Green’s Functions
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Abstract
Consider a Hamiltonian of the system of particles interacting via a pair potential Φ and situated in the entire three-dimensional space ℝ3
$$H = {H_0} + {H_{\text{I}}} = \int {\psi {\text{*}}(x)} {\text{ }}\left( { - \frac{\Delta }{{2m}} - \mu } \right)\psi (x)dx + \frac{g}{2}\int {\psi {\text{*}}(x)\psi {\text{*}}(x\prime )\Phi (x{\text{ - }}x\prime )\psi (x\prime )\psi (x)dxdx\prime .} $$
(14.1)
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
The eigenvector Φ0 that corresponds to the eigenvalue zero of the Hamiltonian H is called the ground state or “physical” vacuum.
$$
H{\Phi _0} = 0.$$
(14.2)
Keywords
Commutation Relation Thermodynamic Limit Evolution Operator Heisenberg Equation Wiener Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
- [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
- [1]Duneau, M. and Souillard, B. Existence of Green’s functions for dilute bose gases, Comm. Math. Phys. (1973), 31,113–125.MathSciNetADSzbMATHCrossRefGoogle Scholar
- [1]Dunford, N. and Schwartz, J. Linear Operators. General Theory, Interscience, 1958.Google Scholar
- [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
- [2]Ginibre, J. Reduced density matrices of quantum gases. II. Cluster property, J. Math. Phys. (1965) 6, No. 2, 252–262.MathSciNetADSCrossRefGoogle Scholar
- [1]Gruber, C. Green’s Functions in Quantum Statistical Mechanics, Thesis, Princeton Univ., 1968 (unpublished).Google Scholar
- [1]Martin, P. and Swinger, J. Theory of many-particle systems, Phys. Rev. (1959), 115, No. 6, 1342.MathSciNetADSzbMATHCrossRefGoogle Scholar
- [1]Petrina, D. Ya. Quantum Field Theory, Vyshcha Shkola, Kiev, 1984.zbMATHGoogle Scholar
- [2]Petrina, D. Ya. Exactly Solvable Models of Quantum Statistical Mechanics, Preprint 18/1992, Politecnico di Torino, Torino, 1992.Google Scholar
- [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
- [1]Ruelle, D. Analyticity of Green’s function for dilute quantum gases, J. Math. Phys. (1971), 12, No. 6, 901–903.MathSciNetADSCrossRefGoogle Scholar
- [2]Ruelle, D. Definition of Green’s function for dilute Fermi gases, Helv. Phys. Acta. (1972),45, No. 2, 215-219.MathSciNetGoogle Scholar
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