. Because we don't even know the physics right. You see it's not just quantum mechanics. People say, "Oh we know quantum systems in biological structures, now we're starting to see that some basic biological systems does depend on quantum. In the first place ALL of chemistry in quantum mechanics. People got used to that, so they don't count that. ha. So let's not count quantum chemistry. We sort of got the hang of that I think. But you have quantum effects which are not just chemical. In photosynthesis, and this is one of the striking things in the last several years, that photosynthesis seems to be a basically quantum process: which is not simply chemical. It's using quantum mechanics in a very basic way.
So you can start saying, "Oh if photosynthesis is based on quantum mechanics, why not the behavior of neurons and things like that?" Maybe there's something which is like photosynthesis in that respect. But what I'm saying even MORE outrageous than that. Because those things are talking about conventional quantum mechanics. Now my argument says that conventional quantum mechanics, if your just following the Schroedinger Equation, that's still computable.
So you've got to beyond that. So you've got to go where quantum mechanics goes wrong in a certain sense. You have to be a little bit careful about that. Because they way people do quantum mechanics is a mixture of two different processes: One of them is the Schroedinger Equation. Which is an equation that Schroedinger wrote down and it tells you the how the state of a system evolves. And it evolves according to this equation, completely deterministic. But it involves into ridiculous situations and this is what Schroedinger was pointing out with his cat. He said, "you follow my equation and you could say that you have a cat which is dead and alive at the same time!" That would be the evolution of the Schroedinger equation that would lead to a state of the cat being dead and alive at the same time. And he's more or less saying, that this is an absurdity.
People now well Schroedinger said you can have a cat that is both dead and alive. It's not that! He was saying, "This is an absurdity. There's something missing." And that the reduction of the state or the collapse of the wavefunction or whatever it is, is something which has to be understood. It's NOT following the Schroedinger equation. It's NOT the way we conventionally do quantum mechanics. There's something MORE than that.
And it's easy to quote authority here, because Einstein, haha; at least three of the greatest physicists of 20th Century, who were very fundamental in developing quantum mechanics, Einstein, one of them, Schroedinger another, Dirac another. You have to look carefully at Dirac's writing. He didn't tend to say this out loud that much cuz he was very cautious about what he said. You find the right place and you see he says, "Quantum Mechanics is a provisional Theory."
We need something which EXPLAINS the collapse of the wavefunction. We need to go BEYOND the theory we have now. I happen to be one of the kinds of people, there's many; there's a whole group of people they're all considered to be a bit mavericks, who believe that quantum mechanics needs to be modified.There's a small minority of those people which are really a minority, who think that the WAY in which it's modified has to be with gravity.
And there's even a smaller minority of those people who think it's a particular way that I think it is, you see. Haha. You see "Quantum Gravity" is already NOT this. Because when you say, "quantum gravity" what you really mean is quantum mechanics applied to gravitational theory. So you see, "Let's take this wonderful formalism of quantum mechanics and make gravity fit into it." So that is what "quantum gravity" is meant to be. Now I'm saying you gotta be more evenhanded: that gravity affects the structure of quantum mechanics too. It's not just you quantize gravity. You've got to Gravitize quantum mechanics. And it's a two-way thing.Recent Sir Roger Penrose Conservation Podcast !! Fascinating!!!
Well it does [infinity physical manifests] in various places. If you're not a mathematician you think, "oh infinity I can't think about that." Mathematicians think about infinity all the time, they get used to the idea and they just play around with different kinds of infinities and it becomes no problem.
But you just have to take my word for it. Now one of the things you see, take Euclidian geometry, it just keeps on going and it goes off to infinity. Now there's other kinds of geometry and this is what's called Hyperbolic geometry. It's a bit like Euclidian geometry, it's a bit different. It's like what Escher was trying to describe in his "Angels and Devils." And he learned about this from Coxeter.
And there infinity by this nice transformation, you squash the infinity down so you can draw it as a nice circle boundary to their universe. Now from our OUTSIDE perspective, we can see there infinity as this boundary.
Now what I'm saying is it's very like that. The infinity that we might experience like those "angels and devils" in their world can be thought as a boundary. Now I found this a very useful way of talking about gravitational radiation and things like that. It was a trick, mathematical trick.
So what I'm saying is that mathematical trick becomes real. That somehow the photons, they need to go somewhere because from their perspective, infinity is just another place. Now this is a difficult idea to get your mind around, so....that's why, one of the reason cosmology is finding a lot of trouble, being taken seriously.
But to me it's not such a wild idea. What's on the other side of that infinity. You have to think why am I allowed to think of this? Because photons don't have any mass. And we in physics have beautiful ways of measuring time. They're incredibly precise clocks. Atomic and nuclear clocks. Unbelievably precise. Why are they so precise? Because the two most famous equations of 20th Century physics - one of them is Einstein's E = mc (squared) - what's that tell us? Energy and mass are equivalent. The other one is even more older than that - still 20th Century, only just: Max Planck E=hv. V is the frequency, h is a constant, again like C; E is energy. Energy and frequency are equivalent.
Put the two together: energy and mass are equivalent: Einstein. Energy and frequency are equivalent: Max Planck. Put them together. Mass and Frequency are equivalent. Absolutely Basic Physical Principle. If you have a Massive Entity - a Massive particle - it is a clock. With a very very precise frequency. You can't directly use it. You scale it down so their atomic and nuclear clocks. But that's the basic principle. You scale it down to something you can actually perceive. But it's the same principle.
If you have mass you have beautiful clocks. But the other side of that coin is if you don't have mass you don't have clocks. If you don't have clocks, you don't have rulers, you don't have scale...
You don't a MEASURE of the scale of space and time.
You do have the structure: what's called the Conformal Structure. It's what the "angels and devils" have. If you look at the eye of the devil, no matter how close to the boundary, it has the same shape, but it has a different size. So you can scale up and you can scale down, but you mustn't change the shape.
So it's basically the same idea but apply it to spacetime now. In the very remote future you have things which don't measure the scale but the shape, if you like, is still there. Now that's in the remote future.
Now I'm going to do the exact opposite. Now I'm going to go way back in the Big Bang. Now you get there, things get hotter and hotter; denser and denser. What's the Universe dominated by? Particles moving around almost at the speed of light. When they get almost at the speed of light, OK they begin to lose the mass too but for a completely opposite reason; they lose the sense of scale as well.
So my crazy idea is the Big Bang and the Remote Future - they seem completely different: One is extremely dense, extremely hot - the other is very very rarefied and very very cold. But if you squash one down by this conformal scale you get the other! So though they look and feel very different - they're really almost the same.
The remote future on the other side, I'm claiming, is that where the photons go? They go into the next Big Bang. You gotta get your mind around that crazy idea....
So I'm saying the other side of our Big Bang, now I'm going back into the Big Bang, there was the Remote Future of a previous Eon. And what I'm saying is that Previous Eon there were signals coming through to us - which we can see and which we DO see.
And these are both signals; the two main signals are to do with Black Holes. One of them is the collisions between black holes. And as they spiral into each other, they release a lot of energy in the form of gravitational waves. Those gravitational waves get through, in a certain form, into the Next Eon.Cancer: Strange Attractors and Complexity Uthamacumaran, Abicumaran. Preprints; Basel, Jun 27, 2019. DOI:10.20944/preprints201906.0288.v1
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