Symmetry breaking

"In my own field of many-body physics, we are, perhaps, closer to our fundamental, intensive underpinnings than in any other science in which non-trivial complexities occur, and as a result we have begun to formulate a general theory of just how this shift from quantitative to qualitative differentation takes place. This formulation called the theory of "broken symmetry," may be of help in making more generally clear the breakdown of the constructive converse of reductionism."

"Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed to a system which has strictly less symmetry. We give a geometric approach to study this phenomenon in the setting of Hamiltonian systems. We provide a method for determining the equilibria and relative equilibria that persist after a symmetry breaking perturbation. In particular a lower bound for the number of each is found, in terms of the equivariant Lyusternik–Schnirelmann category of the group orbit."

"... the phenomena of spontaneous symmetry breaking in the radical sense of non-symmetric behaviour is rather related to the fact that, for non-linear infinitely extended systems (therefore involving infinite degrees of freedom), the solutions of the dynamical problem generically fall into classes of "islands" or "phases", stable under time evolution and characterized by the same behaviour at infinity of the corresponding solutions. Since all physically realizable operations have an inevitable localization in space they cannot change such a behaviour at infinity and therefore starting from the configurations of a given islands one cannot reach the configurations of a different island by physically realizable modifications. The different islands can then be interpreted as describing physically disjoint realizations or different phases, or disjoint physical worlds associated with the given dynamics."

"... Broken symmetries in physics go back to work in the 1950s on superconductivity. From the modern perspective (or at least, a modern perspective) a superconductor is nothing but a place where electromagnetic gauge invariance is spontaneously broken to a discrete subgroup, and from this assumption one can derive all the exact properties of superconductors, such as the Meissner effect, persistent currents, flux quantization, and the formula for the Josephson frequency."

"... a solid differs from a liquid because its crystal structure breaks the translational and rotational symmetries of space. Moreover, solids with different crystal structures should be viewed as different phases of matter because they break these symmetries in different ways. Perhaps more surprisingly, liquids and gases break no such symmetries and so should be viewed as the same phase. When you include further symmetries, such as rotations of spins in a magnet or more subtle quantum counterparts, this classification opens up a wide range of possibilities that allows us to understand almost all the known forms of matter... First, we can be sure that any attempt to change a material from one symmetry class to another will necessarily involve a violent phase transition. Second, it turns out that understanding the symmetries of a system will immediately determine many of its properties, especially at low temperature."