Gauge Symmetry, Unification and Strings 1 Symmetries in Nature
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Gauge Symmetry, Unification and Strings Sunil Mukhi Tata Institute of Fundamental Research, Mumbai 1 Symmetries in nature The term \symmetry" refers to the even balance and proportion of an object. If this definition looks somewhat vague, a more precise one can easily be provided. Symmetry of a shape is the property that its form remains unchanged after a transformation is performed on it. As simple examples, the human figure looks roughly the same when inverted in a mirror, and there are objects like star-shaped fish that look the same when we rotate them about an axis such that each of the arms gets mapped onto the next one. In these examples there is a definite \symmetry operation" that takes the object and trans- forms it such that the final result looks the same as the original. For a regular five-pointed starfish, such an operation is rotation by a fixed angle of 72 degrees (one-fifth of a full turn), and multiples of that. A different rotation, for example a rotation by 5 degrees, will not leave the starfish looking the same as before, but puts it in a different configuration that can be distinguished from the original one. 72◦ rotation −! Figure 1: Symmetry operation on a starfish. (Original photo c Nevit Dilmen.) Things are different for a smooth round ball. Here we can rotate it as much or little as we like and it looks the same as before (we assume nothing is drawn on its surface, so that all points look exactly the same as all others). Moreover there are three independent axes about which it can be rotated. For a smooth egg-shaped surface there is only one axis of symmetry, which { if we orient the egg so that its narrow end points upwards { is the vertical axis. 1 But in either case, the symmetry is present for arbitrarily tiny rotation angles. Rotations by larger angles can be \built up" by combining a large number of tiny rotations. Figure 2: Symmetry axis of an egg. Since an egg only admits rotations about a single axis, two successive symmetry operations have to both be rotations (by two different angles) about the same axis. The end result corresponds to a rotation by the sum of those angles, and the order does not matter. In this case the symmetry is called Abelian. For a sphere, however, we have three different symmetry axes. Rotating any object by some angle around the x-axis and then by another angle around the y-axis, yields a different result from performing the same acts in the reverse order. A symmetry for which the order matters is called non-Abelian. The above examples illustrate some basic features of symmetry transformations on an object. The transformations can be \discrete" as for the starfish, or \continous" as for the ball. We will be interested primarily in the latter kind. The collection of all symmetry transformations on a given system, along with the inverse transformations and the rule for composing two transformations to get a third one, is a mathematical structure called a \group". 2 Local gauge symmetry While symmetries of the kind described above are quite familiar in our everyday experience, there is a refinement of the idea that is very strange at first sight. Suppose we break up the surface of an egg into a hundred little parts, for example by drawing a grid on it. Now we rotate the egg about the vertical axis as before, by a small angle, but let each element of the grid rotate by a different amount. The result, if the surface of the egg were stretchable, would 2 be a rather deformed egg (and with a real egg, the result would surely be much worse!). Thus \local" rotations in space { here \local" means \independently at each point" { are not a symmetry of our world in our everyday experience. A local rotation can be thought of as a rotation by an angle which is a function of where one is located. Surprisingly, this strange operation, along with its generalisations, is actually a symmetry of the world at the microscopic level. We will explore this fact in the rest of this article, and will encounter the even more surprising fact that such local symmetries dictate the way our theories of nature are built, as embodied in the \Standard Model" of particle physics. It is expected that future theories that will take us beyond the domain of the Standard Model will also be governed by such local symmetries, which for historical reasons are more commonly known as \gauge symmetries". 3 Gauge symmetries and general relativity Before engaging with the Standard Model, let us consider the one force in nature that lies outside the domain of this model: gravity. For centuries it was believed that gravity was governed by the inverse-square force law due to Newton. However, Einstein's radical General Theory of Relativity overthrew that in favour of a more complex description. In this description, spacetime is a curved manifold and the force of gravity is a consequence of the geometry of this manifold. Einstein's theory embodies a principle called \general coordinate invariance" according to which coordinates of spacetime are mere tools that we use to label and study physical phenomena, but have no intrinsic significance. We are free to change from one coordinate system to another with no change in the intrinsic geometrical structure and according to Einstein this can lead to no change in the physical laws (the \principle of equivalence"). Now an arbitrary change of coordinate systems amounts to defining new coordinates as arbitrary functions of the old ones. Therefore general coordinate invariance is a kind of local symmetry. The analogy with local rotations of an egg, as discussed in the previous section, can be made stronger. At each point of a manifold one can erect a \tangent space". For this, imagine a tangent plane to each point of a spherical surface, and then (this is the difficult part!) extend the notion to 3+1 dimensional spacetime. Now starting from the general coordinate invariance of general relativity, it can be shown that the laws of physics are unchanged if we perform a spatial rotation or a boost on the tangent space independently at every point 3 of spacetime. Together, rotations and boosts make up the set of \Lorentz transformations" and the Special Theory of Relativity teaches us that these are symmetries of nature. But because these can be performed independently at each point of spacetime, we really have a local, or gauge, symmetry. In this example, and the others to follow, local symmetry is not merely a consequence of the physical laws. In a definite sense it predicts those laws. What this means is that if we require local Lorentz invariance and try to find any theory whose equations possess this invariance, we will be inevitably led to Einstein's theory of gravity { including the very existence of the force as well as the precise set of possible interactions. So, from a modern point of view one can treat local Lorentz invariance as the origin of gravity. In a quantum treatment of gravity, there would be a particle communicating the gravitational force, called the graviton. It is supposed have 2 units of intrinsic angular momentum1 or spin. Although there is no direct evidence for it, it is widely expected that such a particle does exist and is exactly massless, as evidenced by the fact that the gravitational force travels from end to end of the universe. Masslessness of the graviton is actually a prediction from local Lorentz invariance. We will see that gauge symmetry always predicts massless particles in a certain sense, though nature has found interesting and subtle ways to re-interpret that statement! 4 Gauge symmetry in electrodynamics and strong in- teractions After the famous Maxwell equations were discovered, it took a long time for their symmetry properties to be appreciated. Today we know that these equations have a local symmetry under which the wave function of each matter particle acquires a phase, and the photon field shifts by a corresponding amount. The important thing is that this phase is an arbitrary function of spacetime. In the complex plane a phase is just a simple rotation, and in this sense the gauge symmetry of electrodynamics is the one we encountered earlier when we rotated each part of an egg independently2. The Abelian group of phase rotations is called U(1). Here the lesson expounded in the previous section works equally well. By merely postu- lating that the laws of physics are invariant under local phase rotations we find that the 1in units of the scaled Planck constant ¯h. 2We could even rotate these parts differently at different instants of time. 4 theory requires the existence of charged particles as well as the force-carrying particle that communicates between them, the photon. But unlike the graviton, the photon has spin 1. Again gauge invariance implies that the photon must be massless and the electromagnetic force consequently long-range. This is experimentally verified to a very high degree of ac- curacy. So from a modern point of view one can treat local phase rotations as the origin of electrodynamics. Today we know that something very similar is true for the strong interactions3. All known \hadrons" (strongly interacting particles) come in three colours each, and the field theory describing them (discovered in 1970 by Gross, Wilczek and Politzer) is invariant under local rotations of these colours. If the fields describing quarks had been real these rotations would have been exactly like the group of ordinary rotations in space, which we call SO(3) (the \O" stands for \orthogonal" which is a property of the matrices generating such rotations).