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Quantum Hierarchical Systems: Fluctuation Force by Coarse-Graining, Decoherence by Correlation Noise

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From Quantum to Classical

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 204))

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Abstract

While the issues of dissipation, fluctuations, noise and decoherence in open quantum systems (with autocratic divide) analyzed via Langevin dynamics are familiar subjects, the treatment of corresponding issues in closed quantum systems is more subtle, as witnessed by Boltzmann’s explanation of dissipation in a macroscopic system made up of many equal constituents (a democratic system). How to extract useful physical information about a closed democratic system with no obvious ways to distinguish one constituent from another, nor the existence of conservation laws governing certain special kinds of variables, e.g., the hydrodynamic variables—this is the question we raise in this essay. Taking the inspirations from Boltzmann and Langevin, we study (a) how a hierarchical order introduced to a closed democratic system—defined either by substance or by representation, and (b) how hierarchical coarse-graining, executed in a specific order, can facilitate our understanding in how macro-behaviors arise from micro-dynamics. We give two examples in: (a) the derivation of correlation noises in the Bogolyubov-Born-Green-Kirkwood-Yvonne hierarchy and the use of a Boltzmann-Langevin equation to study the decoherence of the lower order correlations; and (b) the derivation of quantum fluctuation forces by ordered coarse-grainings of the relevant variables in the medium, the quantum field and the internal degrees of freedom of an atom.

–Dedicated to the memory of Professor Heinz Dieter Zeh who made invaluable contributions to the foundational issues of quantum mechanics [1] and pioneered the environment-induced quantum decoherence program [2].

–From Quantum to Classical: Essays in Memory of Dieter Zeh, edited by C. Kiefer, in “Fundamental Theories of Physics” (Springer, Cham, 2021).

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Notes

  1. 1.

    Looking up the web, I found this terminology appears most often in relation to biological systems. Ideas akin to this, likely broader in scope and in variety, for the description of real and perceived structures in the physical world and adaptive systems can be found in this interesting paper [8].

  2. 2.

    However, it is important to note that in this case the stochastic field \(\xi ^j_X\) represents the fluctuations of the medium only and does not include the intrinsic fluctuations of the field. After the next level of coarse-graining described below the intrinsic quantum fluctuations of the field will enter which is different from the induced fluctuations from interaction with the dielectric medium.

  3. 3.

    Constructing a stochastic representation of quantum field theory has caught increasing attention. Here are some noteworthy technical points about stochastic equations for the physical propagators, as exemplified in [60], which nicely completes the quest in [59]. In the 1PI theory, the \(\phi ^-\) field vanishes identically on-shell. However, in the stochastic approach we assign a nontrivial source \(\xi _-^{1PI}\) to it. It is by eliminating this auxiliary field that we recover the usual approach, with a single stochastic source \(\tilde{\xi }\) whose self-correlation is given by the noise kernel. Similarly, in the quantum field theory problem the correlator \(G^{--} = \langle \varphi ^- \varphi ^-\rangle \) vanishes identically, as a result of path ordering. However, in the stochastic approach, we consider it as an auxiliary field and couple a source to it. The authors of [59] failed to recognize the violation of the constraint \(G^{--} = 0\), but guessed the right form for the noise self-correlation by introducing a sign change in their expression arising from adding a hermitian conjugation to the second derivative of the 2PI-EA. This was shown to be unnecessary by Calzetta [60] who provided a satisfactory explanation for this sign change: It is due to the elimination of the auxiliary field \(G^{--}\), keeping only the physical degrees of freedom.

References

  1. H.D. Zeh, On the interpretation of measurement in quantum theory. Found. Phys. 1(1), 69–76 (1970). H.D. Zeh, Toward a quantum theory of observation. Found. Phys. 3(1), 109–116 (1973). H.D. Zeh, The Direction of Time (Springer, Berlin, 1989)

    Google Scholar 

  2. E. Joos, H.D. Zeh, The emergence of classical properties through interaction with the environment. Zeitschrift für Physik B Condensed Matter 59(2), 223–243 (1985). D. Giulini, C. Kiefer, H.D. Zeh, Symmetries, super selection rules, and decoherence. Phys. Lett. A 199(5–6), 291–298 (1995). H.D. Zeh, Roots and Fruits of Decoherence Seminaire Poincaré 2, 1–19 (2005). arXiv:quant-ph/0512078v2

  3. H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002). A. Rivas, S.F. Huelga, Open Quantum Systems, vol. 13 (Springer, Berlin, 2012)

    Google Scholar 

  4. E.A. Calzetta, B.-L.B. Hu, Nonequilibrium Quantum Field Theory (Cambridge University Press, 2008)

    Google Scholar 

  5. M. Kardar, R. Golestanian, The friction of vacuum, and other fluctuation-induced forces. Rev. Modern Phys. 71(4), 1233 (1999). P.W. Milonni, An Introduction to Quantum Optics and Quantum Fluctuations (Oxford University Press, 2019)

    Google Scholar 

  6. H.B. Casimir, D. Polder, The influence of retardation on the London-van der Waals forces. Phys. Rev. 73(4), 360 (1948)

    Article  ADS  MATH  Google Scholar 

  7. E. Joos, H.D. Zeh, C. Kiefer, D.J. Giulini, J. Kupsch, I.O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory (Springer Science & Business Media, 2013). M.A. Schlosshauer, Decoherence: and the Quantum-to-Classical Transition (Springer Science & Business Media, 2007)

    Google Scholar 

  8. M. Gell-Mann, J.B. Hartle, Adaptive coarse graining, environment, strong decoherence, and quasiclassical realms. Phys. Rev. A 89(5), 052125 (2014)

    Article  ADS  Google Scholar 

  9. J.P. Paz, S. Sinha, Decoherence and back reaction: the origin of the semiclassical Einstein equations. Phys. Rev. D 44(4), 1038 (1991). C. Kiefer, The semiclassical approximation to quantum gravity, in Canonical Gravity: from Classical to Quantum (Springer, Berlin, 1994), pp. 170–212

    Google Scholar 

  10. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001)

    Google Scholar 

  11. S.L. Adler, Quantum Theory as an Emergent Phenomenon: the Statistical Mechanics of Matrix Models as the Precursor Of Quantum Field Theory (Cambridge University Press, 2004)

    Google Scholar 

  12. G. ’t Hooft, Emergent quantum mechanics and emergent symmetries, in AIP Conference Proceedings American Institute of Physics, vol. 957, no. 1 (2007), pp. 154–163

    Google Scholar 

  13. R.C. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley, 1975) (H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, 1991)

    Google Scholar 

  14. S. Chandrasekhar, Stochastic problems in physics and astronomy. Rev. Modern Phys. 15(1), 1 (1943). N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, vol. 1. (Elsevier, 1992)

    Google Scholar 

  15. M. Gell-Mann, J.B. Hartle, Classical equations for quantum systems. Phys. Rev. D 47(8), 3345 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  16. O. Kübler, H.D. Zeh, Dynamics of quantum correlations. Ann. Phys. 76(2), 405–418 (1973)

    Article  ADS  Google Scholar 

  17. M. Bixon, R. Zwanzig, Boltzmann-Langevin equation and hydrodynamic fluctuations. Phys. Rev. 187(1), 267 (1969). R.F. Fox, G.E. Uhlenbeck, Contributions to nonequilibrium thermodynamics. II. Fluctuation theory for the Boltzmann equation. Phys. Fluids 13(12), 2881–2890 (1970)

    Google Scholar 

  18. M.R. Gallis, G.N. Fleming, Environmental and spontaneous localization. Phys. Rev. A 42(1), 38 (1990)

    Article  ADS  Google Scholar 

  19. L. Diosi, Quantum master equation of a particle in a gas environment. EPL (Europhysics Letters) 30(2), 63 (1995)

    Article  ADS  Google Scholar 

  20. K. Hornberger, Introduction to Decoherence Theory, in Entanglement and Decoherence (Springer, Berlin, 2009), pp. 221–276

    Book  MATH  Google Scholar 

  21. P.J. Dodd, J.J. Halliwell, Decoherence and records for the case of a scattering environment. Phys. Rev. D 67(10), 105018 (2003)

    Article  ADS  Google Scholar 

  22. J. Polonyi, Dissipation and decoherence by a homogeneous ideal gas. Phys. Rev. A 92(4), 042111 (2015)

    Article  ADS  Google Scholar 

  23. C.J. Riedel, W.H. Zurek, Quantum Darwinism in an everyday environment: Huge redundancy in scattered photons. Phys. Rev. Lett. 105(2), 020404 (2010)

    Article  ADS  Google Scholar 

  24. W.H. Zurek, Quantum Darwinism. Nat. Phys. 5, 181–8 (2009). C.J. Riedel, W.H. Zurek, M. Zwolak, The rise and fall of redundancy in decoherence and quantum Darwinism. New J. Phys. 14(8), 083010 (2012)

    Google Scholar 

  25. W. Marshall, C. Simon, R. Penrose, D. Bouwmeester, Towards quantum superpositions of a mirror. Phys. Rev. Lett. 91(13), 130401 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  26. M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, A. Zeilinger, Wave-particle duality of \(C^{60}\) molecules. Nature 401(6754), 680–682 (1999)

    Article  ADS  Google Scholar 

  27. C.H. Chou, B.L. Hu, T. Yu, Quantum Brownian motion of a macroscopic object in a general environment. Phys. A Stat. Mech. Appl. 387(2–3), 432–444 (2008)

    Article  MathSciNet  Google Scholar 

  28. B.L. Hu, J.P. Paz, Y. Zhang, Quantum Brownian motion in a general environment: exact master equation with nonlocal dissipation and colored noise. Phys. Rev. D 45(8), 2843 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. T.A. Brun, L. Mlodinow, Decoherence by coupling to internal vibrational modes. Phys. Rev. A 94(5), 052123 (2016)

    Article  ADS  Google Scholar 

  30. M. Hillery, L. Mlodinow, V. Buek, Quantum interference with molecules: the role of internal states. Phys. Rev. A 71(6), 062103 (2005)

    Article  ADS  Google Scholar 

  31. J.C. Flores, Decoherence from internal degrees of freedom for clusters of mesoparticles: a hierarchy of master equations. J. Phys. A Math. Gen. 31, 8623 (1988)

    Article  ADS  MATH  Google Scholar 

  32. H.K. Park, S.W. Kim, Decoherence from chaotic internal dynamics in two coupled \(\delta \)-function-kicked rotors. Phys. Rev. A 67(6), 060102 (2003)

    Article  ADS  Google Scholar 

  33. W.H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Phys. Rev. D 24, 1516–1525 (1981). W.H. Zurek, Environment-induced superselection rules. Phys. Rev. D 26(8), 1862 (1982). W.H. Zurek, Decoherence, einselection, and the quantum origins of the classical. Rev. Modern Phys. 75(3), 715 (2003)

    Google Scholar 

  34. R.B. Griffiths, Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36(1–2), 219–272 (1984). R.B. Griffiths, Consistent Quantum Theory (Cambridge University Press, 2003)

    Google Scholar 

  35. R. Omnes, Consistent interpretations of quantum mechanics. Rev. Modern Phys. 64(2), 339 (1992). R. Omnes, The Interpretation of Quantum Mechanics, vol. 102 (Princeton University Press, 2018)

    Google Scholar 

  36. M. Gell-Mann, J.B. Hartle, Complexity, entropy and the physics of information. SFI Stud. Sci. Complex. 8, 425 (1990). M. Gell-Mann, J.B. Hartle, in Proceedings of the 4th Drexel Symposium on Quantum Non-Integrability: The Quantum-Classical Correspondence edited by D.-H. Feng, B.-L. Hu (International Press, Boston, 1995)

    Google Scholar 

  37. J.B. Hartle, The quantum mechanics of closed systems, in Directions in General Relativity, vol. 1, ed. by B.-L. Hu, M.P. Ryan, C.V. Vishveshwara (Cambridge University Press, Cambridge, 1993), pp. 104–124. J.B. Hartle, The quasiclassical realms of this quantum universe. Found. Phys. 41(6), 982–1006 (2011)

    Google Scholar 

  38. D.A. Lidar, I.L. Chuang, K.B. Whaley, Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81(12), 2594 (1998)

    Article  ADS  Google Scholar 

  39. J.B. Hartle, R. Laflamme, D. Marolf, Conservation laws in the quantum mechanics of closed systems. Phys. Rev. D 51(12), 7007 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  40. J.J. Halliwell, Decoherent histories and hydrodynamic equations. Phys. Rev. D 58(10), 105015 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  41. J.J. Halliwell, Decoherent histories and the emergent classicality of local densities. Phys. Rev. Lett. 83(13), 2481 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. E.A. Calzetta, B.L. Hu, Influence action and decoherence of hydrodynamic modes. Phys. Rev. D 59(6), 065018 (1999)

    Article  ADS  Google Scholar 

  43. J.J. Halliwell, Decoherence of histories and hydrodynamic equations for a linear oscillator chain. Phys. Rev. D 68(2), 025018 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  44. T.A. Brun, J.B. Hartle, Classical dynamics of the quantum harmonic chain. Phys. Rev. D 60(12), 123503 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  45. E. Calzetta, B.L. Hu, Quantum fluctuations, decoherence of the mean field, and structure formation in the early universe. Phys. Rev. D 52(12), 6770 (1995)

    Article  ADS  Google Scholar 

  46. E. Calzetta, B.L. Hu, Stochastic behavior of effective field theories across the threshold. Phys. Rev. D 55(6), 3536 (1997)

    Article  ADS  Google Scholar 

  47. B.L. Hu, Fluctuation, dissipation and irreversibility in cosmology, in The Physical Origin of Time-Asymmetry, edited by J. J. Halliwell, J. Perez-Mercader, W.H. Zurek (Cambridge University, Cambridge, 1993). arxiv:gr-qc/9302021

  48. M. Gell-Mann, J.B. Hartle, Quasiclassical coarse graining and thermodynamic entropy. Phys. Rev. A 76(2), 022104 (2007)

    Article  ADS  Google Scholar 

  49. E.M. Lifshitz, Sov. Phys. JETP 2(1), 73. 27. I.E. Dzyaloshinskii, E.M. Lifshitz, L.P. Pitaevskii, Adv. Phys. 10(38), 165 (1961). S.Y. Buhmann, Dispersion Forces I: Macroscopic Quantum Electrodynamics and Ground-State Casimir, Casimir-Polder and van der Waals Forces, vol. 247 (Springer, 2013)

    Google Scholar 

  50. S. Shresta, B.L. Hu, Moving atom-field interaction: quantum motional decoherence and relaxation. Phys. Rev. A 68(1), 012110 (2003)

    Article  ADS  Google Scholar 

  51. J.B. Pendry, Shearing the vacuum-quantum friction. J. Phys. Condens. Matter, 9(47), 10301 (1997). J.B. Pendry, Quantum friction- fact or fiction? New J. Phys. 12(3), 033028 (2010). S. Buhmann, Dispersion Forces II: Many-Body Effects, Excited Atoms, Finite Temperature and Quantum Friction, vol. 248 (Springer, 2013). F. Intravaia, R.O. Behunin, D.A. Dalvit, Quantum friction and fluctuation theorems. Phys. Rev. A 89(5), 050101 (2014)

    Google Scholar 

  52. R.O. Behunin, B.L. Hu, Nonequilibrium forces between atoms and dielectrics mediated by a quantum field. Phys. Rev. A 84(1), 012902 (2011)

    Article  ADS  Google Scholar 

  53. R.O. Behunin, B.L. Hu, Nonequilibrium forces between neutral atoms mediated by a quantum field. Phys. Rev. A 82(2), 022507 (2010)

    Article  ADS  Google Scholar 

  54. J. Schwinger, Brownian motion of a quantum oscillator. J. Math. Phys. 2(3), 407–432 (1961). L.V. Keldysh, Diagram technique for nonequilibrium processes. Sov. Phys. JETP 20(4), 1018–1026 (1965)

    Google Scholar 

  55. E. Calzetta, B.L. Hu, Nonequilibrium quantum fields: closed-time-path effective action, Wigner function, and Boltzmann equation. Phys. Rev. D 37(10), 2878 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  56. E. Calzetta, B.L. Hu, Correlations, decoherence, dissipation, and noise in quantum field theory, in Discourses in the Mathematics and its Applications No. 4, edited by S. A. Fulling (Texas A & M University Press, College Station, 1995) Arxiv:hep-th/9501040

  57. E. Calzetta, B.L. Hu, Decoherence of correlation histories. Dir. Gen. Relat. 2, 38–65 (1993)

    MATH  Google Scholar 

  58. C. Anastopoulos, Coarse grainings and irreversibility in quantum field theory. Phys. Rev. D 56(2), 1009 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  59. E. Calzetta, B.L. Hu, Stochastic dynamics of correlations in quantum field theory: from the Schwinger-Dyson to Boltzmann-Langevin equation. Phys. Rev. D 61(2), 025012 (1999)

    Article  ADS  Google Scholar 

  60. E. Calzetta, Fourth-order full quantum correlations from a Langevin-Schwinger-Dyson equation. J. Phys. A Math. Theor. 42(26), 265401 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. P.G. Reinhard, E. Suraud, Stochastic TDHF and the Boltzman-Langevin equation. Ann. Phys. 216(1), 98–121 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. C. Greiner, S. Leupold, Stochastic interpretation of Kadanoff-Baym equations (1998). arXiv:hep-ph/9809296

  63. A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe. Lect. Notes Phys. 246, 107–126 (1986). V. Vennin, A.A. Starobinsky, Correlation functions in stochastic inflation. Eur. Phys. J. C 75(9), 413 (2015)

    Google Scholar 

  64. B.L. Hu, E. Verdaguer, Stochastic gravity: theory and applications. Living Rev. Relat. 11(1), 3 (2008). B.L. Hu, E. Verdaguer, Semiclassical and stochastic gravity: quantum field effects on curved spacetime (Cambridge University Press, 2020)

    Google Scholar 

  65. S.A. Ramsey, B.L. Hu, O (N) quantum fields in curved spacetime. Phys. Rev. D 56(2), 661 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  66. J. Berges, N-particle irreducible effective action techniques for gauge theories. Phys. Rev. D 70(10), 105010 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  67. M.E. Carrington, Techniques for calculations with nPI effective actions, in EPJ Web of Conferences, vol. 95 (EDP Sciences, 2015), p. 04013

    Google Scholar 

  68. E.A. Calzetta, B.L. Hu, Correlation entropy of an interacting quantum field and H theorem for the O (N) model. Phys. Rev. D 68(6), 065027 (2003)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The two key ideas expounded here: the correlation noise, is contained in prior work with Esteban Calzetta begun more than three decades ago, which he satisfactorily completed, while that of hierarchical coarse-graining, with Ryan Behunin, a decade ago and ongoing. I’m thankful for their invaluable contributions which deepened my understanding of these fundamental issues of physics.

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    Bei-Lok Hu

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  1. Institute for Theoretical Physics, University of Cologne, Köln, Nordrhein-Westfalen, Germany

    Claus Kiefer

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Hu, BL. (2022). Quantum Hierarchical Systems: Fluctuation Force by Coarse-Graining, Decoherence by Correlation Noise. In: Kiefer, C. (eds) From Quantum to Classical. Fundamental Theories of Physics, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-030-88781-0_9

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