Spontaneously broken symmetry

Finally it is appropriate to make some remarks about the symmetry aspects of quantum field theories, and to review the ideas implied by the expression spontaneously broken symmetry . The density operator for the field j> can be defined as... 

C. De Concini and G. Vitiello, Spontaneous Breakdown of Symmetry and Group Contractions, Nucl Phys. 116B, 141-156 (1976). C. De Concini and G. Vitiello, Relation Between Projective Geometry and Group Contraction in Spontaneously Broken Symmetries, Phys. Lett. 70B, 355-357 (1977). 

For the discovery of the mechanism of spontaneous broken symmetry in subatomic physics... 

What has happened is that in the spontaneously broken symmetry the gauge boson has acquired mass at the expense of the would-be Goldstone boson, which simply disappears. For each vector gauge field that gets massive we need one complex scalar field, one piece of which becomes unphysical and disappears (it reappears as the longitudinal mode of the vector field) leaving one real scalar physical field, the Higgs boson. 

The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing variables, which usually are either connected with conservation laws or with the Goldstone modes of the spontaneously broken symmetries. To formulate them we wUl follow the traditional approach [65-67] rather than the one based more directly on the principles of hydrodynamics and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously broken and the averaged molecular alignment singles out an axis whose orientation defines the director n, i. e. an object that has the properties of a unit vector with n = -n. The static properties are conveniently expressed in terms of a free energy density whose orientational elastic part is given by [69]... 

This is one of the typical cases when the spontaneously broken symmetry can be relatively easily observed in many-body atomic systems in the thermodynamic limit of phase transitions. If we denote the internal energy of the system in the high-symmetry configuration by U, its free energy by and entropy by S, while those for the low-symmetry SSB phase by E, S, we get for the condition of the phase transition E =E at the temperature T Ui -TS = Ui-TSp and... 

In the nematic phase, one can induce macroscopic orientation by controlling the surface boundary conditions. However, because nematic ordering is the result of a spontaneously broken symmetry, fluctuations of the director n are a soft mode. Indeed a macroscopically oriented nematic phase is much more turbid than a macro-scopically oriented smectic-A phase because of light scattering from orientational fluctuation domains. The layered structure of the smectic-A phase suppresses these orientational fluctuations, and it is this coupling that affects the character of the transition (see [5] for a broad survey of such phase transitions). However, smectic phases exhibit one-dimensional orientational order, characterized by the Landau-Peierls fluctuation of the layer spacing [6, 7]. As a result, the essential features needed to capture the NA transition are ... 

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