Fractional boson gas and fractional poisson measure in infinite dimensions

Abstract

As a consequence of Haag’s theorem, to obtain a non-trivial theory, one either works with a non-Fock representation or with a Fock representation in a finite volume. Calculations in the Fock representation taking the N,V→ ∞limit with the ratio N/V=ρ fixed, show the equivalence of the free Boson gas and the infinitedimensional Poisson measure. The N/V limit provides a way to deal with non-trivial infinite systems using the Fock representation. However, by the very nature of the fixed ρ density limit, it is unable to deal with systems with density fluctuations, a shortcoming that is solved by the use of reducible functionals. A particularly interesting reducible functional is the one associated to the infinite-dimensional fractional Poisson measure which we recall in this work.