The Character of Physical Law

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.

... it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

In fact the total amount that a physicist knows is very little. He has only to remember the rules to get him from one place to another and he is all right...

For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use for measuring energy.

I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.

Electrons, when they were first discovered, behaved exactly like particles or bullets, very simply. Further research showed, from electron diffraction experiments for example, that they behaved like waves. As time went on there was a growing confusion about how these things really behaved ---- waves or particles, particles or waves? Everything looked like both. This growing confusion was resolved in 1925 or 1926 with the advent of the correct equations for quantum mechanics. Now we know how the electrons and light behave. But what can I call it? If I say they behave like particles I give the wrong impression; also if I say they behave like waves. They behave in their own inimitable way, which technically could be called a quantum mechanical way. They behave in a way that is like nothing that you have seen before. Your experience with things that you have seen before is incomplete. The behavior of things on a very tiny scale is simply different. An atom does not behave like a weight hanging on a spring and oscillating. Nor does it behave like a miniature representation of the solar system with little planets going around in orbits. Nor does it appear to be somewhat like a cloud or fog of some sort surrounding the nucleus. It behaves like nothing you have seen before. There is one simplication at least. Electrons behave in this respect in exactly the same way as photons; they are both screwy, but in exactly in the same way…. The difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, "But how can it be like that?" which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of an analogy with something familiar; I will simply describe it. There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possible avoid it, "But how can it be like that?" because you will get 'down the drain', into a blind alley from which nobody has escaped. Nobody knows how it can be like that.

I happen to know this, and I happen to know that, and maybe I know that;and I work everything out from there. Tomorrow I may forgot that this is true, but remember that something else is true, so I can reconstruct it all again. I am never quite sure of where I am supposed to begin or where I am supposed to end. I just remember enough all the time so that as the memory fades and some of the pieces fall out I can put the thing back together again every day

It is absurd that energy can be measured in calories, in ergs, in electron volts, in foot pounds, in B.T.U.s, in horsepower hours, in kilowatt hours–all measuring exactly the same thing. It is like having money in dollars, pounds, and so on; but unlike the economic situation where the ratio can change, these dopey things are in absolutely guaranteed proportion. If

You might say that the only reason for the anti-neutrino is to make the conservation of energy right. But it makes a lot of other things right, like the conservation of momentum and other conservation laws, and very recently it has been directly demonstrated that such neutrinos do indeed exist.

This example illustrates a point. How is it possible that we can extend our laws into regions we are not sure about? Why are we so confident that, because we have checked the energy conservation here, when we get a new phenomenon we can say it has to satisfy the law of conservation of energy ?

For instance, the mass of an object changes when it moves, because of the conservation of energy. Because of the relation of mass and energy the energy associated with the motion appears as an extra mass, so things get heavier when they move. Newton

angular momentum appears in two forms : one of them is angular momentum of motion, and the other is angular momentum in electric and magnetic fields. There is angular momentum in the field around the magnet, although it does not appear as motion, and this has the opposite sign to the spin. If

There are other conservation laws. They are not as interesting as those I have described, and do not deal exactly with the conservation of numbers. Suppose

Suppose we had some kind of device with particles moving with a certain definite symmetry, and suppose their movements were bilaterally symmetrical (fig. 20). Then, following the laws of physics, with all the movements and collisions, you could expect, and rightly, that if you look at the same picture later on it will still be bilaterally symmetrical. So there is a kind of conservation, the conservation of the symmetry character. This should be in the table, but it is not like a number that you measure, and we will discuss it in much more detail in the next lecture. The reason this is not very interesting in classical physics is because the times when there are such nicely symmetrical initial conditions are very rare, and it is therefore a not very important or practical conservation law. But

This is how a rocket works. At first it is standing still, say, in empty space, and then it shoots some gas out of the back, and the rocket goes forward. The point is that of all the stuff in the world, the centre of mass, the average of all the mass, is still right where it was before. The interesting part has moved on, and an uninteresting part that we do not care about has moved back. There

Discovering the laws of physics is like trying to put together the pieces of a jigsaw puzzle. We

Professor Weyl,* the mathematician, gave an excellent definition of symmetry, which is that a thing is symmetrical if there is something that you can do to it so that after you have finished doing it it looks the same as it did before. That is the sense in which we say that the laws of physics are symmetrical; that there are things we can do to the physical laws, or to our way of representing the physical laws, which make no difference, and leave everything unchanged in its effects. It is this aspect of physical laws that is going to concern us in this lecture.

Einstein realized, and Poincaré* too, that the only possible way in which a person moving and a person standing still could measure the speed to be the same was that their sense of time and their sense of space are not the same, that the clocks inside the space ship are ticking at a different speed from those on the ground, and so forth. You

The symmetries of translation in space, delay in time, and so on, were not very deep; but the symmetry of uniform velocity in a straight line is very interesting, and has all kinds of consequences. Furthermore, these consequences are extendable into laws that we do not know. For example, by guessing that this principle is true for the disintegration of a mu meson, we can state that we cannot use mu mesons to tell how fast we are going in a space ship either; and thus we know something at least about mu meson disintegration, even though we do not know why the mu meson disintegrates in the first place.

Of all the conservation laws, that dealing with energy is the most difficult and abstract, and yet the most useful. It

By this time you are probably convinced that all the laws of physics are symmetrical under any kind of change whatsoever, so now I will give a few that do not work. The first one is change of scale. It is not true that if you build an apparatus, and then build another one, with every part made exactly the same, of the same kind of stuff, but twice as big, that it will work in exactly the same way. You

The conservation of energy is a little more difficult, because this time we have a number which is not changed in time, but this number does not represent any particular thing. I

The little cathedral made with matchsticks is attracted to the earth, so to make a comparison the big cathedral should be attracted to an even bigger earth. Too bad. A bigger earth would attract it even more, and the sticks would break even more surely!

I suppose Galileo felt that the discovery of the fact that the laws of nature are not unchanged under change of scale was as important as his laws of motion, because they are both put together in the tome on Two New Sciences.

All physics is rooted in the notion of law, the belief that we live in an ordered universe that can be understood by the application of rational reasoning. But

There is also a rhythm and a pattern between the phenomena of nature which is not apparent to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call Physical Laws. What

But the motion to keep the planet going in a straight line has no known reason. The reason why things coast for ever has never been found out. The law of inertia has no known origin. Although

With energy there is this difference, that there are no blocks, so far as we can tell. Also, unlike the case of the blocks, for energy the numbers that come out are not integers. I

Conservation just means that it does not change.

When we add all the numbers together, from all the different forms of energy, it always gives the same total. But as far as we know there are no real units, no little ballbearings. It

It is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

Suppose that physics, or rather nature, is considered analogous to a great chess game with millions of pieces in it, and we are trying to discover the laws by which the pieces move. The great gods who play this chess play it very rapidly, and it is hard to watch and difficult to see. However, we are catching on to some of the rules, and there are some rules which we can work out which do not require that we watch every move. For instance, suppose there is one bishop only, a red bishop, on the board, then since the bishop moves diagonally and therefore never changes the colour of its square, if we look away for a moment while the gods play and then look back again, we can expect that there will be still a red bishop on the board, maybe in a different place, but on the same colour square. This is in the nature of a conservation law. We do not need to watch the insides to know at least something about the game.

A philosopher once said 'It is necessary for the very existence of science that the same conditions always produce the same results'. Well, they do not. You set up the circumstances, with the same conditions every time, and you cannot predict behind which hole you will see the electron. Yet science goes on in spite of it - although the same conditions do not always produce the same results. <...> What is necessary 'for the very existence of science', and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. <...> If science is to progress, what we need is the ability to experiment, honesty in reporting results - the results must be reported without somebody saying what they would like the results to have been - and finally - an important thing - the intelligence to interpret the results.

On the other hand, in the case of hypnotism, at first it looked as though that also would be impossible, when it was described incompletely. Now that it is known better it is realized that it is not absolutely impossible that hypnosis could occur through normal physiological, though as yet unknown, processes; it does not obviously require some special new kind of force.

This conservation law is very useful in many technical ways. I will give you some very simple examples to show how, knowing the law of conservation of energy and the formulae for calculating energy, we can understand other laws. In other words many other laws are not independent, but are simply secret ways of talking about the conservation of energy. The simplest is the law of the lever (fig. 16). Figure 16

The physicist uses ordinary words in a peculiar manner. To him a conservation law means that there is a number which you can calculate at one moment, then as nature undergoes its multitude of changes, if you calculate this quantity again at a later time it will be the same as it was before, the number does not change. An example is the conservation of energy. There is a quantity that you can calculate according to a certain rule, and it comes out the same answer always, no matter what happens.

I will summarize, then, by saying that electrons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. It is in this sense that the electron behaves sometimes like a particle and sometimes like a wave. It behaves in two different ways at the same time.

The next question was — what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward.

In general we look for a new law by the following process. First we guess it. Then we compute the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In

If it were true that the stars could affect the day that it was good to go to the dentist – in America we have that kind of astrology – then physics theory would be proved wrong, because there is no mechanism understandable in principle from the behaviour of particles which would make this work. That

For example, life itself is supposedly understandable in principle from the movements of atoms, and those atoms are made out of neutrons, protons and electrons. I must immediately say that when we state that we understand it in principle, we only mean that we think that, if we could figure everything out, we would find that there is nothing new in physics which needs to be discovered in order to understand the phenomena of life. Another

all ordinary phenomena can be explained by the actions and the motions of particles. For

frogs are made of the same ‘goup’ as rocks, only in different arrangements. So

An important point about this intelligence is that it should not be sure ahead of time what must be. It can be prejudiced, and say ‘That is very unlikely; I don’t like that’. Prejudice is different from absolute certainty. I do not mean absolute prejudice – just bias. As long as you are only biased it does not make any difference, because if your bias is wrong a perpetual accumulation of experiments will perpetually annoy you until they cannot be disregarded any longer. They can only be disregarded if you are absolutely sure ahead of time of some precondition that science has to have. In fact it is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions, like those of our philosopher.

The most reasonable possibilities often turn out not to be the situation. If science is to progress, what we need is the ability to experiment, honesty in reporting results – the results must be reported without somebody saying what they would like the results to have been – and finally – an important thing – the intelligence to interpret the results. An