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arXiv:1501.03985 · Special Matrices

Another formulation of the Wick’s theorem. Farewell, pairing?

Igor V. Beloussov

Published Jul 28, 2015• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1515/spma-2015-0015

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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.
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.
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.
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.
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.