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Springer optimization and its applications

On Ergodic States, Spontaneous Symmetry Breaking and Quasi-Averages

Walter Felipe Wreszinski, Valentin Anatol'evich Zagrebnov

Published Jan 1, 2023• 0 citations
Publisher
DOI
10.1007/978-3-031-46487-4_17

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It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pulé condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages.
This essay explores the nature of spontaneous symmetry breaking (SSB) in connection with a cluster of interrelated concepts such as Curie's symmetry principle, ergodicity, and chance and stability in classical systems. First, a clarification of the two existing senses of SSB is provided and an argument developed for a proposal for SSB, in which not only the possibilities but also the actual breakings are referred to. Second, a detailed analysis is given of classical SSB that answers the questions: (i) how we are justified in regarding it as a matter of chance, and (ii) why the breakings in it are equally probable. The answer provides some support to the applicability of ergodicity in special systems (such as ours).
We study Bose–Einstein condensation of a charged relativistic boson system as a phenomenon of spontaneous symmetry breaking. By using the Bogoliubov's quasi-average method, it is found that the vacuum expectation values of the scalar fields are nonvanishing, indicating the nonperturbative nature of the condensation. The vacuum of the condensate is explicitly constructed, which is a superposition of states with different charges. A relation is found between the vacuum and the Stagerstam's coherent states of definite charge.