I.1 Normal Symmetry and Classical Physics

Physics theories that are designed to explain certain phenomena are largely related to the initial conditions, which are often arbitrary and unpredictable. Henceforth, symmetry can summarize these regularities in limited sets of rules, or invariance laws, that are independent of the dynamics of the system. Without these limiting symmetries, it is impossible to deduce the laws of nature, at least because it would not be possible to test the theory by repeating the experiments, at different places and times, and yet expect the same results. Additionally, the laws of conservation, that are direct consequences of symmetry, provide the possibility of predicting or calculating how the system evolves in time.

The importance of symmetry in physics is not restricted to the physical properties that can be symmetrical in many geometrical ways, such as the arrangement of atoms in crystalline materials, or meteorological patterns that are employed in forecasting the weather. More importantly is the invariance of the laws themselves, so that the behavior of objects is not affected when their coordinates are transformed by means of some symmetry operations. The two leading theories of Relativity and Quantum Mechanics involve crucial notions of symmetry in their fundamental principles. In Relativity, the speed of light is invariant between all frames of reference, and this defines a set of symmetrical transformations called Poincar group. The invariance of physics laws, under arbitrary differentiable coordinate transformations, is also one of the principles of General Relativity that we shall discuss in Chapter III.

Figure I.1: The atoms in crystalline materials are arranged in a lattice with various symmetries that can be viewed as a regular tiling of a space by a primitive cell.

Although it was not so clearly noticed until the beginning of 20th century, symmetry was also important in Classical Mechanics, for example in the principle of equivalence of inertial frames, or the Galilean invariance. It was later discovered by Noether, symmetry always leads to conservation laws. Although the conservation of momentum and energy were fundamental in Classical Mechanics, no one had noticed that they are consequences of the symmetries that underlay these laws. In Electrodynamics also, Maxwell s equations have both Lorentz invariance and gauge invariance, as we shall discuss further in section II.1.1.

In 1905, with the advent of Special Relativity, where the invariance of the speed of light was one of its primary principle, together with the invariance of physics laws. As a result, space and time obey Lorentz invariance rather than the classical Galilean transformation. After ten years, this was reinforced even more with the Principle of Equivalence in General Relativity, which introduced the local symmetry of space-time that dictated how the dynamics of gravity are linked with the deformations of space-time geometry.

Nonetheless, the importance of symmetry in physics was more evident in the second half of the 20th century after the development of Quantum Field Theory that lead to the Standard Model of elementary particles. All quantum field theories, such as Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD), are based on gauge symmetry, as we shall discuss further in section II.1.

In trying to unify the fundamental interactions, physicists are relentlessly searching for new symmetries of nature. This implies that there must be some intricate patterns of symmetry breaking, so that the current groups are part of some larger local symmetries that could include gravity, such as super-symmetry that has the ability of unifying bosons and fermions, or forces and matter, into one single pattern, as we shall discuss further in Chapter II, where we shall also see that, in addition to the familiar symmetries, String Theory contains other mirror symmetry between geometric objects called Calabi-Yau manifolds, which look geometrically different but are nevertheless equivalent when employed as extra dimensions.