SMF

Maassen Kernels and Self-Similar Quantum Fields

Maassen Kernels and Self-Similar Quantum Fields

K. R. PARTHASARATHY
  • Année : 1996
  • Tome : 236
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 81S25, 60G60
  • Pages : 227-247
  • DOI : 10.24033/ast.341

In his Lecture Notes [Maj] P. Major has outlined a theory of multiple Wiener-Itô integrals with respect to a stationary Gaussian random field $\xi $ over the Schwartz space ${\cal S} (\IR ^d)$ of rapidly decreasing smooth functions in $\IR ^d$. Furthermore, he has exploited the same to construct self-similar random fields subordinate to $\xi $. Here, we observe that the Hilbert space of functions square integrable with respect to the probability measure $P$ of $\xi $ can be identified in a natural way with the Hilbert space of functions square integrable with respect to the symmetric Guichardet measure [Gui] constructed from the spectrum of $\xi $. Under such an identification, multiplication of random variables on the probability space of $\xi $ becomes the twisted convolution of Lindsay and Maassen [Li M 1,2] for Maassen kernels [Maa], [Mey]. The multiple Wiener-Itô integral of Major is described neatly by a twisted version of Meyer's multiplication formula (see (IV.4.1 in [Mey]). Following Lindsay and Parthasarathy [Li P] we introduce the weighted and twisted convolution of Maassen kernels, present a generalization of Meyer's formula and exploit it to construct a family of operator fields whose expectations in the vacuum state exhibit a simultaneous self-similarity property. Such a construction includes Major's examples and at the same time yields a self-similar Clifford field.

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