I.3 Symmetry and Group Theory

Symmetry is the underlying principle behind all patterns in nature, including geology, biology, chemistry and physics, as well as in various arts and synthesized products, such as architecture, pottery, quilting and rugs. In most languages, the word for symmetry gives the sense of harmonious and beautiful proportion and balance. The word symmetry itself is derived from combining the Greek words: syn- , which means together, and metron , which means measure , which were composed into the Latin word symmetros in the 16th century, and it means agreement in dimensions, due to proportion or arrangement .

Symmetry is a geometrical or mathematical feature of some systems where the system remains unchanged under particular transformations. For example, a circle is completely symmetrical with respect to its center, because it remains the same under any rotation or reflection. A square is also symmetrical, but only under a set of some transformations, such as rotating by  degrees, or flipping around its diagonals. Therefore, the family of particular transformations may be continuous, such as the rotations of a circle, or discrete, such as the rotations of a square, the reflection of a bilaterally symmetric figure, or some rotations of regular polygons. Continuous symmetries are described by Lie groups, where the group operations are compatible with smooth functions, that are equivalent to differentiable manifolds. In contrast, discrete symmetries are described by finite groups. In either case, the representation of symmetries by groups can significantly simplify many problems, and this formed the foundation for the fundamental theories of modern physics.

Figure I.6: Reflectional and rotational symmetry in a circle and in hexagon. The symmetry in the circle is continuous, because we can reflect around any line passing from the center, or rotate by any degree, without changing the original shape. In the hexagon, this is only guaranteed under certain reflections and rotations (by 60 degrees) as shown by the dotted and dashed lines, respectively.

As we shall discuss in section III.1.4, one important subclass of continuous symmetries in physics are space-time symmetries, which involve transformations of either space or time, or both together; so we can have spatial symmetries, temporal symmetries, or spatio-temporal symmetries, respectively. This provided elegant representation of Special and General Relativity, as we shall describe in section I.1, and it is one of the primary tools to study Quantum Gravity, as we shall describe further in Chapter II.

Symmetries that hold at all points of space-time are called global, whereas local symmetries have different transformations at different points of space-time. It can be argued that all fundamental symmetries are local, because global symmetries are either broken, such as time reversal invariance, or approximate, such as isotopic spin invariance, or they are the remnants of spontaneously broken local symmetries, which form the basis for gauge theories that we shall discuss in section II.1.1.

Spontaneous symmetry breaking where the ground-state breaks the system s symmetry plays the main role in generation of cooper pairs in superconductivity, Goldstone Bosons, acquisition of mass through Higgs mechanism in the Standard Model and generation of phonons in crystals. On the other hand, explicit symmetry breaking occurs when the defining equations of motion represented by the Lagrangian break the symmetry.

Symmetries in physical systems are defined in terms of conserved Noether Currents of the associated Lagrangian. In electrodynamic systems, global symmetry is defined through conservation of charges, which is reflected in gauge symmetry. However, loss of charges from a radiating system can be interpreted as localized loss of the Noether current which implies that electrodynamic symmetry has been locally broken.