Relative entropy and proximity of quantum field theories

  • Vijay Balasubramanian
  • Jonathan J. Heckman
  • Alexander Maloney
Open Access
Regular Article - Theoretical Physics


We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.


Conformal and W Symmetry Statistical Methods Renormalization Group 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    S. Kullback and R.A. Leibler, On Information and Sufficiency, Ann. Math. Statist. 22 (1951) 79.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V. Balasubramanian, Statistical inference, occams razor and statistical mechanics on the space of probability distributions, Neur. Comp. 9 (1997) 349 [cond-mat/9601030] [INSPIRE].CrossRefzbMATHGoogle Scholar
  3. [3]
    S.-I. Amari and H. Nagaoka, Translations of Mathematical Monographs. Vol. 191: Methods of Information Geometry, American Mathematical Society, Providence U.S.A. (2000).Google Scholar
  4. [4]
    T.M. Cover and J.A. Thomas, Elements of Information Theory, Wiley, New York U.S.A. (1991).CrossRefzbMATHGoogle Scholar
  5. [5]
    N.N. Chentsov, Translations of Mathematical Monographs. Vol. 53: Statistical Decision Rules and Optimal Inference, American Mathematical Society, Providence U.S.A. (1982).Google Scholar
  6. [6]
    J. Calmet and X. Calmet, Distance between physical theories based on information theory, Mod. Phys. Lett. A 26 (2011) 319 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].ADSGoogle Scholar
  8. [8]
    H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    D.C. Brody and A. Ritz, On the symmetry of real space renormalization, Nucl. Phys. B 522 (1998) 588 [hep-th/9709175] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    S.M. Apenko, Information theory and renormalization group flows, Physica A 391 (2012) 62 [arXiv:0910.2097] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    J.C. Gaite and D. O’Connor, Field theory entropy and the renormalization group, Phys. Rev. D 54 (1996) 5163 [hep-th/9511090] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    C. Beny and T.J. Osborne, Information geometric approach to the renormalisation group, arXiv:1206.7004 [INSPIRE].
  13. [13]
    C. Bény and T.J. Osborne, Renormalisation as an inference problem, arXiv:1310.3188 [INSPIRE].
  14. [14]
    M. Blau, K.S. Narain and G. Thompson, Instantons, the information metric and the AdS/CFT correspondence, hep-th/0108122 [INSPIRE].
  15. [15]
    S.-J. Rey and Y. Hikida, Black hole as emergent holographic geometry of weakly interacting hot Yang-Mills gas, JHEP 08 (2006) 051 [hep-th/0507082] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    A.B. Zamolodchikov, “Irreversibilityof the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730.ADSMathSciNetGoogle Scholar
  17. [17]
    D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    E. Barnes, E. Gorbatov, K.A. Intriligator, M. Sudano and J. Wright, The Exact superconformal R-symmetry minimizes tau(RR), Nucl. Phys. B 730 (2005) 210 [hep-th/0507137] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    M.R. Douglas, Spaces of Quantum Field Theories, J. Phys. Conf. Ser. 462 (2013) 012011 [arXiv:1005.2779] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    A.N. Schellekens and S. Yankielowicz, Extended Chiral Algebras and Modular Invariant Partition Functions, Nucl. Phys. B 327 (1989) 673 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    J.O. Andersen, E. Braaten and M. Strickland, The Massive thermal basketball diagram, Phys. Rev. D 62 (2000) 045004 [hep-ph/0002048] [INSPIRE].ADSGoogle Scholar
  25. [25]
    J.J. Heckman, Statistical Inference and String Theory, arXiv:1305.3621 [INSPIRE].
  26. [26]
    A. Hebecker, AdS/CFT for accelerator physics, Phys. Rev. D 88 (2013) 125025 [arXiv:1305.6311] [INSPIRE].ADSGoogle Scholar
  27. [27]
    J. Heckman and H. Verlinde, Covariant non-commutative space-time, Nucl. Phys. B 894 (2015) 58 [arXiv:1401.1810] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Vijay Balasubramanian
    • 1
    • 2
    • 3
  • Jonathan J. Heckman
    • 4
  • Alexander Maloney
    • 5
  1. 1.David Rittenhouse LaboratoriesUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.CUNY Graduate Center, Initiative for the Theoretical SciencesNew YorkU.S.A.
  3. 3.Theoretische Natuurkunde, Vrije Universiteit Brussel, and International Solvay InstitutesBrusselsBelgium
  4. 4.Department of PhysicsUniversity of North Carolina at Chapel HillChapel HillU.S.A.
  5. 5.Department of PhysicsMcGill UniversityMontrealCanada

Personalised recommendations