Feynman Physics Lectures V1 Ch52 1962-06-01 Symmetry in Physical Laws

Feynman Physics Lectures V1 Ch52 1962-06-01 Symmetry in Physical Laws

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Unformatted text preview: 52 Symmetry in Physical Laws 52—1 Symmetry operations The subject of this chapter is what we may call symmetry in physical laws. We have already discussed certain features of symmetry in physical laws in con- nection with vector analysis (Chapter 11), the theory of relativity (Chapter 16), and rotation (Chapter 20). Why should we be concerned with symmetry? In the first place, symmetry is fascinating to the human mind, and everyone likes objects or patterns that are in some way symmetrical. It is an interesting fact that nature often exhibits certain kinds of symmetry in the objects we find in the world around us. Perhaps the most symmetrical object imaginable is a sphere, and nature is full of spheres— stars, planets, water droplets in clouds. The crystals found in rocks exhibit many different kinds of symmetry, the study of which tells us some important things about the structure of solids. Even the animal and vegetable worlds show some degree of symmetry, although the symmetry of a flower or of a bee is not as perfect or as fundamental as is that of a crystal. But our main concern here is not with the fact that the objects of nature are often symmetrical. Rather, we wish to examine some of the even more remarkable symmetries of the universe—the symmetries that exist in the basic laws themselves which govern the operation of the physical world. First, what is symmetry? How can a physical law be “symmetrical”? The problem of defining symmetry is an interesting one and we have already noted that Weyl gave a good definition, the substance of which is that a thing is symmetrical if there is something we can do to it so that after we have done it, it looks the same as it did before. For example, a symmetrical vase is of such a kind that if we reflect or turn it, it will look the same as it did before. The question we wish to consider here is what we can do to physical phenomena, or to a physical situation in an experiment, and yet leave the result the same. A list of the known operations under which various physical phenomena remain invariant is shown in Table 52—1. 52—2 Symmetry in space and time The first thing we might try to do, for example, is to translate the phenomenon in space. If we do an experiment in a certain region, and then build another ap- paratus at another place in space (or move the original one over) then, whatever went on in one apparatus, in a certain order in time, will occur in the same way if we have arranged the same condition, with all due attention to the restrictions that we mentioned before: that all of those features of the environment which make it not behave the same way have also been moved over—we talked about how to define how much we should include in those circumstances, and We shall not go into those details again. In the same way, we also believe today that displacement in time will have no effect on physical laws. (That is, as far as we know today—all of these things are as far as we know today!) That means that if we build a certain apparatus and start it at a certain time, say on Thursday at 10:00 am, and then build the same appara- tus and start it, say, three days later in the same condition, the two apparatuses will go through the same motions in exactly the same way as a function of time no matter what the starting time, provided again, of course, that the relevant features of the environment are also modified appropriately in time. That symmetry means, 52—1 52—1 Symmetry operations 52—2 Symmetry in space and time 52—3 Symmetry and conservation laws 52—4 Mirror reflections 52—5 Polar and axial vectors 52—6 Which hand is right? 52—7 Parity is not conserved! 52—8 Antimatter 52—9 Broken symmetries Table 52—1 Symmetry Operations Translation in space Translation in time Rotation through a fixed angle Uniform velocity in a straight line (Lorentz transformation) Reversal of time Reflection of space Interchange of identical atoms or identical particles Quantum-mechanical phase Matter-antimatter (charge conjugation) of course, that if one bought General Motors stock three months ago, the same thing would happen to it if he bought it now! We have to watch out for geographical differences too, for there are, of course, variations in the characteristics of the earth’s surface. So, for example, if we measure the magnetic field in a certain region and move the apparatus to some other region, it may not work in precisely the same way because the magnetic field is different, but we say that is because the magnetic field is associated with the earth. We can imagine that if we move the whole earth and the equipment, it would make no difference in the operation of the apparatus. Another thing that we discussed in considerable detail was rotation in space: if we turn an apparatus at an angle it works just as well, provided we turn every- thing else that is relevant along with it. In fact, we discussed the problem of sym- metry under rotation in space in some detail in Chapter 11, and we invented a mathematical system called vector analysis to handle it as neatly as possible. On a more advanced level we had another symmetry—the symmetry under uniform velocity in a straight line. That is to say—a rather remarkable effect—that if we have a piece of apparatus working a certain way and then take the same ap- paratus and put it in a car, and move the whole car, plus all the relevant surround- ings, at a uniform velocity in a straight line, then so far as the phenomena inside the car are concerned there is no difference: all the laws of physics appear the same. We even know how to express this more technically, and that is that the mathe- matical equations of the physical laws must be unchanged under a Lorentz [rans- formarion. As a matter of fact, it was a study of the relativity problem that concen- trated physicists’ attention most sharply on symmetry in physical laws. Now the above-mentioned symmetries have all been of a geometrical nature, time and space being more or less the same, but there are other symmetries of a different kind. For example, there is a symmetry which describes the fact that we can replace one atom by another of the same kind; to put it difierently, there are atoms of the same kind. It is possible to find groups of atoms such that if we change a pair around, it makes no difference—the atoms are identical. Whatever one atom of oxygen of a certain type will do, another atom of oxygen of that type will do. One may say, “That is ridiculous, that is the definition of equal types!” That may be merely the definition, but then we still do not know whether there are any “atoms of the same type”; the fact is that there are many, many atoms of the same type. Thus it does mean something to say that it makes no difierence if we replace one atom by another of the same type. The so-called elementary particles of which the atoms are made are also identical particles in the above sense—all electrons are the same; all protons are the same; all positive pions are the same; and so on. After such a long list of things that can be done without changing the phe- nomena, one might think we could do practically anything; so let us give some examples to the contrary, just to see the difference. Suppose that we ask: “Are the physical laws symmetrical under a change of scale?” Suppose we build a certain piece of apparatus, and then build another apparatus five times bigger in every part, will it work exactly the same way? The answer is, in this case, no! The wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelength of light emitted by a gas of sodium atoms five times in volume is not five times longer, but is. in fact exactly the same as the other. So the ratio of the wavelength to the size of the emitter will change. Another example: we see in the newspaper, every once in a while pictures of a great cathedral made with little matchsticks—a tremendous work of art by some retired fellow who keeps gluing matchsticks together. It is much more elaborate and wonderful than any real cathedral. If we imagine that this wooden cathedral were actually built on the scale of a real cathedral, we see where the trouble is; it would not last—the whole thing would collapse because of the fact that scaled-up matchsticks are just not strong enough. “Yes,” one might say, “but we also know that when there is an influence from the outside, it also must be changed in pro- portion!” We are talking about the ability of the object to withstand gravitation. So what we should do is first to take the model cathedral of real matchsticks and 52-2 the real earth, and then we know it is stable. Then we should take the larger cathe- dral and take a bigger earth. But then it is even worse, because the gravitation is increased still more! Today, of course, we understand the fact that phenomena depend on the scale on the grounds that matter is atomic in nature, and certainly if we built an appara- tus that was so small there were only five atoms in it, it would clearly be something we could not scale up and down arbitrarily. The scale of an individual atom is not at all arbitrary—it is quite definite. The fact that the laws of physics are not unchanged under a change of scale was discovered by Galileo. He realized that the strengths of materials were not in exactly the right proportion to their sizes, and he illustrated this property that we were just discussing, about the cathedral of matchsticks, by drawing two bones, the bone of one dog, in the right proportion for holding up his weight, and the imaginary bone of a “super dog” that would be, say, ten or a hundred times bigger—that bone was a big, solid thing with quite different proportions. We do not know whether he ever carried the argument quite to the conclusion that the laws of nature must have a definite scale, but he was so impressed with this dis- covery that he considered it to be as important as the discovery of the laws of motion, because he published them both in the same volume, called “On Two New Sciences.” Another example in which the laws are not symmetrical, that we know quite well, is this: a system in rotation at a uniform angular velocity does not give the same apparent laws as one that is not rotating. If we make an experiment and then put everything in a space ship and have the space ship spinning in empty space, all alone at a constant angular velocity, the apparatus will not work the same way because, as we know, things inside the equipment will be thrown to the outside, and so on, by the centrifugal or coriolis forces, etc. In fact, we can tell that the earth is rotating by using a so—called Foucault pendulum, without looking outside. Next we mention a very interesting symmetry which is obviously false, i.e., reversibility in time. The physical laws apparently cannot be reversible in time, because, as we know, all obvious phenomena are irreversible on a large scale: “The moving finger writes, and having writ, moves on.” So far as we can tell, this irreversibility is due to the very large number of particles involved, and if we could see the individual molecules, we would not be able to discern whether the machinery was working forward or backwards. To make it more precise: we build a small apparatus in which we know what all the atoms are doing, in which we can watch them jiggling. Now we build another apparatus like it, but which starts its motion in the final condition of the other one, with all the velocities precisely reversed. It will then go through the same motions, but exactly in reverse. Putting it another way: if we take a motion picture, with sufficient detail, of all the inner works of a piece of material and shine it on a screen and run it backwards, no physicist will be able to say, “That is against the laws of physics, that is doing something wrong!” If we do not see all the details, of course, the situation will be perfectly clear. If we see the egg splattering on the sidewalk and the shell cracking open, and so on, then we will surely say, “That is irreversible, because if we run the moving picture backwards the egg will all collect together and the shell will go back together, and that is obviously ridiculous!” But if we look at the individual atoms themselves, the laws look completely reversible. This is, of course, a much harder discovery to have made, but apparently it is true that the fundamental physical laws, on a microscopic and fundamental level, are completely reversible in time! 52—3 Symmetry and conservation laws The symmetries of the physical laws are very interesting at this level, but they turn out, in the end, to be even more interesting and exciting when we come to quantum mechanics. For a reason which we cannot make clear at the level of the present discussion—a fact that most physicists still find somewhat staggering, a most profound and beautiful thing, is that, in quantum mechanics, for each of the rules of symmetry there is a corresponding conservation law; there is a definite 52—3 connection between the laws of conservation and the symmetries of physical laws. We can only state this at present, without any attempt at explanation. The fact, for example, that the laws are symmetrical for translation in space when we add the principles of quantum mechanics, turns out to mean that mo- mentum is conserved. That the laws are symmetrical under translation in time means, in quantum mechanics, that energy is conserved. Invariance under rotation through a fixed angle in space corresponds to the conservation of angular momentum. These connections are very interesting and beautiful things, among the most beautiful and profound things in physics. Incidentally, there are a number of symmetries which appear in quantum mechanics which have no classical analog, which have no method of description in classical physics. One of these is as follows: If ii; is the amplitude for some process or other, we know that the absolute square of w is the probability that the process will occur. Now if someone else were to make his calculations, not with this to, but with a W which differs merely by a change in phase (let A be some constant, and multiply eiA times the old i/x), the absolute square of W, which is the probability of the event, is then equal to the absolute square of III: W = we“; W12 = W. (52.1) Therefore the physical laws are unchanged if the phase of the wave function is shifted by an arbitrary constant. That is another symmetry. Physical laws must be of such a nature that a shift in the quantum-mechanical phase makes no difl‘er- ence. As we have just mentioned, in quantum mechanics there is a conservation law for every symmetry. The conservation law which is connected with the quan- tum-mechanical phase seems to be the conservation of electrical charge. This is altogether a very interesting business! 52—4 Mirror reflections Now the next question, which is going to concern us for most of the rest of this chapter, is the question of symmetry under reflection in space. The problem is this: Are the physical laws symmetrical under reflection? We may put it this way: Suppose we build a piece of equipment, let us say a clock, with lots of wheels and hands and numbers; it ticks, it works, and it has things wound up inside. We look at the clock in the mirror. How it looks in the mirror is not the question. But let us actually build another clock which is exactly the same as the first clock looks in the mirror—every time there is a screw with a right-hand thread in one, we use a screw with a left-hand thread in the corresponding place of the other; where one is marked “2” on the face, we mark a “S” on the face of the other; each coiled spring is twisted one way in one clock and the other way in the mirror- image clock; when we are all finished, we have two clocks, both physical, which bear to each other the relation of an object and its mirror image, although they are both actual, material objects, we emphasize. Now the question is: If the two clocks are started in the same condition, the springs wound to corresponding tight- nesses, will the two clocks tick and go around, forever after, as exact mirror images? (This is a physical question, not a philosophical question.) Our intuition about the laws of physics would suggest that they would. We would suspect that, at least in the case of these clocks, reflection in space is one of the symmetries of physical laws, that if we change everything from “right” to “left” and leave it otherwise the same, we cannot tell the difference. Let us, then, suppose for a moment that this is true. If it is true. then it would be impossible to distinguish “right” and “left” by any physical phenomenon, just as it is, for example, impossible to define a particular absolute velocity by a physical phe- nomenon. So it should be impossible, by any physical phenomenon, to define absolutely what we mean by “right” as opposed to “left,” because the physical laws should be symmetrical. Of course, the world does not have to be symmetrical. For example, using what we may call “geography,” surely “right” can be defined. For instance, we stand 52-4 in New Orleans and look at Chicago, and Florida is to our right (when our feet are on the ground!). So we can define “right” and “left” by geography. Of course, the actual situation in any system does not have to have the symmetry that we are talking about; it is a question of whether the laws are symmetrical—in other words, whether it is against the physical laws to have a sphere like the earth with “left- handed dirt” on it and a person like ourselves standing looking at a city like Chicago from a place like New Orleans, but with everything the other way around, so Florida is on the other side. It clearly seems not impossible, not against the physical laws, to have everything changed left for right. Another point is that our definition of “right” should not depend on history. An easy way to distinguish right from left is to go to a machine shop and pick up a screw at random. The odds are it has a right-hand thread—not necessarily, but it is much more likely to have a right—hand thread than a left-hand one. This is a question of history or convention, or the way things happen to be, and is again not a question of fundamental laws. As we can well appreciate, everyone could have started out making left-handed screws! So we must try to find some phenomenon in which “right hand” is involved fundamentally. The next possibility we discuss is the fact that polarized light rotates its plane of polarization as it goes through, say, sugar water. As we saw in Chapter 33, it rotates, let us say, to the right in a certain sugar solution. That is a way of defining “right-hand,” because we may dissolve some sugar in the water and then the polarization goes to the right. But sugar has come from living things, and if we try to make the sugar artificially, then we discover that it does not rotate the plane of polarization! But if we then take that same sugar which is made artificially and which does not rotate the plane of polarization, and put bacteria in it (they eat some of the sugar) and then filter out the bacteria, we find that we still have sugar left (almost half as much as we had before), and this time it does rotate the plane of polarization, but the other way! It seems very confusing, but is easily explained. Fig. 52—1. (0) L-ulanine (left), and (b) D-alanine (right). Take another example: One of the substances which is common to all living creatures and that is fundamental to life is protein. Proteins consist of chains of amino acids. Figure 52—1 shows a model of an amino acid that comes out of a protein. This amino acid is called alanine, and the molecular arrangement would look like that in Fig. 52-1(a) if it came out of a protein of a real living thing. On the other hand, if we try to make alanine from carbon dioxide, ethane, and am- monia (and we can make it, it is not a complicated mol...
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