A representation of the moment measures of the general ideal Boe gas

Abstract

Key words Infinitely divisible point processes, integration by parts formula, random KMM-measure, permanental and determinantal point processes MSC (2010) 35K55, 35K65 Dedicated to the 65th anniversary of Professor Suren Poghosyan We reconsider the fundamental work of Fichtner [2] and exhibit the permanental structure of the ideal Bose gas again, using a new approach which combines a characterization of infinitely divisible random measures (due to Kerstan, Kummer and Matthes [4], [6] and Mecke [9], [10]) with a decomposition of the moment measures into its factorial measures due to Krickeberg [5].To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain "loop" integrals. This representation can be considered as a point process analogue of the old idea of Symanzik [15] that local times and self-crossings of the Brownian motion can be used as a tool in quantum field theory.Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Lévy-measure belonging to some large class of measures containing that of the classical ideal Bose gas considered by Fichtner.It is well-known that the calculation of moments of higher order of point processes is notoriously complicated. See for instance Krickeberg's calculations for the Poisson or the Cox process in [5].Relations to the work of Shirai, Takahashi [12] and Soshnikov [14] on permanental and determinantal processes are outlined. An integration by parts formula for infinitely divisible random measuresThe aim here is to characterize infinitely divisible random measures on a general state space from the point of view of its Campbell measure. It is shown that such random measures are characterized by some integration by parts formula. Our proof seems to be a more direct approach if compared to the one of Kallenberg [3], Matthes et al. [4], [6], [7] and Wegmann [16], where in principle such results can be found already. The main ideas of the following reasoning can already be found in the seminal work of Mecke [8]-[10]. The integration by parts formulaIn the sequel we freely use the notions of random measure theory and refer for their definitions to the monographies of Kallenberg [3] and Matthes et al. [7].X denotes a Polish state space, B(X) resp. B 0 (X) its Borel resp. bounded Borel sets. M(X) is the vaguely Polish space of locally finite measures on X (i.e., of Radon measures on X); M .. (X) denotes the measurable subspace of Radon point measures and M . (X) the measurable subspace of simple Radon point measures on X. A random element ξ in or a law P on M(X) is called a random measure on X. Random elements in or their laws on M .. (X) resp. M . (X) are called point processes resp. simple point processes in X. *

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