I.3.2 Breaking of Symmetry

Although symmetry is one decisive indication of beauty and harmony, excessive symmetry is rather boring and not very attractive at all. If (everything in) the World was completely symmetrical in simple and direct ways, such as spherical shapes, it would have been much easier to describe in mathematics, but this would not make the complex and stimulating world we like to live in and love. In general, symmetrical objects are more beautiful if they also have enough complexity and exciting features that make them more fascinating.

As we introduced at the top of this chapter, because of the fractal dimensions and chaotic behavior of the system, some small fluctuations around a critical point can utterly decide the fate of the system, by determining which branch of bifurcation is followed. For large systems, this means that some individual parts, or sub-systems, may develop in many ways that may be close or sometimes completely different from each other. This symmetry breaking is the fundamental reason for pattern formation, but this applies not only with respect to physical shapes and properties of familiar objects, but also to the laws of physics themselves. That s why symmetry breaking is usually classified as explicit and spontaneous , respectively. More specifically, the first case happens when the equations of motion, the Lagrangian or the Hamiltonian, become no more invariant, while the spontaneous breaking of symmetry occurs when only the ground state of the system becomes no more invariant under some circumstances, so the symmetry is broken for perturbations around the vacuum state even though the entire Lagrangian retains that symmetry, and this breaking may also be dynamical, so that symmetry may be restored when the parameters change.

It is very important to notice that, by definition, spontaneous symmetry breaking requires the existence of a symmetric probability distribution, where any pair of outcomes have the same probability; so that the underlying laws are invariant under a symmetry transformation between the two pairs. As we shall discuss further in Chapter II, this is very important in studying super-symmetry, because it implies that the physical and psychical worlds are essentially identical, because we can transform from one to the other by a simple Wick rotation, by , because their corresponding arrows of time are orthogonal.

Pierre Curie was the first to study the role of symmetry breaking in his famous 1894 article, in which he explains symmetry breaking as the lowering of the original symmetry group of the medium to the sub-group of the phenomenon, by the action of some cause [[10]]. Therefore, symmetry breaking is what creates the phenomenon, thus we can describe symmetry breaking in terms of the relations between the original group and its resulting subgroups.

Symmetry is the supreme secret of nature, but the clearly apparent wide varieties can only be achieved with the various mechanisms of symmetry breaking or violation, which play the role of hiding the original symmetry behind the complex and interwoven factors and parameters of the current condition. In many situations, the initial deviation could be tiny and unnoticed, in which case the system would remain approximately symmetric, and the violation will cause a small correction that can still be covered by approximate conservation laws, but the laws themselves are still invariant. However, in other situations, symmetry may hide in a more profound way, where the laws themselves may change according to different transformations.