Just how gauge fixing invalidates the Goldstone, Salam, and Weinberg theorem was shown in a paper by Gerald Guralnik, Carl Hagen, and Tom Kibble, 7 7. The Lorentz and gauge symmetries that are built into the classical theory still control the quantum physics, but they do so through a delicate choreography. In quantizing the theory, you need to choose arbitrarily one of the equivalent quantum descriptions of the physics that’s what is meant by fixing a gauge. Instead, gauge symmetry is a redundancy you can formulate the quantum theory in a way that is manifestly Lorentz invariant and manifestly gauge invariant, but that formulation actually represents an infinite number of copies of the same physical system. Gauge symmetries, in fact, are not symmetries in the usual sense of relating seemingly distinct physical processes or configurations. Where did the extra degree of freedom come from? It’s the Goldstone mode! Anderson concluded that “these two types of bosons seem capable of ‘cancelling each other out’ and leaving finite mass bosons only.” Anderson’s work made it clear that gauge theories and symmetry breaking have a special relationship what remained was to understand it. Being a massive spin-1 boson, the plasmon has an extra longitudinal polarization compared with a propagating photon, which also is a spin-1 boson but with only two transverse polarizations. Anderson started with the simple London theory of the Meissner effect, rewrote the equations in a relativistic form more palatable to particle physicists, and showed that they describe what is, in effect, a massive photon he called it a plasmon. Superconductors also exhibit the famous Meissner effect, the expulsion of external magnetic fields that makes possible magnetic levitation. It has a symmetry-breaking condensate-the Cooper pairs-but no Goldstone boson. Consider the Bardeen-Cooper-Schrieffer superconductor that was Nambu and Goldstone’s original inspiration. In 1962 Anderson (shown in the right-hand panel of figure 1 ) realized that the twin problems of massless Goldstone bosons and massless gauge bosons were related. If nature employed gauge theories beyond electromagnetism, then where were all the massless cousins of the photon? As it turned out, Pauli had developed the same construction on his own, but he had abandoned it when he realized that the symmetry of the theory, called a gauge symmetry, would force the gauge bosons of such models to be exactly massless, just as the photon is. Yang gave a seminar on his new idea at the Institute for Advanced Study in Princeton, New Jersey, where Wolfgang Pauli verbally attacked him. In their theory, new forces are mediated by new particles called gauge bosons in a way similar to the way electromagnetic forces are mediated by photons. Yang and Robert Mills produced a mathematically elegant generalization of electromagnetism. If no new physics intervenes, an unlucky quantum fluctuation will eventually spark a cosmic catastrophe.Ī similar embarrassment involving massless particles had already been festering in the particle-physics community for some years. The measured mass of the Higgs boson implies that the symmetry-breaking vacuum is metastable. Cosmologists are trying to understand the symmetry-breaking Higgs phase transition, which took place early in the history of the universe, and whether that event explains the excess of matter over antimatter.
Now that the Higgs boson (or something much like it) has been found at the Large Hadron Collider (LHC see Physics Today, September 2012, page 12), particle experimentalists are searching for more kinds of Higgs bosons and working to find out if the Higgs boson interacts with the dark matter that holds the universe together. And just as a particular glucose orientation breaks an underlying rotation symmetry, a nonvanishing vacuum expectation value of the Higgs boson field, as we will describe, breaks symmetries that would otherwise forbid masses for elementary particles.
But in quantum field theory, the ground state, or vacuum, behaves like a many-body system. It may seem that the above discussion has no relevance to particle physics in general or to the Higgs boson in particular.
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