Green’s Functions

  • D. Ya. Petrina
Part of the Mathematical Physics Studies book series (MPST, volume 17)


Consider a Hamiltonian of the system of particles interacting via a pair potential Φ and situated in the entire three-dimensional space ℝ3
$$H = {H_0} + {H_{\text{I}}} = \int {\psi {\text{*}}(x)} {\text{ }}\left( { - \frac{\Delta }{{2m}} - \mu } \right)\psi (x)dx + \frac{g}{2}\int {\psi {\text{*}}(x)\psi {\text{*}}(x\prime )\Phi (x{\text{ - }}x\prime )\psi (x\prime )\psi (x)dxdx\prime .} $$
Here, μ is a chemical potential, ψ*(x) and ψ(x) are operators of creation and annihilation independently of the type of statistics, and g is a coupling constant. Suppose that the frame of reference is chosen so that the lowest eigenvalue of the Hamiltonian is equal to zero
$$ H{\Phi _0} = 0.$$
The eigenvector Φ0 that corresponds to the eigenvalue zero of the Hamiltonian H is called the ground state or “physical” vacuum.


Commutation Relation Thermodynamic Limit Evolution Operator Heisenberg Equation Wiener Measure 
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  1. [1]
    Abrikosov, A. A., Gor’kov, L. P., and Dzyaloshinsky, I. E. Methods of Quantum Field Theory in Statistical Physics [in Russian], Fizmatgiz, Moscow 1962.Google Scholar
  2. [1]
    Duneau, M. and Souillard, B. Existence of Green’s functions for dilute bose gases, Comm. Math. Phys. (1973), 31,113–125.MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. [1]
    Dunford, N. and Schwartz, J. Linear Operators. General Theory, Interscience, 1958.Google Scholar
  4. [1]
    Ginibre, J. Reduced density matrices of quantum gases. I. Limit of infinite volume, J. Math. Phys. (1965), 6, No. 2, 238–251.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. [2]
    Ginibre, J. Reduced density matrices of quantum gases. II. Cluster property, J. Math. Phys. (1965) 6, No. 2, 252–262.MathSciNetADSCrossRefGoogle Scholar
  6. [1]
    Gruber, C. Green’s Functions in Quantum Statistical Mechanics, Thesis, Princeton Univ., 1968 (unpublished).Google Scholar
  7. [1]
    Martin, P. and Swinger, J. Theory of many-particle systems, Phys. Rev. (1959), 115, No. 6, 1342.MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. [1]
    Petrina, D. Ya. Quantum Field Theory, Vyshcha Shkola, Kiev, 1984.zbMATHGoogle Scholar
  9. [2]
    Petrina, D. Ya. Exactly Solvable Models of Quantum Statistical Mechanics, Preprint 18/1992, Politecnico di Torino, Torino, 1992.Google Scholar
  10. [1]
    Petrina, D. Ya. and Yatsyshin, V. P. On a model Hamiltonian in the theory of superconductivity, Teor. Mat. Fiz. (1972), 10, 283.CrossRefGoogle Scholar
  11. [1]
    Ruelle, D. Analyticity of Green’s function for dilute quantum gases, J. Math. Phys. (1971), 12, No. 6, 901–903.MathSciNetADSCrossRefGoogle Scholar
  12. [2]
    Ruelle, D. Definition of Green’s function for dilute Fermi gases, Helv. Phys. Acta. (1972),45, No. 2, 215-219.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • D. Ya. Petrina
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

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