Physical Review

Conservation Laws and Correlation Functions

Gordon Baym, Leo P. Kadanoff

Published Oct 15, 1961• 1,659 citations

Abstract

In describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations has been developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator $G$, defined in the presence of a nonlocal external scalar field $U$. Approximations for $G(U)$ are obtained by replacing the ${G}_{2}(U)$ which appears in these equations by various functionals of $G(U)$. If the approximation for ${G}_{2}(U)$ satisfies certain simple symmetry conditions, then the $G(U)$ thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L$, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal $f$-sum rule.Several examples of conserving approximations are described. The Hartree approximation, ${G}_{2}(U)=G(U)G(U)$, generates the random-phase approximation for $L$. The Hartree-Fock approximation for $G(U)$ leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for $G(U)$ is obtained by expanding the self-energy to first order in the many-particle scattering matrix $T(U)$. This $T$ is obtained by summing ladder diagrams in which the sides of the ladder are composed of $G(U)'\mathrm{s}$. The resulting $L$ equation, which involves coefficients proportional to ${|T|}^{2}$, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential, ${V}_{s}(U)$. This potential is obtained by summing bubbles composed of two $G(U)'\mathrm{s}$. The resulting $L$ equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to ${|{V}_{s}|}^{2}$.

DOI: 10.1103/physrev.124.287

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Conserving approximations: two-particle Green's function

Gianluca Stefanucci, Robert van Leeuwen

Cambridge University Press eBooksMar 6, 20130 citations

Theory of Many-Particle Systems. I

Paul C. Martin, Julian Schwinger

Physical ReviewSep 15, 19591,813 citations

Renormalization and Statistical Mechanics in Many-Particle Systems. I. Hamiltonian Perturbation Method

Duane C. Wallace

Physical ReviewDec 2, 196631 citations

Establishing conservation laws in pair-correlated many-body theories: T -matrix approaches

Yan He, Kathryn Levin

arXiv:1308.6793Physical Review BJan 6, 201413 citations

Equations of Motion with Particle–Particle Interactions and Approximations

Norman J. M. Horing

Jul 27, 20170 citations

Quantum transport: Self-consistent approximations

Dmitry N. Voskresensky

Functional methods: Generating Φ {ϕ, G } functional of Baym on Schwinger-Keldysh contour; symmetries and conservation laws; thermodynamic consistency. (See [1]. Renormalization scheme can be found in [2].) Generalized kinetic approach: gradient approximation. (See [3]. Gradient expansion diagram technique see in [4].) Some helpful examples. (See [5].)

Physics of Particles and NucleiNov 22, 20080 citations

Physical Review

Conservation Laws and Correlation Functions

Gordon Baym, Leo P. Kadanoff

Published Oct 15, 1961• 1,659 citations

Publisher

Abstract

G

, defined in the presence of a nonlocal external scalar field

U

. Approximations for

G(U)

are obtained by replacing the

{G}_{2}(U)

which appears in these equations by various functionals of

G(U)

. If the approximation for

{G}_{2}(U)

satisfies certain simple symmetry conditions, then the

G(U)

thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as

{(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L

, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal

f

-sum rule.Several examples of conserving approximations are described. The Hartree approximation,

{G}_{2}(U)=G(U)G(U)

, generates the random-phase approximation for

L

. The Hartree-Fock approximation for

G(U)

leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for

G(U)

is obtained by expanding the self-energy to first order in the many-particle scattering matrix

T(U)

. This

T

is obtained by summing ladder diagrams in which the sides of the ladder are composed of

G(U)'\mathrm{s}

. The resulting

L

equation, which involves coefficients proportional to

{|T|}^{2}

, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential,

{V}_{s}(U)

. This potential is obtained by summing bubbles composed of two

G(U)'\mathrm{s}

. The resulting

L

equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to

{|{V}_{s}|}^{2}

DOI: 10.1103/physrev.124.287

Related papers

Conserving approximations: two-particle Green's function

Gianluca Stefanucci, Robert van Leeuwen

Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.Introduction In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving . A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n (x, t ) and a current density J (x, t ) which satisfy the continuity equation, or a total momentum P ( t ) and a Lorentz force F ( t ) which is the time-derivative of P ( t ), or a total energy E ( t ) and a total power which is the time-derivative of E ( t ), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.

Cambridge University Press eBooksMar 6, 20130 citations

Theory of Many-Particle Systems. I

Paul C. Martin, Julian Schwinger

This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.

Physical ReviewSep 15, 19591,813 citations

Renormalization and Statistical Mechanics in Many-Particle Systems. I. Hamiltonian Perturbation Method

Duane C. Wallace

A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where $\mathcal{H}$ is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order $m$ , and also to satisfy the appropriate boson commutators and fermion anticommutators to order $m$ , then the problem is solved to order $m$ . In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of $\mathcal{H}$ and are orthonormal, all to order $m$ . A procedure is otained for finding the ( $m+1$ )-order contributions to the particle creation energies, in terms of the $m$ -order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where $\mathcal{H}$ is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order $m$, and also to satisfy the appropriate boson commutators and fermion anticommutators to order $m$, then the problem is solved to order $m$. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of $\mathcal{H}$ and are orthonormal, all to order $m$. A procedure is otained for finding the ($m+1$)-order contributions to the particle creation energies, in terms of the $m$-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.

Physical ReviewDec 2, 196631 citations

Establishing conservation laws in pair-correlated many-body theories: T -matrix approaches

Yan He, Kathryn Levin

We address conservation laws associated with current, momentum, and energy and show how they can be satisfied within many body theories which focus on pair correlations. Of interest are two well known $T$ -matrix theories which represent many body theories which incorporate pairing in the normal state. The first of these is associated with the Nozieres Schmitt-Rink theory, while the second involves the $T$ matrix of a BCS-Leggett-like state as identified by Kadanoff and Martin. $T$ -matrix theories begin with an ansatz for the single particle self energy and are to be distinguished from $\ensuremath{\Phi}$-derivable theories which introduce an ansatz for a particular contribution to the thermodynamical potential. Conservation laws are equivalent to Ward identities which we address in some detail here. Although $\ensuremath{\Phi}$-derivable theories are often referred to as ``conserving theories,'' a consequence of this work is the demonstration that these two $T$ -matrix approaches similarly can be made to obey all conservation laws. When simplifying approximations are made in $\ensuremath{\Phi}$-derivable or other theories, one has to take care that the end results are not incompatible with conservation.We address conservation laws associated with current, momentum, and energy and show how they can be satisfied within many body theories which focus on pair correlations. Of interest are two well known $T$-matrix theories which represent many body theories which incorporate pairing in the normal state. The first of these is associated with the Nozieres Schmitt-Rink theory, while the second involves the $T$ matrix of a BCS-Leggett-like state as identified by Kadanoff and Martin. $T$-matrix theories begin with an ansatz for the single particle self energy and are to be distinguished from $\ensuremath{\Phi}$-derivable theories which introduce an ansatz for a particular contribution to the thermodynamical potential. Conservation laws are equivalent to Ward identities which we address in some detail here. Although $\ensuremath{\Phi}$-derivable theories are often referred to as ``conserving theories,'' a consequence of this work is the demonstration that these two $T$-matrix approaches similarly can be made to obey all conservation laws. When simplifying approximations are made in $\ensuremath{\Phi}$-derivable or other theories, one has to take care that the end results are not incompatible with conservation.

arXiv:1308.6793Physical Review BJan 6, 201413 citations

Equations of Motion with Particle–Particle Interactions and Approximations

Norman J. M. Horing

Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrixAbstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix

Jul 27, 20170 citations

Quantum transport: Self-consistent approximations

Dmitry N. Voskresensky

Physics of Particles and NucleiNov 22, 20080 citations

G

, defined in the presence of a nonlocal external scalar field

U

. Approximations for

G(U)

are obtained by replacing the

{G}_{2}(U)

which appears in these equations by various functionals of

G(U)

. If the approximation for

{G}_{2}(U)

satisfies certain simple symmetry conditions, then the

G(U)

thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as

{(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L

, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal

f

-sum rule.Several examples of conserving approximations are described. The Hartree approximation,

{G}_{2}(U)=G(U)G(U)

, generates the random-phase approximation for

L

. The Hartree-Fock approximation for

G(U)

leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for

G(U)

is obtained by expanding the self-energy to first order in the many-particle scattering matrix

T(U)

. This

T

is obtained by summing ladder diagrams in which the sides of the ladder are composed of

G(U)'\mathrm{s}

. The resulting

L

equation, which involves coefficients proportional to

{|T|}^{2}

{V}_{s}(U)

. This potential is obtained by summing bubbles composed of two

G(U)'\mathrm{s}

. The resulting

L

equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to

{|{V}_{s}|}^{2}

\mathcal{H}

is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order

m

, and also to satisfy the appropriate boson commutators and fermion anticommutators to order

m

, then the problem is solved to order

m

. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of

\mathcal{H}

and are orthonormal, all to order

m

. A procedure is otained for finding the (

m+1

)-order contributions to the particle creation energies, in terms of the

m

-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.

T

-matrix theories which represent many body theories which incorporate pairing in the normal state. The first of these is associated with the Nozieres Schmitt-Rink theory, while the second involves the

T

matrix of a BCS-Leggett-like state as identified by Kadanoff and Martin.

T

-matrix theories begin with an ansatz for the single particle self energy and are to be distinguished from $\ensuremath{\Phi}$-derivable theories which introduce an ansatz for a particular contribution to the thermodynamical potential. Conservation laws are equivalent to Ward identities which we address in some detail here. Although $\ensuremath{\Phi}$-derivable theories are often referred to as ``conserving theories,'' a consequence of this work is the demonstration that these two

T

-matrix approaches similarly can be made to obey all conservation laws. When simplifying approximations are made in $\ensuremath{\Phi}$-derivable or other theories, one has to take care that the end results are not incompatible with conservation.