The Universe Within

by Neil Turok

  Imagine you are given two identical boxes, each in the shape of a tetrahedron — a pyramid with four equilateral triangular sides. One side is a shiny metal base, and the other three are red, green, and blue. The coloured sides are actually small doors. Each of them can be opened to look inside the pyramid. Whenever you open a door, you see a coin lying inside on the base, showing either heads or tails.

  Upon playing with the boxes, you notice that the bases are magnetic and stick together base to base. When the boxes are stuck together like this, the doors are held tightly shut, and there is a soft hum indicating the state of the boxes is being set.

  Now you and a friend pull the two boxes apart. This is the analogue of the Einstein–Podolsky–Rosen experiment. You each take a box and open one of its doors. First, you both look through the red door of your box. You see heads and your friend sees tails. So you repeat the experiment. You put the boxes together, pull them apart, and each of you opens the red door. After doing this many times, you conclude that each result is entirely random — half the time your coin shows heads, and half the time it shows tails. But whatever you get, your friend gets exactly the opposite. You try taking the boxes very far apart before opening them, and the same thing happens. You cannot predict your own result, but whatever that result turns out to be, it allows you to predict your partner’s finding with certainty. Somehow, even though each box gives an apparently random result, the two boxes always give opposite results.

  It’s strange, but so far there is no real contradiction with a local, classical picture of the world. You could imagine that there is a little machine that makes a random choice of how to program the two boxes when they are placed base to base. If it programs the first box to show heads when the red door is opened, it will program the second box to show tails. And vice versa. This programming trick will happily reproduce everything you have seen so far.

  Now you go further with the experiment. You decide that you will both open only the green door. And you find the same thing as before — each of you gets heads or tails half the time, and each result is always the opposite of the other. The same happens with the blue door.

  Still, there is no real contradiction with a classical picture of the world. All that is required to reproduce what you have seen is that when the two bases are held together, one box is programmed randomly and the other box is given exactly the opposite program. For example, if the first box is programmed to give heads/heads/tails when you open the red, green, or blue door, then the other is programmed tails/tails/heads when you open the red, green, or blue door. If the first box is programmed heads/heads/heads, the second is programmed tails/tails/tails. And so on. This arrangement would explain everything you have seen so far.

  Now you try something different. Again, you put the two bases together and pull the boxes apart. But now, each of you chooses at random which door to open — either red, green, or blue — and records the result. Doing this again and again, many times, you find that half the time you agree and half the time you disagree. Initially, it seems like sanity has been restored: the boxes are each giving a random result. But wait! Comparing your results more carefully, you see that whenever you and your partner happen to open the same-coloured door, you always disagree. So there is still a strong connection between the two boxes, and their results are not really independent at all. The question is: could the boxes possibly have been programmed to always disagree when you open the same-coloured door but to disagree only half the time when you each open a door randomly?

  Imagine, for example, that the boxes were programmed to give, say, heads/heads/tails for your box and tails/tails/heads for your friend’s. You pick one of three doors, at random, and your friend does the same. So there are nine possible ways for the two of you to make your choices: red–red, red–green, red–blue, green–red, green–green, green–blue, blue–red, blue–green, and blue–blue. In five of them you will get the opposite results, with one seeing heads and the other tails, but in four you will agree. What about if the boxes were programmed heads/heads/heads and tails/tails/tails? Well, then you would always disagree. Since every other program looks like one of these two cases, we can safely conclude that however the boxes are programmed, if you open the doors randomly there is always at least a five-ninths chance of your disagreeing on the result. But that isn’t what you found in the experiment: you disagreed half the time.

  As you may have already guessed, quantum theory predicts exactly what you found. You agree half the time and disagree half the time. The actual experiment is to take two widely separated Einstein–Podolsky–Rosen particles in a spin zero state and measure their spins along one of three axes, separated by 120 degrees. The axis you choose is just like the door you pick in the pyramidal box. Quantum theory predicts that when you pick the same measurement axis for the two particles, their spins always disagree. Whereas if you pick different axes, they agree three-quarters of the time and disagree one-­quarter of the time. And if you pick axes randomly, you agree half the time and disagree half the time. As we have just argued with the boxes, such a result is impossible in a local, classical theory.50

  Before drawing this conclusion, you might worry that the particles might somehow communicate with each other, for example by sending a signal at the speed of light. So that, for example, if you chose different measurement axes, the particles would correlate their spins so that they agreed three-quarters of the time and disagreed one-quarter of the time, just as predicted by quantum mechanics. Experimentally, you can eliminate this possibility by ensuring that at the moment you choose the measurement axis, the particles are so far apart that no signal could have travelled between them, even at the speed of light, in time to influence the result.

  In 1982, the French physicists Alain Aspect, Philippe Grangier, and Gérard Roger conducted experiments in which the setting for the measurement axis of Einstein–Podolsky–Rosen particles was chosen while the particles were in flight. This was done in such a way as to exclude any possible communication between the measured particles regarding this choice. Their results confirmed quantum theory’s prediction, showing that the world works in ways we cannot possibly explain using classical notions. Some physicists were moved to call this physics’ greatest-ever discovery.

  Although the difference between five-ninths and one-half may sound like small change, it is a little like doing a very long sum and finding that you have proven that 1,000 equals 1,001 (I am sure this has happened to all of us many times, while doing our taxes!). Imagine you checked and checked again, and could not find any mistake. And then everyone checked, and the world’s best computers checked, and everyone agreed with the result. Well, then by subtracting 1,000, you would have proven that 0 equals 1. And with that, you can prove any equation to be right and any equation to be wrong. So all of mathematics would disappear in a puff of smoke. Bell’s argument, and its experimental verification, caused all possible classical, local descriptions of the world similarly to disappear.

  These results were a wake-up call, emphasizing that the quantum world is qualitatively different from any classical world. It caused people to think carefully about how we might utilize these differences in the future. In Chapter Five, I will describe how the quantum world allows us to do things that would be impossible in a classical world. It is opening up a whole new world of opportunity ahead of us — of quantum computers, communication, and, perhaps, perception — whose capabilities will dwarf what we have now. As those new technologies come on stream, they may enable a more advanced form of life capable of comprehending and picturing the functioning of the universe in ways we cannot. Our quantum future is awesome, and we are fortunate to be living at the time of its inception.

  · · ·

  OVER THE COURSE OF the twentieth century, in spite of Einstein’s qualms, quantum theory went from one triumph to the next. Curie’s radioactivity was understood to be due to quantum tunnelling: a particle trapped inside an at
omic nucleus is occasionally allowed to jump out of it, thanks to the spreading out in space of its probability wave. As the field of nuclear physics was developed, it was understood how nuclear fusion powers the sun, and nuclear energy became accessible on Earth. Particle physics and the physics of solids, liquids, and gases were all built on the back of quantum theory. Quantum physics forms the foundation of chemistry, explaining how molecules are held together. It describes how real solids and materials behave and how electricity is conducted through them. It explains superconductivity, the condensation of new states of matter, and a host of other extraordinary phenomena. It enabled the development of transistors, integrated circuits, lasers, LEDs, digital cameras, and all the modern gadgetry that surrounds us.

  Quantum theory also led to rapid progress in fundamental physics. Paul Dirac combined Einstein’s theory of relativity with quantum mechanics into a relativistic equation for the electron, in the process predicting the electron’s antiparticle, the positron. Then he and others worked out how to describe electrons interacting with Maxwell’s electromagnetic fields — a framework known as quantum electrodynamics, or QED. The U.S. physicists Richard Feynman and Julian Schwinger and the Japanese physicist Sin-Itiro Tomonaga used QED to calculate the basic properties and interactions of elementary particles, making predictions whose accuracy eventually exceeded one part in a trillion.

  Following a suggestion from Paul Dirac, Feynman also developed a way of describing quantum theory that connected it directly to Hamilton’s action formalism. What Feynman showed was that the evolution in time of Schrödinger’s wavefunction could be written using only Euler’s e, the imaginary number i, Planck’s constant h, and Hamilton’s action principle. According to Feynman’s formulation of quantum theory, the world follows all possible histories at once, but some are more likely than others. Feynman’s description gives a particularly nice account of the “double-slit” experiment: it says that the particle or photon follows both paths to the screen. You add up the effect of the two paths to get the Schrödinger wavefunction, and it is the interference between the two paths that creates the pattern of probability for the arrival of particles or photons at various points on the screen. Feynman’s wonderful formulation of quantum theory is the language I shall use in Chapter Four to describe the unification of all known physics.

  As strange as it is, quantum theory has become the most successful, powerful, and accurately tested scientific theory of all time. Although its rules would never have been discovered without many clues from experiment, quantum theory represents a triumph of abstract, mathematical reasoning. In this chapter, we have seen the magical power of such thinking to extend our intuition well beyond anything we can picture. I emphasized the role of the imaginary number i, the square root of minus one, which revolutionized algebra, connected it to geometry, and then enabled people to construct quantum theory. To a large extent, the entry of i is emblematic of the way in which quantum theory works. Before we observe it, the world is in an abstract, nebulous, undecided state. It follows beautiful mathematical laws but cannot be described in everyday language. According to quantum theory, the very act of our observing the world forces it into terms we can relate to, describable with ordinary numbers.

  In fact, the power of i runs deeper, and it is profoundly related to the notion of time. In the next chapter, I will describe how Einstein’s theory of special relativity unified time with space into a whole called “spacetime.” The German mathematician Hermann Minkowski clarified this picture, and also noticed that if he started with four dimensions of space, instead of three, and treated one of the four space coordinates as an imaginary number — an ordinary number times i — then this imaginary space dimension could be reinterpreted as time. Minkowski found that in this way, he could recover all the results of Einstein’s special relativity, but much more simply.51

  It is a remarkable fact that this very same mathematical trick, of starting with four space dimensions and treating one of them as imaginary, not only explains all of special relativity, it also, in a very precise sense, explains all of quantum physics! Imagine a classical world with four space dimensions and no time. Imagine that this world is in thermal equilibrium, with its temperature given by Planck’s constant. It turns out that if we calculate all the properties of this world, how all quantities correlate with each other, and then we perform Minkowski’s trick, we reproduce all of quantum theory’s predictions. This technique, of representing time as another dimension of space, is extremely useful. For example, it is the method used to calculate the mass of nuclear particles — like the proton and the neutron — on a computer, in theoretical studies of the strong nuclear force.

  Similar ideas, of treating time as an imaginary dimension of space, are also our best clue as to how the universe behaves in black holes or near the big bang singularity. They underlie our understanding of the quantum vacuum, and how it is filled with quantum fluctuations in every field. The vacuum energy is already taking over the cosmos and will control its far future. So, the imaginary number i lies at the centre of our current best efforts to face up to the greatest puzzles in cosmology. Perhaps, just as i played a key role in the founding of quantum physics, it may once again guide us to a new physical picture of the universe in the twenty-first century.

  Mathematics is our “third eye,” allowing us to see and understand how things work in realms so remote from our experience that they cannot be visualized. Mathematicians are often viewed as unworldly, working in a dreamed-up, artificial setting. But quantum physics teaches us that, in a very real sense, we all live in an imaginary reality.

  PHOTO INSERT

  Glenlair, James Clerk Maxwell’s home in Dumfries and Galloway, Scotland. The house and grounds were Maxwell’s childhood playground, providing many stimuli to the budding young scientist. (back to text)

  A field, the key mathematical concept introduced in Maxwell’s theory of electromagnetism. The arrows show the direction and strength of the field, and the grid of grey lines the coordinates, for each point in space. (back to text)

  Maxwell’s diagrams showing the machinery for magnetic fields and electric currents. On the left, the hexagonal cells are “vortices,” representing a magnetic field. The particles between them carry an electric current. On the right is the magnetic field of a current in a wire. (back to text)

  The School of Athens by Raphael. Plato (left) and Aristotle (right) stand in the central arch. Seated at front centre, thinking and writing, is Heraclitus. To the left of him are Parmenides, Hypatia, Pythagoras, and Anaximander. At front right, using a compass, is Euclid. (back to text)

  The 1927 Fifth International Solvay Conference, held at the height of the quantum revolution which overturned the classical world-view. (back to text)

  The double-slit experiment. A laser (top) shines light of a single wavelength on two slits in a partition (middle). The light waves from each slit spread out and interfere, producing a pattern of stripes on the screen (bottom). (back to text).

  Emergence of a region of the universe from the big bang. (back to text)

  The temperature of the cosmic microwave background radiation, measured across the sky by the Differential Microwave Radiometer on nasa’s cobe satellite. Hotter is red, colder is blue. At top left is the original picture, showing the asymmetry due to Earth’s motion. Removing the effect of the motion produces the picture at lower left, showing the Milky Way as a band across the middle. Removing the galaxy’s emission produces the picture at right, showing the primordial density variations in the universe. (back to text)

  The cluster of galaxies known as Abell 2218, two billion light years away from us. The stretched-out “arcs” of light are the images of galaxies behind the cluster, lensed and distorted by the cluster’s gravitational field. The distortion can be used to measure the distribution of mass within the cluster, revealing the existence of a substantial quantity of “d
ark matter” in addition to the visible galaxies, stars, and hot gas. (back to text)

  The African Institute for Mathematical Sciences in Cape Town, South Africa. (back to text)

  A group of aims students at the 2008 launch of the Next Einstein Initiative, a plan to create fifteen aims centres across Africa within a decade. The two men in suits are Michael Griffin (left), head of nasa, and Mosibudi Mangena (right), the South African Minister for Science and Technology at the time. The woman in the head scarf at the picture’s centre is Esra, from Sudan. The man at front right is Yves, from Cameroon. (back to text)

  The formula that summarizes all the known laws of physics. (back to text)

  The atlas detector at the Large Hadron Collider at cern in Geneva, Switzerland. This giant apparatus (notice the person standing on the platform) is used to detect and analyze the spray of particles produced by the collision of two particle beams at its centre. (back to text)

  Artist’s impression of a typical collision at the Large Hadron Collider, revealing the existence of the Higgs boson. The discovery confirmed theorists’ predictions, made a half-century ago, that the vacuum is permeated with a Higgs field. (back to text)

  THREE

  WHAT BANGED?

  “The known is finite, the unknown infinite; intellectually we stand on an islet