SYMMETRY

When we talk at an intuitive level about “symmetries of nature” we usually have in mind objects that have perfectly symmetrical shapes; for example, the geometric symmetry of crystals or the spherical shapes and motions of the planets. Those symmetries supposedly reflect the inner simplicity and harmony of the universe. The history of the concept of symmetry starts with the ancient Greeks and has developed in various ways to include the notions of beauty, harmony, and unity. One of the best examples of the power of symmetry arguments in the history of science comes from kepler’s Mysterium Cosmographicum (1596). Because he believed that God created the solar system according to a mathematical pattern, kepler attempted to correlate the distances of the planets from the sun with the radii of spherical shells that were inscribed within and circumscribed around a nest of solids. The goal was to find an agreement between the observed ratios of the radii of the planets and the ratios calculated from the geometry of the nested solids. Although the latter ratios disagreed with empirical data he went on to search for deeper mathematical harmonies in the solar system and succeeded in formulating his three laws of planetary motion; a corrected version of which later formed the foundation of Newtonian mechanics. But this is not just a historical peculiarity: belief in the mathematical harmony of the universe still holds a prominent place in various branches of physics. However, as Herman Weyl (1952) remarked “we no longer seek this harmony in static forms like regular solids, but in dynamic laws.” The statement expressed a shifting away from thinking about symmetry in terms of objects or phenomena to focus instead on the symmetry of laws. So, what exactly is the connection between symmetries and laws? Symmetry in physics involves the notion of invariance. If something remains unchanged (invariant) under a particular operation or transformation, we say that it is symmetric under that operation. For example, a cylinder is invariant under rotations about its axis, and a sphere, which has a greater degree of symmetry, is invariant under rotations about any axis through its centre. The two examples also exhibit a reflection symmetry, meaning that they look the same in a mirror. When we speak about laws, however, what is important is that they behave in the same way with respect to a variety of possible reference frames. Einstein’s principle of relativity is an example of

this. It states that the laws of physics (and the behavior of light) must be the same for any two observers moving with a constant velocity relative to one another. This equivalence of different points of view was extended in his theory of general relativity to incorporate all possible observers including those that are rotating and accelerating. What this means, in physical terms, is that the inertial effects of acceleration or rotation (e.g., the forces an astronaut feels during blast off) can be attributed to either your own motion or the presence of different gravitational forces. This conclusion, expressed more formally in Einstein’s principle of equivalence, states that the laws of gravity are such that the apparent forces due to any kind of motion are indistinguishable from gravitational forces. In that sense we can see how symmetries are related to the dynamical properties of physical systems; in other words the symmetries describe how systems or phenomena react to forces. The connection between symmetries and dynamics helps to reveal the connection between symmetries of laws and symmetries of objects. If we think of our example of the sphere, the rotation under which it remained invariant can be described by the mathematical equation that governs the sphere. Because the equation does not depend in any way on the angles of rotation, we say that both the equation and the sphere are invariant under rotation. In order to fully investigate the physical consequences of symmetry it is necessary to learn about the specific transformations or sets of transformations that leave a particular object or function invariant. The theory that deals with this is called “group theory” where a group is defined as a mathematical structure or set of elements that can be transformed into each other by means of certain operations. The set of all transformations that leaves an object or law invariant forms the symmetry group of that object. We can then make the connection between laws and objects more specific by saying that a physical object/phenomenon obeys a certain symmetry if its laws are invariant under any transformation of the corresponding symmetry group. For example, space is symmetric under translations – no point in space is privileged over any other – and, consequently, that invariance under spatial translation means that the laws of physics are the same in London as in Toronto. Similarly, if physical laws are independent of time (time-invariance) then experimental results will be the same regardless of when the experiment is performed. A further way in which laws and symmetries are connected involves the link between invariance under a symmetry operation and the existence of conservation laws in physics, laws stating that the total amount of some quantity is constant and does not change with time. A well-known example is the conservation of energy, which says that energy cannot be created or destroyed but only transformed from one form to another. A theorem first proved by Emmy Noether in 1918 states that for every symmetry of the laws of physics there is a corresponding conservation law. (The reverse is also true although that wasn’t part of her original theorem.) What this means is that for any invariance in a particular symmetry group there is a corresponding physical quantity that is conserved under the applicable transformation. For example, the conservation of energy and momentum is associated with the impossibility of measuring an absolute position in time and space, respectively, which is in turn associated with the homogeneity of time and space; in other words, every moment in time and every point in space is as good as any

other. Put slightly differently, because of time-invariance the laws of physics predict the same evolution of identical processes regardless of when they occur, which in turn implies that the conservation of energy is built into the laws describing the process. Invariance under spatial rotations implies conservation of angular momentum, which is the product of the mass, velocity, and position of a particle. The link between symmetry and conserved quantities points to a slightly more precise definition of Noether’s theorem: for every symmetry of the equations of motion of a system there is a quantity that is conserved by its dynamics. However, when we say that equations or laws of a theory are unchanged under specific transformations, we say that they are “covariant.” The technically precise use of the term “invariance” involves reference to specific objects or things that remain unchanged under certain transformations. And, in the case of conservation laws, the thing that remains invariant is the conserved quantity. The notion of symmetrical laws or equations becomes important in the discussion of hidden and local symmetries; so now let us turn our attention to some of the different kinds of symmetries in order to give us a better understanding of the way symmetry functions in modern physics. The symmetries important for physics can be divided into the following categories: global and local, continuous and discrete, as well as geometrical and internal. Global symmetries deal with transformations that are not affected by position in space and time. They can be either geometrical, reflecting the homogeneity of space and time or internal which refers to the intrinsic nature of particles (like the conservation of various charges) rather than their position or motion. The symmetries mentioned above (translation through space, translation through time, as well as rotation about an axis) are all geometrical symmetries. But, we can also have global internal symmetries which involve the transformation of several field components into one another in such a way that the physical situation remains unchanged; that is, each component is rotated to the same degree, with the total field energy remaining constant. In the case of local internal symmetries the rotation of field components varies from point to point so that a rotation at one position does not necessarily correspond to a rotation at another. An example of a continuous symmetry is the rotation of a circle; it is a continuous operation describable by groups that possess an infinite number of elements. In contrast to continuous symmetries there are also discrete symmetries, instances of which include the rotations of a square or a triangle. Spatial reflections (things looking the same in a mirror) are also discrete symmetries where the associated transformation group contains only two elements, reflection of spatial coordinates and identity. Discrete internal symmetries involve invariance under charge conjugation where there is an exchange of particles with their anti-particles. Continuous internal symmetries govern specific properties of particles and the continuous transformation of quantized fields. This is an extension of the ordinary geometrical symmetries; so, for example, the U(1) group governs the continuous rotations of a circle and also describes the symmetry of a single field.