This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices.Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices.Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.

Physical ReviewAug 15, 19621,367 citations

Conservation Laws and Correlation Functions

Gordon Baym, Leo P. Kadanoff

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

Physical ReviewOct 15, 19611,659 citations

Correlation functions, Bell's inequalities and the fundamental conservation laws

C. S Unnikrishnan

I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum-mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates Bell's inequalities. Taken together with Bell's theorem, this result has far-reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden-variable theories are incompatible with fundamental symmetries and conservation laws.I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum-mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates Bell's inequalities. Taken together with Bell's theorem, this result has far-reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden-variable theories are incompatible with fundamental symmetries and conservation laws.

arXiv:quant-ph/0407041Europhysics Letters (EPL)Feb 1, 200511 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}

, 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}