Skip to main content
SpringerLink
Log in

Information Geometry and Quantum Fields

  • Conference paper
  • First Online:
Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 361))

Abstract

We explore the information geometry associated with a variety of simple physical systems: the massless scalar \(\phi ^4\) theory and the 2d classical Ising model. We distill out a number of general lessons with important implications for the application of information geometry to holography: a geometry may be defined both on the states of one theory as well as on a family of theories and this geometry reflects the symmetries, stability, and phase structure of the physical theory. We describe a family of connections, the full physical content of which remains to be studied.

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 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 279.99
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

Similar content being viewed by others

Notes

  1. 1.

    It is possible to extend the Fisher metric on probability distributions, which have to be normalized via \(\int d \mu (x) \, p(x ; \theta ) = 1\), to non-negative measures by relaxing the normalization condition. In that setting, the metric is often called the Fisher-Shahshahani metric in honor of the work of Shahshahani [29]. The denormalization of a statistical manifold is also discussed in Sect. 2.6 of Amari’s textbook [27] (see also [30]). However, we will not pursue this topic here.

  2. 2.

    There is a more general concept in the mathematical literature of a restricted Markov morphism (see Definition 4.4 of [31]), which encompasses the definition of symmetry given above and generalizes it to transformations among families of distributions and not just within one family.

  3. 3.

    We will work in Euclidean signature here, but a-posteriori Wick rotation applies these same arguments to AdS spacetime.

  4. 4.

    Note that while (7) is correct in any dimension, (9) is the correct expression only in dimension \(d=4\). Of course, this is fine because (6) is the solution to the Euler-Lagrange equations of motion only in dimension \(d=4\).

  5. 5.

    In the isotropic case, one sets \(J=K=1\) and leaves the temperature as the remaining variable. The Fisher metric for the isotropic case with an external magnetic field was studied in [39].

  6. 6.

    Auto-parallel flow lines correspond to geodesics only for the Levi-Civita connection.

References

  1. Erdmenger, J., Grosvenor, K.T., Jefferson, R.: Information geometry in quantum field theory: lessons from simple examples. SciPost Phys. 8(5), 073 (2020). https://doi.org/10.21468/SciPostPhys.8.5.073. [arXiv:2001.02683 [hep-th]]

    Article  MathSciNet  Google Scholar 

  2. Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96 (2006). https://doi.org/10.1103/PhysRevLett.96.181602. [arXiv:hep-th/0603001 [hep-th]]

  3. Hubeny, V.E., Rangamani, M., Takayanagi, T.: A Covariant holographic entanglement entropy proposal. JHEP 07, 062 (2007). https://doi.org/10.1088/1126-6708/2007/07/062. [arXiv:0705.0016 [hep-th]]

    Article  MathSciNet  Google Scholar 

  4. Engelhardt, N., Wall, A.C.: Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime. JHEP 01, 073 (2015). https://doi.org/10.1007/JHEP01(2015)073. [arXiv:1408.3203 [hep-th]]

    Article  Google Scholar 

  5. Almheiri, A., Dong, X., Harlow, D.: Bulk locality and quantum error correction in AdS/CFT. JHEP 04, 163 (2015). https://doi.org/10.1007/JHEP04(2015)163. [arXiv:1411.7041 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  6. Pastawski, F., Yoshida, B., Harlow, D., Preskill, J.: Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015). https://doi.org/10.1007/JHEP06(2015)149. [arXiv:1503.06237 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  7. Harlow, D.: The Ryu-Takayanagi formula from quantum error correction. Commun. Math. Phys. 354(3), 865–912 (2017). https://doi.org/10.1007/s00220-017-2904-z. [arXiv:1607.03901 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  8. Harlow, D.: TASI lectures on the emergence of bulk physics in AdS/CFT. PoS TASI2017, 002 (2018). https://doi.org/10.22323/1.305.0002. [arXiv:1802.01040 [hep-th]]

    Article  Google Scholar 

  9. Swingle, B.: Entanglement renormalization and holography. Phys. Rev. D 86, (2012). https://doi.org/10.1103/PhysRevD.86.065007. [arXiv:0905.1317 [cond-mat.str-el]]

  10. Bao, N., et al.: Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence. Phys. Rev. D 91(12), 125036 (2015). https://doi.org/10.1103/PhysRevD.91.125036. [arXiv:1504.06632 [hep-th]]

    Article  MathSciNet  Google Scholar 

  11. Hayden, P., Nezami, S., Qi, X.L., Thomas, N., Walter, M., Yang, Z.: Holographic duality from random tensor networks. JHEP 11, 009 (2016). https://doi.org/10.1007/JHEP11(2016)009. [arXiv:1601.01694 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  12. Miyaji, M., Numasawa, T., Shiba, N., Takayanagi, T., Watanabe, K.: Distance between Quantum States and Gauge-Gravity Duality. Phys. Rev. Lett. 115(26), 261602 (2015). https://doi.org/10.1103/PhysRevLett.115.261602. [arXiv:1507.07555 [hep-th]]

    Article  Google Scholar 

  13. Banerjee, S., Erdmenger, J., Sarkar, D.: Connecting fisher information to bulk entanglement in holography. JHEP 08, 001 (2018). https://doi.org/10.1007/JHEP08(2018)001. [arXiv:1701.02319 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  14. Susskind, L.: Computational complexity and black hole horizons. Fortsch. Phys. 64, 24–43 (2016). https://doi.org/10.1002/prop.201500092. [arXiv:1403.5695 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  15. Brown, A.R., Roberts, D.A., Susskind, L., Swingle, B., Zhao, Y.: Holographic complexity equals bulk action? Phys. Rev. Lett. 116(19), 191301 (2016). https://doi.org/10.1103/PhysRevLett.116.191301. [arXiv:1509.07876 [hep-th]]

    Article  Google Scholar 

  16. Chapman, S., Marrochio, H., Myers, R.C.: Complexity of formation in holography. JHEP 01, 062 (2017). https://doi.org/10.1007/JHEP01(2017)062. [arXiv:1610.08063 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  17. Fisher, R.A.: Theory of statistical estimation. Math. Proc. Cambridge Philos. Soc. 22(5), 700 (1925). https://doi.org/10.1017/S0305004100009580

    Article  MATH  Google Scholar 

  18. Souriau, J.M.: Thermodynamique Relativiste des Fluides, Volume 35. Rendiconti del Seminario Matematico, Università Politecnico di Torino; Torino, Italy, pp. 21-34 (1978)

    Google Scholar 

  19. Ruppeiner, G.: Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605–659 (1995). https://doi.org/10.1103/RevModPhys.67.605. [erratum: Rev. Mod. Phys. 68 (1996), 313-313]

    Article  MathSciNet  Google Scholar 

  20. Barbaresco, F.: Geometric theory of heat from Souriau Lie groups thermodynamics and Koszul Hessian geometry: applications in information geometry for exponential families. Entropy 18(11), 386 (2016)

    Article  Google Scholar 

  21. Souriau, J.M.: Milieux continus de dimension 1, 2 ou 3: Statique et dynamique. In: Proceedings of the 13eme Congrès Frana̧is de Mécanique; Poitiers, Francy, 1–5 September 1997, pp. 41–53 (1997)

    Google Scholar 

  22. Janke, W., Johnston, D.A., Kenna, R.: The information geometry of the spherical model. Phys. Rev. E 67, (2003). https://doi.org/10.1103/PhysRevE.67.046106. [arXiv:cond-mat/0210571 [cond-mat]]

  23. Dolan, B.P., Johnston, D.A., Kenna, R.: The information geometry of the one-dimensional Potts model. J. Phys. A 35, 9025–9036 (2002). https://doi.org/10.1088/0305-4470/35/43/303. [arXiv:cond-mat/0207180 [cond-mat]]

    Article  MathSciNet  MATH  Google Scholar 

  24. Heckman, J.J.: Statistical inference and string theory. Int. J. Mod. Phys. A 30(26), 1550160 (2015). https://doi.org/10.1142/S0217751X15501602. [arXiv:1305.3621 [hep-th]]

    Article  MATH  Google Scholar 

  25. Amari, S.-I., Kurata, K., Nagaoka, H.: Information geometry of Boltzmann machines. IEEE Trans. Neural Netw. 3(2), 260 (1992)

    Article  Google Scholar 

  26. Amari, S.-I.: Information geometry of neural networks – an overview. In: Ellacott, S.W., Mason, J.C., Anderson, I.J. (eds.) Mathematics of Neural Networks. Operations Research/Computer Science Interfaces Series, vol. 8, pp. 15–23. Springer, Boston (1997). https://doi.org/10.1007/978-1-4615-6099-9_2

    Chapter  Google Scholar 

  27. Amari, S.-I., Nagaoka, H.: Methods of Information Geometry, vol. 191, 2nd edn. Americal Mathematical Society, Oxford (2007)

    Book  Google Scholar 

  28. Blau, M., Narain, K.S., Thompson, G.: Instantons, the information metric, and the AdS/CFT correspondence. [arXiv:hep-th/0108122 [hep-th]]

  29. Shahshahani, S.: A New Mathematical Framework for the Study of Linkage and Selection. American Mathematical Society, Providence (1979)

    Book  Google Scholar 

  30. Ay, N., Jost, J., Vân Lê, H., Schwachhöfer, L.: Information Geometry. Springer, Heidelberg (2017)

    Book  Google Scholar 

  31. Ay, N., Jost, J., Vân Lê, H., Schwachhöfer, L.: Information geometry and sufficient statistics. Probab. Theory Relat. Fields 162(1–2), 327–364 (2015). [arXiv:1207.6736 [math.ST]]

    Article  MathSciNet  Google Scholar 

  32. Dorey, N., Khoze, V.V., Mattis, M.P., Vandoren, S.: Yang-Mills instantons in the large N limit and the AdS/CFT correspondence. Phys. Lett. B 442, 145–151 (1998). https://doi.org/10.1016/S0370-2693(98)01233-7. [arXiv:hep-th/9808157 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  33. Suzuki, Y., Takayanagi, T., Umemoto, K.: Entanglement wedges from the information metric in conformal field theories. Phys. Rev. Lett. 123(22), 221601 (2019). https://doi.org/10.1103/PhysRevLett.123.221601. [arXiv:1908.09939 [hep-th]]

    Article  MathSciNet  Google Scholar 

  34. Kusuki, Y., Suzuki, Y., Takayanagi, T., Umemoto, K.: Looking at shadows of entanglement wedges. [arXiv:1912.08423 [hep-th]]

  35. Matsueda, H.: Geometry and dynamics of emergent spacetime from entanglement spectrum. [arXiv:1408.5589 [hep-th]]

  36. Matsueda, H.: Geodesic distance in fisher information space and holographic entropy formula. [arXiv:1408.6633 [hep-th]]

  37. Clingman, T., Murugan, J., Shock, J.P.: Probability density functions from the fisher information metric. [arXiv:1504.03184 [cs.IT]]

  38. Hitchin, N.J.: The geometry and topology of moduli spaces. In: Francaviglia, M., Gherardelli, F. (eds.) Global Geometry and Mathematical Physics. Lecture Notes in Mathematics, vol. 1451, pp. 1–48. Springer, Heidelberg (1990). https://doi.org/10.1007/BFb0085064

    Chapter  MATH  Google Scholar 

  39. Brody, D.C., Ritz, A.: Geometric phase transitions. [arXiv:cond-mat/9903168]

  40. Janke, W., Johnston, D.A., Kenna, R.: Information geometry and phase transitions. Phys. A 336, 181 (2004). https://doi.org/10.1016/j.physa.2004.01.023. [arXiv:cond-mat/0401092 [cond-mat]]

    Article  MathSciNet  MATH  Google Scholar 

  41. Carollo, A., Valenti, D., Spagnolo, B.: Geometry of quantum phase transitions. Phys. Rept. 838, 1–72 (2020). https://doi.org/10.1016/j.physrep.2019.11.002. [arXiv:1911.10196 [quant-ph]]

    Article  MathSciNet  MATH  Google Scholar 

  42. Mera, B.: Information geometry in the analysis of phase transitions. Spin 1, 2 (2019)

    Google Scholar 

  43. Kar, S.: The geometry of RG flows in theory space. Phys. Rev. D 64 (2001). https://doi.org/10.1103/PhysRevD.64.105017. [arXiv:hep-th/0103025 [hep-th]]

  44. Bény, C., Osborne, T.J.: Information-geometric approach to the renormalization group. Phys. Rev. A 92(2), 022330 (2015). https://doi.org/10.1103/PhysRevA.92.022330. [arXiv:1206.7004 [quant-ph]]

    Article  MathSciNet  Google Scholar 

  45. Balasubramanian, V., Heckman, J.J., Maloney, A.: Relative Entropy and Proximity of Quantum Field Theories. JHEP 05, 104 (2015). https://doi.org/10.1007/JHEP05(2015)104. [[arXiv:1410.6809 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  46. Jefferson, R., Myers, R.C.: Circuit complexity in quantum field theory. JHEP 10, 107 (2017). https://doi.org/10.1007/JHEP10(2017)107. [arXiv:1707.08570 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

  47. Dolan, B.P., Lewis, A.: Renormalization group flow and parallel transport with nonmetric compatible connections. Phys. Lett. B 460, 302–306 (1999). https://doi.org/10.1016/S0370-2693(99)00792-3. [arXiv:hep-th/9904119 [hep-th]]

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is supported by the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147, Project-id No. 39085490) through the Hallwachs-Röntgen fellowship program.

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Physik komplexer Systeme and Würzburg-Dresden Cluster of Excellence ct.qmat, Nöthnitzer Str. 38, 01187, Dresden, Germany

    Kevin T. Grosvenor

Authors
  1. Kevin T. Grosvenor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin T. Grosvenor .

Editor information

Editors and Affiliations

  1. Thales Land & Air Systems, Technical Directorate, Thales, Limours, France

    Frédéric Barbaresco

  2. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Grosvenor, K.T. (2021). Information Geometry and Quantum Fields. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_17

Download citation

Publish with us

Policies and ethics

Search

Navigation