Simple analytic evaluation of the integrals in the Gaussian model of BoseEinstein correlations
Abstract
A generating function for products of Hermite polynomials is used to significantly simplify the evaluation of the integrals occurring in the Gaussian model of multiple particle production. These integrals are crucial for studies of multiparticle effects in BoseEinstein correlations.
PACS numbers 25.75.Gz, 13.65.+i
BoseEinstein correlations, Gaussian
model.
1. BoseEinstein correlations in multiple particle production processes are a field of intense study. For recent reviews see [1][4]. An important subfield is the study of multiparticle effects on multiplicity and momenta distributions. Here exactly soluble models are a convenient starting point. A soluble model, which is particularly popular, is the Gaussian model [5]  [13]. In this model all the integrations are Gaussian and consequently can be performed analytically. The generic integral is
(1) 
where
(2) 
with
(3) 
The constants and are real and positive. For simplicity we start with the onedimensional case. In the Gaussian model the transition to three dimensions is trivial. It is seen that the integrals are Gaussian and consequently can be performed analytically. One needs, however, a closed formula valid for any . This is more difficult to obtain. Zimányi and Csörgö [6, 7] following some suggestions from Pratt [5] constructed a system of recurrence relations for three series of parameters and after a rather lengthy calculation got a formula for the integrals, which can be written in a fairly simple form [12]. Wiedemann [13] applied successfully the MATHEMATICA program directly to the integrals . In the present note we point out that a very short derivation of the formula for the integrals can be given, if a mathematical identity for Hermite polynomials is made use of. This work is an extension of the method used in [8, 9].
2. An identity, valid for , is
(4) 
A convenient starting point to check this formula is the definition of the Hermite polynomials in the form [14]^{2}^{2}2In this edition of the tables the power of two is mistakenly written as instead of . In order to check the formula it is enough to expand the binomial into powers of , integrate term by term and compare the coefficients with the formulae given in [15].
(5) 
After substituting it into (4) the summation over gives an exponential and it is enough to perform a double Gaussian integral to get the righthand side. The function is defined by this formula.
3. For any orthonormal set of functions and any set of nonnegative numbers satisfying the normalization condition
(6) 
the function
(7) 
can be interpreted as a welldefined singleparticle density matrix in the momentum representation. The functions are its eigenfunctions and the numbers are the corresponding eigenvalues. Let us choose in particular for the functions the energy eigenfunctions in the momentum representation of a onedimensional harmonic oscillator and for the numbers a geometric progression [8, 9]:
(8) 
It is assumed here that , so that the eigenvalues are correctly normalized. The parameter is positive. Its interpretation in terms of the mechanical parameters of the oscillator is of no importance here. This choice corresponds to a single particle density matrix which using (4) can be written in the form
(9) 
or substituting the definition of the function and replacing the parameters by the parameters according to (3)
(10) 
This density matrix differs only by notation from the density matrix (2). The correspondence between the parameters is
(11) 
Following [12] we have introduced the notation
(12) 
where the inequality follows from the uncertainty principle.
4. In order to calculate the function let us note first that it is the matrix element of the th power of the matrix . Therefore, it has the same eigenfunctions and its eigenvalues are th powers of the corresponding eigenvalues of matrix . Thus
(13) 
Substituting the definition of function and changing variables from to we find
(14) 
It is convenient to rewrite this formula in the form [12]
(15) 
By comparison with the previous form with
(16) 
and
(17) 
5. Let us add some comments.

As mentioned, the transition from the onedimensional case to dimensions is very simple. The density matrix (2) gets replaced by a product
(18) Consequently, the integrals from (1) factorize and one obtains for each a product of expressions (15), in general with different values of and for each factor.

For the harmonic oscillator the energy eigenfunctions in the momentum representation and in the coordinate representation have the same form. Besides the replacement of by , the interpretation of the parameter changes, but this is irrelevant for the present analysis. Thus, the results given in the present paper can be used, after some changes in notation, also for the study of BoseEinstein correlations in ordinary space.

Since the energy eigenvalues of the harmonic oscillator are equidistant, the corresponding Boltzmann factors form a geometrical progression. Thus, formula (9) as well as the corresponding formula in the coordinate representation, can be used to calculate the canonical density matrix for the harmonic oscillator. According to Landau and Lifszyc [16], this calculation for the diagonal elements of the density matrix in the coordinate representation has been first performed by F. Bloch in year 1932.
The author thanks Andrzej Bialas for collaboration and stimulating discussions.
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