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Let's forget about the pairing

Игорь Иванович Белоусов, Beloussov, I.V.

Published Dec 1, 2014• 0 citations
OpenAlex

Abstract

The algebraic formulation of a Wick's theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory and statistical physics by the methods of symbolic mathematics using computers.

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The algebraic formulation of a Wick's theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory and statistical physics by the methods of symbolic mathematics using computers.
Abstract The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known computer algebra systems.
The matrix element of a general many-body operator between two Slater determinants is calculated explicitly. For this, a split-and-pair method is introduced that provides a convenient expression of Wick's theorem and simplifies many-body calculations. The same method is used to determine the generating function of the matrix elements of many-body operators. The split-and-pair method allows also for the diagonalization of the density correlation operators $:n({x}_{1})\ensuremath{\cdots}n({x}_{k}):$, where $n(x)={\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\ensuremath{\psi}(x)$ is the density operator. The relation between the split-and-pair method and quantum group theory is clarified.
In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.
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.