Skip to main content
SpringerLink
Log in

Entropy Distribution of Localised States

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is \({\pi E}\) , with E the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius r, the relative entropy is non linear, the asymptotic mean radial entropy density is \({\pi E}\) and Bekenstein’s bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal U(1)-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Araki H.: Expansional in Banach algebras. Ann. Sci. École Norm. Sup. (4) 6, 67–84 (1973)

    Article  MathSciNet  Google Scholar 

  2. Araki H.: Relative entropy of states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)

    Article  MathSciNet  Google Scholar 

  3. Araki H., Zsido L.: Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Rev. Math. Phys. 17, 491–543 (2005)

    Article  MathSciNet  Google Scholar 

  4. Balakrishnan, S., Faulkner, T., Khandker, Z.U., Wang, H.: A general proof of the quantum null energy condition, arXiv:1706.09432 [hep-th]

  5. Bekenstein J.D.: Universal upper bound on the entropy-to-energy ratio for bounded systems. Phys. Rev. D 23, 287 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bisognano J., Wichmann E.: On the duality condition for a Hermitean scalar field. J. Math. Phys. 16, 985 (1975)

    Article  ADS  Google Scholar 

  7. Blanco D., Casini H.: Localization of negative energy and the Bekenstein bound. Phys. Rev. Lett. 111, 221601 (2013)

    Article  ADS  Google Scholar 

  8. Borchers H.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41, 3604 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  9. Bousso R., Fisher Z., Koeller J., Leichenauer S., Wall A. C.: Proof of the quantum null energy condition. Phys. Rev. D 93, 024017 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. Bratteli, O., Robinson, D.: Operator Algebras and quantum statistical mechanics, I & II, Berlin: Springer (1987 & 1997)

  11. Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  12. Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  13. Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proceedings Supplement) 5, 20–56 (1988)

    Article  MathSciNet  Google Scholar 

  14. Casini, H., Testé, E., Torroba, G.: Modular Hamiltonians on the null plane and the Markov property of the vacuum state, arXiv:1703.10656 [hep-th]

  15. Connes A.: Une classification des facteurs de type III. Ann. Sci. Ec. Norm. Sup. 6, 133–252 (1973)

    Article  MathSciNet  Google Scholar 

  16. Connes A., Størmer E.: Homogeneity of the state space of factors of type III1. J. Funct. Anal. 28(2), 187–196 (1978)

    Article  Google Scholar 

  17. Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics, I. Commun. Math. Phys. 23, 199–230 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  18. Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. II. Commun. Math. Phys. 35, 49–85 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  19. Epstein H., Glaser V., Jaffe A.: Nonpositivity of the energy density in quantized field theories. Nuovo Cimento 36, 1016 (1965)

    Article  MathSciNet  Google Scholar 

  20. Fewster C.J., Hollands S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577 (2005)

    Article  MathSciNet  Google Scholar 

  21. Ford L.H., Roman T.A.: Averaged energy conditions and quantum inequalities. Phys. Rev. D 51, 4277 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  22. Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148(3), 521–551 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  23. Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  24. Haag R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, New York (1996)

    MATH  Google Scholar 

  25. Hartman, T.: Bounds on energy, entropy, and transport, Talk at the Strings 2018 conference, https://indico.oist.jp/indico/event/5/picture/114.pdf

  26. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time, Cambridge Monographs on Mathematical Physics Cambridge University Press, Cambridge (2011)

  27. Hislop P.D., Longo R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71–85 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  28. Hollands, S.: Relative entropy close to the edge, arXiv:1805.10006 [hep-th]

  29. Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  30. Kawahigashi Y., Longo R.: Noncommutative spectral invariants and black hole entropy. Commun. Math. Phys. 257, 193–225 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  31. Leichenauer, S., Levine, A., Shahbazi-Moghaddam, A.: Energy is entanglement, arXiv:1802.02584 [hep-th]

  32. Longo, R.: Algebraic and modular structure of von Neumann algebras of Physics, Proceedings of Symposia in Pure Math. 38, Part 2, 551 (1982)

  33. Longo R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126(2), 217–247 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  34. Longo R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130(2), 285–309 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  35. Longo R.: An analogue of the Kac-Wakimoto formula and black hole conditional entropy. Commun. Math. Phys. 186, 451–479 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  36. Longo, R.: The Bisognano–Wichmann theorem for charged states and the conformal boundary of a black hole, Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, 1999), Electron. J. Differ. Equ. Conf. 4 (2000), 159–164

  37. Longo R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  38. Longo R.: On Landauer’s principle and bound for infinite systems. Commun. Math. Phys. 363, 531–560 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  39. Longo R., Roberts J.E.: A theory of dimension. K-Theory 11, 103–159 (1997)

    Article  MathSciNet  Google Scholar 

  40. Longo R., Xu F.: Comment on the Bekenstein bound. J. Geom. Phys. 130, 113–120 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  41. Longo R., Xu F.: Relative entropy in CFT. Adv. Math. 337, 139–170 (2018)

    Article  MathSciNet  Google Scholar 

  42. Ohya M., Petz D.: “Quantum entropy and its use”, Texts and Monographs in Physics. Springer, Berlin (1993)

    MATH  Google Scholar 

  43. Takesaki, M.: “Theory of operator algebras”, I & II, Springer, New York (2002 & 2003)

  44. Verch R.: The averaged null energy condition for general quantum field theories in two-dimensions. J. Math. Phys. 41, 206–217 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  45. Weiner M.: Conformal covariance and positivity of energy in charged sectors. Commun. Math. Phys. 265, 493–506 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  46. Wiesbrock H.-W.: Half-sided modular inclusions of von Neumann algebras. Commun. Math. Phys. 157, 83 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  47. Witten E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  48. Witten, E.: Notes on some entanglement properties of quantum field theory, arXiv:1803.04993 [hep-th]

Download references

Acknowledgments

This paper is the follow up of a question privately set to the author by Edward Witten at the Okinawa Strings 2018 conference. The author warmly thanks him for sharing his insight and constant encouragement. We wish to thank also Hirosi Ooguri and the conference organisers for the kind invitation, and Nima Lashkari for comments. We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Rome, Italy

    Roberto Longo

Authors
  1. Roberto Longo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Longo.

Additional information

Communicated by Y. Kawahigashi

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

R. Longo: Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Longo, R. Entropy Distribution of Localised States. Commun. Math. Phys. 373, 473–505 (2020). https://doi.org/10.1007/s00220-019-03332-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03332-8

Search

Navigation