A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.
A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.
In this paper, we develop a time-dependent approach to many-body perturbation theory for closed shells based on the resolvent of the Schrödinger equation. We introduce a quantity S(t) = i<φ‖exp [−i(H−E0)t]‖ψ≳/<φ‖ψ≳, where φ and ψ are, respectively, the unperturbed and exact wave functions for the system and E0 is the unperturbed energy. The Fourier transform of S(t), S(ω) = <φ‖(ω+E0− H)−1‖ψ≳/ <φ‖ψ≳, is a matrix element of the resolvent containing the exact function ψ and, thus, has a pole at ω = (E−E0), the correlation energy. Starting from a time-dependent perturbation expansion of S(t) via the Gellman–Low adiabatic theorem, we have obtained a Dyson-like equation: S−1(ω) = S−10(ω)+Σ<φ‖ψ≳ for S(ω). Such a derivation requires judicious grouping of terms of the perturbation series for S(t). It has been shown that specific regroupings of the terms of S(t) into appropriate ’’top’’ and ’’bottom’’ parts and corresponding time-integration procedures yield a Σ which generates the Brillouin–Wigner (BW) or Rayleigh–Schrödinger (RS) energy series. Σ for both the cases are identified as the ’’top’’ parts. The characteristic features of the BW series, namely, the appearance of disconnected and ω-dependent diagrams is to be contrasted with that of the RS series, which contains connected and ω-independent diagrams only, and both emerge naturally as a consequence of the dissection procedure into appropriate top and bottom parts.
In this paper, we develop a time-dependent approach to many-body perturbation theory for open shells based on the resolvent of the Schrödinger equation. We introduce, analogous to the closed-shell case, quantities Sij(t) = i<φi‖e−i(H−Ec)t ‖ψj>/<ψj0‖ψj>, where ψj0 and ψj are the jth unperturbed and exact functions, respectively. The ψj0’s can be expressed in terms of ’’model space’’ functions φi = Ωi+‖φ>, where the Ωi+ are appropriate creation/annihilation operator products acting on a conveniently chosen closed-shell vacuum φ. These φi’s are not necessarily degenerate with respect to the unperturbed Hamiltonian H0. Ec is the exact (correlated) energy of the vacuum φ. The Fourier transforms Sij(ω) of Sij(t) have the form Sij (ω)=<φi‖(ω +Ec−H)−1‖ψj>/<ψj0‖ψ j> and thus have poles at energy differences (Ej−Ec), i.e., relative to the exact vacuum energy. Using the time-dependent perturbation expansion of S(t), we obtain a Dyson-like equation N̄−1(ω)=N̄0–1(ω)+Σ, where N̄ is defined as N̄ij (ω) = <φi‖(ω+Ec−H)−1‖φj> and N̄0 is the corresponding unperturbed component. Knowledge of the combining coefficients Cji in ψj0=ΣJCjiφi is thus not required for finding the poles. We arrive at the Dyson-like equation by first eliminating closed diagrams and then regrouping the remaining terms in the perturbation series for S into ’’top’’ and ’’bottom’’ parts. Regrouping appropriate to the Brillouin–Wigner (BW) case together with an associated time-integration procedure yields ΣBW which consists of disconnected and ω-dependent diagrams. This is shown to yield the open-shell BW series in the Bloch–Horowitz form. An alternative regrouping procedure and use of the ’’folding technique’’ of Johnson and Baranger leads to a ΣRS which is ω-independent, Hermitian, contains connected diagrams only, and is, thus, size-consistent.
Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.
Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
We have analyzed Dyson's U-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the U matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our U matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our U matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our U(s) matrix, which is the U matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.