Colloquium

Speaker: Professor Luigi Accardi (Universita di Roma Tor Vergata, Italy)
Title: Nonlinear Weyl relations and the quadratic Fock functor
Time: 2010-08-05 (Thu.)  16:10 - 17:00
Place: Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
Abstract: What is usually called second quantization should be more precisely called linear quantization. In fact, even in the linear case, the term second quantization is used in a variety of inequivalent meanings. In physics the most frequent use of this term is as a synonim of Boson Fock quantization.

The program of nonlinear second quantization, i.e. of the construction of renormalized higher powers of white noise, is best illustrated by the quadratic analogue of Boson Fock quantization. This is now relatively well understood, even if the theory is by far not developed as in the linear case.

In particular the three non standard Meixner classes of Levy processes emerge as vacuum distributions of the generalized field operators, exactly in the same way as the two standard ones (Wiener and Poisson) emerge in the first order case.

If time allows, I will also mention a different type of second quantization, whose existence has been recently established, which in some sense interpolates between the first and the second order.

In this case the analogue of the Meixner classes have not yet been identified, even if their characteristic functions (and in some cases their Levy--Khintchin forms) are explicitly known.

This is a joint work of L. Accardi with A. Dhahri (for the Galilei algebra and its extensions) and with H. Rebei and H. Ouerdiane (for $sl(2,\mathbb R)$).
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