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Annales Henri Poincaré

, Volume 20, Issue 11, pp 3509–3541 | Cite as

A Nonrelativistic Quantum Field Theory with Point Interactions in Three Dimensions

  • Jonas LampartEmail author
Article

Abstract

We construct a Hamiltonian for a quantum-mechanical model of nonrelativistic particles in three dimensions interacting via the creation and annihilation of a second type of nonrelativistic particles, which are bosons. The interaction between the two types of particles is a point interaction concentrated on the points in configuration space where the positions of two different particles coincide. We define the operator, and its domain of self-adjointness, in terms of co-dimension-three boundary conditions on the set of collision configurations relating sectors with different numbers of particles.

Notes

Acknowledgements

I am grateful to Stefan Keppeler, Julian Schmidt, Stefan Teufel and Roderich Tumulka for many interesting discussions on the subject of interior-boundary conditions.

References

  1. 1.
    Behrndt, J., Micheler, T.: Elliptic differential operators on Lipschitz domains and abstract boundary value problems. J. Funct. Anal. 267(10), 3657–3709 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: Stability for a system of N fermions plus a different particle with zero-range interactions. Rev. Math. Phys. 24(07), 1250017 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: A class of Hamiltonians for a three-particle fermionic system at unitarity. Math. Phys. Anal. Geom. 18(1), 32 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dell’Antonio, G., Figari, R., Teta, A.: Hamiltonians for systems of N particles interacting through point interactions. Ann. Inst. H. Poincaré Phys. Théor. 60(3), 253–290 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Eckmann, J.-P.: A model with persistent vacuum. Commun. Math. Phys. 18(3), 247–264 (1970)MathSciNetCrossRefADSzbMATHGoogle Scholar
  6. 6.
    Grusdt, F., Demler, E.: New theoretical approaches to Bose polarons. In: Stringari, S., Roati, R., Inguscio, M., Ketterle, W. (eds.) Quantum Matter at Ultralow Temperatures. IOS Press, Amsterdam (2016)Google Scholar
  7. 7.
    Grusdt, F., Shchadilova, Y.E., Rubtsov, A.N., Demler, E.: Renormalization group approach to the Fröhlich polaron model: application to impurity-BEC problem. Sci. Rep. 5, 12124 (2015)CrossRefADSGoogle Scholar
  8. 8.
    Griesemer, M., Wünsch, A.: On the domain of the Nelson Hamiltonian. J. Math. Phys. 59(4), 042111 (2018)MathSciNetCrossRefADSzbMATHGoogle Scholar
  9. 9.
    Lévy-Leblond, J.-M.: Galilean quantum field theories and a ghostless Lee model. Commun. Math. Phys. 4(3), 157–176 (1967)MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. 10.
    Lampart, J.: The Renormalised Bogoliubov–Fröhlich Hamiltonian. arXiv preprint arXiv:1909.02430 (2019)
  11. 11.
    Lampart, J., Schmidt, J.: On Nelson-type Hamiltonians and abstract boundary conditions. Commun. Math. Phys. 367(2), 629–663 (2019)MathSciNetCrossRefADSzbMATHGoogle Scholar
  12. 12.
    Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21(2), 12 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Moser, T., Seiringer, R.: Stability of a Fermionic N+1 particle system with point interactions. Commun. Math. Phys. 356(1), 329–355 (2017)MathSciNetCrossRefADSzbMATHGoogle Scholar
  14. 14.
    Moser, T., Seiringer, R.: Stability of the 2+ 2 fermionic system with point interactions. Math. Phys. Anal. Geom. 21(3), 19 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moshinsky, M.: Boundary conditions for the description of nuclear reactions. Phys. Rev. 81, 347–352 (1951)CrossRefADSzbMATHGoogle Scholar
  16. 16.
    Moshinsky, M.: Boundary conditions and time-dependent states. Phys. Rev. 84, 525–532 (1951)MathSciNetCrossRefADSzbMATHGoogle Scholar
  17. 17.
    Moshinsky, M.: Quantum mechanics in Fock space. Phys. Rev. 84, 533 (1951)MathSciNetCrossRefADSzbMATHGoogle Scholar
  18. 18.
    Moshinsky, M., López Laurrabaquio, G.: Relativistic interactions by means of boundary conditions: the Breit–Wigner formula. J. Math. Phys. 32, 3519–3528 (1991)MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5(9), 1190–1197 (1964)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Schrader, R.: On the existence of a local Hamiltonian in the Galilean invariant Lee model. Commun. Math. Phys. 10(2), 155–178 (1968)MathSciNetCrossRefADSzbMATHGoogle Scholar
  21. 21.
    Schmidt, J.: On a direct description of pseudorelativistic Nelson Hamiltonians. arXiv preprint arXiv:1810.03313 (2018)
  22. 22.
    Schmidt, J.: The massless Nelson Hamiltonian and its domain. arXiv preprint arXiv:1901.05751 (2019)
  23. 23.
    Teufel, S., Tumulka, R.: New type of Hamiltonians without ultraviolet divergence for quantum field theories. arXiv preprint arXiv:1505.04847 (2015)
  24. 24.
    Teufel, S., Tumulka, R.: Avoiding ultraviolet divergence by means of interior–boundary conditions. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds.) Quantum Mathematical Physics, pp. 293–311. Birkhäuser, Basel (2016)CrossRefGoogle Scholar
  25. 25.
    Thomas, L.E.: Multiparticle Schrödinger Hamiltonians with point interactions. Phys. Rev. D 30, 1233–1237 (1984)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Vlietinck, J., Casteels, W., Van Houcke, K., Tempere, J., Ryckebusch, J., Devreese, J.T.: Diagrammatic Monte Carlo study of the acoustic and the Bose–Einstein condensate polaron. New J. Phys. 17(3), 033023 (2015)CrossRefADSGoogle Scholar
  27. 27.
    Wünsch, A.: Self-adjointness and domain of a class of generalized Nelson models. Ph.D. thesis, Universität Stuttgart, (March 2017)Google Scholar
  28. 28.
    Yafaev, D.R.: On a zero-range interaction of a quantum particle with the vacuum. J. Phys. A: Math. Gen. 25(4), 963 (1992)MathSciNetCrossRefADSzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CNRS & Laboratoire interdisciplinaire Carnot de Bourgogne (UMR 6303)Université de Bourgogne Franche-ComtéDijon CedexFrance

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