Part 3 – Roger starts to unravel the universe
Sir Roger Penrose, Oxford Univ
Is he the most brilliant man ever?
This is a complex topic. I highly suggest you read the articles in sequence starting from the first:
Here is the series on Roger Penrose and Reality:
Part III The start of Twistors
“He’s one of the very few people I’ve met in my life who — without reservation — I call a genius. Roger is the kind of person who has something original to say — something you’ve never heard before — on almost any subject that comes up.” Lee Smolin, professor of physics Waterloo University
Lee Smolin is no slouch. It’s fair to say that Lee is a genius and has met a few geniuses. If you read a history of Roger you will discover he contributed almost as much as Einstein to general relativity. He worked with Hawking on black hole collapse showing that black holes really existed and were likely to be common.
I became more interested in him because of his controversial ideas about quantum brain and the nature of consciousness. He claims the brain is a quantum computer.
Roger invented Spinors and Twistors (among many other things.) but Spinors and Twistors were rarely used or discussed until recently.
It started to change in 2004 after Edward Witten discovered that Twistors could be used in string theory instead of possibly the 11 dimensional framework they had come up with could be reduced to the 5 dimensions of Twistor space. After that a series of breakthroughs in the use of Twistors in other parts of physics sparked what has become known as “The Twistor Revolution.”
Twistors most significant win has been what is called the Amplituhedron, an abstract geometrical object that lives in Twistor space. Without it it is infinitely harder to calculate what it going to happen in quantum mechanics. This has made Twistors more or less mandatory now to be able to do quantum mechanics. There are other startling revelations from using Twistor theory that I will wait to describe.
I will try to show you how Twistor space is the most natural way to understand our reality.
It is unfortunate that mathematics today is full of complex and obscure references to individuals and much of mathematics now consists of lots of terms that are obscure to anyone trying to understand it. I believe math could use a makeover that would rethink the terminology. I will try to avoid the terminology and explain things as simple as possible.
What motivated Roger?
He actually didn’t do that well in school. His math teacher in high school realized that Roger needed more time on tests and so he gave him 2 periods to finish tests. Roger said he has a terrible memory so he needs to work things out for himself often from basic principles. He says he was terrible at arithmetic and it just took him more time than the other kids.
Roger has an early interest was in geometry and so he invented the Mandelbrot curve, a fractal geometry and loved working on geometry, problems of visualization.
M.C Escher was inspired by Penrose’s work in cohomology (shapes that are possible and impossible). The famous staircase paintings for instance were inspired by Penrose showing him a diagram of an impossible staircase from the algebraic geometry he was working on.
He decided to major in Algebraic geometry because he thought it would have lots of geometry in it, but it turns out it was more algebra than geometry. He considered this sort of a bait and switch. 🙂 He preferred to do things graphically rather than algebraically.
He did his PHD thesis using graphical representations for the algebra and operations. When he put the paper to the university to be accepted he transformed his “graphical formulas” to standard algebra worried it would not be accepted with his invented graphical notation. The point is Roger thinks about things in graphical terms unlike many other scientists who simply apply the algebra never thinking much about the “reality” or how it would work, how it looks.
As I mentioned about the Copenhagen Interpretation physicists had largely prior to Roger given up trying to understand quantum mechanics. Roger wouldn’t accept this. This means many scientists ignore issues that bothered Roger a lot. Roger started to look at things from the beginning that others had just passed by just like he did in high school but now it paid off in spades.
I could go on and on. You can read more of his biography here. This is not the intent of my blog here. To explain the TWISTOR universe I think I needed to start by explaining how Roger is uncommon and led him to make the amazing discoveries and theories he has.
One of the first things Roger did was to try to understand spinning particles – Spinors
Things like the collapse of the wave-function that for many physicists is just deadly confusion they would table and move on to other issues, Roger couldn’t let go. He had to understand it graphically. He said that it was “lazy” that Physicists didn’t try to understand these things more in terms of imagining how we create a mental image of these things.
In Penroses own words: (On the origin of Twistor Theory):
“The very fact that quantum behaviour is so hard to picture in the normal way had seemed to me to argue strongly that the normal space-time picture of things, even at that level, is inappropriate physically. Indeed, there was nothing really new in this either, inadequacy of space-time pictures being very much a part of the standard quantum-mechanical philosophy. However, I felt that one ought to try to be more positive than this, in actually providing a picture of objective reality, albeit one perhaps radically different from the usual one.”
There are more than 100 particles that have been discovered. One of the fundamental characteristics of particles is spin. In fact we categorize all particles by the amount of spin they have. Particles with integer spins are called Bosons and Fermions are particles with 1/2 spins. All particles are Fermions or Bosons.
The notion of a spinning particle is odd in many ways. It’s not obvious why particles spin and what particle spin means at the level of electron? It can’t be like a curve ball in baseball or a spinning earth and yet many particles do act as if they are spinning magnets.
Other characteristics of particles are not so easily anthropomorphized. How do we anthropomorphize “baryon” number which is a conserved characteristic. Each conserved quantity must represent at a fundamental level some kind of fundamental unbreakable, real thing because by being conserved it means it can’t be broken or destroyed. It is immutable for all time so something as weird as baryon must correspond to something physical that really exists.
So, let’s ask what is reality, really? What are the fundamental things that really exist not the things we see at the macro level that are simply conveniences of our senses.
Penrose invented Spinors to describe the geometry and physics of spinning particles in the 1960s. Spinors use an imaginary dimension. Twistor space consists of Spinors, Twistor space has 2 complex dimensions.
One of the most interesting things is that particles always spin at multiples of 1/2 spin in discrete jumps not unlike how energy is always made in jumps of the planck energy (as discovered by Einstein). 1/2 simply corresponds to the increment of angular momentum using planck constants. It’s arbitrary since it is like other things like this simply a ratio of one thing to another.
We tend to anthropomorphize these characteristics we’ve given particles. It’s impossible not to think of a baseball spinning when we think of a particle spinning. As you can imagine this analogy doesn’t really work very well.
A spinning magnet is very visual compelling idea. However, an idea is hard to do math on. It turns out the simplest way to describe the spinning magnet is with a complex number. The combination of the real and imaginary portions describe an angle of spin.
Here is a graphical representation of a spinning electron for instance.
It is a good idea to start with particles to understand one of the most basic and perplexing things about reality. All particles spin for some reason. We classify all particles by the amount of spin they have so understanding this perplexing property would be a first step.
Interestingly, spinning particles can be represented by using a complex number to represent the angle of the spin. When saying complex numbers I mean: A + Bi where i = -1^0.5. The geometric equivalent of the complex number was a Riemann sphere which has as one of its axis a complex dimension.
The angles of the spinning can be translated to a second number that you can conveniently put with the i part of the complex number.
Roger invented Spinors to represent spinning particles and it worked. It turns out that Spinors rotate differently than what we imagine. They rotate through 720 degrees as in the graphic below.
So, it seems a convenient mathematical trick that you can represent this using a complex dimension and all the math works out when you combine these things, multiply or add them together but that’s just a math coincidence right? It is assumed there is no actual complex dimension that the spinning particle inhabits.
Mathematically this is convenient. However, what does it mean to have a complex dimension in reality? When Penrose did this I don’t think even he imagined at the time that reality had a complex dimension. Nonetheless, inventing Spinors had a positive effect on Rogers career. Spinors did help people visualize spinning particles and more important were found useful to calculate things.
You can think of the complex dimension as the angle of spin of a particle rather than try to imagine what a complex dimension might be.
Roger and other physicists were confused about what the significance of the fact that complex numbers keep appearing in quantum mechanics, not just in spin but everywhere in quantum mechanics.
In Newtonian physics before Einstein and the 20th century we had formulas that said if you launch a ball with such and such a force it will go here in 10 seconds. According to Newtons law it went exactly there. In quantum physics it could go 100 miles from there. It’s likely it will go close to where Newtons formula said but in fact in the quantum world we really live in, it could go almost anywhere. Everything is a probability and in some universe the ball does go 100 miles from where it would in Newtonian world.
So, quantum mechanics doesn’t tell us where the particles will go exactly. It tells us with what probability it will go here or go there. Those probabilities can be calculated very accurately with Schroedinger’s equation and nature follows the Schroedinger equation to MORE digits of accuracy than Newton’s law ever did! So, that’s amazing.
When we calculate the value of Schroedinger’s equation we don’t get the probability but something else we called the amplitude. The amplitude is a complex number. The reason for this is that particles interact in a wave-like manner adding and subtracting from each other depending on the frequency and magnitude of a wave. We saw above that we can represent periodic wavelike spinning things with a complex dimension. So, the easiest way to manipulate how the particles interact it turns out is with a complex number.
So, where do we get the “probability?”
You take the complex number called the amplitude that is the solution to Schroedinger’s equation and do a simple mathematical trick of reversing the sign of the second part(which is then called the conjugate), multiply it by the original the result was a pure real number with no complex part and turns out to be the number which if you threw that ball a trillion times it would show up there exactly that number of times (in proportion to the number of course).
This is the math trick physicists depend on:
(a+bi) * (a-bi) = a^2 – b^2*i^2 (i^2 = -1 of course) so
(a+bi) * (a-bi) = a^2 + b^2 = probability particle appears here
The following graphic represents the probability distribution for an electron in a higher orbital in a hydrogen atom where the calculated probability (a^2 + b^2) is > 0.02. The color is used to show the complex parts of the amplitude.
These “amplitude” numbers that come from Schroedinger’s equation can be added or multiplied to represent different physical things that happen in the real world. The math of complex numbers works perfectly to reproduce the physics we see. The result is exactly what we observe to an unbelievable level of precision.
A success of sorts. It works but at a huge cost. We had to turn our entire conception of the universe on its head. Worse than that, we’ve had to give up all conception of the universe and think of the world as a set of equations.
The 4rth multi-verse ( you may skip)
One theory of physics says that any consistent system of mathematics that describes a possible world actually exists. Thus there may be infinite universes with different sets of theories and we happen to inhabit the one that includes complex numbers. There may be universes that don’t have them. Who knows.
If imaginary numbers are necessary to compute the answers to simple algebraic equations then nature would naturally have an imaginary dimension because it is constantly solving such equations in the process of the world simply working the way it does.
The point is that to our surprise in most cases when the math produces a result that seems crazy, we have found that nature actually does what we thought was crazy. Physicists have been caught trying to out-elegant reality too many times. So, some now believe that math is the ultimate reality and that all possible mathematical realities exist somewhere.
The 20th century is replete with accounts of physicists including Einstein himself who would reject the mathematical results of his theories. Initially when his General Relativity predicted the existence of black holes he denied they were actually possible. When people showed how they would arise he tried like other physicists to show how they weren’t possible. When his general relativity showed the universe wouldn’t be stable and would be contracting Einstein believing in the steady state as most physicists did at the time introduced a fudge factor to keep the universe at a stable steady state. It was only a few laters that we discovered the universe was expanding and Einstein realized he didn’t need his fudge factor. Amazingly, the fudge factor turns out to be real too. So, amazingly frequently in physics when the math says X is possible, we discover that X is true.
Another way to think about complex dimensions
It is extremely hard to understand what a complex dimension means or looks like.
Roger is in a sense trying to give us back a conceptual geometry to re-imagine our world. In the case of Spinors we could imagine the complex mathematical description of a spinning particle with a geometrical object called a spinor.
If we think of the amplitudes as something spinning then the complex dimension of amplitudes may be thought of as just a the angle of the spinning particle in another way, not the spinning of electric charge but maybe something else. In this case the spinning is happening related to the probability of the particle being in a time and place. Call this property that is spinning the X factor. So, the particle is spinning in magnetic property and the X property.
If this X property is related to the interaction with other particles then possibly dark matter is partially composed of particles with 0 spin in this dimension and therefore they don’t interact with other matter with spinning X property. If that’s too complex forget it. It’s my thought. Consider instead that thinking of particles as simply spinning and not having 2 complex dimensions is a more graphical way to think of the world.
In Penrose Twistor theory, Twistors have 5 dimensions 2 complex dimensions (4 including the real and complex parts). So, you have to imagine something spinning in multiple ways simultaneously. Hence it is twisting.