- Dual superconducting model
quantum gauge theory, the dual superconducting model is a proposed explanation of confinement as the dual of a superconductor. A superconductor, at least BCS superconductors, is another name for the Higgs mechanism.
Suppose a gauge theory has monopoles. Monopoles are particles. The simplest case is a U(1) theory. The monopole charges are quantized as
integers. And the monopole charge is conserved. So we can describe the monopoles using a quantum field with a global U(1) (not related to the gauge U(1)) symmetry. At weak couplings, the monopoles are really massive and so there are few monopoles around and it is really energetically costly to create a monopole-antimonopole pair out of the vacuum. But as the coupling goes up, the monopole masses keep going down. Suddenly, we reach a stage where creating a monopole-antimonopole pair "reduces" the energy(or free energy, as the case may be). This signals the onset of an instability and we get a monopole condensate breaking the monopole U(1) symmetry.
It is easy to understand this.
Electromagnetismadmits a duality interchanging electric and magnetic fieldand electric charges and monopoles. Actually, the symmetry can be extended to an arbitrary E-B rotation, but that is irrelevant here. So let us implement this duality. We get a U(1) theory with an electrically charged particle and its antiparticle of the opposite charge. What happens if the energy goes "down" when a particle-antiparticle pair is created? We get a Higgs condensate, or if you prefer, a BCS condensate. This is nothing other than the Higgs mechanism. What happens to magnetic fields in superconductors/Higgs condenstates? They get quantized into vortices called magnetic flux tubes. So what happens if you introduce a monopole-antimonopole pair into a superconductor? We get a flux tube connecting both of them because the magnetic field cannot spread out. This flux tube has a constant tension and as you separate the monopole-antimonopole pair, the flux tube lengthens and more energy is stored within the flux tube and we get an approximately linear potential for large separations.
Back to the original monopole condensate: dually, we would expect to see vortices corresponding to electric flux tubes. Let us introduce an electrically charged particle-antiparticle pair. An electric flux tube connects both of them and we get a linear potential at large distances. We get confinement.
Maximum Abelian gauge
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