Consequences of the convexity of Legendre transformations (generalized Goldstone theorem)

Some consequences of the already proven convexity of Legendre transformations are analyzed. Different possible behaviors of a system are considered; in particular, degeneracy and a phase transition. A generalization of Goldstone's theorem [1, 2] is proved: in every theory with continuous degeneracy of the solution, zero-mass particles are present. Spontaneous breaking of continuous symmetry automatically leads to Continuous degeneracy, which explains why zero-mass particles arise in theories with broken symmetry.