Vacuum state of the Dirac field in de Sitter space and entanglement entropy

  • Sugumi KannoEmail author
  • Misao Sasaki
  • Takahiro Tanaka
Open Access
Regular Article - Theoretical Physics


We compute the entanglement entropy of a free massive Dirac field between two causally disconnected open charts in de Sitter space. We first derive the Bunch-Davies vacuum mode functions of the Dirac field. We find there exists no supercurvature mode for the Dirac field. We then give the Bogoliubov transformation between the Bunch-Davies vacuum and the open chart vacua that makes the reduced density matrix diagonal. We find that the Dirac field becomes more entangled than a scalar field as m 2 /H 2 becomes small, and the difference is maximal in the massless limit.


Classical Theories of Gravity Effective Field Theories 


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  1. [1]
    A. Einstein, B. Podolsky and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935) 777 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Aspect, P. Grangier and G. Roger, Experimental tests of realistic local theories via Bell’s theorem, Phys. Rev. Lett. 47 (1981) 460 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    A. Aspect, J. Dalibard and G. Roger, Experimental test of Bell’s inequalities using time varying analyzers, Phys. Rev. Lett. 49 (1982) 1804 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Sasaki, T. Tanaka and K. Yamamoto, Euclidean vacuum mode functions for a scalar field on open de Sitter space, Phys. Rev. D 51 (1995) 2979 [gr-qc/9412025] [INSPIRE].
  5. [5]
    J. Maldacena and G.L. Pimentel, Entanglement entropy in de Sitter space, JHEP 02 (2013) 038 [arXiv:1210.7244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Kanno, J.P. Shock and J. Soda, Entanglement negativity in the multiverse, JCAP 03 (2015) 015 [arXiv:1412.2838] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Kanno, J.P. Shock and J. Soda, Quantum discord in de Sitter space, Phys. Rev. D 94 (2016) 125014 [arXiv:1608.02853] [INSPIRE].ADSGoogle Scholar
  8. [8]
    S. Kanno, Impact of quantum entanglement on spectrum of cosmological fluctuations, JCAP 07 (2014) 029 [arXiv:1405.7793] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Kanno, Cosmological implications of quantum entanglement in the multiverse, Phys. Lett. B 751 (2015) 316 [arXiv:1506.07808] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Kanno, A note on initial state entanglement in inflationary cosmology, Europhys. Lett. 111 (2015) 60007 [arXiv:1507.04877] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    I. Fuentes-Schuller and R.B. Mann, Alice falls into a black hole: Entanglement in non-inertial frames, Phys. Rev. Lett. 95 (2005) 120404 [quant-ph/0410172] [INSPIRE].
  12. [12]
    P.M. Alsing, I. Fuentes-Schuller, R.B. Mann and T.E. Tessier, Entanglement of Dirac fields in non-inertial frames, Phys. Rev. A 74 (2006) 032326 [quant-ph/0603269] [INSPIRE].
  13. [13]
    A. Datta, Quantum discord between relatively accelerated observers, Phys. Rev. A 80 (2009) 052304 [arXiv:0905.3301].ADSCrossRefGoogle Scholar
  14. [14]
    R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
  15. [15]
    A.R. Liddle and M. Cortês, Cosmic microwave background anomalies in an open universe, Phys. Rev. Lett. 111 (2013) 111302 [arXiv:1306.5698] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    S. Kanno, M. Sasaki and T. Tanaka, A viable explanation of the CMB dipolar statistical anisotropy, PTEP 2013 (2013) 111E01 [arXiv:1309.1350] [INSPIRE].
  17. [17]
    T. Kobayashi, M. Cortês and A.R. Liddle, A separate universe view of the asymmetric sky, JCAP 05 (2015) 029 [arXiv:1501.05864] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    C. Byrnes, G. Domènech, M. Sasaki and T. Takahashi, Strongly scale-dependent CMB dipolar asymmetry from super-curvature fluctuations, JCAP 12 (2016) 020 [arXiv:1610.02650] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D. Yamauchi, T. Fujita and S. Mukohyama, Is there supercurvature mode of massive vector field in open inflation?, JCAP 03 (2014) 031 [arXiv:1402.2784] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    A.A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section, Phys. Rept. 266 (1996) 1 [hep-th/9505061] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Theoretical Physics and History of ScienceUniversity of the Basque CountryBilbaoSpain
  2. 2.IKERBASQUE, Basque Foundation for ScienceBilbaoSpain
  3. 3.Center for Gravitational Physics, Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
  4. 4.Department of PhysicsKyoto UniversityKyotoJapan

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