Research PaperNo Access

Classical and quantum energies for interacting magnetic systems

Laboratoire de Mathématiques de Reims, UMR CNRS 9008, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France

E-mail Address: laurent.amour@univ-reims.fr

Search for more papers by this author

and

J. Nourrigat

Laboratoire de Mathématiques de Reims, UMR CNRS 9008, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France

E-mail Address: jean.nourrigat@univ-reims.fr

Search for more papers by this author

https://doi.org/10.1142/S0129055X22500179Cited by:0 (Source: Crossref)

Abstract

The purpose of this paper is to give different interpretations of the first non-vanishing term (quadratic) of the ground state asymptotic expansion for a spin system in quantum electrodynamics, as the spin magnetic moments go to 0. One of the interpretations makes a direct link with some classical physics laws. A central role is played by an operator $A_{M}$ $A_{M}$ acting only in the finite-dimensional spin state space and making the connections with the different interpretations, and also being in close relation with the multiplicity of the ground state.

Keywords:

AMSC: 81V10

References

1. L. Amour, L. Jager and J. Nourrigat, Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance, J. Math. Phys. 60(7) (2019) 071503. Crossref, Web of Science, Google Scholar
2. L. Amour, L. Jager and J. Nourrigat, Ground state photon number at large distance, Rep. Math. Phys. 85(2) (2020) 227–238. Crossref, Web of Science, ADS, Google Scholar
3. L. Amour and J. Nourrigat, Lindblad approximation and spin relaxation in quantum electrodynamics, J. Phys. A Math. Theor. 53 (2020) 245204. Crossref, Web of Science, ADS, Google Scholar
4. A. Arai, A new asymptotic perturbation theory with applications to models of massless quantum fields, Ann. Henri Poincaré 15(6) (2014) 1145–1170. Crossref, Web of Science, ADS, Google Scholar
5. A. Arai and M. Hirokawa, On the existence and uniqueness of ground states of a generalized spin-boson model, J. Funct. Anal. 151(2) 1997 455–503. Crossref, Web of Science, Google Scholar
6. C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Processus d’interaction entre photons et atomes, Savoirs actuels (EDP Sciences/CNRS Editions, 2001). Google Scholar
7. R. P. Feynman, R. B. Leighton and M. L. Sands, The Feynman Lectures on Physics, Vol. 2, Mainly Electromagnetism and Matter (Addison-Wesley, 1964). Google Scholar
8. C. Gérard, On the existence of ground states for massless Pauli–Fierz Hamiltonians, Ann. Henri Poincaré 1 (2000) 443–459. Crossref, Web of Science, ADS, Google Scholar
9. J. Glimm and A. Jaffe, Statistical Mechanics and Quantum Field Theory, Les Houches 1970, eds. C. DeWitt and R. Stora (Gordon and Breach, New York, 1971). Google Scholar
10. J. Glimm and A. Jaffe, Quantum Field Theory and Statistical Mechanics (Birkhäuser, Boston, MA, 1985). Google Scholar
11. L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal. 10 (1972) 52–109. Crossref, Google Scholar
12. L. Gross, The relativistic polaron without cutoffs, Comm. Math. Phys. 31 (1973) 25–73. Crossref, Web of Science, ADS, Google Scholar
13. D. Hasler and I. Herbst, Ground states in the spin boson model, Ann. Henri Poincaré 12(4) (2011) 621–677. Crossref, Web of Science, ADS, Google Scholar
14. F. Hiroshima, Self-adjointness of the Pauli–Fierz Hamiltonian for arbitrary values of coupling constants, Ann. Henri Poincaré 3 (2002) 171–201. Crossref, Web of Science, ADS, Google Scholar
15. F. Hiroshima, Multiplicity of ground states in quantum field models: Applications of asymptotic fields, J. Funct. Anal. 224 (2005) 431–470. Crossref, Web of Science, Google Scholar
16. F. Hiroshima, Localization of the number of photons of ground states in nonrelativistic QED, Rev. Math. Phys. 15(3) (2003) 271–312. Link, Web of Science, Google Scholar
17. M. Hübner and H. Spohn, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. H. Poincaré Sect. A 62(3) (1995) 289–323. Web of Science, Google Scholar
18. E. Lieb and M. Loss, A note on polarization vectors in quantum electrodynamics, Comm. Math. Phys. 252 (2004) 477–483. Crossref, Web of Science, ADS, Google Scholar
19. M. Loss, T. Miyao and H. Spohn, Kramers degeneracy theorem in nonrelativistic QED, Lett. Math. Phys. 89 (2009) 21–31. Crossref, Web of Science, ADS, Google Scholar
20. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, London, 1978). Google Scholar
21. F. A. Reuse, Electrodynamique et Optique Quantiques (Presses Polytechniques et Universitaires Romandes, Lausanne, 2007). Google Scholar
22. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Company, Evanston, IL, Elmsford, NY 1961). Google Scholar
23. H. Spohn, Ground state(s) of the spin-boson Hamiltonian, Comm. Math. Phys. 123 (1989) 277–304. Crossref, Web of Science, ADS, Google Scholar