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Journal of Mathematical Physics

Extension of Wick’s theorem by means of the state operator formalism

Masao Nomura

Published Apr 1, 1985• 14 citations
Publisher

Abstract

Reordering of operators in the second quantization formalism is described by using state operators as basic operators, where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation. The commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage (cfp), the factor specific to state operator (de) composition. Wick’s theorem, prevalently applied to a string of creation and annihilation operators, is extended to a string of state operators. Similar extension is done for the contraction theorem.

DOI
10.1063/1.526560

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Reordering of operators in the second quantization formalism is described by using state operators as basic operators, where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation. The commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage (cfp), the factor specific to state operator (de) composition. Wick’s theorem, prevalently applied to a string of creation and annihilation operators, is extended to a string of state operators. Similar extension is done for the contraction theorem.
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 for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator Oˆ\^OOˆ, which leaves invariant the 2N2N2N-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group SO(2N,C)\mathrm{SO}(2N,C)SO(2N,C) for the case of fermion, or the group Sp(2N,C)\mathrm{Sp}(2N,C)Sp(2N,C) for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.
Treatment of a general operator in the shell model bases is investigated as a reformulation and a slight generalization of a previous work. First, reordering of an operator into normal products is presented by the use of matrix elements in a few-body configurations instead of the use of the Wick theorem. Second, the commutation relation and the particle-hole relation in the many-body bases are presented each of which is equivalent to the cfp relation arising from the completeness of basic functions. Next, the operator whose matrix element fulfills particle-hole symmetry is constructed from an arbitrary operator. In turn, the operator is expressed as a sum of particle-hole symmetrized operators.
A generalization of normal ordering and of Wick’s theorem with respect to an arbitrary reference function Φ as some generalized “physical vacuum” is formulated in a different (but essentially equivalent) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any reference state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as physical vacuum must be contained as a special case. The derivation is largely based on the concepts of “Quantum Chemistry in Fock space,” which means that particle-number-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of a generalized Wick theorem are derived, that hold for non-particle-number-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the “residual n-particle density matrices” λ, which are the irreducible (non-factorizable) parts of the conventional n-particle density matrices γ, in the sense of a cumulant expansion for the density. A spinfree formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration reference function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary reference function, and with general implications of the new formalism, that is related to “internally contracted multireference configuration interaction.” The present generalization of normal ordering is not only valid for arbitrary reference functions, but also if the reference state is an ensemble state.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)