Positive Particle Frequencies in Some Time-Periodic Universes

Letters in Mathematical Physics

Volumn 36, Issue 3, 1996, Pages 277-290

  Droz Vincent, Philippe ^a  

^a CNRS   (France)

Author keywords

Klein Gordon equation; Particle frequencies; Time periodic universes

Indexed keywords

EID: 0007348355     PISSN: 03779017     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00943280     Document Type: Article

References (18)

1. Berger, M., Gauduchon, P. and Mazet, E.: Lecture Notes in Math. 194, Springer-Verlag, Berlin, Heidelberg, New York, 1971; Gallot, S., Hulin, D. and Lafontaine, J.: Riemannian Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1987, Chap. IV D, pp. 196-202.
2. Berger, M., Gauduchon, P. and Mazet, E.: Lecture Notes in Math. 194, Springer-Verlag, Berlin, Heidelberg, New York, 1971; Gallot, S., Hulin, D. and Lafontaine, J.: Riemannian Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1987, Chap. IV D, pp. 196-202.
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4. Lichnerowicz, A.: Bull. Soc. Math. France 92 (11) (1964). Contribution to De Witt and De Witt (eds), Relativity, Groups and Topology, Les Houches 1963, Science Publishers Inc., New York, 1964.
5. Lichnerowicz, A.: Bull. Soc. Math. France 92 (11) (1964). Contribution to De Witt and De Witt (eds), Relativity, Groups and Topology, Les Houches 1963, Science Publishers Inc., New York, 1964.
6. Ashtekar, A. and Magnon, A.: Proc. Roy. Soc. London A 346 (1975), 375.
7. As the literature on the subject is enormous, an extensive list of references is impossible. See, for instance Fulling, S. Thesis, Princeton University (1972); Parker, L. and Fulling, S.: Phys. Rev, D 9 (1974), 341; Woodhouse, N.: Phys. Rev. Lett. 36 (1976), 999. For a recent survey, see Fulling, S.: Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press, 1989.
8. As the literature on the subject is enormous, an extensive list of references is impossible. See, for instance Fulling, S. Thesis, Princeton University (1972); Parker, L. and Fulling, S.: Phys. Rev, D 9 (1974), 341; Woodhouse, N.: Phys. Rev. Lett. 36 (1976), 999. For a recent survey, see Fulling, S.: Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press, 1989.
9. As the literature on the subject is enormous, an extensive list of references is impossible. See, for instance Fulling, S. Thesis, Princeton University (1972); Parker, L. and Fulling, S.: Phys. Rev, D 9 (1974), 341; Woodhouse, N.: Phys. Rev. Lett. 36 (1976), 999. For a recent survey, see Fulling, S.: Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press, 1989.
10. As the literature on the subject is enormous, an extensive list of references is impossible. See, for instance Fulling, S. Thesis, Princeton University (1972); Parker, L. and Fulling, S.: Phys. Rev, D 9 (1974), 341; Woodhouse, N.: Phys. Rev. Lett. 36 (1976), 999. For a recent survey, see Fulling, S.: Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press, 1989.
11. Moreno, C.: J. Math. Phys. 18 (1977), 2153.
12. Moreno, C.: Rep. Math. Phys. 17 (1980), 333.
13. Floquet, G.: Ann. E.N.S. 12 (1883), 47; Ince, E. L.: Ordinary Differential Equations, Dover, New York, 1956; Hochstadt, H.: Differential Equations: A Modern Approach, Dover, New York, 1975, Chap. 5, pp. 191-197, see especially p. 193.
14. Floquet, G.: Ann. E.N.S. 12 (1883), 47; Ince, E. L.: Ordinary Differential Equations, Dover, New York, 1956; Hochstadt, H.: Differential Equations: A Modern Approach, Dover, New York, 1975, Chap. 5, pp. 191-197, see especially p. 193.
15. Floquet, G.: Ann. E.N.S. 12 (1883), 47; Ince, E. L.: Ordinary Differential Equations, Dover, New York, 1956; Hochstadt, H.: Differential Equations: A Modern Approach, Dover, New York, 1975, Chap. 5, pp. 191-197, see especially p. 193.
16. In the conventional terminology of linear differential equations, 'stable, unstable' simply stand for bounded, unbounded.
17. It is clear that essential frequency may differ only by some multiple of 2π/T from the 'naive frequency' one could have defined by requiring 9 to stay within the interval [-π, +π]. We do not know for which analytic form of B(t), and for which values of μ and λ, it may happen that this difference vanishes. In particular, we cannot discard the possibility that naive and essential frequencies have opposite signs.
18. Liapounoff, A. M.; Ann. Fac. Sci. Toulouse 9 (1907), 203-469, (originally in Russian, Kharkov 1892).