Spontaneous symmetry breaking is widespread and fundamental in physics and science. The most famous occurrence is that it is the mechanism responsible for the importance of Higgs physics, the reason quarks and electrons are allowed to have mass, and for the vacuum of our universe not being nothing. The notion is widespread in condensed matter physics, and indeed was first understood there. But it is much broader, potentially leading to confusion between theories and solutions in many areas.
The basic idea can be explained simply and generally. Suppose a theory is stated in terms of an equation, X times Y =16. For simplicity consider only positive integer values of X, Y as solutions. Then there are three solutions, X=1 and Y=16, X=2 and Y=8, and X=Y=4. What is important is that the theory (XY=16) is symmetric if we interchange X and Y, but some solutions are not. The most famous example is that the theory of the solar system has the Sun at the center and is spherically symmetric, but the planetary orbits are ellipses, not symmetric. The spherical symmetry of the theory misled people to expect circular orbits for centuries. Whenever a symmetric theory has non-symmetric solutions, which is common, it is called spontaneous symmetry breaking.
In this example above, as often in nature, there are several solutions, so more information is needed, either theoretical or experimental, to determine nature’s solution. We could measure one of X or Y and the other is determined. Improving the theory leads to an interesting case. Suppose there is an additional theory equation, X+Y=10, also symmetric if we interchange X and Y so the theory remains symmetric. But now there is a unique solution, X=2, Y=8, and it is not symmetric. In fact, there are no symmetric solutions.
Magnetism is a familiar real world example. The equations describing individual iron atoms don’t distinguish different directions in space. But when a piece of iron is cooled below about 770°C, it spontaneously develops a magnetic field in some direction. The original symmetry between different directions is broken. Describing this is how the name “spontaneous symmetry breaking” originated. In physics what happens is understood—known electromagnetic forces tend to make the spins of individual atoms become parallel, and each spin is a little magnet.
Normally we expect all fields (such as electromagnetic fields) to be zero in the ground state or vacuum or the universe. Otherwise they add energy, and the universe will naturally settle in the state of minimum energy. Now we have learned that the universe is in a lower state of energy when the Higgs field is non-zero than when it is zero, a non-symmetric result, and that is essential for understanding how electrons and quarks get mass. Nature’s solution is a state of reduced symmetry.
In many fields we make theories to describe and explain phenomena. But the behavior of systems is described by the solutions to the theories, not by the theories alone. We saw here that trying to deduce the properties of the solutions, and the behavior of phenomena in sciences and social sciences, and the world in general from the form of the theory can be completely misleading. Another way to view the situation is the reverse perspective: the properties of the theory (such as its symmetries) may be hidden when we only observe the non-symmetric solutions. If it’s described by equations it’s easy to see this, but it’s true much more generally. These ideas should be much better known.