Summary
The nature of asymmetric ground states in invariant theories is investigated in detail for the cases in which Coulomb-type interactions are absent from the original equations of interacting non-relativistic fields. It is shown that such ground states can be created only by the Bose-Einstein condensation of massless Bose fields in the stationary states (thus requiring the presence of such fields). Energy differences are seen to be forbidden in observed symmetry multiplets. The considerations, predominantly algebraic, which lead to these results are shown to provide a good qualitative picture of the behaviour of systems with asymmetric ground states. As illustrations, the examples of ferromagnetism, neutral superconductivity and superfluidity are treated from the same unified viewpoint.
Riassunto
Si studia dettagliatamente la natura degli stati fondamentali asimmetriei nelle teorie invarianti nei casi in cui le interazioni coulombiane sono assenti dalle equazioni originali dei campi relativistici interagenti. Si dimostra che tali stati fondamentali possono essere creati solo dalla condensazione di Bose-Einstein di campi di Bose privi di massa negli stati stazionari (richiedendo così la presenza di tali campi). Si vede che sono proibite differenze di energia nei multipletti di simmetria osservati. Si dimostra che le considerazioni, prevalentemente algebriche, che portano a questi risultati danno una buona rappresentazione qualitativa del comportamento dei sistemi con stati fondamentali asimmetrici. A scopo illustrativo, si trattano dallo stesso punto di vista unificato gli esempi del ferromagnetismo, della superconduttività neutra e della superfluidità.
Реэюме
Подробно исследуется природа асимметричных основных состояний в инвариантных теориях в тех случаях, в которых отсутствуют вэаимодействия Кулоновского типа в основных уравнениях вэаимодействуюших нерелятивистских полей. Покаэано, что такие основные состояния могут обраэовываться только посредством конденсации Боэе-Эйнщтейна для беэмассовых Боэе полей в стационарных состояниях (тем самым требуя наличия таких полей). Покаэывается, что раэности знергий эапрешены в наблюдаемых мультиплетах симметрии. Покаэано, что рассмотрения, в основном алгебраические, которые ведут к зтим реэультатам, дают хорощую качественную картину поведения систем с асимметричными основными состояниями. Как иллюстрации, с единой точки эрения рассматриваются примеры ферромагнетиэма, нейтральной сверхпроводимости и сверхтекучести.
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References
R. N. Sen andH. Umezawa:Limitations on the Spontaneous Breakdown of Symmetries, reported at theColloquium on l’Extension du Groupe de Poincaré aux symétries internes des Particules Elémentaires, Gif-sur-Yvette (S-et-O.) (April 1966). A detailed account will be published elsewhere.
L. Leplae, R. N. Sen andH. Umezawa:Problems of Fundamental Physics, (Kyoto, 1965), p. 673; also published asProgr. Theor. Phys. Suppl., Extra Number (1965).
A detailed study of such B-E condensations has been made byH. Ezawa:Journ. Math. Phys.,6, 380 (1965); see alsoH. Araki andE. J. Woods:Journ. Math. Phys.,4, 637 (1963).
W. Heisenberg:Zeits. Naturforsch.,14a, 441 (1959); earlier works are cited there.
This remarkable aspect of the dynamical rearrangement has already been noticed in several examples. See ref. (5–7).
H. Umezawa:Nuovo Cimento,38, 1415 (1965);40, 450 (1965).
K. Nakagawa, R. N. Sen andH. Umezawa:Nuovo Cimento,42, 565 (1966).
L. Leplae andH. Umezawa:Nuovo Cimento,44 B, 410 (1966).
Recently very interesting considerations on the spontaneous breakdown of symmetry in systems with Coulomb-type potentials have been presented by several authors. See, for example, ref. (8).
G. S. Guralnik, C. R. Hagen andT. W. B. Kibble:Phys. Rev. Lett.,13, 585 (1964);N. G. Deshpande andS. A. Bludman:Spontaneously Broken Symmetries and the Question of Massless Particles in a Model Field Theory, University of Pennsylvania Preprint (1966);P. W. Higgs:Spontaneous Symmetry Breakdown without Massless Bosons, University of North Carolina Preprint (1966).
H. Wagner:Long-Wavelength Excitations and the Goldstone Theorem in Many-Particle Systems with Broken Symmetries, Max-Planck-Institut für Physik and Astrophysik (München) Preprint (1966).
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Supported in part by the Air Force Office of Scientific Research under contract no. AF 61(052)-826 through the European Office of Aerospace Research (OAR), USAF.
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Leplae, L., Sen, R.N. & Umezawa, H. Asymmetric ground states in invariant many-body theories. Nuovo Cimento B (1965-1970) 49, 1–31 (1967). https://doi.org/10.1007/BF02710990
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DOI: https://doi.org/10.1007/BF02710990