Fundamental

by Tim James

  The ripple you initially created in the magnetic field is thus carried along by both the electric and magnetic fields vibrating into each other at cross purposes.

  We call these ripples ‘electromagnetic waves’ and represent them in diagrams as ribbons rising and falling, with the electric and magnetic field values pointing at ninety degrees, as shown in the diagram below (the electric field is rippling on the vertical axis E, and the magnetic field is rippling on the horizontal axis M):

  It seems impossible to separate the magnetic and electric fields because when one vibrates so does the other, so sometimes we refer to them collectively as a single ‘electromagnetic field’ rather than two interlocking fields.

  As you can imagine though, the mathematics needed to describe these overlapping fields is fiendishly complicated. Every point in space needs to be assigned a value telling us how that point will influence an incoming particle.

  We imagine tiny arrows (called vectors) at every point of the field pointing in the direction a particle will be pushed. Then, we have to include an electric-field vector at right angles to each magnetic one, which also gets a say in an incoming particle’s movement.

  Now just imagine what happens if you move one of the magnets and create a wave in the electromagnetic field. Keeping track of an infinite number of arrows all changing direction and size, rotating at right angles to each other, is no simple task, especially for Faraday who never studied maths in school and could not do much beyond simple fractions.

  It was the young Scottish physicist James Clerk Maxwell who leapt to Faraday’s aid, just as people were starting to call field theory into question, and worked out the maths needed for his fields to be tested properly.

  Maxwell’s equations helped predict how electric and magnetic fields would interact precisely, and the equations matched the data. Faraday’s feelings about fields were finally given fidelity, which was frankly about freaking time.

  The Maxwell equations also made a startling prediction. It turns out that electromagnetic waves ought to propagate through space at a very specific speed: 299,792,458 m/s. Look familiar? It is, of course, the speed of light.

  LIGHT ME UP

  One Saturday evening in 1846 (11 April to be precise), Faraday was accompanying his friend Charles Wheatstone to the Royal Institution lecture hall. Wheatstone was due to deliver a public lecture but decided to have a panic attack instead and ran out of the building, leaving Faraday on his own. In order to avoid disappointing the audience Faraday decided to wing it and improvised a lecture of his own about an idea he had recently been pondering.1

  He speculated that as particles dance around inside an object they generate electromagnetic waves which travel outward into space until they get intercepted by our eyeballs, notifying us to the particle’s presence.

  The match between Maxwell’s equations and the measured experimental value of lightspeed is too close to be a simple coincidence. Faraday’s guess was apparently correct. The medium responsible for light waves (which both René Descartes and Thomas Young had argued for) was the electromagnetic field itself.

  If we vibrate an electron (a particle with electric and magnetic properties) we can create a wave that travels outward at the speed of light. We would not see it, however, because the beam would be too low in energy for our eyes, but a radio antenna might be able to pick it up.

  If we vibrate the electron faster, millions of times a second, the electromagnetic distortions would become visible, making the electron glow red then orange, then yellow, green, blue, indigo and finally violet.

  If we got it going faster still, the waves would become so energetic they would become invisible once more, bypassing our eyes the way a high-pitched dog whistle bypasses our ears. The electron would now be emitting ultraviolet and X-rays.

  In a very real sense, devices such as mobile phones, radio transmitters, wi-fi hubs, Bluetooth emitters, microwave ovens, infrared remotes and X-ray scanners are all just torches. The frequency of their light (a measure of how fast their wavefunction is oscillating) might be too low or high for our eyes to see, but all electromagnetic waves are the same thing.

  There is no conceptual difference between a torch beam passing through a sheet of glass and an X-ray beam passing through a sheet of human skin. It is the energy of the wave and the gaps between electron shells of the material, which determine whether the wave is reflected or goes through. The principle is identical in both instances.

  Humans see a mere fraction of the colours carried about in the electromagnetic field and Faraday opened our eyes to how blind we are. The stars you see at night are not just emitting visible light; they are also giving out radio, microwave, infrared, ultraviolet, X-ray and gamma waves too.

  The very fact we can see the stars at night also tells us that the electromagnetic field must exist in space as well as on Earth, otherwise there would be no medium to transfer the energy to us. The electromagnetic field floods the cosmos from edge to edge. You’re sitting in it right now.

  When your eyes perceive words on this page it is because electrons in the page’s surface are jumping between energy levels, disturbing the electromagnetic field around them. These electromagnetic disturbances move through the field towards your face and finally reach your eyeballs where they are absorbed by electrons at the back of your retina, causing them to send currents down an optic nerve to your brain. Vibrations in the electromagnetic field are, in a very literal sense, the only things you have ever seen.

  ICE CREAM AND BED SHEETS

  This is all very classical so far, what with fields rippling smoothly, etc. But we know, thanks to Planck and Einstein, that light energy is split into chunks: photons. So, in the late 1920s the reclusive English physicist Paul Dirac decided to invent a way of explaining the electromagnetic field in quantum terms – a ‘quantum field theory’.

  It is always unwise to retro-diagnose a person as having a certain medical disposition after they are dead but it is very likely Paul Dirac was on the autistic spectrum. He had no concern for social nonsense, no desire for publicity or glamour, understood people’s words as they were literally meant and was so reserved his students jokingly invented the ‘Dirac unit’ to refer to a speaking rate of one word per hour.2

  So, here is another joke. Did you ever hear about the magic tractor? It went down the road and turned into a field.

  That joke functions (barely) because the idea of a tractor literally turning into a field is, absurd. A lumpy object such as a tractor cannot be transformed into a smooth field because objects and fields are different things. They cannot be interconverted. But in quantum field theory this is exactly what we do.

  A photon is a particle (it holds itself together) but since it has wavelike character (and a wave is an oscillation in a background medium) there must be some field out of which the photon is being vibrated into existence.

  In classical terms we think of electromagnetic fields undulating up and down, but in quantum terms we think of energy as an isolated blip in the field, like a spike on a heart monitor.

  The electromagnetic field is a placid backdrop to our universe, but if one particular region gets agitated it can bunch up and form a twisty little knot of concentrated energy. This packet of field disturbance is a ‘quantum of the electromagnetic field’. A lump in the backdrop. A photon.

  Drawing field quanta in three dimensions is hard so most analogies imagine fields as a flat surface with particles appearing out of them as in the diagram above.

  One helpful approach is to picture a smooth bed sheet pulled across a mattress. If we pluck at a point on the sheet we can create a tiny fabric mountain, representing a quantum of the field. A bed sheet particle, if you will.

  These field quanta (particles) can be transferred from one point to the next, giving the impression of moving through empty space. We just have to remember that the field is actually a three-dimensional fabric that surrounds us in all directions and photons are excited out of it.

&nbs
p; I sometimes like to think of it as being analogous to scooping a ball of ice cream from a tub. The flat surface of the ice cream prior to our scoop is the undisturbed electromagnetic field and when we carve little spheres out of it they are the photons we detect in experiments.

  Really, we should start calling the electromagnetic field the photon field, but you know how much physicists love clinging to outdated terminology (looking in your direction, ‘spin’).

  YOU ARE NOT MADE OF PARTICLES

  Paul Dirac was able to explain mathematically how we could extract photons from the photon/electromagnetic field, and showed that since every type of particle has wave character, every type of particle can be thought of as a quantum in its own field.3

  There is an electron field everywhere in the universe, overlapping with the electromagnetic field, and when we disturb it we get an electron bubbling out. All particles, including those in your body, are really excited vibrations of invisible underlying fields.

  The particles you are made of and the empty space surrounding you are not separate at all. You are made from packets of energy floating through fields made of nothingness. Whether you consider this fact disturbing or beautiful is up to you.

  CHAPTER TWELVE Lines and Wiggles

  ONE THEORY TO RULE THEM ALL

  Paul Dirac’s hope was that quantum field theories would one day be able to explain every conceivable phenomenon in physics. Every particle would be treated as a quantum in a field, and the interactions between fields would underpin the interactions between particles. The idea was a potential game-changer for physics. But unfortunately it was so complicated it turned out to be a game very few people could play.

  The mathematics of quantum field theory is difficult. Crazy difficult. There is even a $1 million prize offered by the Clay Mathematics Institute to anyone who can solve one of quantum field theory’s tougher challenges (called the Yang Mills Existence Mass Gap Problem if you fancy a go this weekend).

  To try and make headway with something so intractable, Dirac suggested we start small and only consider the two simplest particles/fields: electrons and photons. He called the interaction between them ‘quantum electro-dynamics’, or QED for short, and his hope was that if we could work out a fully detailed QED theory, we could start adding in other particles and go from there.

  At the end of his 1930 book The Principles of Quantum Mechanics, Dirac concluded with the words, ‘it seems that some essentially new physical ideas are here needed’. Quite an understatement, but hardly surprising given the reserved nature of Dirac himself.

  His words drifted into the physics community like a gentle challenge on the breeze, and eventually snagged in the mind of a man who was by all accounts his polar opposite. Science’s most charismatic and colourful rogue: Richard Phillips Feynman.

  LITTLE DRUMMER MAN

  Born in New York to a uniform salesman and his wife, Richard Feynman showed enormous talent for physics from his first exposure. His official IQ was clocked at 123 (reasonable, while not astounding) but by the time he was an adult Feynman was regarded as the most gifted scientist on the planet, comparable even to Einstein.

  To give you some idea of how clever he was, in 1958 when NASA launched the Explorer II satellite, something failed during the ascent and it never reached orbit. Feynman bet the NASA engineers he could calculate where the satellite would land, faster than their computer. Not only did he win the bet, he calculated the answer more accurately. Twice.1

  Feynman was also the soul of any party he was invited to and entertained his numerous friends with safe-cracking tricks, bongo-drumming skills and circus juggling. He had red carpets laid out for him at weekly lectures and spent his free time hanging out in topless bars, doing calculations on napkins or drawing sketches of the dancers and sometimes the men watching.2

  A skilled raconteur and an unashamed prankster, Feynman was the Han Solo of physics. But, above all else, he possessed the finest mind of his generation.

  He began his studies at MIT before moving to Princeton (earning a perfect entrance-exam score) to complete his doctorate under John Archibald Wheeler, who also guided Hugh Everett on the many worlds interpretation.

  Halfway through his PhD, he was recruited by Robert Oppenheimer to help the American military design an atom bomb and was described by Oppenheimer as ‘the most brilliant young physicist here’,3 which is saying something considering ‘here’ referred to Los Alamos National Laboratory – an establishment set up exclusively to house the smartest scientists in the world.

  After the war, Feynman completed a post-doctorate at Cornell and then took a professorship at Caltech where he sought to shed the unpleasant taste in his mouth left from helping with the bomb. He decided to commit his life to three things only: thinking, teaching and taking care of students.4

  The teaching and taking care of students bit was easy. Feynman’s nickname was ‘the great explainer’ since his lectures were so good they were attended not just by freshmen but senior colleagues who found themselves learning their own subject better from hearing his take on them. That just left the third focus: thinking. And what he decided to think about was Dirac’s challenge.

  I’M DRAWN THAT WAY

  Coming up with a fully detailed quantum field theory for electrons and photons had big problems. Quite literally. A lot of the calculations yielded infinite answers or needed an infinite number of inputs to get to an answer, which obviously cannot be right for a finite universe (see Appendix IV for a slightly more detailed look).

  It is hard to say what made Feynman different to other geniuses of the same era, but I personally think it comes down to him being a physicist first and a mathematician second.

  Not to undersell him, he was a mathematical virtuoso second to none, but to him the equations were a language and not the final goal. You had to keep a focus on the stuff they described and not get bogged down in the symbols. Since the mathematical language being used by everyone to tackle QED was cumbersome and only generating partially correct answers, Feynman decided to invent a new type of maths to make life simpler.

  Start by picturing an electron minding its own business as it travels through the universe. In quantum field theory terms, we have to describe this as a quantum of energy in the electron field propagating from one place to another. We describe its trajectory with an equation called (reasonably enough) a propagator.

  In Feynman’s new mathematical system, we replace the electron propagator equation with nothing more complicated than an arrowed line.

  (NB: Strictly speaking this symbol represents ‘electron in motion’ and does not have to represent a straight line from bottom to top; it could also be going in a curve or circling a nucleus.)

  But now, as the electron moves on its intended course, an incoming photon approaches and gets absorbed into it, knocking it somewhere new. In quantum field theory terms we need to describe a quantum of the photon field meeting our electron and an energy transfer taking place between the two fields.

  We represent a photon propagator with a wiggly line and draw the interaction like so:

  Reading from the bottom we see an electron propagating through its field, interacting with the photon field (absorbing a photon) and then blasting off in some new direction as it absorbs the energy. Or, just as easily, it could be describing the reverse process – an electron emitting a photon and recoiling in another direction, like someone’s hand flying backward as they fire a gun.

  The point where the three lines join is the ‘vertex’ of the diagram and is handled mathematically with something called a coupling constant. A coupling constant is just a number that measures how easy it is for two fields to exchange energy. The higher the number, the more likely two quanta (particles) are to interact.

  The whole thing looks blindingly simple, but that is where the power of Feynman’s approach lies. Feynman diagrams cut out pages and pages of excessive maths jargon and whittle it all down to the essentials. Take the incoming electron propagator,
the photon propagator, the outgoing electron propagator and the coupling constant, multiply them together and you get a prediction for how an electron and photon will interact. A quantum electrodynamics theory which worked.

  CHARGE EXPLAINED… AT LAST

  An ordinary beam of light is made of photons obeying certain laws about direction, velocity and energy. But if two electrons pass each other they can exchange a photon like footballers passing a ball and, due to Heisenberg uncertainty, we cannot tell which way the photon actually moves. We can say a photon exchange takes place, but not which electron receives and which donates: that would give us too much information about momentum and position.

  These photon exchanges that occur between electrons are like temporary photon ripples rather than permanent beams of light passing between, so they are clearly not your average photons.

  Imagine two boats moving across the surface of a lake and passing within close proximity. As they do so, the wakes generated from each boat will meet in the space between, creating a temporary water disturbance that pushes both of them apart. The boats never touch, but this momentary fluctuation between them in the water field allows them to exchange energy, deflecting at angles rather than passing in a straight line.

  The toy boats (viewed from above in this diagram) represent electrons and the bulge in the water (the concentric circles between them) is the exchanged photon, which exists for only a moment as energy is transferred.

  We call these momentary ripples in the electromagnetic field ‘virtual photons’ to distinguish them from the actual, permanent photons that comprise beams of light. The same way you might call the wave that pushes two boats apart a swelling of the water, rather than a permanent wave, which roams the ocean independently.