Articles Comments

Stan Bentvelsen » onderwijs » Symmetry of physics laws

Symmetry of physics laws

Feynman’s lecture on physics
Dit is de reproductie van hoofdstuk 52 uit de beroemde serie `Lectures on Physics’ van Feynman (`The Feynman Lectures on Physics’, R.P. Feynman, R.B. Leighton and M. Sands, Addison Wesley publishing, 1963). Deze serie is gebaseerd op een serie colleges die Feynman heeft gehouden in 1961-1963 op Caltech. Hoewel deze colleges jaren geleden zijn gegeven, zijn ze voor het grootste gedeelte nog altijd actueel – en het is zeker aan te raden om ook andere hoofdstukken te lezen. Een aantal passages zijn aangepast of weggelaten omdat ze niet meer relevant zijn.

Richard Phillips Feynman (May 11, 1918 – February 15, 1988) was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics (he proposed the parton model). For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sin-Itiro Tomonaga, received the Nobel Prize in Physics in 1965. He developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world.

He assisted in the development of the atomic bomb and was a member of the panel that investigated the Space Shuttle Challenger disaster. In addition to his work in theoretical physics, Feynman has been credited with pioneering the field of quantum computing, and introducing the concept of nanotechnology. He held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology.

Feynman was a keen popularizer of physics through both books and lectures, notably a 1959 talk on top-down nanotechnology called There’s Plenty of Room at the Bottom and The Feynman Lectures on Physics. Feynman also became known through his semi-autobiographical books (Surely You’re Joking, Mr. Feynman!} and What Do You Care What Other People Think?) and books written about him, such as Tuva or Bust!
(bron wikipedia)

 

  • Symmetry operations

The subject of this chapter is what we may call symmetry in physical laws. 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 here:

  • Translation in space
  • Translation in time
  • Rotation through a fixed angle
  • Uniform velocity in a straight
  • Lorentz transformation
  • Reversal of time
  • Reflection of space
  • Interchange of identical atoms or identical particles
  • Quantum-mechanical phase
  • Matter-antimatter (charge conjugation)

 

  • 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 apparatus 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 a.m., and then build the same apparatus 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, 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 is rotation in space: if we turn an apparatus at an angle it works just as well, provided we turn everything else that is relevant along with it. In fact, we discussed the problem of symmetry 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 have a 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 apparatus and put it in a car, and move the whole car, plus all the relevant surroundings, 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 mathematical equations of the physical laws must be unchanged under a Lorentz transformation. As a matter of fact, it was a study of the relativity problem that concentrated 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 differently, 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 difference 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 phenomena, 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 proportion!’ 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 the real earth, and then we know it is stable. Then we should take the larger cathedral 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 apparatus 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 discovery
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.

  •  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 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 momentum 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.

  • 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 `$\triangleleft$’ on the face, we mark a `$\triangleright$’ 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 tightnesses, 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 phenomenon. 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 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!

  • Magnet reflection

Now we go further. If the laws of physics are symmetrical, we should find that if some demon were to sneak into all the physics laboratories and replace the word ‘right’ for ‘left’ in every book in which ‘right-hand rules’ are given, and instead we were to use all ‘left-hand rules,’ uniformly, then it should make no difference whatever in the physical laws.This is not so obvious, as you can see in an example with magnets. Suppose that we have two magnets. One is a magnet with the coils going around a certain way, and with current in a given direction. The other magnet looks like the reflection of the first magnet in a mirror-the coil will wind the other way, everything that happens inside the coil is exactly reversed, and the current goes as shown. Now, from the laws for the production of magnetic fields, which we most likely learned in high school, it turns out that the magnetic field is as shown in the figure. In one case the pole is a south magnetic  pole, while in the other magnet the current is going the other way and the magnetic field is reversed-it is a north magnetic pole. So we see that when we go from right to left we must indeed change from north to south!

Never mind changing north to south; these too are mere conventions. Let us talk about phenomena. Suppose, now, that we have an electron moving through one field, going into the page. Then, if we use the Lorentz formula for the magnetic force, we find that the electron will deviate in the indicated direction according to the physical law. So the phenomenon is that we have a coil with a current going in a specified sense and an electron curves in a certain way – that is the physics – never mind how we label everything.

Now let us do the same experiment with a mirror: we send an electron through in a corresponding direction and now the force is reversed, if we calculate it from the same rule, and that is very good because the corresponding motions are then mirror images!

  • Which hand is right?

So the fact of the matter is that in studying any phenomenon the net result is that the phenomena always look symmetrical. In short, therefore, we cannot tell right from left if we also are not able to tell north from south. However, it may seem that we can tell the north pole of a magnet. The north pole of a compass needle, for example, is one that points to the north. But of course that is again a local property that has to do with geography of the earth; that is just like talking about in which direction is Chicago, so it does not count. If we have seen compass needles, we may have noticed that the north-seeking pole is a sort of bluish color. But that is just due to the man who painted the magnet. These are all local, conventional criteria.

However, if a magnet were to have the property that if we looked at it closelyenough we would see small hairs growing on its north pole but not on its south pole, if that were the general rule, or if there were any unique way to distinguish the north from the south pole of a magnet, then we could tell which of the two cases we actually had, and that would be the end of the law of reflection symmetry.

To illustrate the whole problem still more clearly, imagine that we were talking to a Martian, or someone very far away, by telephone. We are not allowed to send him any actual samples to inspect; for instance, if we could send light, we could send him right-hand circularly polarized light and say, ‘That is right-hand light just watch the way it is going.’ But we cannot give him anything, we can only talk to him. He is far away, or in some strange location, and he cannot see anything we can see. For instance, we cannot say, ‘Look at Ursa major; now see how those stars are arranged. What we mean by ‘right’ is … ‘ We are only allowed to telephone him.

Now we want to tell him all about us. Of course, first we start defining numbers, and say, ‘Tick, tick, two, tick, tick, tick, three . .. ,’ so that gradually he can understand a couple of words, and so on. After a while we may become very familiar with this fellow, and he says, ‘What do you guys look like?’ We start to describe ourselves, and say, ‘Well, we are six feet tall.’ He says, ‘Wait a minute, what is six feet?’ Is it possible to tell him what six feet is? Certainly! We say, ‘You know about the diameter of hydrogen atoms-we are 17,000,000,000 hydrogen atoms high!’ That is possible because physical laws are not variant under change of scale, and therefore we can define an absolute length. And so we define the size of the body, and tell him what the general shape is-it has prongs with five bumps sticking out on the ends, and so on, and he follows us along, and we finish describing how we look on the outside, presumably without encountering any particular difficulties. He is even making a model of us as we go along. He says, ‘My, you are certainly very handsome fellows; now what is on the inside?’ So we start to describe the various organs on the inside, and we come to the heart, and we carefully describe the shape of it, and say, ‘Now put the heart on the left side.’ He says, ‘Duhhh-the left side?’ Now our problem is to describe to him which side the heart goes on without his ever seeing anything that we see, and without
our ever sending any sample to him of what we mean by ‘right’-no standard right-handed object. Can we do it?

  • Parity is not conserved!

It turns out that the laws of gravitation, the laws of electricity and magnetism,  nuclear forces, all satisfy the principle of reflection symmetry, so these laws, or anything derived from them, cannot be used. But associated with the many particles that are found in nature there is a phenomenon called beta decay, or weak decay.

In 1952 Miss Wu from Columbia carried out an experiment as follows. Using a very strong magnet at a very low temperature, it turns out that a certain isotope of cobalt, which disintegrates by emitting an electron, is magnetic, and if the temperature is low enough that the thermal oscillations do not jiggle the atomic magnets about too much, they line up in the magnetic field. So the cobalt atoms will all line up in this strong field. They then disintegrate, emitting an electron, and it was discovered that when the atoms were lined up in a field whose B vector points upward, most of the electrons were emitted in a downward direction.

If one is not really ‘hep’\footnote{`hep’ stand for `high energy physics’. Feynman remarks here that the observation is really strange, but is perhaps not appreciate by high energy physicists!} to the world, such a remark does not sound like anything of significance, but if one appreciates the problems and interesting things in the world, then he sees that it is a most dramatic discovery: When we put cobalt atoms in an extremely strong magnetic field, more disintegration electrons go down than up. Therefore if we were to put it in a corresponding experiment in a ‘mirror,’ in which the cobalt atoms would be lined up in the opposite direction, they would spit their electrons up, not down; the action is unsymmetrical. The magnet has grown hairs! The south pole of a magnet is of such a kind that the electrons in a .a-disintegration tend to go away from it; that distinguishes, in a physical way, the north pole from the south pole.

In short, we can tell a Martian where to put the heart: we say, ‘Listen, build yourself a magnet, and put the coils in, and put the current on, and then take some cobalt and lower the temperature. Arrange the experiment so the electrons go from the foot to the head, then the direction in which the current goes through the coils is the direction that goes in on what we call the right and comes out on the left.’ So it is possible to define right and left, now, by doing an experiment of this kind.

Now that is the rule, but today we do not really understand the whys and wherefores of it. Why is this the right rule, what is the fundamental reason for it, and how is it connected to anything else? At the moment we have been so shocked by the fact that this thing is unsymmetrical that we have not been able to recover enough to understand what it means with regard to all the other rules. However, the subject is interesting, modern, and still unsolved, so it seems appropriate that we discuss some of the questions associated with it.

NB Now we know this violation of parity, the left-right symmetry, is described by the weak force in the Standard Model of elementary particles.

Filed under: onderwijs

Geef een reactie