This chapter contains both good news and a warning. The good news is that there will certainly be more than one fully correct way to look at quantum. In a sense, we have an infinity of choices. The warning is that it is perilously easy to make a choice that hinders progress rather than helping it, and we shall look at some pertinent cautionary tales.

There is an ancient idea that events here on Earth are merely the actualization of a game being played between gods. It appears in Greek legends, in the Norse sagas of the Vikings, and in folk tales from cultures all around the world. Most people no longer believe in a panoply of gods who can control human beings like pieces on a board, but in the past century the metaphor was revived by the great physicist Richard Feynman. He wrote:

As usual with Feynman’s insights, this opens up a rich vein of thought, going far beyond the immediate purpose for which he used the metaphor. One such development is remarkably empowering when it comes to interpreting quantum physics.

Many readers will have heard of game theory, a field of mathematics whose applications include finding optimal strategies in such fields as business, military conflict, and deterrence, where the decisions of others must be taken into account. Rather more obscure is games theory, which concerns itself with real-life games such as chess and bridge—usually games involving boards, cards, dice, and suchlike accessories. Yet games theory, too, has practical applications. For example, a lateral-thinking approach to proving a mathematical theorem is this. Imagine a game played between two mathematicians: A, who wants to prove the theorem, and B, who wants to refute it. If we can prove that A has a guaranteed winning strategy for the game, then the theorem is proved, without needing to work through the details of all the various moves A and B can make.

An important insight from games theory—indeed, the foundation on which the whole field is based—is the fact that many games that appear quite different are, in fact, algorithmically the same game. A trivial example is the different editions of the game Monopoly that are played in different countries. In each country the properties are labeled after districts of a great city. Thus in the American edition, the most expensive property is called Boardwalk; in the English, it is Mayfair, once London’s most fashionable area; in the French, it is the Rue de la Paix. But the rents and prices of the property remain the same in each case, so of course this relabeling makes no difference to the game. Nor does the fact that in the English version the prices are nominally in pounds, and the French in euros. In the old French version the prices were in francs, and because a dollar is worth about 10 francs, the numerical prices were all multiplied by exactly 10 with respect to the American version. The addition of an extra zero to all the currency bills and prices likewise had no effect on the play.

In the case of Monopoly, it is not hard for adults to distinguish the

algorithmic part of the game from the story component. When you get a card informing you that you have won a beauty contest, or that it is your birthday, the significant information is of course how many points (money) are to be transferred from whose account to whose. But children can have difficulty even at this level. A few years ago, on a long airplane flight, I found myself sitting next to a charming lady whose son was happily occupied with a book on the game Pokemon, then a worldwide fad among preteens. I chatted with the mother and we agreed that the game separated the world sharply by age group. Whereas children were obsessed with it, almost no adults—even those who were parents of young children—had any idea of even the basic rules or objective of the game. The kid overheard our conversation, and decided that it was his duty to educate us, with amusing but unproductive results. He was trying his best to describe how the game worked, but could think of no other way to start than with the story preamble, telling us how the Pokemon characters (represented by cards) were mouse-sized creatures carried in bottles on a special belt. When it came to the cards themselves, realizing that a grown-up approach was called for, he skipped embarrassedly past childish-looking creatures until he found something that fit the bill. “You’ll like this one,” he said proudly. “It’s the Great Green Brain-Blaster, it’s really good….”

Desperate though he was to describe the essence of the game, he found it impossible to reach the required level of abstraction: “What really matters is the points number on each corner of the card. You add the ones in the top left-hand corners together, then subtract….” Before we laugh too hard, we should reflect that adults, too, can find the distinction between story and essence harder than it seems.

Let us embark on a field trip. The idea is simple: to find some gods playing a game, observe them for a while, and figure out what the game is. A couple of thousand years ago, we would have climbed Mount Olympus but this being the 21st century, we will fly off in a spaceship until we discover some promising-looking gods, as shown in Figure 7-1.

We realize that the game these alien gods are playing might in fact

FIGURE 7-1 These aliens are playing a game that involves speaking words in alternation. By patient observation we discover that nine different words are used, and each is used a maximum of once per game. The players take turns to utter a word until the last one to speak wins. Games are a minimum of five and a maximum of nine words long (because all the permissible syllables have then been used). Nine-syllable games sometimes end in a draw with neither player winning, although shorter ones never do. We see that the rules are consistent. If a given sequence of syllables wins on one occasion, it does so on any subsequent occasion.

be quite simple, but unfortunately they are playing it in their heads, without use of board or counters. Yet it is vital that we learn the rules because in due course we want to be able to play against these gods and beat them. We might start by tabulating all the different possible games, as shown in Table 7-1, in which the first player’s moves are in

capitals and the second’s in lowercase. Unfortunately the table will be rather long; it could have up to nine-factorial entries, that is, 9×8×7×6×5×4×3×2×1 = 362,880. In practice it will be less than that, because many games end before a full nine moves have been played, but we will still be looking at a book the size of a telephone directory. As an aid to playing the game—telling us what move to make next to have the best chance of winning, even against a randomly playing god—it will be pretty much useless, at least without the aid of a computer.

Fortunately, patterns in the data soon become apparent. For example, not all the words appear equally potent. Whichever god says ag often wins that game. Flump, choo, nis, and doh are not nearly so useful; wibble, fizz, gah, and zig come somewhere in between. Eventually you work out the secret: there are eight “magic triples” of syllables, namely,

xig, flump, wibble

ni, ag, choo

gah, doh, fizz

xig, ni, gah

flump, ag, doh

wibble, choo, fizz

xig, ag, fizz

wibble ag, gah

The first god to include a complete magic triple in the words it has spoken—it does not matter in what order the words of the triple are called, or whether it says other words in between them—wins. This is in a sense a complete description of the game, and it is certainly more compact than the telephone-directory-length list of all possible games. Moreover, because the list of winning strings exhibits certain symmetries, the expedition’s mathematician could find ways to code the information still more compactly. But you still have no feel for what is going on in the gods’ heads as they play, and little confidence that you will win when the time comes for you to leave the ship and challenge one of them yourself.

Then the expedition’s physicist comes to you. “I have it!” he shouts triumphantly. “The aliens are playing a simple game with sticks. Here is my interpretation.

“They start with an imaginary pile of nine sticks measuring from 1 unit to 9 units in length. Each alien claims a stick from the pile by calling its length. I have cracked the code for the alien number system.” He writes down the following table:

flump = 1

gah = 2

choo = 3

fizz = 4

ag = 5

xig = 6

ni = 7

wibble = 8

doh = 9

“Each alien takes a stick in turn, adding it to his personal collection, until he has a set from which three sticks add up to exactly 15 units in length. Let me remind you of the first game we saw, the game that went: XIG flump WIBBLE ni AG choo DOH gah. After those moves, the first alien has chosen sticks of length 6, 8, 5, and 9 units. No three of these add to 15. The second alien has sticks of length 1, 7, 3, and 2 units. No three of these add to 15, either. But then the first alien

says FIZZ and claims the 4-stick. Now 6 plus 5 plus 4 equals 15, and he duly wins. I have solved the mystery: the aliens are playing the one-dimensional game Fill-the-Gap.”

You are in the middle of congratulating him when the cabin boy bursts in.

“Captain, I have solved it,” he shouts. “The aliens are playing a simple two-dimensional game! I have cracked the code for the alien position system.” He shows you the following table:

“Really, sir, all the aliens are doing is playing tick-tack-toe. Remember the first game we saw, the game that went XIG flump WIBBLE ni AG choo DOH gah? After those moves the board looks like this, where the first alien writes X and the second O:

“So far, neither alien has a line of three. But then the first alien calls ‘FIZZ’ and wins with a diagonal line.”

You scratch your head, completely baffled. Both of them seem to have an equally strong case. To whom are you going to give the bottle of whiskey you have promised as a prize? Are the aliens really playing a one-dimensional or a two-dimensional game?

As you ponder the matter, there comes a knock at the door: It is the expedition’s archaeologist.

“I think I can help,” he says. “I remembered the famous Rosetta stone, which carried the same message in three languages. It inspired me to draw a tick-tack-toe board labeled as follows.”

“Why, of course,” you exclaim, “it is the famous magic square: one whose every row, column, and diagonal add to 15. With this board, you can see instantly how the numerical and tick-tack-toe interpretations of the alien game are really one and the same thing. It was obvious that this had to be possible, when you think about it!”

The archaeologist departs rather hurt, but your problem of who wins the prize is not solved. So you call a meeting of the entire crew. The physicist and the cabin boy present their respective interpretations to general applause. But then the expedition’s anthropologist stands up.

“I have a better interpretation,” he says. “These aliens are gods who would never bother with anything as trivial as tick-tack-toe. What matters to gods is worshippers. These gods are obviously picking sa-

cred triads of worshippers from a species that has three sexes, male, female, and neuter; and three hair colors, black, blonde, and red. They are calling the names of a group of nine people who between them possess every combination of these characteristics.

“It follows from universal aesthetic laws that a god would want a triad of worshippers who are either as alike as possible, or as different as possible. Either three people all of the same sex, but who must have different hair colors; or three people all with the same hair color, but who must be of different sex; or, at the opposite extreme, three people each of different sex and different hair color than the others. A group of the latter type must, however, include ag. I can tell from universal psycholinguistic principles that xig, flump, and wibble are male; ni, ag, and choo female; and gah, do, and fizz neuter. Xig, ni, and gah are blondes; flump, ag, and doh redheads; and wibble, choo, and fizz dark-haired. Ag is uniquely important because her fiery hair and femininity symbolize the importance of contrast. I can also tell from universal aesthetic principles that the gods are imagining their worshippers gathering in a vestibule lined with red velvet, and choo is wearing a bronze amulet.”

Before you can comment on this, someone stands up and beats you to it. He is the expedition’s cultural relativist.

“You are all talking ze rubbish!” he says in a French accent. “You cannot possibly know what is going on in ze minds of zese alien gods. All you are doing eez projecting your own cultural prejudices. Trying to find what game zese aliens are playing is a futile exercise! You might as well choose any story.”

You can sympathize with his sentiments when it comes to the anthropologist’s absurdly rococo tale with its wealth of unverifiable detail. But surely he is being a bit hard on the other two interpretations? Then it occurs to you that they are, indeed, also colored by cultural subjectivity. The fact that the cabin boy used X’s and O’s as symbols on the tick-tack-toe board was certainly culturally determined; and you need two kinds of symbols or objects that are easily distinguishable from one another to play tick-tack-toe sensibly, whatever choice you make. Even the adds-to-15 system was culturally influenced. Why choose positive integers from 1 to 9? You could number

the squares instead from 0 to 8; then each line would add to 12. Or you could number the squares from −4 to +4, so that each line adds to zero. And why choose consecutive integers at all…? The only way to avoid spurious cultural overtones is to stick to the aridity of the mathematician’s minimalist algorithm, and not attempt to visualize.

Certainly, the cultural relativist is mistaken to claim that any story will do, because stories contain statements that can be falsified as well as ones that cannot. The anthropologist might have gone a bit over the top about the red velvet, but the eight winning triads described by his theory are the correct ones. There might be many correct tales to choose from, but there are even more incorrect ones; for example, any tale that predicted xig, choo, and doh were a winning triad would be wrong.

But it still looks as if you will have to split the bottle of whiskey not merely three ways, but potentially infinite ways, because games theory tells you that there is no end to the supply of correct games that can be invented. Then it occurs to you that there is a point to all this. You want to go out there and kick some alien ass, preferably all by yourself without the help of a computer. From that point of view, there is no question which interpretation of those you have seen is the winner. Human beings have evolved to be extremely good at processing two-dimensional patterns—and relatively weak at arithmetic and abstract logic. In this instance, tick-tack-toe should be your choice of arena. It is the cabin boy who should get the whiskey.

In the world of real physics, of course, we are not dealing with a two-player game like tick-tack-toe. Real physics is more like playing solitaire, seeing what cards turn up. But an immensely empowering insight follows from our study of games: There are bound to be many equally valid ways to look at the universe. We are thus free to pick whichever one we find most comfortable and useful to work with.

Note that I have not claimed—although many have—that the current interpretations of quantum theory are as equivalent as the superficially different forms of tick-tack-toe above. The question of whether they are experimentally distinguishable is one that we will address in the later chapters. The point I am making is that whatever further experimental discoveries there may be, we will always have a choice of

ways to visualize the universe. It is our duty to our successors, to those who must go out and battle the laws of physics on territory beyond that currently explored, to make that choice the best one we can at every stage.

It is wonderful to know that with sufficient ingenuity, there is almost no limit to the number of stories we can use to describe the universe. But there is a downside to such ingenuity. Its application can also twist a bad story, one that is not a good way to look at things, so as to make it irrefutable. We will look at two great cautionary examples from the history of science. Both are now generally described as disproved theories, but I will argue that they are merely inept interpretations. They cannot be proved wrong and it is only too plausible that if their proponents had been a little more ingenious, they might still be accepted wisdom—and our understanding of physics would be immeasurably poorer.

The first of these old interpretations is the concept of phlogiston, sometimes also referred to as calistogen. Phlogiston was postulated as an invisible substance that permeated all solids—and indeed all liquids and gases. It conferred the property of heat; the more phlogiston an object contained, the hotter it was. To phlogiston, all substances were porous, so that whenever you put a hot object in contact with a cold one, phlogiston flowed naturally from the hot to the cold until both had the same temperature, just like water flowing to equalize its level or gas to equalize its pressure.

Before the age of machinery, phlogiston really worked very well as an explanation of heat. Different substances differed in the amount of phlogiston they could contain per unit volume. In modern terms, we would say that they have different specific heats. Substances also differed in how readily phlogiston could flow through them; in modern terms, they have different thermal conductivities. That was natural enough; different substances also have different capacities to absorb and permit the flow of ordinary liquids through them (contrast what happens when a sponge, a book, and a house brick are placed in a

bucket of water). Phlogiston was compressible, but it possessed some kind of volume, because substances expand when they are heated. Indestructible phlogiston explains why heat is conserved—and to early scientists, it did seem to be conserved, because devices for turning heat into mechanical work functioned at extremely low efficiencies.

One problem with phlogiston was that it did not appear to have any detectable weight. But a far more serious difficulty became apparent with the start of the industrial age—and that was the apparent ability of machines to create new phlogiston. A turning shaft can generate unlimited heat at a point by means of friction, and this works even if the shaft is made of an insulating material so that little or no heat can flow along it. This simple fact was the downfall of the concept of phlogiston.

How lucky that its defenders were not as clever as modern philosophers of physics. If they were, they could have easily explained the apparent problem away. Because, of course, the shaft must have some device at the other end to turn it; for example, a steam turbine takes steam in at a high temperature and ejects it at a lower one. At that end of the device, heat is consumed and phlogiston is apparently vanishing. The process could be explained by the hypothesis of phlogiston tunneling—assuming that phlogiston just undetectably and instantly jumps from one place to another. Does this remind you of something?

Nowadays, we can even in a sense verify that phlogiston has weight. Einstein’s famous E = mc² equation predicts that energy has mass, and this includes heat energy. If you take two otherwise identical objects, each containing exactly the same number of atoms, the hot object does in fact weigh slightly more than the cold one. The difference was simply too small for 19th-century instruments to measure. We arguably had a very near miss with getting stuck with the notion of phlogiston, and failing to progress to the more general concept of energy.

A second famous example of a flawed scientific paradigm is the notion of epicycles. Ancient astronomers, trying to figure out the motion

of the planets in the heavens, were handicapped by not one, but two, false assumptions. The first was that Earth was itself stationary at the center of the motion. The second was that the planets, being perfect heavenly objects, must move in circles. Because the planets obviously did not move in simple circles, their motion was described in terms of epicycles. For each planet, an invisible pivot point did move in a perfect circle, and the planet moved (on an invisible arm) about the pivot point in a second smaller circle. This idea could crudely approximate the motions of the actual planets as seen in the sky, but for greater accuracy it was necessary to postulate second, third, and even fourth epicycles.

Then came Copernicus, Galileo, and Kepler. As everyone knows, the new hypothesis, that the Sun was the true center of the system, with the other planets including Earth orbiting around it, displaced the old assumption. What is not so well appreciated is that the idea of epicycles need not have died at that point. Kepler discovered that the planets do not orbit the Sun in perfect circles, but in ellipses; and they do not move at constant speed, but faster when they are nearer the Sun, and slower when they are farther away. But this kind of motion can be explained quite well in terms of epicycles. If we assume that each planet has an epicycle whose diameter is equal to the difference between the planet’s nearest and farthest distances from the Sun, and that the direction of the epicycle is retrograde—that is, it turns in the direction opposite to the motion of the main arm—then the planet will indeed move fastest at its closest approach to the Sun, and more slowly as the distance increases. We are lucky that Kepler was a stickler for accuracy, and rejected this tempting fudge.

How fortunate it is that he did not have access to modern mathematical techniques and computers. Because we now know that just as a technique called Fourier analysis can approximate a two-dimensional graph as a sum of an infinite series of sine waves (a technique often used in applied mathematics and engineering), so any three-dimensional motion—not just elliptical orbits—can be approximated to any desired degree of accuracy as the sum of an infinite series of circular motions. A sufficiently clever mathematician could even work out a formula for predicting the epicycles of an object, like a comet or

spacecraft, entering our solar system for the first time. If the mathematics of Kepler’s day had been advanced enough, we might have been stuck with the concept of epicycles.

This would have produced an odd puzzle when relativity was discovered, because in some circumstances (planets orbiting a neutron star, for example), the imaginary pivot points of the epicycles could perfectly well be moving faster than light. Scientists would struggle to explain how, although the invisible arms propelling them moved faster than light, the motions of the epicycles fortunately always seemed to cancel at the right moments, so that the actual planets never broke the speed limit.

Do epicycles remind you at all of the imaginary waves of quantum?

But enough of the negative. Before we pick our favorite story of quantum, let us look at the approaches that have worked well in developing the other aspects of the scientific world-picture we accept today.