Abstract
It is well-known that two complete, orthonormal sets of solutions of the Klein-Gordon equation are related by invertible Bogolubov transformations and that the Bogolubov coefficients therefore satisfy certain identities. We show that the converse is false, namely, that the fact that the Bogolubov coefficients defined by two sets of solutions satisfy these identities doesnot imply that either set can be expanded in terms of the other. Several simple counterexamples are given.
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Bogolubov, N. (1947).J. Phys. USSR,11, 23; Bogoliubov, N. N. (1958).ZA. Eksp. Teor. Fiz.,34, 58 (translated in (1958).Sov. Phys. JETP,7, 51); Bogoljubov, N. N. (1958).Nuovo Cimento,7, 794; Bogoliubov, N. N., Tolmachev, V. V., and Shirkov, D. V. (1959).A New Method in the Theory of Superconductivity (Consultants Bureau Inc., New York) (translated from Russian). [Note: The author's name has been transliterated as it appeared in the (translated) article. However, he himself chose “Bogolubov” (see Bogolubov, N. N., Logunov, A. A., and Todonov, I. T. (1975).Introduction to Axiomatic Quantum Field Theory (W. A. Benjamin Inc., Reading, Massachusetts), p. 633].
Birrell, N. D., and Davies, P. C. W. (1982).Quantum Fields in Curved Space (Cambridge University Press, Cambridge).
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Dray, T., Manogue, C.A. Bogolubov transformations and completeness. Gen Relat Gravit 20, 957–965 (1988). https://doi.org/10.1007/BF00760094
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DOI: https://doi.org/10.1007/BF00760094