What This Chapter Is About

The titles of most of the chapters in this book convey, I hope, something to the reader. The title of this chapter is an exception. The idea that there is a quantity called action, which takes its least value when the equations of motion are obeyed, is now one of the foundations of classical (as opposed to quantum) physics, and even in quantum physics the action is a basic quantity. This idea is not easily expressed in everyday terms, but I think I would be guilty of some sort of distortion if I omitted it from this book. So what is it all about?

In Section 4.4, I explained Fermat's principle of least time as applied to light rays. The principle says that the path taken by a light ray between two given points is such as to make the time taken by the light a minimum. The principle of least action is an extension of this sort of idea to the motion of particles or to any other timevarying system, like the electromagnetic field. The difference from the case of light is that it is now not just the time that is a minimum, but some less obvious quantity, called action. One needs some rule to decide what the action is for a given system. This rule, indirectly, defines the forces operating in the system.

I will use the word least throughout, but as I will explain in Section 6.3, this is not quite accurate. In some cases, we need a property a little more general than being least (i.e., minimum), that is, the property of being stationary.

Waves

This chapter is about the nature of light. Section 4.8 describes one of the great unifications of physics: the demonstration that light is just part of electricity and magnetism.

The first thing to be explained is the wave nature of light, so I begin by saying what is meant by a wave. We are all familiar with water waves, but I will define a wave in a general way.

In a wave, a shape propagates over a long distance, but matter (or whatever the wave is “in”) moves only locally. For example, if a stone is dropped into the middle of a sizeable pond, waves may be propagated to the edge of the pond. But the actual water is only moving locally. For example, the surface goes up and down.

We need to define one or two terms. The simplest sort of wave is what is called a simple harmonic wave (the name comes from the connection with musical notes). Here the shape is like that of a corrugated surface. To define it mathematically, we can imagine doing the following. (See Figure 4.1.)

Take a wheel with a peg on its side. Take a vertical pen with a slot in its stem and with the peg in the slot. Let the wheel rotate at a steady rate, so that the pen is moved from side to side (remaining always vertical). Let the nib of the pen rest on a long sheet of paper (under the wheel), which is being pulled at a constant speed in the direction of the wheel's axis.

The object of this appendix is to explain the property of Brownian motion mentioned in Section 2.5. We will start with a simple model situation (whose relevance may not be immediately apparent). Two people toss a coin for money. If a head comes up, he pays her one pound. If a tail comes up, she pays him one pound. On average, if the game is played many times, each player's winnings and losses will tend to cancel out.

Now suppose that each day the players play a sequence of a certain fixed number of tosses. Call this number n. At the end of the day, they note how many pounds he has paid her. Call this amount the winnings, W. Note that if he has won, W is a negative number. Now square this: W2. (Never mind what a “square pound” means.) Note that the square of a number is always positive, whether the number itself is positive or negative. Now average over lots of days. The average value of W will be zero (if it were not, we would suspect that the coin was biased). But the average value of W2 is certainly not zero, because each value of W2 is positive (or perhaps zero). In fact, it can be shown that the average value of W2 is just the number n (the number of games each day).

I will illustrate this with two examples. First suppose n = 2. The possible outcomes of the games are HH, HT, TH, TT, where H stands for head and T for tail. For each of these outcomes, the values of W (the winnings) are 2, 0, 0, −2.

Newton's Principia is written (in Latin) in a lofty, austere way, as if to allow the reader no opportunity to disagree. The Opticks published (in English) 17 years later is more human. It ends with several Queries, in which Newton speculates, without claiming certitude.

I have ventured to borrow Newton's word as the title of this chapter. It is meant to be a warning to the reader that I am now venturing off the fairly well-beaten track followed in the preceding 14 chapters. The speculations that follow are not mine, of course. I have chosen ideas that seem to me to have attracted the attention of the greatest numbers of physicists. I don't suppose any of them is exactly right as it stands. Some may be completely wrong. But I hope that some of them have some truth in them. Only time, and experiment, will tell.

Hidden Dimensions: Charge as Geometry

I begin with a speculation that is almost certainly wrong. My excuses are that it is very pretty and that string theory (Section 15.3) makes use of some of the same ideas.

Around 1915, there seemed to be two beautiful theories: Maxwell's electromagnetism and Einstein's gravity. An obvious dream would be to try to unify them and, in the process, perhaps to find a geometrical basis for electromagnetism similar to Einstein's geometrical theory of gravity. With hindsight, we now know this vision to have been a mirage, because electromagnetism is only a part of the electroweak forces, but even so let us look at one intriguing idea that came up in those days.

The Problem

In 1906, Einstein, having the previous year explained Brownian motion (see Appendix C), invented his special theory of relativity (see Chapter 6) and, having begun the quantum theory of light, was promoted to technical expert second class in the patent office at Bern. In 1907 he began his great struggle to create a new theory of gravity, which culminated in his “general theory of relativity” in 1915.

Why was Einstein dissatisfied with Newton's theory of gravity? It had agreed with observations for more than two hundred years, predicting, for example, the existence of the planet Neptune (the work of Adams and Leverrier), which was discovered in 1846. There was one exception: a small discrepancy in the orbit of Mercury, announced by Leverrier in 1859. Several attempts had been made to explain this (for example, as due to a planet, Vulcan, between Mercury and the Sun), but none was generally accepted. So far as I know, this problem with Mercury was not important in motivating Einstein.

Einstein saw two problems with Newtonian gravity (Chapter 1). First: it was not consistent with Einstein's special theory of relativity – his account of spacetime. Second: it offered no explanation of the equality of inertial and gravitational mass (see Section 1.7). I will explain these two points in turn.

In Newton's theory, gravity is transmitted instantaneously; indeed the theory is formulated in the context of an absolute time.

Introduction

Heat is just random motion; more accurately, heat is any random process that involves energy. This short sentence needs a lot of explanation to make sense of it. It certainly requires a definition of energy. More difficult, and even controversial, is the word random.

The general idea that heat is a form of motion goes back at least to Galileo. It was a widespread notion in the seventeenth century, favoured by, for example, Robert Boyle (1627–1691). It fitted in with the Newtonian world view, and Newton himself believed (roughly correctly) that heat caused substances to expand because their atoms vibrated more.

There was an opposing theory that heat was a sort of “subtle fluid”, often called caloric (the word was coined by the great French chemist Antoine-Laurent Lavoisier). This went along with the idea that there were also electric and magnetic “fluids”. The caloric theory also suggested that, if the fluid could not be created or destroyed, then there should be, in some sense, “conservation of heat”. This idea opened the way to a quantitative theory, with heat being something that could be measured. It was not obvious how to make the motion theory of heat quantitative in the same way, and throughout the eighteenth century the caloric theory seemed to many people to be more fruitful.

Not until the second half of the nineteenth century was the motion theory of heat fully understood and established.

Electric Charges

William Gilbert (1504–1603), physician to Queen Elizabeth, coined the adjective electric from the Greek word for amber. It had been known in antiquity that a piece of amber rubbed with a cloth acquired the power to attract small objects. Many other electrically insulating substances, like glass and plastics, behave similarly. An inflated rubber balloon, after being rubbed on clothing, will stick to the ceiling. Metals and damp substances are unsuitable, because any electricity generated on them leaks away immediately.

During the course of the seventeenth century, people realized that electrified objects can repel as well as attract one another. One may easily perform the following experiment at home. Cut two pieces of cooking foil about one centimetre square. Glue each of them to the end of a piece of cotton and hang them up so they are next to each other. Rub a pen on wool and bring it up to the pieces of foil. As soon as the pen touches them they jump away, then left to themselves they hang a little apart. Now rub a sherry glass on the wool and move it near the foils. It will attract them (perhaps rather weakly).

Charles-François du Fay (1698–1739), superintendent of gardens to the king of France, discovered that electric charge made by rubbing resinous material (we would use plastic nowadays) attracts that made by rubbing glass. He inferred the existence of two kinds of electricity, which we now call “positive” and “negative”. Glass rubbed with wool or cat's fur gets a positive charge. Plastics so rubbed get a negative charge.

Seeing the Very Small

Lenses were used for microscopic purposes in the second half of the seventeenth century. The Dutch microscopist, van Leeuwenhoek identified blood capillaries, red blood cells, spermatazoa and bacteria. In the nineteenth century, the cell structure of plants and animals was established.

But optical microscopes, using light of wavelength about four to eight times 10-7 metre, cannot resolve smaller distances than these and certainly cannot be used to study atoms or even large molecules. To do better, one must use electromagnetic radiation of shorter wavelength, like X-rays, or beams of particles that have, according to quantum theory, wave functions with wavelengths determined by the momenta of the particles. X-rays were discovered by Röntgen in Würzburg in 1895. Their ability to penetrate opaque bodies, like hands, was of course the initial cause of excitement, but it is not our concern here. In 1912, von Laue in Berlin showed that X-rays incident on a crystal produce (on a photographic plate) a pattern of spots. The spots (that is, positions of maximum intensity) appear where the waves scattered by individual atoms add up constructively (where there phases are equal). This is the phenomenon of interference described in Section 4.6. It works because the wavelength of X-rays is comparable to the atomic separation in crystals.

From the X-ray “diffraction patterns” (of spots), it is possible to deduce the arrangement of atomic positions in the crystal. The Braggs, father and son (William and Lawrence), developed these methods.

This appendix gives a brief explanation of two important mathematical ideas: vectors and complex numbers.

First, vectors. There are many physical quantities that naturally have a direction associated with them, in addition to their magnitude. Examples are velocities, forces, electric fields. These are vectorial quantities, and mathematicians say that they can be “represented by a vector”. But the simplest example of a vector is the geometrical displacement (in a straight line) from one point to another. (The Latin word vector means “carrier”, as in the usage insect vector). We may use this example to illustrate the mathematical properties of vectors.

Mathematicians use symbols to denote vectors, and it is conventional to use boldface type, v, E, and so on, to emphasize that they are not ordinary numbers. Are there mathematical operations that can be performed on vectors, addition, multiplication, and so forth? There is a very simple and natural definition of addition. It is illustrated in Figure A.3.

Any vector (in ordinary three-dimensional space) can be made up as the sum of three vectors, one in each of three specified independent directions. Thus a vector needs three ordinary numbers in order to specify it – the lengths of the three “component” vectors.

Multiplication of vectors is a more complicated question. There is no way to define an operation of “multiplication” acting on a pair of vectors, which has all the properties of multiplication of ordinary numbers (like x × y = y × x).

Newton's law of gravitation (Section 1.7) has an inverse-square dependence on the distance. Double the distance means a quarter the force, and so on. This particular dependence on distance (as opposed to, say, an inverse cube or some other rate of decrease) has several special features, which I will explain in this appendix.

First of all, there is an interesting mathematical analogy with another physical situation. I stress that it is only an analogy: it is not meant to be a physical explanation of gravity, or anything like that.

The analogous situation is this. Imagine a large tank of still water. In the middle, put a small device near the center that sucks out water at a steady rate. (See Figure A.1.)

Where the water goes does not concern us: we can imagine it being drawn out through a narrow pipe that does not significantly interfere with the rest of the water in the tank. As a consequence of sucking out of the water, there must be a flow of water from distant parts of the tank towards the device, which I shall henceforth call (for want of a better word) an extractor. We also want to arrange that the sucking action of the extractor is the same in all directions. Then the flow in the tank will be the same in all directions. We want to work out the speed at which the water in the tank is flowing towards the extractor, and in particular how this speed depends upon the distance from it. Imagine any sperical surface in the fluid with its centre at the extractor.

Cooling and Freezing

We are always ignorant about the detailed microscopic state of a macroscopic lump of matter. What we know about it is generally of a statistical nature. Statements about its temperature, pressure, magnetism and so on, are statements about average properties. The entropy is defined as a measure of our ignorance. All this was explained in Chapter 2.

There is one exception. If we could get the lump of matter to the absolute zero of temperature, it would (in principle) be in a single quantum state: the state of minimum energy. The entropy (as well as the temperature) would be zero. This would be very interesting, because we would be studying the quantum theory of macroscopic things, not just of atoms. It is very difficult to get near enough to the absolute zero of temperature to achieve this single quantum state. But fortunately nature does provide us with many examples of interesting large-scale quantum effects that occur when bodies are made cold enough. Some of these have always been familiar; others were total surprises when they were discovered in the twentieth century.

When water is cooled, ice crystals form. This happens at a welldefined temperature (for a given pressure). There is a qualitative difference between water and ice. In a crystal, the molecules are arranged in a regular array, held in place by the forces between them. Knowledge about the positions of molecules at one point tells us something about their positions at macroscopic distances away (say, millimetres).