52 Symmetry in Physical Laws 52-1 Symmetry operations The subject of this chapter is what we may call symmetry in physical laws. 52-1 Symmetry operations We have already discussed certain features of symmetry in physical laws in con- 52-2 Symmetry in space and time nection with vector analysis(Chapter 11),the theory of relativity (Chapter 16),and rotation (Chapter 20). 52-3 Symmetry and conservation Why should we be concerned with symmetry?In the first place,symmetry is laws fascinating to the human mind,and everyone likes objects or patterns that are in 524 Mirror reflections 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 52-5 Polar and axial vectors 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 52-6 Which hand is right? different kinds of symmetry,the study of which tells us some important things about 52-7 Parity is not conserved! the structure of solids.Even the animal and vegetable worlds show some degree of 52-8 Antimatter symmetry,although the symmetry of a flower or of a bee is not as perfect or as fundamental as is that of a crystal. 52-9 Broken symmetries 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. Table 52-1 52-2 Symmetry in space and time Symmetry Operations 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- Translation in space paratus at another place in space (or move the original one over)then,whatever Translation in time went on in one apparatus,in a certain order in time,will occur in the same way if Rotation through a fixed angle we have arranged the same condition,with all due attention to the restrictions that Uniform velocity in a straight we mentioned before:that all of those features of the environment which make it line (Lorentz transformation) not behave the same way have also been moved over-we talked about how to Reversal of time define how much we should include in those circumstances,and we shall not go into those details again. Reflection of space In the same way,we also believe today that displacement in time will have no Interchange of identical atoms effect on physical laws.(That is,as far as we know today-—all of these things are or identical particles as far as we know today!)That means that if we build a certain apparatus and start Quantum-mechanical phase it at a certain time,say on Thursday at 10:00 a.m.,and then build the same appara- Matter-antimatter (charge conjugation) 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