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Two classes of generalized functions used in nonlocal field theory

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Abstract

We elucidate the relation between the two ways of formulating causality in nonlocal quantum field theory: using analytic test functions belonging to the space S0 (which is the Fourier transform of the Schwartz space \(\mathcal{D}\)) and using test functions in the Gelfand-Shilov spaces S α 0 . We prove that every functional defined on S0 has the same carrier cones as its restrictions to the smaller spaces S α 0 . As an application of this result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular generalized functions of tempered growth and obtain the corresponding extension of Vladimirov’s algebra of functions holomorphic in a tubular domain.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 195–210, May, 2005.

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Soloviev, M.A. Two classes of generalized functions used in nonlocal field theory. Theor Math Phys 143, 651–663 (2005). https://doi.org/10.1007/s11232-005-0096-8

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Keywords

  • nonlocal quantum fields
  • causality
  • Wightman functions
  • analytic functionals
  • Hörmander’s estimates
  • Paley-Wiener-Schwartz-type theorems