cannot be detected externally. A more rigorous generalization of the Faddeev–Popov procedure is given by BRST quantization. Spontaneous symmetry-breaking Spontaneous symmetry breaking is a mechanism whereby the symmetry of the Lagrangian is violated by the system described by it. To illustrate the mechanism, consider a linear sigma model containing real scalar fields, described by the Lagrangian density: where and are real parameters. The theory admits an global symmetry: The lowest energy state (ground state or vacuum state) of the classical theory is any uniform field satisfying Without loss of generality, let the ground state be in the -th direction: The original fields can be rewritten as: and the original Lagrangian density as: where . The original global symmetry is no longer manifest, leaving only the subgroup . The larger symmetry before spontaneous symmetry breaking is said to be "hidden" or spontaneously broken. Goldstone's theorem states that under spontaneous symmetry breaking, every broken continuous global symmetry leads to a massless field called the Goldstone boson. In the above example, has continuous symmetries (the dimension of its Lie algebra), while has . The number of broken symmetries is their difference, , which corresponds to the massless fields . On the other hand, when a gauge (as opposed to global) symmetry is spontaneously broken, the resulting Goldstone boson is "eaten" by the corresponding gauge boson by becoming an additional degree of freedom for the gauge boson. The Goldstone boson equivalence theorem states that at high energy, the amplitude for emission or absorption of a longitudinally polarized massive gauge boson becomes equal to the amplitude for emission or absorption of the Goldstone boson that was eaten by the gauge boson. In the QFT of ferromagnetism, spontaneous symmetry breaking can explain the alignment of magnetic dipoles at low temperatures. In the Standard Model of elementary particles, the W and Z bosons, which would otherwise be massless as a result of gauge symmetry, acquire mass through spontaneous symmetry breaking of the Higgs boson, a process called the Higgs mechanism. Supersymmetry All experimentally known symmetries in nature relate bosons to bosons and fermions to fermions. Theorists have hypothesized the existence of a type of symmetry, called supersymmetry, that relates bosons and fermions. The Standard Model obeys Poincaré symmetry, whose generators are the spacetime translations and the Lorentz transformations . In addition to these generators, supersymmetry in (3+1)-dimensions includes additional generators , called supercharges, which themselves transform as