Geodesic
Consequences of the convexity of Legendre transformations (generalized Goldstone theorem)
Teoretičeskaâ i matematičeskaâ fizika, Tome 15 (1973) no. 3, pp. 320-331

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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. 
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     author = {A. N. Vasil'ev},
     title = {Consequences of the convexity of {Legendre} transformations (generalized {Goldstone} theorem)},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {320--331},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1973_15_3_a4/}
}
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A. N. Vasil'ev. Consequences of the convexity of Legendre transformations (generalized Goldstone theorem). Teoretičeskaâ i matematičeskaâ fizika, Tome 15 (1973) no. 3, pp. 320-331. http://geodesic.mathdoc.fr/item/TMF_1973_15_3_a4/