A representation of the moment measures of the general ideal Bose gas (Q2888904)

The aim of the paper is to exhibit the permanental structure of the ideal Bose gas, by employing a new approach which combines a characterization of infinitely divisible random measures with a decomposition of the moment measures into its factorial measures. This is a reformulation of the famous fundamental work of \textit{K.-H. Fichtner} [Math. Nachr. 151, 59--67 (1991; Zbl 0728.60052)]. As a matter of fact, the authors exhibit the moment measures of all orders of the general ideal Bose gas in terms of loop integrals. This representation is nothing but a point process analogue of the old idea of Symanzik from 1969 for Euclidean quantum field theory.NEWLINENEWLINEWe can characterize infinitely divisible random measures on a general state space from the point of view of its Cambell measure. It is shown that such random measures are characterized by some integration by parts formula. Their proof is due to a direct approach, and its leading philosophy can be found in the seminal work of \textit{J. Mecke} [Z.\ Wahrscheinlichkeitstheor.\ Verw.\ Geb.\ 9, 36--58 (1967; Zbl 0164.46601)].NEWLINENEWLINETheorem 1. Let \(\alpha\) be an element in the space \({\mathcal M}(X)\) of locally finite measures on a Polish state space \(X\), and let \(L\) be a measure on \(Y= {\mathcal M}(X) \setminus \{0 \}\) of first order. Then the following conditions are equivalent for a random measure \(\tilde{\xi}\): \newline (a) \(\tilde{\xi}\) is infinitely divisible for \(( \alpha, L)\); \newline (b) the Laplace transform of \(\tilde{\xi}\) is given by NEWLINE\[NEWLINE {\mathcal L}_P(f) = \exp \{ - ( \alpha(f) + L( 1 - \exp\{ - \zeta_f \}) ) \} \qquad \text{for} \quad f \in F_+(X), \tag{1} NEWLINE\]NEWLINE where \(F_+(X)\) denotes the collection of measurable, non-negative numerical functions, and \(\nu(f)\) is an integral of \(f\) with respect to a measure \(\nu\) and \(\zeta_f\) : \(\mu \mapsto \mu(f)\) is defined on \(Y\) \(=\) \({\mathcal M}(X) \setminus \{ 0 \}\); \newline (c) \(\tilde{\xi}\) is a solution of the integration by parts formula NEWLINE\[NEWLINE {\mathcal C}_P = {\mathcal C}_L \star P + \alpha \otimes P, \tag{2} NEWLINE\]NEWLINE where \({\mathcal C}_P\) is the Campbell measure of a random measure \(P\) on \(X\), defined by NEWLINE\[NEWLINE {\mathcal C}_P (h) = \int_{ {\mathcal M}(X) } \int_X h(x, \mu) \mu(dx) P( d \mu), \qquad \text{for} \quad h \in F_+ \equiv F_+(X \times {\mathcal M}(X) ), \tag{3} NEWLINE\]NEWLINE and the operation \(\star\) is a version of the convolution of \({\mathcal C}_L\) and \(P\), defined by NEWLINE\[NEWLINE {\mathcal C}_L \star P(h) = \int h(x, k + \nu ) {\mathcal C}_L( dx, d \nu) P( d k), \qquad h \in F_+( X \times {\mathcal M}(X)); \tag{4} NEWLINE\]NEWLINE (d) \(\tilde{\xi}\) is the random KMM-measure \(\xi\) in \(X\) for \((\alpha, L)\). \newline In particular, (2) has \(\xi\) as a unique solution, which is of first order.NEWLINENEWLINEThe above theorem asserts that the transformation \(\xi\) gives a one-to-one correspondence between Poisson processes on \(Y\) with an intensity measure \(L\) of first order and the set of infinitely divisible random measures on \(X\) with the Lévy measure \(L\).NEWLINENEWLINENext consider a large class of infinitely divisible point processes called general ideal Bose gases with a specified class of Lévy measure \(L\). Let \(\rho\) be a Radon measure on \(X\), and \({\mathcal B}_m^x(d x_1 \cdots d x_{m-1})\) on \(X^{m-1}\) \((m \geq 1)\) be a measurable family of finite measures. To compute the Campbell measure of \(L\), the measure \({\mathcal B}^x\) \(=\) \(\sum_{m=1}^{\infty}\) \(z^m {\mathcal B}_m^x\) on the sum space \({\mathbb X}\) \(=\) \(\sum_{n \geq 0} X^n\) is introduced. Then note that the Campbell measure \({\mathcal C}_L\) of \(L\) is given by NEWLINE\[NEWLINE {\mathcal C}_L(h) = \int_X \int_{ {\mathbb X} } h(x, \mu_y + \mu_x) {\mathcal B}^x(dy) \rho (dx), \qquad h \in F_+. \tag{5} NEWLINE\]NEWLINE The next theorem is the main result concerning the general ideal Bose gas \({\mathfrak J}_L\) \(=\) \(\xi P_L\).NEWLINENEWLINETheorem 2. let \({\mathcal B}_m^x\), \(x \in X\), \(m \geq 1\), be a family of kernels satisfying the assumptions: (i) \(\rho(dx)\) \({\mathcal B}_m^x(d x_1\) \(\cdots\) \(d x_{m-1})\) is a cyclic invariant Radon measure on \(X^m\) for \(m \geq 1\); (ii) \(L\) is of first order. Then the distribution \({\mathfrak J}_L\) of the general ideal Bose gas \(\xi\) is the unique solution of the following integration by parts formula: NEWLINE\[NEWLINE {\mathcal C}_{ {\mathfrak J}_L }(h) = \int h(x, k + \mu_y + \delta_x) {\mathcal B}^x( dy) \rho( dx) {\mathfrak J}_L(d k), \qquad h \in F_+. \tag{6} NEWLINE\]NEWLINENEWLINENEWLINEEq.(6) means that, given the cyclic invariant measure \(\rho(dx) {\mathcal B}^x(dy)\) on \(X \times {\mathbb X}\), the corresponding random KMM process is characterized as the unique solution of the integration by parts formula (6). When we denote by \({\mathcal S}_k^{cy}\) the set of all permutations of \([k]\) \(=\) \(\{ 1, \dots, k \}\) consisting of one cycle, NEWLINE\[NEWLINE {\mathcal C}_k^x = \sum_{n=1}^{\infty} z^n \cdot {\mathcal B}_{k,n}^x \qquad \text{with} \quad {\mathcal C}_0^x = {\mathcal B}^x ({\mathbb X} ), \qquad {\mathcal B}_{k,n}^x = \sum_{ R \in {\mathcal P}_k( [n-1]) } {\mathcal B}_{R,n}^x, NEWLINE\]NEWLINE \({\mathcal B}_{R,n}^x\) is the image of \({\mathcal B}_n^x\) under the projection \((x_1, \dots, x_{n-1}) \mapsto (x_j)_{j \in R}\), and \({\mathcal P}_k\) means building subsets of cardinality \(k\), then the factorial moment measures of \(L\) are given by NEWLINE\[NEWLINE \check{\nu}_L^k(d x_1 \cdots d x_k) = \sum_{ \omega \in {\mathcal S}_k^{cy} } {\mathcal C}_{k-1}^{x_1} ( d x_{ \omega(1)} \cdots d x_{ \omega^{k-1}(1) } ) \rho( d x_1). \tag{7} NEWLINE\]NEWLINE Consider the case in which the permutation \(\sigma\) is decomposed into its cycles: \(\sigma\) \(=\) \(\omega_1 \cdots \omega_n\), and \(i_j \in \omega_j\) and \(| \omega_j |\) denotes the cyclic length. If we further assume that \(L\) is of any order, then the factorial moment measures of the associated general ideal Bose gas \({\mathfrak J}_L\) can be represented, by means of the measures \({\mathcal C}_k^x\) defined as in (7), as NEWLINE\[NEWLINE \begin{aligned} \check{\nu}_{ {\mathfrak J}_L }^k ( d x_1 \cdots d x_k ) = \sum_{ \sigma \in {\mathcal S}_k } {\mathcal C}_{ | \omega_1 | -1}^{ x_{ i_1} } ( d x_{ \omega_1}(i_1) &\cdots d x_{ \omega_1^{ | \omega_1 | -1}(i_1) } ) \rho( d x_{i_1} ) \cdots \tag{8} \\ &{\mathcal C}_{ | \omega_n | -1}^{ x_{ i_n} } ( d x_{ \omega_n}(i_n) \cdots d x_{ \omega_n^{ | \omega_n | -1}(i_n) } ) \rho( d x_{ i_n} ). \end{aligned} NEWLINE\]