Why Beauty is Truth Page 27

  Kaluza had not set out to unify gravity and electromagnetism. For some reason best known to him, he had been messing around with five-dimensional gravity, a kind of mathematician’s warmup exercise, working out how Einstein’s field equations would look if space had that absurd extra dimension.

  In four dimensions the Einstein equations have ten “components”—they boil down to ten separate equations describing ten separate numbers. These numbers jointly constitute the metric tensor, which describes the curvature of space-time. In five dimensions there are fifteen components, hence fifteen equations. Ten of them reproduce Einstein’s standard four-dimensional theory, which is no surprise; four-dimensional space-time is embedded in five-dimensional space-time, so you would naturally expect the four-dimensional version of gravity to be embedded in the five-dimensional one. What about the remaining five equations? They could have been just some peculiar structure with no significance for our own world. But they weren’t. Instead, they were very familiar, and that’s what amazed Einstein. Four of Kaluza’s remaining equations were precisely Maxwell’s equations for the electromagnetic field, the ones that hold in our four-dimensional space-time.

  The one remaining equation described a very simple kind of particle, which played an insignificant role. But no one, least of all Kaluza, had expected both Einstein’s theory of gravity and Maxwell’s theory of electromagnetism to emerge spontaneously from the five-dimensional analogue of gravity alone. Kaluza’s calculation seemed to be saying that light is a vibration in an extra, hidden dimension of space. You could put gravity and electromagnetism together into a seamless whole, but only by supposing that space is really four-dimensional and space-time is five-dimensional.

  Einstein agonized over Kaluza’s paper, because there was absolutely no reason to imagine that space-time has an extra dimension. But eventually he decided that however strange the idea might seem, it was so beautiful, and potentially so far-reaching, that it should be published. After dithering for two years, Einstein accepted Kaluza’s paper for a major physics journal. Its title was “On the unity of problems of physics.”

  All this talk of extra dimensions probably sounds rather vague and mystical. It is a concept associated with Victorian spiritualists, who invoked the fourth dimension as a convenient place to hide everything that didn’t make sense in the familiar three. Where do spirits live? In the fourth dimension. Where does ectoplasm come from? The fourth dimension. Theologians even placed God and His angels there until they realized that the fifth was better and the sixth better still, and that finally only an infinite dimension would really do for an omniscient and omnipresent entity.

  All great fun, but not terribly scientific. So it is worth digressing to clarify the underlying mathematics. The main point is that the “dimension” of some mathematical or physical setup is the number of distinct variables needed to describe it.

  Scientists spend a lot of time thinking about variables—quantities that are subject to change. Experimental scientists spend even more time measuring them. “Dimension,” which is just a geometric way to refer to a variable, has turned out to be so useful that it is now built into science and mathematics as a standard way of thinking and is considered to be entirely prosaic and unremarkable.

  Time is a nonspatial variable, so it provides a possible fourth dimension, but the same goes for temperature, wind speed, or the lifespan of termites in Tanzania. The position of a point in three-dimensional space depends on three variables—its distances east, north, and upward relative to some reference point, using negative numbers for the opposite directions. By analogy, anything that depends on four variables lives in a four-dimensional “space,” and anything that depends on 101 variables lives in a 101-dimensional space.

  Any complex system is inherently multidimensional. The weather conditions in your backyard depend on temperature, humidity, three components of wind velocity, barometric pressure, intensity of rainfall—that’s seven dimensions already, and there are plenty of others we might include. I bet you didn’t realize you had a seven-dimensional backyard. The state of the nine (well, eight; alas, poor Pluto!) planets in the solar system is determined by six variables for each planet—three positional coordinates and three components of velocity. So our solar system is a 54- (I mean 48)-dimensional mathematical object; and many more if you include satellites and asteroids. An economy with a million different commodities, each having its own price, lives in a million-dimensional space. Electromagnetism, which requires only six extra numbers to characterize the local states of the electric and magnetic fields, is child’s play by comparison. Examples like these abound. As science became interested in systems with large numbers of variables, it was forced to come to grips with extravagantly multidimensional spaces.

  The formal mathematics of multidimensional spaces is purely algebraic, based on “obvious” generalizations from low-dimensional spaces. For example, every point in the plane (a two-dimensional space) can be specified by two coordinates, and every point in three-dimensional space can be specified by three coordinates. It is a short step to define a point in four-dimensional space as a list of four coordinates, and more generally to define a point in n-dimensional space as a list of n coordinates. Then n-dimensional space itself (or n-space for short) is just the set of all such points.

  Similar algebraic machinations let you work out the distance between any two points in n-space, the angle between any two lines, and so on. From there on out, it’s a matter of imagination: most sensible geometric shapes in two or three dimensions have straightforward analogues in n dimensions, and the way to find them is to describe the familiar shapes using the algebra of coordinates and then extend that description to n coordinates.

  To get a feel for n-space, we must somehow equip ourselves with n-dimensional spectacles. We can borrow a trick from the English clergyman and schoolmaster Edwin Abbott Abbott, who in 1884 wrote a short book called Flatland. It is about the adventures of A. Square, who lived in the two-dimensional space of a Euclidean plane. Abbott does not tell us what the initial “A” stands for: I am convinced it should be “Albert,” for reasons explained in my sequel Flatterland, and I will make that assumption here. Albert Square, a sensible sort, did not believe in the absurd notion of the third dimension until, one fateful day, a sphere passed through his planar universe and flung him into realms he could never have imagined.

  Flatland was a satirical look at Victorian society embedded in a parable about the fourth dimension based on a transdimensional analogy. It’s the analogy, not the satire, that concerns us here. Having successfully imagined yourself as a two-dimensional creature living in a plane, blissfully unaware of the greater reality of 3-space, it is not so hard to imagine yourself as a three-dimensional creature living in 3-space, blissfully unaware of the greater reality of 4-space. Suppose Albert Square, sitting happily in Flatland, wants to “visualize” a solid sphere. Abbott achieves this by making such a sphere pass through the plane of Flatland, moving perpendicular to it so that Albert sees its cross-sections. First he sees a point, which grows to a circular disk. The disk expands until he is seeing the equator of the sphere, after which it shrinks again to a point and then vanishes.

  The sphere encounters Flatland.

  Actually, Albert sees these disks edge-on, as line segments with graded shading, but his visual sense interprets this image as a disk, just as our stereo vision interprets a flat image as being solid.

  By analogy, we can “see” a “hypersphere,” the four-dimensional analogue of a solid sphere, as a point that grows to form a sphere, expands until we see the “equator,” and then shrinks back to a point before disappearing.

  Could space really have more than three dimensions? Not fancy mathematical fictions corresponding to nonspatial variables, but real physical space? After all, how can you fit the fourth dimension in? Everything’s filled up already.

  If you think that, you didn’t listen to Albert Square, who would have argued exactly the same way abo
ut the plane. Ignoring our parochial prejudices, it seems that in principle, space might have been 4-dimensional, or million-dimensional, or whatever. Everyday observation, however, informs us that in our particular universe the good Lord settled on three dimensions for space, plus one for time.

  Or did He? Whatever physics teaches us, one lesson is to be wary of everyday observation. A chair feels solid, but it’s mostly empty space. Space looks flat, but according to relativity it’s curved. Quantum physicists think that on very small scales space is a kind of quantum foam, mostly holes. And devotees of the “many worlds” interpretation of quantum uncertainty believe our universe is one of an infinite variety of coexisting universes, and that we occupy just a wafer-thin slice of a vast multiverse. If common sense can mislead us about those things, maybe it’s wrong about the dimensionality of space or time.

  The hypersphere encounters Spaceland.

  From a distance (above), a hosepipe looks one-dimensional. From close up (below) it has two additional dimensions.

  Kaluza had a simple explanation for the extra dimension his theory assigned to space-time. The traditional dimensions point along straight lines, long enough to observe, billions of light years long, in fact. The new dimension, Kaluza suggested, is very different: it curls up tightly into a circle much smaller than an atom. The ripples that constitute light waves can move round the circle, because they, too, are much smaller than atoms, but matter cannot move in that direction because there isn’t enough room.

  This isn’t such a silly idea. If you look at a hosepipe from a distance, the pipe looks like a curve, which is one-dimensional. Only from close up does it become clear that the pipe is really three-dimensional, with a small two-dimensional cross-section. This hidden structure in new dimensions explains something that you can observe from a distance: how the hose is able to carry water. The cross-section just needs to have the right shape, with a central hole. Now imagine that the thickness of the hose is less than the size of an atom. You would have to look extraordinarily closely to notice the extra dimensions. The incredibly thin hosepipe would no longer be able to carry water, but anything sufficiently small could still travel along it.

  So it may be possible to perceive the effect of extra dimensions without perceiving the dimensions themselves. That means that hidden dimensions of space-time are an entirely scientific suggestion: their presence can in principle be tested—but by inference, rather than by a direct use of the senses. Most scientific tests work by inference—if you could see the cause of some phenomenon directly, you wouldn’t need theories or experiments. No one has ever seen an electromagnetic field, for example. They have seen sparks and watched compass needles swing to point north, and (if they are scientists) they have inferred that a field must be responsible.

  Kaluza’s theory gained a certain popularity because it was the only known idea that held out hope of a unified field theory. In 1926, another mathematician, Oskar Klein, improved Kaluza’s theory with the suggestion that quantum mechanics might explain why that fifth dimension curled up so tightly. In fact, its size should be of a similar order of magnitude to Planck’s constant: the “Planck length” of 10–35 meters.

  For a while, physicists were attracted to Kaluza–Klein theory, as it became known. But the impossibility of directly demonstrating the presence of that extra dimension preyed on their minds. By definition, Kaluza–Klein theory was consistent with every known phenomenon in gravitation and electromagnetism. You could never disprove it with standard experiments. But it didn’t really add anything; it didn’t predict anything new that could be tested. The same problem bedevils many attempts to unify existing laws. What’s testable is already known, and what’s new isn’t testable. The initial enthusiasm began to wane.

  The deathblow for Kaluza–Klein theory—not whether it was right, but whether it was worth spending precious research time on—was the explosive growth of a much sexier theory, one in which you really could make new predictions and do experiments to test them. This was quantum theory, then in its first flush of youth.

  By the 1960s, though, quantum mechanics was running out of steam. Early progress had given way to deep puzzles and inexplicable observations. Quantum theory’s success was undeniable, and it would shortly lead to the “standard model” of fundamental particles. But it was becoming ever more difficult to find new questions that had any chance of being answered. Genuinely novel ideas were too hard to test; testable ideas were mere extensions of existing ones.

  One very elegant underlying principle had emerged from all the research: the key to the structure of matter on very tiny scales is symmetry. But the important symmetries for fundamental particles are not the rigid motions of Euclidean space, not even the Lorentz transformations of relativistic space-time. They include “gauge symmetries” and “supersymmetries.” And there are other kinds of symmetry, too, more like the symmetries studied by Galois, which act by permuting a discrete set of objects.

  How can there be different kinds of symmetry?

  Symmetries always form a group, but there are many different ways in which a group can act. It might act by rigid motions such as rotations, by permuting components, or by reversing the flow of time. Particle physics led to the discovery of a new way for symmetries to act, called gauge symmetries. The term is a historical accident, and a better name would be local symmetries.

  Suppose that you are traveling in another country—let us call it Duplicatia—and you need money. The Duplicatian currency is the pfunnig, and the exchange rate is two pfunnigs to the dollar. You find this confusing until you notice a very simple and obvious rule for translating dollar transactions into pfunnigs. Namely, everything costs twice as many pfunnigs as you would expect to pay in dollars.

  This is a kind of symmetry. The “laws” of commercial transactions are unchanged if you double all the numbers. To compensate for the numerical difference, though, you have to pay in pfunnigs, not in dollars. This “invariance under change of monetary scale” is a global symmetry of the rules for commercial transactions. If you make the same change throughout, the rules are invariant.

  But now . . . Just across the border, in neighboring Triplicatia, the local currency is the boodle, and these are valued at three to the dollar. When you take a day trip to Triplicatia, the corresponding symmetry requires all sums to be multiplied by three. But again the laws of commerce remain invariant.

  Now we have a “symmetry” that differs from one place to another. In Duplicatia, it is multiplication by two; in Triplicatia, by three. You would not be surprised to find that on visiting Quintuplicatia the corresponding multiple is five. All of these symmetry operations can be applied simultaneously, but each is valid only in the corresponding country. The laws of commerce are still invariant, but only if you interpret the numbers according to the correct local currency.

  This local rescaling of currency transactions is a gauge symmetry of the laws of commerce. In principle, the exchange rate could be different at every point of space and time, but the laws would still be invariant provided you interpreted all transactions in terms of the local value of the “currency field.”

  Quantum electrodynamics combines special relativity and electromagnetism. It was the first physical unification since Maxwell’s, and it is based on a gauge symmetry of the electromagnetic field.

  We have seen that electromagnetism is symmetric under the Lorentz group of special relativity. This group consists of global space-time symmetries; that is, its transformations must be applied consistently throughout the entire universe if we want to preserve Maxwell’s equations. However, Maxwellian electromagnetism also has a gauge symmetry, which is vital to quantum electrodynamics. This symmetry is a change of phase in light.

  Any wave consists of regular wobbles. The maximum size of the wobble is the amplitude of the wave. The time at which the wave hits that maximum is called its phase; the phase tells you when and where the peak value occurs. What really matters is not the absolute phase of any given
wave but the difference in phases between two distinct waves. For example, if the phase difference of two otherwise identical waves is half the period (the time between maximum heights), then one wave hits its maximum exactly out of step with the other one, so the peaks of one coincide with the troughs of the other.

  When you walk along the street, you left foot is half a period out of phase with your right foot. When an elephant walks along the street, successive feet hit the ground at phases 0, ¼, ½, and ¾ of the full period; first the left rear, then the left front, then the right rear, then the right front. You can appreciate that if we started counting from 0 at a different foot, we would get different numbers—but the phase differences would still be 0, ¼, ½, and ¾. So relative phases are well defined and physically meaningful.

  Suppose a beam of light is passing through some complicated system of lenses and mirrors. The way it behaves turns out not to be sensitive to the overall phase. A change of phase is equivalent to a small time delay in making observations, or a resetting of the observer’s clock. It does not affect the geometry of the system or the path of the light. Even if two light waves overlap, nothing changes, provided both waves have their phases shifted by the same amount.

  Effect of a phase shift on a wave.

  So far, “change phase” is a global symmetry. But if an alien experimentalist somewhere in the Andromeda Galaxy changed the phase of light in one of its experiments, we would not expect to notice any effect inside a terrestrial laboratory. So the phase of light can be changed at will at all locations in space and time, and the laws of physics should remain invariant. The possibility of changing the phase arbitrarily at each point of space-time, with no global constraint to make the same change everywhere, is a gauge symmetry of Maxwell’s equations, and it carries over into the quantum version of those equations, quantum electrodynamics.