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Part of the book series: Mathematical Physics and Applied Mathematics ((MPAM,volume 10))

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Abstract

Both in physics and in mathematics itself, the classical notion of a function, characterized by assigning functional values at all possible values of the arguments, was long ago felt to be restrictive. Important notions of classical physics such as the density of a point mass or charge, are not built in to this ordinary notion of function. In quantum mechanics and quantum field theory, such “singular” or “improper” functions as δ(x) (the Dirac delta function), D(x) (the commutation function), etc., have been systematically used; these arose, in particular, in the so-called commutation relations which have played an important role in quantum theory. In mathematics itself, the restrictiveness of the classical notion of function was felt, for example, in attempts to define the Green’s function for equations of hyperbolic type (in particular, for the wave equation), in the normalization of eigenfunctions corresponding to the continuous spectrum of an operator, and in a number of other questions.

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© 1990 Kluwer Academic Publishers

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Bogolubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T., Gould, G.G. (1990). The Technique of Generalized Functions. In: Bogolubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T. (eds) General Principles of Quantum Field Theory. Mathematical Physics and Applied Mathematics, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0491-0_2

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  • DOI: https://doi.org/10.1007/978-94-009-0491-0_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6707-2

  • Online ISBN: 978-94-009-0491-0

  • eBook Packages: Springer Book Archive

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