Superselection
[1] It was originally introduced by Gian Carlo Wick, Arthur Wightman, and Eugene Wigner to impose additional restrictions to quantum theory beyond those of selection rules.[2] A superselection sector is a concept used in quantum mechanics when a representation of a *-algebra is decomposed into irreducible components.All the wave functions within a single superselection sector satisfy A large physical system with infinitely many degrees of freedom does not always visit every possible state, even if it has enough energy.In a solid, different rotations and translations which are not lattice symmetries define superselection sectors.If a string is wound around a circular wire, the total number of times it winds around never changes under local fluctuations.Both of these path integrals have the property that large changes in an effectively infinite system require an improbable conspiracy between the fluctuations.But when a superconductor fills space, or equivalently in a Higgs phase, electric charge is still globally conserved but no longer defines the superselection sectors.In this case, the superselection sectors of the vacuum are labeled by the direction of the Higgs field.This suggests a deep relationship between symmetry breaking directions and conserved charges.Below the phase transition temperature, an infinite ising model can be in either the mostly-plus or the mostly-minus configuration.If it starts in the mostly-plus phase, it will never reach the mostly-minus, even though flipping all the spins will give the same energy.By changing the temperature, the system acquired a new superselection rule--- the average spin.An Ising model on a finite lattice will eventually fluctuate from the mostly plus to the mostly minus at any nonzero temperature, but it takes a very long time.The amount of time is exponentially small in the size of the system measured in correlation lengths, so for all practical purposes the flip never happens even in systems only a few times larger than the correlation length., and the energy or action only depends on combinations which are symmetric under rotations of these components into each other, the contributions with the lowest dimension are (summation convention): and define the action in a quantum field context or free energy in the statistical context.So as t moves toward more negative values in either context, the field has to choose some direction to point.In the ordered phase, there is still a little bit of symmetry--- rotations around the axis of the breaking.In the disordered phase, the superselection sectors are described by the representation of SO(3) under which a given configuration transforms globally.There is a mass gap, or a correlation length, which separates configurations with a nontrivial SO(3) transformations from the rotationally invariant vacuum.This is true until the critical point in t where the mass gap disappears and the correlation length is infinite.In addition, pairs of winding configurations with opposite topological charge can be produced copiously as the transition is approached from below.When the winding number is zero, so that the field everywhere points in the same direction, there is an additional infinity of superselection sectors, each labelled by a different value of the unbroken SO(2) charge.But there is no mass gap for all the superselection sectors labeled by zero because there are massless Goldstone bosons describing fluctuations in the direction of the condensate.fields), the superselection sectors are labelled by a nonnegative integer (the absolute value of the topological charge).If the Higgs t parameter is varied so that it does not acquire a vacuum expectation value, the universe is now symmetric under an unbroken SU(2) and U(1) gauge group.Below the phase transition, only electric charge defines the superselection sector.Consider the global flavour symmetry of QCD in the chiral limit where the masses of the quarks are zero.This is not exactly the universe in which we live, where the up and down quarks have a tiny but nonzero mass, but it is a very good approximation, to the extent that isospin is conserved.