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Combinatorics of Boson Normal Ordering: the Dobihski Formula Revisited

Karol A. Pensonj, A. I. Solomon

Published Nov 30, 2003• 0 citations
Publisher

Abstract

We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobihski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.

DOI
10.1201/9781482269079-114

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Ordering of some boson operator functions

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Journal of Physics A Mathematical and General•Nov 11, 1983•26 citations
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobihski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a†a)_{n,λ}, using recurrence relations for the coefficients and an operator action on number states. Here a† and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick’s theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.
We address a systematic combinatorial approach to the anti-normal ordering problem. In this way, we use the Stirling numbers and their generating function, the so-called Bell polynomials, together with the operational methods to anti-normal the operator eλa†a{e^{λ{a^†}a}}eλa†a. In fact, we exploit the new theorem given by Shähandeh et al. in a special case of anti-normal ordering.
The problem of normal ordering of the following boson operator functions (a+a + ) m , (a r +a + ) m , (a+N) m , (a 2 +N) m , where N=a + a, is solved explicitly. Simple algebraic methods are used. As an intermediate result the ordering formulae for the symmetrised products of boson operators are presented. There is agreement with earlier results of other authors and generalisation of these results. A new representation of the generalised Stirling numbers used for the ordering of the binomial (a 2 +N) m is given and several principal properties of these numbers are derived.