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Theoretical and Mathematical Physics

Hamiltonians of quantum statistics and the model Hamiltonian of the theory of superconductivity

D. Ya. Petrina

Published Sep 1, 1970• 14 citations
Publisher
DOI
10.1007/bf01038305

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•Oct 13, 2011•22 citations
In many exactly solvable models of quantum statistical mechanics, an interaction Hamiltonian is equal to an integral of a product of operators of annihilation and creation and a potential divided by a volume of a system raised to a certain power. For example, the interaction Hamiltonian of the BCS model of superconductivity has the form HI,M=g2V∬∬Φ(x1−x2,x′1−x′2)ψ+(x2)ψ(x′1)ψ(x′2)dx1dx2dx′1dx′2, {H_{I,M}} = \frac{g}{{2V}}\iint {\iint {\Phi ({x_1} - {x_2},{{x'}_1} - {{x'}_2}){\psi ^ + }({x_2})\psi ({{x'}_1})\psi ({{x'}_2}})d{x_1}d{x_2}d{{x'}_1}d{{x'}_2},} HI,M​=2Vg​∬∬Φ(x1​−x2​,x′1​−x′2​)ψ+(x2​)ψ(x′1​)ψ(x′2​)dx1​dx2​dx′1​dx′2​, (1) where integration is carried out over the whole three-dimensional Euclidean space ℝ3, V is the volume of ℝ3, ψ+ and ψ- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions Φ(x1−x2,x1′−x2′)=Φ(x1−x2,x′1−x′2‾)=Φ(x2′−x1′,x2−x2),Φ(x2−x1,x1′−x2′)=Φ(x2−x1,x2′−x1′) \Phi ({x_1} - {x_2},{x'_1} - {x'_2}) = \Phi (\overline {{x_1} - {x_2},{{x'}_1} - {{x'}_2}} ) = \Phi ({x'_2} - {x'_1},{x_2} - {x_2}),\Phi ({x_2} - {x_1},{x'_1} - {x'_2}) = \Phi ({x_2} - {x_1},{x'_2} - {x'_1}) Φ(x1​−x2​,x1′​−x2′​)=Φ(x1​−x2​,x′1​−x′2​​)=Φ(x2′​−x1′​,x2​−x2​),Φ(x2​−x1​,x1′​−x2′​)=Φ(x2​−x1​,x2′​−x1′​) (2) and is a test function with respect to x 1-x 2 and x1′−x2′ {x'_1} - {x'_2} x1′​−x2′​. The dependence on spin is omitted here.
Introduction. 1: Evolution of States of Quantum Systems of Finitely Many Particles. 1. Principal Concepts of Quantum Mechanics. 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles. 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation. Mathematical Supplement I. References. 2: Evolution of States of Infinite Quantum Systems. 4. Bogolyubov Equations for Statistical Operators. 5. Solution of the Bogolyubov Equations. 6. Gibbs Distributions. Mathematical Supplement II. Mathematical Supplement III. References. 3: Thermodynamic Limit. 7. Thermodynamic Limit for Statistical Operators. 8. Statistical Operators in the Case of Quantum Statistics. 9. Bogolyubov's Principle of Weakening of Correlations. Mathematical Supplement IV. References. 4: Mathematical Problems in the Theory of Superconductivity. 10. Frohlich Model. 11. Bogolyubov's Compensation Principle for 'Dangerous' Diagrams. Compensation Equations. 12. Bardeen--Cooper--Schrieffer (BCS) Hamiltonian. 13. Microscopic Theory of Superfluidity. Mathematical Supplement V. References. 6: Green's Functions. 14. Green's Functions. Equations for Green's Functions. 15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity. 16. Green's Functions in the Thermodynamic Limit. References. 6: Exactly Solvable Models. 17. Description of the Hamiltonians of Model Systems. 18. Functional Spaces of Translation--Invariant Function. 19. ModelHamiltonians in the Spaces of Translation Invariant Functions. 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs. 21. Equations for Green's Functions and their Solutions. Mathematical Supplement VI. References. 7: Quasiaverages. Theorem on Singularities of Green's Functions of 1/q2-Type. 22. Quasiaverages. 23. Green's Functions and their Spectral Representations. 24. Theorem on Singularities of Green's Functions of 1/q2-Type. References. Subject Index.