Symmetries and Symmetry Breaking

Abstract

Symmetries play an important role in modern physics. For instance, the Standard Model of high-energy physics is to a large extent specified by its local symmetry group U(1)×. SU(2)= SU(3). Apart from the esthetic beauty of having a symmetric description of physical phenomena, the presence of a symmetry in the problem also often leads to useful practical simplifications. For example, the energy-level structure of a Hamiltonian, and in particular the degeneracies, can often be understood on the basis of the symmetry of the Hamiltonian without doing any calculations. Another use of symmetries was seen in Chap. 3, where the rotational symmetry of an isotropic harmonic potential allows for a separation of variables in the Schro¨dinger equation and an expansion of the wavefunction into spherical harmonics. This last example illustrates the fact that symmetries lead to conservation laws, in this case the conservation of angular momentum.

In condensed-matter physics symmetries also play a crucial role, because many phase transitions can be seen as a spontaneous breakdown of symmetry, i.e. the Hamiltonian has a certain symmetry which is not shared by a particular ground state of the many-body system. If this occurs, the application of the symmetry operation onto this ground state leads to another eigenstate of the system with the same energy. As a result, the ground state must be degenerate. This leads to the fundamental problem of how the system chooses a particular ground state when it goes through a phase transition. In atomic gases, the answer to this question is related to the interesting phenomena of phase diffusion that we discuss at the end of this chapter. In addition, we have seen in the previous chapters how phase transitions can be described by making use of a Landau free-energy functional. In the case of the Ising model, we explicitly showed that the Landau free energy is an effective Hamiltonian that comes about by integrating out the microscopic degrees of freedom. Moreover, starting from a microscopic action, we demonstrated how the Hubbard-Stratonovich transformation can be used to introduce the order parameter into the problem and to obtain the effective long-wavelength action of the system. In this chapter, we formalize the concept of effective actions and study some of their symmetry properties.