by Brian Cox
We’ll first state the rule without any justification. We will come back to discuss just why the rule looks like it does in a few paragraphs, but for now we should treat it as one of the rules in a game. Here’s the rule: at a time t in the future, a clock a distance x from the original clock has its hand wound in an anti-clockwise direction by an amount proportional to x2; the amount of winding is also proportional to the mass of the particle m and inversely proportional to the time t. In symbols, this means we are to wind the clock hand anti-clockwise by an amount proportional to mx2/t. In words, it means that there is more winding for a more massive particle, more winding the further away the clock is from the original, and less winding for a bigger step forward in time. This is an algorithm – a recipe if you like – that tells us exactly what to do to work out what a given arrangement of clocks will look like at some point in the future. At every point in the universe, we draw a new clock with its hand wound around by an amount given by our rule. This accounts for our assertion that the particle can, and indeed does, hop from its initial position to each and every other point in the Universe, spawning new clocks in the process.
To simplify matters we have imagined just one initial clock, but of course at some instant in time there might already be many clocks, representing the fact that the particle is not at some definite location. How are we to figure out what to do with a whole cluster of clocks? The answer is that we are to do what we did for one clock, and repeat that for each and every one of the clocks in the cluster. Figure 4.2 illustrates this idea. The initial set of clocks are represented by the little circles, and the arrows indicate that the particle hops from the site of every initial clock to the point X, ‘depositing’ a new clock in the process. Of course, this delivers one new clock to X for every initial clock, and we must add all these clocks together in order to construct the final, definitive clock at X. The size of this final clock’s hand gives us the chance of finding the particle at X at the later time.
It is not so strange that we should be adding clocks together when several arrive at the same point. Each clock corresponds to a different way that the particle could have reached X. This addition of the clocks is understandable if we think back to the double-slit experiment; we are simply trying to rephrase the wave description in terms of clocks. We can imagine two initial clocks, one at each slit. Each of these two clocks will deliver a clock to a particular point on the screen at some later time, and we must add these two clocks together in order to obtain the interference pattern.2 In summary therefore, the rule to calculate what the clock looks like at any point is to transport all the initial clocks to that point, one by one, and then add them together using the addition rule we encountered in the previous chapter.
Figure 4.2. Clock hopping. The open circles indicate the locations of the particle at some instant in time; we are to associate a clock with each point. To compute the probability to find the particle at X we are to allow the particle to hop there from all of the original locations. A few such hops are indicated by the arrows. The shape of the lines does not have any meaning and it certainly does not mean that the particle travels along some trajectory from the site of a clock to X.
Since we developed this language in order to describe the propagation of waves, we can also think about more familiar waves in these terms. The whole idea, in fact, goes back a long way. Dutch physicist Christiaan Huygens famously described the propagation of light waves like this as far back as 1690. He did not speak about imaginary clocks, but rather he emphasized that we should regard each point on a light wave as a source of secondary waves (just as each clock spawns many secondary clocks). These secondary waves then combine to produce a new resultant wave. The process repeats itself so that each point in the new wave also acts as a source of further waves, which again combine, and in this way a wave advances.
We can now return to something that may quite legitimately have been bothering you. Why on earth did we choose the quantity mx2/t to determine the amount of winding of the clock hand? This quantity has a name: it is known as the action, and it has a long and venerable history in physics. Nobody really understands why Nature makes use of it in such a fundamental way, which means that nobody can really explain why those clocks get wound round by the amount they do. Which somewhat begs the question: how did anyone realize it was so important in the first place? The action was first introduced by the German philosopher and mathematician Gottfried Leibniz in an unpublished work written in 1669, although he did not find a way to use it to make calculations. It was reintroduced by the French scientist Pierre-Louis Moreau de Maupertuis in 1744, and subsequently used to formulate a new and powerful principle of Nature by his friend, the mathematician Leonard Euler. Imagine a ball flying through the air. Euler found that the ball travels on a path such that the action computed between any two points on the path is always the smallest that it can be. For the case of a ball, the action is related to the difference between the kinetic and potential energies of the ball.3 This is known as ‘the principle of least action’, and it can be used to provide an alternative to Newton’s laws of motion. At first sight it’s a rather odd principle, because in order to fly in a way that minimizes the action, the ball would seem to have to know where it is going before it gets there. How else could it fly through the air such that, when everything is done, the quantity called the action is minimized? Phrased in this way, the principle of least action sounds teleological – that is to say things appear to happen in order to achieve a pre-specified outcome. Teleological ideas generally have a rather bad reputation in science, and it’s easy to see why. In biology, a teleological explanation for the emergence of complex creatures would be tantamount to an argument for the existence of a designer, whereas Darwin’s theory of evolution by natural selection provides a simpler explanation that fits the available data beautifully. There is no teleological component to Darwin’s theory – random mutations produce variations in organisms, and external pressures from the environment and other living things determine which of these variations are passed on to the next generation. This process alone can account for the complexity we see in life on Earth today. In other words, there is no need for a grand plan and no gradual assent of life towards some sort of perfection. Instead, the evolution of life is a random walk, generated by the imperfect copying of genes in a constantly shifting external environment. The Nobel-Prize-winning French biologist Jacques Monod went so far as to define a cornerstone of modern biology as ‘the systematic or axiomatic denial that scientific knowledge can be obtained on the basis of theories that involve, explicitly or not, a teleological principle’.
As far as physics is concerned, there is no debate as to whether or not the least action principle actually works, for it allows calculations to be performed that correctly describe Nature and it is a cornerstone of physics. It can be argued that the least action principle is not teleological at all, but the debate is in any case neutralized once we have a grasp of Feynman’s approach to quantum mechanics. The ball flying through the air ‘knows’ which path to choose because it actually, secretly, explores every possible path.
How was it discovered that the rule for winding the clocks should have anything to do with this quantity called the action? From a historical perspective, Dirac was the first to search for a formulation of quantum theory that involved the action, but rather eccentrically he chose to publish his research in a Soviet journal, to show his support for Soviet science. The paper, entitled ‘The Lagrangian in Quantum Mechanics’, was published in 1933 and languished in obscurity for many years. In the spring of 1941, the young Richard Feynman had been thinking about how to develop a new approach to quantum theory using the Lagrangian formulation of classical mechanics (which is the formulation derived from the principle of least action). He met Herbert Jehle, a visiting physicist from Europe, at a beer party in Princeton one evening, and, as physicists tend to do when they’ve had a few drinks, they began discussing research ideas. Jehle remembered Dirac’s obs
cure paper, and the following day they found it in the Princeton Library. Feynman immediately started calculating using Dirac’s formalism and, in the course of an afternoon with Jehle looking on, he found that he could derive the Schrödinger equation from an action principle. This was a major step forward, although Feynman initially assumed that Dirac must have done the same because it was such an easy thing to do; easy, that is, if you are Richard Feynman. Feynman eventually asked Dirac whether he’d known that, with a few additional mathematical steps, his 1933 paper could be used in this way. Feynman later recalled that Dirac, lying on the grass in Princeton after giving a rather lacklustre lecture, simply replied, ‘No, I didn’t know. That’s interesting.’ Dirac was one of the greatest physicists of all time, but a man of few words. Eugene Wigner, himself one of the greats, commented that ‘Feynman is a second Dirac, only this time human.’
To recap: we have stated a rule that allows us to write down the whole array of clocks representing the state of a particle at some instant in time. It’s a bit of a strange rule – fill the Universe with an infinite number of clocks, all turned relative to each other by an amount that depends on a rather odd but historically important quantity called the action. If two or more clocks land at the same point, add them up. The rule is built on the premise that we must allow a particle the freedom to jump from any particular place in the Universe to absolutely anywhere else in an infinitesimally small moment. We said at the outset that these outlandish ideas must ultimately be tested against Nature to see whether anything sensible emerges. To make a start on that, let’s see how something very concrete, one of the cornerstones of quantum theory, emerges from this apparent anarchy: Heisenberg’s Uncertainty Principle.
Heisenberg’s Uncertainty Principle
Heisenberg’s Uncertainty Principle is one of the most misunderstood parts of quantum theory, a doorway through which all sorts of charlatans and purveyors of tripe4 can force their philosophical musings. He presented it in 1927 in a paper entitled ‘Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik’, which is very difficult to translate into English. The difficult word is anschaulich, which means something like ‘physical’ or ‘intuitive’. Heisenberg seems to have been motivated by his intense annoyance that Schrödinger’s more intuitive version of quantum theory was more widely accepted than his own, even though both formalisms led to the same results. In the spring of 1926, Schrödinger was convinced that his equation for the wavefunction provided a physical picture of what was going on inside atoms. He thought that his wavefunction was a thing you could visualize, and was related to the distribution of electric charge inside the atom. This turned out to be incorrect, but at least it made physicists feel good during the first six months of 1926: until Born introduced his probabilistic interpretation.
Heisenberg, on the other hand, had built his theory around abstract mathematics, which predicted the outcomes of experiments extremely successfully but was not amenable to a clear physical interpretation. Heisenberg expressed his irritation to Pauli in a letter on 8 June 1926, just weeks before Born threw his metaphorical spanner into Schrödinger’s intuitive approach. ‘The more I think about the physical part of Schrödinger’s theory, the more disgusting I find it. What Schrödinger writes about the Anschaulichkeit of his theory … I consider Mist.’ The translation of the German word mist is ‘rubbish’ or ‘bullshit’ … or ‘tripe’.
What Heisenberg decided to do was to explore what an ‘intuitive picture’, or Anschaulichkeit, of a physical theory should mean. What, he asked himself, does quantum theory have to say about the familiar properties of particles such as position? In the spirit of his original theory, he proposed that a particle’s position is a meaningful thing to talk about only if you also specify how you measure it. So you can’t ask where an electron actually is inside a hydrogen atom without describing exactly how you’d go about finding out that information. This might sound like semantics, but it most definitely is not. Heisenberg appreciated that the very act of measuring something introduces a disturbance, and that as a result there is a limit on how well we can ‘know’ an electron. Specifically, in his original paper, Heisenberg was able to estimate what the relationship is between how accurately we can simultaneously measure the position and the momentum of a particle. In his famous Uncertainty Principle, he stated that if Δx is the uncertainty in our knowledge of the position of a particle (the Greek letter Δ is pronounced ‘delta’, so Δx is pronounced ‘delta x’) and Δp is the corresponding uncertainty in the momentum, then
where h is Planck’s constant and the ‘∼’ symbol means ‘is similar in size to’. In words, the product of the uncertainty in the position of a particle and the uncertainty in its momentum will be roughly equal to Planck’s constant. This means that the more accurately we identify the location of a particle, the less well we can know its momentum, and vice versa. Heisenberg came to this conclusion by contemplating the scattering of photons off electrons. The photons are the means by which we ‘see’ the electron, just as we see everyday objects by scattering photons off them and collecting them in our eyes. Ordinarily, the light that bounces off an object disturbs the object imperceptibly, but that is not to deny our fundamental inability to absolutely isolate the act of measurement from the thing one is measuring. One might worry that it could be possible to beat the limitations of the Uncertainty Principle by devising a suitably ingenious experiment. We are about to show that this is not the case and the Uncertainty Principle is absolutely fundamental, because we are going to derive it using only our theory of clocks.
Deriving Heisenberg’s Uncertainty Principle from the Theory of Clocks
Rather than starting with a particle at a single point, let us instead think about a situation where we know roughly where the particle is, but we don’t know exactly where it is. If we know that a particle is somewhere in a small region of space then we should represent it by a cluster of clocks filling that region. At each point within the region there will be a clock, and that clock will represent the probability that the particle will be found at that point. If we square up the lengths of all the clock hands at every point and add them together, we will get 1, i.e. the probability to find the particle somewhere in the region is 100 per cent.
In a moment we are going to use our quantum rules to perform a serious calculation, but first we should come clean and say that we have failed to mention an important addendum to the clock-winding rule. We didn’t want to introduce it earlier because it is a technical detail, but we won’t get the correct answers when it comes to calculating actual probabilities if we ignore it. It relates to what we said at the end of the previous paragraph.
If we begin with a single clock, then the hand must be of length 1, because the particle must be found at the location of the clock with a probability of 100 per cent. Our quantum rule then says that, in order to describe the particle at some later time, we should transport this clock to all points in the Universe, corresponding to the particle leaping from its initial location. Clearly we cannot leave all of the clock hands with a length of 1, because then our probability interpretation falls down. Imagine, for example, that the particle is described by four clocks, corresponding to its being at four different locations. If each one has a size of 1 then the probability that the particle is located at any one of the four positions would be 400 per cent and this is obviously nonsense. To fix this problem we must shrink the clocks in addition to winding them anti-clockwise. This ‘shrink rule’ states that after all of the new clocks have been spawned, every clock should be shrunk by the square root of the total number of clocks.5 For four clocks, that would mean that each hand must be shrunk by √4, which means that each of the four final clocks will have a hand of length ½. There is then a (½)2 = 25 per cent chance that the particle will be found at the site of any one of the four clocks. In this simple way we can ensure that the probability that the particle is found somewhere will always total 100 per cent. Of course, there may be a
n infinite number of possible locations, in which case the clocks would have zero size, which may sound alarming, but the maths can handle it. For our purposes, we shall always imagine that there are a finite number of clocks, and in any case we will never actually need to know how much a clock shrinks.
Let’s get back to thinking about a Universe containing a single particle whose location is not precisely known. You can treat the next section as a little mathematical puzzle – it may be tricky to follow the first time through, and it may be worth rereading, but if you are able to follow what is going on then you’ll understand how the Uncertainty Principle emerges. For simplicity, we’ve assumed that the particle moves in one dimension, which means it is located somewhere on a line. The more realistic three-dimensional case is not fundamentally different – it’s just harder to draw. In Figure 4.3 we’ve sketched this situation, representing the particle by a line of three clocks. We should imagine that there are many more than this – one at every possible point that the particle could be – but this would be very hard to draw. Clock 3 sits at the left side of the initial clock cluster and clock 1 is at the right side. To reiterate, this represents a situation in which we know that the particle starts out somewhere between clocks 1 and 3. Newton would say that the particle stays between clocks 1 and 3 if we do nothing to it, but what does the quantum rule say? This is where the fun starts – we are going to play with the clock rules to answer this question.
Figure 4.3. A line of three clocks all reading the same time: this describes a particle initially located in the region of the clocks. We are interested in figuring out what the chances are of finding the particle at the point X at some later time.