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Homogeneous canonical transformations in phase space: On necessary conditions for the existence of symplectic actions of simple lie groups on differentiable manifolds

Однородные канониче ские преобразования в фазовом пространстве. Необхо димые условия сущест вования симплектито вых действий для прос тых групп Ли на ди симплектитовых дейс твий для простых груп п Ли на дифференцируе мых множествах. дифференцируемых мн ожествах

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Il Nuovo Cimento B (1971-1996)

Summary

The problem of the existence of homogeneous canonical transformations in phase space is dealt with in the framework of differential geometry. Necessary theorems for the existence of symplectic actions of simple Lie groups on differentiable manifolds are proved.

Riassunto

Il problema della esistenza di trasformazioni canoniche omogenee sullo spazio delle fasi è trattato con il formalismo della geometria differenziale. Si dimostrano teoremi necessari per la esistenza di azioni simplettiche di gruppi di Lie semplici su varietà differenziali.

Резюме

Рассматривается про блема существования однородных канониче ских преобразований в фазовом пространст ве в рамках дифференц иальной геометрии. пространстве в рамка х дифференциальной г еометрии. Доказывают ся необходимые теоре мы для существования симплектитовых дейс твий для простых груп п Ли на д Доказываются необхо димые теоремы для сущ ествования симплект итовых действий для п ростых групп Ли на диф ференцируемых множе ствах. симплектитовых дейс твий для простых груп п Ли на дифференцируе мых множествах. дифференцируемых мн ожествах.

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References

  1. Our notations follow as closely as possible those of R. Abraham and J. E. Marsden :Foundations of Mechanics (New York, 1967).

  2. A. Simoni, P. Zaccaria and B. Vitale:Nuovo Cimento,51 A, 448 (1967); F. Bunchaft:Nuovo Cimento,57 A, 689 (1968).

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  4. The foregoing facts are well known. A clear exposition of them can be founde.g. in F. Brickell and R. S. Clark:Differentiable Manifolds, Sect.,13’ 2 (London, 1970).

  5. V. Bargmann:Ann. Math.,59, 1 (1954); P. J. Hilton and U. Stammbach:A Course on Homological Algebra (New York, 1970).

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  7. Given an algebraA overR, by its complexificationAc we mean the∼-linear space whose elements are the∼-linear finite combinations of elements ofA, together with the product inherited fromA, which is taken to be∼-bilinear. Conversely, given an algebraA over⊂, we consider the algebraAR overR which, as a set, coincides withA. A real form of 1 is a subalgebraA0 ofAR such thatAR =A0 ⊕ iA0. Notice that a real algebra is in general only one of the real forms of its complexification. For a detailed treatment see: S. Helgason:Differential Geometry and Symmetric Spaces, Chap. III-6 (London, 1962).

  8. See for instance: S. Helgason:op. cit.

  9. See: F. Brickell and R. S. Clark:op. cit., Chap. 11.

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Authors and Affiliations

  1. Istituto di Fisica Teorica dell’ Università, Napoli

    G. Caratù, G. Marmo, A. Simoni, B. Vitale & F. Zaccaria

  2. Sezione di Napoli, Istituto Nazionale di Fisica Nucleare, Italia

    G. Caratù, G. Marmo, A. Simoni, B. Vitale & F. Zaccaria

Authors
  1. G. Caratù
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  2. G. Marmo
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  3. A. Simoni
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  4. B. Vitale
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  5. F. Zaccaria
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Caratù, G., Marmo, G., Simoni, A. et al. Homogeneous canonical transformations in phase space: On necessary conditions for the existence of symplectic actions of simple lie groups on differentiable manifolds. Nuovo Cim B 19, 228–238 (1974). https://doi.org/10.1007/BF02895645

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