Accounting for the Heisenberg and Pauli principles in the kinetic approach to neutrino oscillations


While oscillations of solar neutrinos are usually studied using the single-particle quantum-mechanical approach, flavor conversions of supernovae neutrinos are typically analyzed using the kinetic equation for the matrix of densities due to the necessity of including also the scattering processes. Using the Wigner formulation of quantum mechanics we show the equivalence of the quantum-mechanical and kinetic approaches in the limit of collision-less neutrino propagation (in a background medium). Based on this observation we also argue that solutions of the kinetic equation account for the Heisenberg uncertainty principle and the related effect of wave packet separation (for single neutrinos), as well as the Pauli exclusion principle, if the initial conditions are consistent with these fundamental quantum principles. Such initial conditions can be constructed e.g. by identifying the matrix of densities with the (reduced) single-particle Wigner function computed using initial conditions for the neutrino wave function. Hence the neutrino momentum uncertainty is an integral part of the initial conditions for the matrix of densities, that may have an impact on the phenomenology of supernovae neutrinos via the effect of wave packet separation.

A preprint version of the article is available at ArXiv.


  1. [1]

    J. Davis, Raymond, D.S. Harmer and K.C. Hoffman, Search for neutrinos from the Sun, Phys. Rev. Lett. 20 (1968) 1205 [INSPIRE].

  2. [2]

    Super-Kamiokande collaboration, Determination of solar neutrino oscillation parameters using 1496 days of Super-Kamiokande I data, Phys. Lett. B 539 (2002) 179 [hep-ex/0205075] [INSPIRE].

  3. [3]

    SNO collaboration, Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. 89 (2002) 011301 [nucl-ex/0204008] [INSPIRE].

  4. [4]

    J.N. Bahcall, M.H. Pinsonneault and S. Basu, Solar models: current epoch and time dependences, neutrinos, and helioseismological properties, Astrophys. J. 555 (2001) 990 [astro-ph/0010346] [INSPIRE].

  5. [5]

    Kamiokande-II collaboration, Observation of a neutrino burst from the Supernova SN 1987a, Phys. Rev. Lett. 58 (1987) 1490 [INSPIRE].

  6. [6]

    R.M. Bionta et al., Observation of a neutrino burst in coincidence with supernova SN 1987a in the Large Magellanic Cloud, Phys. Rev. Lett. 58 (1987) 1494 [INSPIRE].

    ADS  Google Scholar 

  7. [7]

    E.N. Alekseev, L.N. Alekseeva, V.I. Volchenko and I.V. Krivosheina, Possible Detection of a Neutrino Signal on 23 February 1987 at the Baksan Underground Scintillation Telescope of the Institute of Nuclear Research, JETP Lett. 45 (1987) 589 [Pisma Zh. Eksp. Teor. Fiz. 45 (1987) 461] [INSPIRE].

  8. [8]

    P. Antonioli et al., SNEWS: the Supernova Early Warning System, New J. Phys. 6 (2004) 114 [astro-ph/0406214] [INSPIRE].

  9. [9]

    L. Wolfenstein, Neutrino oscillations in matter, Phys. Rev. D 17 (1978) 2369 [INSPIRE].

    ADS  Google Scholar 

  10. [10]

    S.P. Mikheyev and A. Smirnov, Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos, Sov. J. Nucl. Phys. 42 (1985) 913 [Yad. Fiz. 42 (1985) 1441] [INSPIRE].

  11. [11]

    G. Sigl and G. Raffelt, General kinetic description of relativistic mixed neutrinos, Nucl. Phys. B 406 (1993) 423 [INSPIRE].

    ADS  Google Scholar 

  12. [12]

    H.A. Bethe, Energy production in stars, Phys. Rev. 55 (1939) 434.

    ADS  MATH  Google Scholar 

  13. [13]

    C. Giunti and C.W. Kim, Fundamentals of neutrino physics and astrophysics, Oxford University Press, Oxford U.K. (2007).

    Google Scholar 

  14. [14]

    BOREXINO collaboration, First Direct Experimental Evidence of CNO neutrinos, arXiv:2006.15115 [INSPIRE].

  15. [15]

    C. Pena-Garay and A. Serenelli, Solar neutrinos and the solar composition problem, arXiv:0811.2424 [INSPIRE].

  16. [16]

    P.C. de Holanda and A. Smirnov, Solar neutrinos: the SNO salt phase results and physics of conversion, Astropart. Phys. 21 (2004) 287 [hep-ph/0309299] [INSPIRE].

  17. [17]

    B. Pontecorvo, Inverse beta processes and nonconservation of lepton charge, Sov. Phys. JETP 7 (1958) 172 [Zh. Eksp. Teor. Fiz. 34 (1957) 247] [INSPIRE].

  18. [18]

    Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].

    ADS  MATH  Google Scholar 

  19. [19]

    E.K. Akhmedov and J. Kopp, Neutrino oscillations: quantum mechanics vs. quantum field theory, JHEP 04 (2010) 008 [Erratum ibid. 10 (2013) 052] [arXiv:1001.4815] [INSPIRE].

  20. [20]

    R.S.L. Hansen and A.Y. Smirnov, The Liouville equation for flavour evolution of neutrinos and neutrino wave packets, JCAP 12 (2016) 019 [arXiv:1610.00910] [INSPIRE].

    MathSciNet  Google Scholar 

  21. [21]

    E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749 [INSPIRE].

    ADS  MATH  Google Scholar 

  22. [22]

    A. Kartavtsev, Relating quantum mechanics and kinetics of neutrino oscillations, JHEP 01 (2020) 138 [arXiv:1404.5626] [INSPIRE].

    ADS  Google Scholar 

  23. [23]

    Super-Kamiokande collaboration, Solar neutrino measurements in Super-Kamiokande-IV, Phys. Rev. D 94 (2016) 052010 [arXiv:1606.07538] [INSPIRE].

  24. [24]

    Borexino collaboration, The Borexino detector at the Laboratori Nazionali del Gran Sasso, Nucl. Instrum. Meth. A 600 (2009) 568 [arXiv:0806.2400] [INSPIRE].

  25. [25]

    Particle Data Group collaboration, Review of particle physics, Phys. Rev. D 98 (2018) 030001.

  26. [26]

    E.K. Akhmedov, M.A. Tortola and J.W.F. Valle, A simple analytic three flavor description of the day night effect in the solar neutrino flux, JHEP 05 (2004) 057 [hep-ph/0404083] [INSPIRE].

  27. [27]

    V. Barger, K. Whisnant, S. Pakvasa and R.J.N. Phillips, Matter effects on three-neutrino oscillations, Phys. Rev. D 22 (1980) 2718.

    ADS  Google Scholar 

  28. [28]

    A.S. Dighe and A.Y. Smirnov, Identifying the neutrino mass spectrum from the neutrino burst from a supernova, Phys. Rev. D 62 (2000) 033007 [hep-ph/9907423] [INSPIRE].

  29. [29]

    A. Smirnov, The MSW effect and solar neutrinos, hep-ph/0305106 [INSPIRE].

  30. [30]

    P.C. de Holanda and A. Smirnov, Homestake result, sterile neutrinos and low-energy solar neutrino experiments, Phys. Rev. D 69 (2004) 113002 [hep-ph/0307266] [INSPIRE].

  31. [31]

    P.C. de Holanda, W. Liao and A. Smirnov, Toward precision measurements in solar neutrinos, Nucl. Phys. B 702 (2004) 307 [hep-ph/0404042] [INSPIRE].

  32. [32]

    A.N. Ioannisian and A. Smirnov, Neutrino oscillations in low density medium, Phys. Rev. Lett. 93 (2004) 241801 [hep-ph/0404060] [INSPIRE].

  33. [33]

    S. Goswami and A.Y. Smirnov, Solar neutrinos and 1-3 leptonic mixing, Phys. Rev. D 72 (2005) 053011 [hep-ph/0411359] [INSPIRE].

  34. [34]

    M. Blennow and A.Y. Smirnov, Neutrino propagation in matter, Adv. High Energy Phys. 2013 (2013) 972485 [arXiv:1306.2903] [INSPIRE].

    Google Scholar 

  35. [35]

    M. Maltoni and A.Y. Smirnov, Solar neutrinos and neutrino physics, Eur. Phys. J. A 52 (2016) 87 [arXiv:1507.05287] [INSPIRE].

    ADS  Google Scholar 

  36. [36]

    A.Y. Smirnov, Solar neutrinos: oscillations or no-oscillations?, arXiv:1609.02386 [INSPIRE].

  37. [37]

    E.K. Akhmedov and A.Y. Smirnov, Paradoxes of neutrino oscillations, Phys. Atom. Nucl. 72 (2009) 1363 [arXiv:0905.1903] [INSPIRE].

    ADS  Google Scholar 

  38. [38]

    A.M. Dziewonski and D.L. Anderson, Preliminary reference Earth model, Phys. Earth Plan. Int. 25 (1981) 297.

    ADS  Google Scholar 

  39. [39]

    A. Ioannisian, A. Smirnov and D. Wyler, Scanning the Earth with solar neutrinos and DUNE, Phys. Rev. D 96 (2017) 036005 [arXiv:1702.06097] [INSPIRE].

  40. [40]

    E. Lisi and D. Montanino, Earth regeneration effect in solar neutrino oscillations: An Analytic approach, Phys. Rev. D 56 (1997) 1792 [hep-ph/9702343] [INSPIRE].

  41. [41]

    P. Bakhti and A.Y. Smirnov, Oscillation tomography of the Earth with solar neutrinos and future experiments, Phys. Rev. D 101 (2020) 123031 [arXiv:2001.08030] [INSPIRE].

    ADS  Google Scholar 

  42. [42]

    V. Antonelli, L. Miramonti, C. Pena Garay and A. Serenelli, Solar neutrinos, Adv. High Energy Phys. 2013 (2013) 351926 [arXiv:1208.1356] [INSPIRE].

    Google Scholar 

  43. [43]

    T. Stirner, G. Sigl and G. Raffelt, Liouville term for neutrinos: flavor structure and wave interpretation, JCAP 05 (2018) 016 [arXiv:1803.04693] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    K. Husimi, Some formal properties of the density matrix, Proc. Phys. Met. Soc. Jpn. 22 (1940) 264.

    MATH  Google Scholar 

  45. [45]

    M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner, Distribution functions in physics: Fundamentals, Phys. Rept. 106 (1984) 121 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  46. [46]

    E. Cancellieri, P. Bordone, and C. Jacoboni, Effect of the Pauli exclusion principle in the many-electron Wigner function, cond-mat/0703185.

  47. [47]

    E. Cancellieri, P. Bordone, and C. Jacoboni, Effect of symmetry in the many-particle wigner function, Phys. Rev. B 76 (2007) 214301.

    ADS  Google Scholar 

  48. [48]

    A. Kartavtsev, G. Raffelt and H. Vogel, Neutrino propagation in media: flavor-, helicity-, and pair correlations, Phys. Rev. D 91 (2015) 125020 [arXiv:1504.03230] [INSPIRE].

    ADS  Google Scholar 

  49. [49]

    J.T. Pantaleone, Neutrino oscillations at high densities, Phys. Lett. B 287 (1992) 128 [INSPIRE].

    ADS  Google Scholar 

  50. [50]

    H. Duan, G.M. Fuller and Y.-Z. Qian, Collective neutrino oscillations, Ann. Rev. Nucl. Part. Sci. 60 (2010) 569 [arXiv:1001.2799] [INSPIRE].

    ADS  Google Scholar 

  51. [51]

    A. Mirizzi et al., Supernova neutrinos: production, oscillations and detection, Riv. Nuovo Cim. 39 (2016) 1 [arXiv:1508.00785] [INSPIRE].

    ADS  Google Scholar 

  52. [52]

    S. Chakraborty, R. Hansen, I. Izaguirre and G. Raffelt, Collective neutrino flavor conversion: Recent developments, Nucl. Phys. B 908 (2016) 366 [arXiv:1602.02766] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    S. Horiuchi and J.P. Kneller, What can be learned from a future supernova neutrino detection?, J. Phys. G 45 (2018) 043002 [arXiv:1709.01515] [INSPIRE].

  54. [54]

    I. Tamborra, L. Huedepohl, G. Raffelt and H.-T. Janka, Flavor-dependent neutrino angular distribution in core-collapse supernovae, Astrophys. J. 839 (2017) 132 [arXiv:1702.00060] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    S. Abbar, H. Duan, K. Sumiyoshi, T. Takiwaki and M.C. Volpe, On the occurrence of fast neutrino flavor conversions in multidimensional supernova models, Phys. Rev. D 100 (2019) 043004 [arXiv:1812.06883] [INSPIRE].

  56. [56]

    M. Delfan Azari et al., Linear analysis of fast-pairwise collective neutrino oscillations in core-collapse supernovae based on the results of Boltzmann simulations, Phys. Rev. D 99 (2019) 103011 [arXiv:1902.07467] [INSPIRE].

    ADS  Google Scholar 

  57. [57]

    T. Morinaga, H. Nagakura, C. Kato and S. Yamada, Fast neutrino-flavor conversion in the preshock region of core-collapse supernovae, Phys. Rev. Res. 2 (2020) 012046 [arXiv:1909.13131] [INSPIRE].

  58. [58]

    M. Delfan Azari et al., Fast collective neutrino oscillations inside the neutrino sphere in core-collapse supernovae, Phys. Rev. D 101 (2020) 023018 [arXiv:1910.06176] [INSPIRE].

  59. [59]

    S. Abbar, H. Duan, K. Sumiyoshi, T. Takiwaki and M.C. Volpe, Fast neutrino flavor conversion modes in multidimensional core-collapse supernova models: the role of the asymmetric neutrino distributions, Phys. Rev. D 101 (2020) 043016 [arXiv:1911.01983] [INSPIRE].

  60. [60]

    H. Nagakura, T. Morinaga, C. Kato and S. Yamada, Fast-pairwise collective neutrino oscillations associated with asymmetric neutrino emissions in core-collapse supernova, arXiv:1910.04288 [INSPIRE].

  61. [61]

    L. Johns, H. Nagakura, G.M. Fuller and A. Burrows, Neutrino oscillations in supernovae: angular moments and fast instabilities, Phys. Rev. D 101 (2020) 043009 [arXiv:1910.05682] [INSPIRE].

  62. [62]

    R. Glas et al., Fast neutrino flavor instability in the neutron-star convection layer of three-dimensional supernova models, Phys. Rev. D 101 (2020) 063001 [arXiv:1912.00274] [INSPIRE].

  63. [63]

    F. Capozzi, G. Raffelt and T. Stirner, Fast neutrino flavor conversion: collective motion vs. decoherence, JCAP 09 (2019) 002 [arXiv:1906.08794] [INSPIRE].

  64. [64]

    M. Chakraborty and S. Chakraborty, Three flavor neutrino conversions in supernovae: slow & fast instabilities, JCAP 01 (2020) 005 [arXiv:1909.10420] [INSPIRE].

    ADS  Google Scholar 

  65. [65]

    S. Shalgar, I. Padilla-Gay and I. Tamborra, Neutrino propagation hinders fast pairwise flavor conversions, JCAP 06 (2020) 048 [arXiv:1911.09110] [INSPIRE].

    ADS  Google Scholar 

  66. [66]

    F. Capozzi, M. Chakraborty, S. Chakraborty and M. Sen, Fast flavor conversions in supernovae: the rise of mu-tau neutrinos, arXiv:2005.14204 [INSPIRE].

  67. [67]

    S. Shalgar and I. Tamborra, Dispelling a myth on dense neutrino media: fast pairwise conversions depend on energy, arXiv:2007.07926 [INSPIRE].

  68. [68]

    I. Padilla-Gay, S. Shalgar and I. Tamborra, Multi-dimensional solution of fast neutrino conversions in binary neutron star merger remnants, arXiv:2009.01843 [INSPIRE].

  69. [69]

    S. Shalgar and I. Tamborra, Dispelling a myth on dense neutrino media: fast pairwise conversions depend on energy, arXiv:2007.07926 [INSPIRE].

  70. [70]

    R.F. Sawyer, Speed-up of neutrino transformations in a supernova environment, Phys. Rev. D 72 (2005) 045003 [hep-ph/0503013] [INSPIRE].

  71. [71]

    R.F. Sawyer, The multi-angle instability in dense neutrino systems, Phys. Rev. D 79 (2009) 105003 [arXiv:0803.4319] [INSPIRE].

    ADS  Google Scholar 

  72. [72]

    F. Capozzi, B. Dasgupta, A. Mirizzi, M. Sen and G. Sigl, Collisional triggering of fast flavor conversions of supernova neutrinos, Phys. Rev. Lett. 122 (2019) 091101 [arXiv:1808.06618] [INSPIRE].

  73. [73]

    S. Yamada, Boltzmann equations for neutrinos with flavor mixings, Phys. Rev. D 62 (2000) 093026 [astro-ph/0002502] [INSPIRE].

  74. [74]

    A. Vlasenko, G.M. Fuller and V. Cirigliano, Neutrino quantum kinetics, Phys. Rev. D 89 (2014) 105004 [arXiv:1309.2628] [INSPIRE].

    ADS  MATH  Google Scholar 

  75. [75]

    V. Cirigliano, G.M. Fuller and A. Vlasenko, A new spin on neutrino quantum kinetics, Phys. Lett. B 747 (2015) 27 [arXiv:1406.5558] [INSPIRE].

    ADS  MATH  Google Scholar 

  76. [76]

    A. Vlasenko, G.M. Fuller and V. Cirigliano, Prospects for neutrino-antineutrino transformation in astrophysical environments, arXiv:1406.6724 [INSPIRE].

  77. [77]

    D.N. Blaschke and V. Cirigliano, Neutrino quantum kinetic equations: the collision term, Phys. Rev. D 94 (2016) 033009 [arXiv:1605.09383] [INSPIRE].

  78. [78]

    S.A. Richers, G.C. McLaughlin, J.P. Kneller and A. Vlasenko, Neutrino quantum kinetics in compact objects, Phys. Rev. D 99 (2019) 123014 [arXiv:1903.00022] [INSPIRE].

    ADS  Google Scholar 

  79. [79]

    J. Kersten and A.Y. Smirnov, Decoherence and oscillations of supernova neutrinos, Eur. Phys. J. C 76 (2016) 339 [arXiv:1512.09068] [INSPIRE].

    ADS  Google Scholar 

  80. [80]

    E. Akhmedov, J. Kopp and M. Lindner, Collective neutrino oscillations and neutrino wave packets, JCAP 09 (2017) 017 [arXiv:1702.08338] [INSPIRE].

    ADS  Google Scholar 

  81. [81]

    E. Akhmedov and A. Mirizzi, Another look at synchronized neutrino oscillations, Nucl. Phys. B 908 (2016) 382 [arXiv:1601.07842] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  82. [82]

    G.G. Raffelt and I. Tamborra, Synchronization versus decoherence of neutrino oscillations at intermediate densities, Phys. Rev. D 82 (2010) 125004 [arXiv:1006.0002] [INSPIRE].

    ADS  Google Scholar 

  83. [83]

    G. Raffelt, G. Sigl, and L. Stodolsky, Quantum statistics in particle mixing phenomena, Phys. Rev. D 45 (1992) 1782

    ADS  Google Scholar 

  84. [84]

    C. Volpe, D. Väänänen and C. Espinoza, Extended evolution equations for neutrino propagation in astrophysical and cosmological environments, Phys. Rev. D 87 (2013) 113010 [arXiv:1302.2374] [INSPIRE].

    ADS  Google Scholar 

  85. [85]

    D. Väänänen and C. Volpe, Linearizing neutrino evolution equations including neutrino-antineutrino pairing correlations, Phys. Rev. D 88 (2013) 065003 [arXiv:1306.6372] [INSPIRE].

  86. [86]

    J. Serreau and C. Volpe, Neutrino-antineutrino correlations in dense anisotropic media, Phys. Rev. D 90 (2014) 125040 [arXiv:1409.3591] [INSPIRE].

    ADS  Google Scholar 

  87. [87]

    C. Volpe, Neutrino quantum kinetic equations, Int. J. Mod. Phys. E 24 (2015) 1541009 [arXiv:1506.06222] [INSPIRE].

    ADS  Google Scholar 

  88. [88]

    A. Ioannisian and S. Pokorski, Three neutrino oscillations in matter, Phys. Lett. B 782 (2018) 641 [arXiv:1801.10488] [INSPIRE].

    ADS  Google Scholar 

  89. [89]

    S. Mikheyev and A. Smirnov, Resonant neutrino oscillations in matter, Prog. Part. Nucl. Phys. 23 (1989) 41.

    ADS  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to A. Kartavtsev.

Additional information

Publisher’s Note

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

ArXiv ePrint: 2007.13736

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kartavtsev, A. Accounting for the Heisenberg and Pauli principles in the kinetic approach to neutrino oscillations. J. High Energ. Phys. 2020, 135 (2020).

Download citation


  • Neutrino Physics
  • Solar and Atmospheric Neutrinos