Nonlinear Realization of Global Symmetries

The concept of symmetry is of central importance in modern physics. Plenty information can be extracted from the symmetry of a physical system. In field theory, the symmetry plays an essential role in defining some very basic concept like particle, charge, etc. Symmetry also put strong constraint on the properties of fields/particles, as well as their interactions and correlations. In quantum field theory, Wigner’s theorem tells us that symmetry transformations act as unitary or anti-unitary operators acting on the Hilbert space. Under the action of symmetry operators, states in Hilbert space are arranged into representations (or more rigorously, projective representations) of the symmetry group G. There are basically two ways of symmetry realization in Hilbert space. One is such that the vacuum state is invariant under the action of all symmetry transformations, which is known to be the Wigner-Weyl realization, the other contains degenerate vacua, which can transforms into each other under some of the symmetry transformation. This is known as Nambu-Goldstone realization. It can be shown that in the Wigner-Weyl realization, excited states above vacuum in Hilbert space are organized into linear representations of the symmetry group. Furthermore, the field operators corresponding to these states lie in the same linear representations. Thus, from the viewpoint of fields, we may say the symmetry is linearly realized in Wigner-Weyl realization. On the other hand, in Nambu-Goldstone realization, the vacua are invariant only under the action of a subgroup H of full symmetry group G. When acting a group element outside H on a vacuum state, it will move to another vacuum. Intuitively, if this action is applied locally, i.e., with different group elements on different spacetime points, a special massless excitation will be generated, which is called the Goldstone mode. The existence of Goldstone modes Nambu-Goldstone realization is established by the famous Nambu-Goldstone’s theorem. Besides Goldstone modes, there can of course be other particle states in the theory. However, the states within the same representation of the full symmetry group G may have different mass. Thus we see that the spectrum of Nambu-Goldstone realization is very different from Wigner-Weyl realization. Meanwhile, the field operators corresponding to these states (both Goldstone modes and other particles) do not form linear representation of G. Therefore we may say the symmetry is nonlinearly realized in Nambu-Goldstone realization. ∗E-mail: [email protected]