Advertisement

Il Nuovo Cimento (1955-1965)

, Volume 25, Issue 2, pp 287–299 | Cite as

The mathematical structure of the bardeen-cooper-schrieffer model

Article

Summary

The BCS-model for an infinitely extended superconductor is analysed mathematically. The reason why the model is solvable becomes evident in the present formulation. It is shown that the ground states with unsharp particle number belong to irreducible representations, those with sharp particle number to reducible representations of the basic operator algebra. The connection between uniqueness of the ground state, irreducibility and linked cluster decomposition is reviewed.

Riassunto

Si analizza matematicamente il modello di Bardeen-Cooper-Schrieffer per un superconduttore infinitamente esteso. Il motivo per cui il modello è risolvibile diviene evidente nella presente formulazione. Si dimostra che gli stati fondamentali con un numero di particelle non ben definito appartengono a rappresentazioni irriducibili, quelli con un ben definito numero di particelle a rappresentazioni riducibili dell’algebra degli operatori fondamentali. Si riesamina la connessione fra l’unicità dello stato fondamentale, l’irriducibilità e la decomposizione deì cluster legati.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

references

  1. (1).
    J. Bardeen, L. N. Cooper andJ. R. Schrieffer:Phys. Rev.,108, 1175 (1957).MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. (2).
    N. N. Bogoliubov:Physica,26, 1 (1960).ADSCrossRefGoogle Scholar
  3. (3).
    For this reasonB. Zumino (Werner Heisenberg und die Physik unserer Zeit (1961), p. 234) suggested the use of a nonseparable Hilbert space in this context. We shall, however, avoid this. Compare theorem (a) in Sect.4 below.Google Scholar
  4. (6).
    The connection between the form of the Hamiltonian and the representation of the canonical commutator algebra has been studied for some Bose-field models byF. Coester andR. Haag:Phys. Rev.,117, 1137 (1960) andH. Araki:Journ. Math. Phys.,1, 492 (1960). In these models the representation was uniquely determined by the Hamiltonian.MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. (7).
    H. Borchers:Nuovo Cimento,24, 214 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  6. (8).
    D. Ruelle:On the asymptotic condition in quantum field theory, preprint.Google Scholar
  7. (9).
    K. Hepp, R. Jost, D. Ruelle andD. Steinmann:Helv. Phys. Acta,34, 542 (1960).Google Scholar
  8. (10).
    The letter alternative would mean that the linear momentum of the system cannot be expressed in terms of the basic quantitiesψ, ψ*. In the field-theoretic discussion of ref. (7) and (8) it is shown that this alternative can be excluded by virtue of the restriction of the energy-momentum spectrum to the positive cone.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica 1962

Authors and Affiliations

  • R. Haag
    • 1
  1. 1.Department of PhysicsUniversity of IllinoisUrbana

Personalised recommendations