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Journal of Physics A Mathematical and Theoretical

General ordering problem and its combinatorial roots

Farid Shähandeh, M. R. Bazrafkan

Published Mar 27, 2012• 10 citations
Publisher

Abstract

We introduce the general method of converting a given operator function into its s -ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. This will also unify the two essentially distinct notions of 'taking an operator into some ordered form' and 'reordering, or formally, ordering an operator'. In this way, we reduce the general ordering problem into a purely combinatorial one. Finally, we show the application of the theorem through two generic examples from both quantum optics and field theory.

DOI
10.1088/1751-8113/45/15/155204

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We introduce the general method of converting a given operator function into its s -ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. This will also unify the two essentially distinct notions of 'taking an operator into some ordered form' and 'reordering, or formally, ordering an operator'. In this way, we reduce the general ordering problem into a purely combinatorial one. Finally, we show the application of the theorem through two generic examples from both quantum optics and field theory.
We introduce the general method of converting a given function of boson operators into its s -ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. In this way, we reduce the general ordering problem of boson operators into a purely combinatorial one. We show the applications of the theorem through some generic examples from quantum optics, namely ordering of the Fock space projectors, together with some simple extensions of the previously given quantum optical lemmas.
The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing relations among different orderings is demonstrated by Wick's theorem (which relates time ordering to normal ordering), which played a crucial role in the development of quantum field theory. We prove the general ordering theorem (GOT), which establishes a relation among any pair of orderings, that act on operators satisfying generic (i.e., operatorial) commutation relations. We expose the working principles of the GOT by simple examples, and we demonstrate its potential by recovering two famous algebraic theorems as special instances: the Magnus expansion and the Baker-Campbell-Hausdorff formula. Remarkably, the GOT establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.
Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. Blasiak, quant-ph/0507206]. In this paper, we apply the linear representation of noncrossing partitions to contractions of normally ordered forms. In this way, we define the notion of noncrossing normal ordering. This appear to be a natural mathematical concept and it results indeed to be linked with well-known combinatorial objects. We explicitely give the noncrossing normally ordered form of the functions (a^{r}(a^{dagger})^{s})^n) and (a^{r}+(a^{dagger})^{s})^n, plus various special cases. We establish bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. Physical applications of the noncrossing normal ordering are desiderata.
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.
We give a general expression for the normally ordered form of a function F[ŵ(a,a†)] where ŵ is a function of boson creation and annihilation operators satisfying [a,a†]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional quantum field theory defined by F(ŵ). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type Hamiltonians.
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.