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Spontaneous symmetry breaking: variations on a theme

Spontaneous symmetry breaking (SSB) is a widespread phenomenon in several areas of physics. In this paper I wish to illustrate some situations where spontaneous symmetry breaking presents non obvious aspects. The first example is taken from molecular physics and is related to the paradox of existence of chiral molecules. The second case refers to a Dirac field in presence of a magnetic field. Gusynin, Miransky and Shovkovy have shown that Nambu-Jona-Lasinio (NJL) models, where SSB of chiral symmetry takes place for the nonlinear coupling over a certain threshold, in presence of a magnetic field exhibit SSB for any value of the coupling. They called this phenomenon magnetic catalysis. I will discuss this problem in 2 + 1 dimensions from an operatorial point of view and show that the basic phenomenon is a double pairing induced by the magnetic field in the vacuum for any value of the mass. This pairing provides the environment responsible for chiral symmetry breaking in NJL models with very weak nonlinearities. The third case illustrates briefly SSB in stationary nonequilibrium states and its possible relevance in natural phenomena.