Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Book cover Classical Analogies in the Solution of Quantum Many-Body Problems

Part of the book series: Springer Theses ((Springer Theses))

  • 389 Accesses

Abstract

In this thesis, we will solve or extend the solution of three quantum many-body problems. These problems in addition to their practical value in the context of recent developments in condensed matter physics, such as zero-energy modes in topological systems, strongly interacting Bose condensates, and many-body localized systems, serve an auxiliary purpose as well. Each model is connected to a seemingly unrelated analogous classical system that elucidates the underlying physics or helps us address a broader issue in modern physics, such as the emergence of classical degrees of freedom, space-time, and quantum chaos. Before introducing these three problems, we will give a very brief qualitative review of the relation between classical and quantum mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. K. Vogtmann, A. Weinstein, V.I. Arnol’d, Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics (Springer, New York, 1997)

    Google Scholar 

  2. H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics (Addison Wesley, Reading, 2002)

    MATH  Google Scholar 

  3. R.M. Wald, General Relativity (University of Chicago Press, Chicago, 2010)

    MATH  Google Scholar 

  4. B. Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, 2009)

    Book  Google Scholar 

  5. R.K. Pathria, P.D. Beale, Statistical Mechanics (Elsevier Science, Amsterdam, 1996)

    MATH  Google Scholar 

  6. R. Kubo, Thermodynamics: An Advanced Course with Problems and Solutions (North-Holland, Amsterdam, 1976)

    Google Scholar 

  7. J.A. Wheeler, W.H. Zurek, Quantum Theory and Measurement. Princeton Legacy Library (Princeton University Press, Princeton, 1983)

    Book  Google Scholar 

  8. W.H. Zurek, Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  9. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals. International Series in Pure and Applied Physics (McGraw-Hill, New York, 1965)

    Google Scholar 

  10. D.J. Griffiths, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2016)

    Google Scholar 

  11. P.A.M. Dirac, On the analogy between classical and quantum mechanics. Rev. Mod. Phys. 17, 195–199 (1945)

    Article  ADS  MathSciNet  Google Scholar 

  12. G.A. Baker, Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space. Phys. Rev. 109, 2198–2206 (1958)

    Article  ADS  MathSciNet  Google Scholar 

  13. J.D. Trimmer, The present situation in quantum mechanics: a translation of Schrödinger’s “cat paradox” paper. Proc. Am. Philos. Soc. 124(5), 323–338 (1980)

    Google Scholar 

  14. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad, J. Chiaverini, D.B. Hume, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, R. Reichle, D. J. Wineland, Creation of a six-atom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005)

    Article  ADS  Google Scholar 

  15. M. Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267–1305 (2005)

    Article  ADS  Google Scholar 

  16. A. Altland, B. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2006)

    Book  Google Scholar 

  17. J. Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity. Oxford Master Series in Physics (Oxford University Press, Oxford, 2006)

    Google Scholar 

  18. A.J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems. Oxford Graduate Texts (Oxford University Press, Oxford, 2006)

    Book  Google Scholar 

  19. N.B. Kopnin, Theory of Nonequilibrium Superconductivity. International Series of Monographs on Physics (Clarendon Press, Oxford, 2001)

    Google Scholar 

  20. M.V. Berry, Semiclassical mechanics of regular and irregular motion, in Les Houches Lecture Series, ed. by G. Iooss, R.H.G. Helleman, R. Stora, vol. 36 (North Holland, Amsterdam, 1983), pp. 171–271

    Google Scholar 

  21. M. Berry, Quantum chaology, not quantum chaos. Phys. Scr. 40(3), 335 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  22. M.V. Berry, N.L. Balazs, M. Tabor, A. Voros, Quantum maps. Ann. Phys. 122(1), 26–63 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  23. J. Maldacena, S.H. Shenker, D. Stanford, A bound on chaos. J. High Energy Phys. 2016(8), 106 (2016)

    Google Scholar 

  24. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 2002)

    Book  Google Scholar 

  25. M.V. Berry, Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?), in Quantum Mechanics: Scientific Perspectives on Divine Action, ed. by R.J. Russell, P. Clayton, K. Wegter-McNelly, J. Polkinghorne (CTNS Publications, Berkeley, 2001), pp. 41–54

    Google Scholar 

  26. S. Fishman, D.R. Grempel, R.E. Prange, Chaos, quantum recurrences, and Anderson localization. Phys. Rev. Lett. 49, 509–512 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  27. C. Tian, A. Kamenev, A. Larkin, Weak dynamical localization in periodically kicked cold atomic gases. Phys. Rev. Lett. 93, 124101 (2004)

    Article  ADS  Google Scholar 

  28. D.R. Grempel, S. Fishman, R.E. Prange, Localization in an incommensurate potential: an exactly solvable model. Phys. Rev. Lett. 49, 833–836 (1982)

    Article  ADS  Google Scholar 

  29. R.E. Prange, D.R. Grempel, S. Fishman, Solvable model of quantum motion in an incommensurate potential. Phys. Rev. B 29, 6500–6512 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Srednicki, Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994)

    Article  ADS  Google Scholar 

  31. L. D’Alessio, Y. Kafri, A. Polkovnikov, M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65(3), 239–362 (2016)

    Article  ADS  Google Scholar 

  32. J.M. Deutsch, Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991)

    Article  ADS  Google Scholar 

  33. D.M. Basko, I.L. Aleiner, B.L. Altshuler, Metal insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321(5), 1126–1205 (2006)

    Article  ADS  Google Scholar 

  34. V. Oganesyan, D.A. Huse, Localization of interacting fermions at high temperature. Phys. Rev. B 75(15), 155111 (2007)

    Google Scholar 

  35. A. Pal, D.A. Huse, Many-body localization phase transition. Phys. Rev. B 82(17), 174411 (2010)

    Google Scholar 

  36. J.H. Bardarson, F. Pollmann, J.E. Moore, Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett. 109, 017202 (2012)

    Article  ADS  Google Scholar 

  37. S. Iyer, V. Oganesyan, G. Refael, D.A. Huse, Many-body localization in a quasiperiodic system. Phys. Rev. B 87, 134202 (2013)

    Article  ADS  Google Scholar 

  38. B. Bauer, C. Nayak, Area laws in a many-body localized state and its implications for topological order. J. Stat. Mech: Theory Exp. 2013(9), P09005 (2013)

    Article  MathSciNet  Google Scholar 

  39. B. Bauer, C. Nayak, Analyzing many-body localization with a quantum computer. Phys. Rev. X 4(4), 041021 (2014)

    Google Scholar 

  40. J.Z. Imbrie, On many-body localization for quantum spin chains. J. Stat. Phys. 163(5), 998–1048 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  41. A. Chandran, I.H. Kim, G. Vidal, D.A. Abanin, Constructing local integrals of motion in the many-body localized phase. Phys. Rev. B 91, 085425 (2015)

    Article  ADS  Google Scholar 

  42. M.V. Berry, Incommensurability in an exactly-soluble quantal and classical model for a kicked rotator. Phys. D: Nonlinear Phenom. 10(3), 369–378 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  43. G. Hooft, A mathematical theory for deterministic quantum mechanics. J. Phys.: Conf. Ser. 67(1), 012015 (2007)

    Google Scholar 

  44. L.P. Kadanoff, G. Baym, D. Pines, Quantum Statistical Mechanics. Advanced Books Classics Series (Perseus Books, New York, 1994)

    Google Scholar 

  45. L.V. Keldysh, Diagram technique for nonequilibrium processes. J. Exp. Theor. Phys. 20(4), 1018 (1965)

    Google Scholar 

  46. O.V. Konstantinov, V.I. Perel’, A diagram technique for evaluating transport quantities. J. Exp. Theor. Phys. 12(1), 142 (1961)

    Google Scholar 

  47. J. Schwinger, Brownian motion of a quantum oscillator. J. Math. Phys. 2(3), 407–432 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  48. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011)

    Book  Google Scholar 

  49. E. Calzetta, B.L. Hu, Closed-time-path functional formalism in curved spacetime: application to cosmological back-reaction problems. Phys. Rev. D 35, 495–509 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  50. B.L. Hu, E. Verdaguer, Stochastic gravity: a primer with applications. Classical Quantum Gravity 20(6), R1 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  51. J. Rammer, H. Smith, Quantum field-theoretical methods in transport theory of metals. Rev. Mod. Phys. 58, 323–359 (1986)

    Article  ADS  Google Scholar 

  52. J. Rammer, H. Smith, Quantum field-theoretical methods in transport theory of metals. Rev. Mod. Phys. 58, 323–359 (1986)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Physics, UNSW Australia, Sydney, NSW, Australia

    Aydın Cem Keser

Authors
  1. Aydın Cem Keser
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Keser, A.C. (2018). Introduction. In: Classical Analogies in the Solution of Quantum Many-Body Problems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-00488-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation