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arXiv:1710.09248

Notes on Wick's theorem in many-body theory

Luca Guido Molinari

Published Oct 25, 2017• 9 citations
arXivHTMLPDF

Abstract

In this pedagogical note I present the operator form of Wick's theorem, i.e. a procedure to bring a product of 1-particle creation and destruction operators to normal order, with respect to some reference many-body state. Both the static and the time-ordered cases are presented. For the latter, in particular, I provide a simple proof.

DOI
10.48550/arxiv.1710.09248

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In this pedagogical note I present the operator form of Wick's theorem, i.e. a procedure to bring a product of 1-particle creation and destruction operators to normal order, with respect to some reference many-body state. Both the static and the time-ordered cases are presented. For the latter, in particular, I provide a simple proof.
Abstract Wick’s theorem, known for yielding normal ordered from time-ordered bosonic fields, may be generalized for a simple relationship between any two orderings that we define over canonical variables, in a broader sense than before. In this broad class of orderings, the general Wick theorem follows from the Baker–Campbell–Hausdorff identity. We point out that, generally, the characteristic function does not induce an unambigous scheme to order the multiple products of the canonical operators although the value of the ordered product is unique. We construct a manifold of different schemes for each value of s of s -orderings of Cahill and Glauber.
The multireference normal order theory, introduced by Kutzelnigg and Mukherjee [J. Chem. Phys. 107, 432 (1997)], is defined explicitly, and an algebraic proof is given for the corresponding contraction rules for a product of any two normal ordered operators. The proof does not require that the contractions be cumulants, so it is less restricted. In addition, it follows from the proof that the normal order theory and corresponding contraction rules hold equally well if the contractions are only defined up to a certain level. These relaxations enable us to extend the original normal order theory. As a particular example, a quasi-normal-order theory is developed, in which only one-body contractions are present. These contractions are based on the one-particle reduced density matrix.
A generalization to all spin magnitudes of the drone-fermion representation (previously known for spin \textonehalf{}), is presented, and the resulting eigenstates investigated. Linear combinations of products of fermion operators acting on a vacuum state can always be found which correspond to spin states which are eigenvectors of S2{\mathrm{S}}^{2}S2 and Sz{S}^{z}Sz. It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.
Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.