Advertisement

Communications in Mathematical Physics

, Volume 372, Issue 1, pp 281–341 | Cite as

Anomalies in Time-Ordered Products and Applications to the BV–BRST Formulation of Quantum Gauge Theories

  • Markus B. FröbEmail author
Article
  • 92 Downloads

Abstract

We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul–Tate and the full BRST differential in the BV–BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an \(L_\infty \) algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras.

Notes

Acknowledgements

It is a pleasure to thank Chris Fewster, Atsushi Higuchi, Stefan Hollands, Kasia Rejzner, Mojtaba Taslimi Tehrani and Jochen Zahn for discussions on (algebraic) quantum field theory, Igor Khavkine for comments on \(L_\infty \) algebras and a critical reading of the manuscript, Paweł Duch for pointing out a mistake and a simplification in the proof of Theorem 10, and the anonymous referee for a careful reading of the manuscript and for pointing out various mistakes and typos. This work is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 702750 “QLO-QG”.

References

  1. 1.
    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Unruh, W.G.: Notes on black-hole evaporation. Phys. Rev. D 14, 870 (1976)CrossRefADSGoogle Scholar
  3. 3.
    Crispino, L.C.B., Higuchi, A., Matsas, G.E.A.: The Unruh effect and its applications. Rev. Mod. Phys. 80, 787 (2008). arXiv:0710.5373 CrossRefADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Planck collaboration, Ade, P.A.R., et al.: Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016). arXiv:1502.01589
  5. 5.
    Planck collaboration, Ade, P.A.R..: Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. Astron. Astrophys. 594, A17 (2016). arXiv:1502.01592
  6. 6.
    Planck collaboration, Ade. P.A.R., et al.: Planck 2015 results. XX. Constraints on inflation. Astron. Astrophys. 594, A20 (2016). arXiv:1502.02114
  7. 7.
    Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003). arXiv:math-ph/0112041 CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. 8.
    Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001). arXiv:gr-qc/0103074 CrossRefADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002). arXiv:gr-qc/0111108 CrossRefADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008). arXiv:0705.3340 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime (updated version). arXiv:0705.3340v4
  12. 12.
    Fredenhagen, K., Rejzner, K.: Batalin–Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697 (2013). arXiv:1110.5232 CrossRefADSMathSciNetzbMATHGoogle Scholar
  13. 13.
    Rejzner, K.: Remarks on local symmetry invariance in perturbative algebraic quantum field theory. Ann. H. Poincaré 16, 205 (2015). arXiv:1301.7037 CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Dütsch, M., Boas, F.M.: The Master Ward Identity. Rev. Math. Phys. 14, 977 (2002). arXiv:hep-th/0111101 CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Dütsch, M.: Proof of perturbative gauge invariance for tree diagrams to all orders. Ann. Phys. (Leipzig) 14, 438 (2005). arXiv:hep-th/0502071 CrossRefADSMathSciNetzbMATHGoogle Scholar
  16. 16.
    Brennecke, F., Dütsch, M.: Removal of violations of the Master Ward Identity in perturbative QFT. Rev. Math. Phys. 20, 119 (2008). arXiv:0705.3160 CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    de Medeiros, P., Hollands, S.: Superconformal quantum field theory in curved spacetime. Class. Quantum Gravity 30, 175015 (2013). arXiv:1305.0499 CrossRefADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Taslimi Tehrani, M.: Self-consistency of conformally coupled ABJM theory at the quantum level. JHEP 11, 153 (2017). arXiv:1709.08532 ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Brunetti, R., Fredenhagen, K., Rejzner, K.: Quantum gravity from the point of view of locally covariant quantum field theory. Commun. Math. Phys. 345, 741 (2016). arXiv:1306.1058 CrossRefADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Brunetti, R., Fredenhagen, K., Hack, T.-P., Pinamonti, N., Rejzner, K.: Cosmological perturbation theory and quantum gravity. JHEP 08, 032 (2016). arXiv:1605.02573 CrossRefADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Fröb, M.B.: Gauge-invariant quantum gravitational corrections to correlation functions. Class. Quantum Gravity 35, 055006 (2018). arXiv:1710.00839 CrossRefADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Fröb, M.B., Lima, W.C.C.: Propagators for gauge-invariant observables in cosmology. Class. Quantum Gravity 35, 095010 (2018). arXiv:1711.08470 CrossRefADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Fredenhagen, K., Rejzner, K.: Perturbative algebraic quantum field theory. In: Proceedings, Winter School in Mathematical Physics: Mathematical Aspects of Quantum Field Theory: Les Houches, France, January 29—February 3, 2012, p. 17. Springer (2015). arXiv:1208.1428.  https://doi.org/10.1007/978-3-319-09949-1_2
  24. 24.
    Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. Phys. Rep. 574, 1 (2015). arXiv:1401.2026 CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.), Advances in Algebraic Quantum Field Theory, p. 125. Springer International Publishing, Cham, (2015). arXiv:1504.00586.  https://doi.org/10.1007/978-3-319-21353-8_4
  26. 26.
    Fredenhagen, K., Rejzner, K.: Quantum field theory on curved spacetimes: axiomatic framework and examples. J. Math. Phys. 57, 031101 (2016). arXiv:1412.5125 CrossRefADSMathSciNetzbMATHGoogle Scholar
  27. 27.
    Hack, T.-P.: Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-21894-6 CrossRefzbMATHGoogle Scholar
  28. 28.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society Publishing House, Zürich, Switzerland (2007)CrossRefzbMATHGoogle Scholar
  29. 29.
    Fröb, M.B., Taslimi Tehrani, M.: Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges. Phys. Rev. D 97, 025022 (2018). arXiv:1708.00444 CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Hörmander, L.: The analysis of linear partial differential operators I, 2nd edn. Springer, Berlin (2003).  https://doi.org/10.1007/978-3-642-61497-2 CrossRefzbMATHGoogle Scholar
  31. 31.
    Brunetti, R., Dütsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13, 1541 (2009). arXiv:0901.2038 CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Fredenhagen, K., Rejzner, K.: Batalin–Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93 (2012). arXiv:1101.5112 CrossRefADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space–time. Commun. Math. Phys. 179, 529 (1996)CrossRefADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633 (1996). arXiv:gr-qc/9510056 CrossRefADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000). arXiv:math-ph/9903028 CrossRefADSMathSciNetzbMATHGoogle Scholar
  36. 36.
    Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005). arXiv:gr-qc/0404074 CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Zahn, J.: Locally covariant charged fields and background independence. Rev. Math. Phys. 27, 1550017 (2015). arXiv:1311.7661 CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Hollands, S., Wald, R.M.: On the renormalization group in curved spacetime. Commun. Math. Phys. 237, 123 (2003). arXiv:gr-qc/0209029 CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16, 1291 (2004). arXiv:hep-th/0403213 CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Khavkine, I., Melati, A., Moretti, V.: On Wick polynomials of boson fields in locally covariant algebraic QFT. Ann. H. Poincaré 20, 929 (2019). arXiv:1710.01937 CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219, 5 (2001). arXiv:hep-th/0001129 CrossRefADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Achilles, R., Bonfiglioli, A.: The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin. Arch. Hist. Exact Sci. 66, 295 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Dütsch, M., Fredenhagen, K.: The Master Ward Identity and generalized Schwinger–Dyson equation in classical field theory. Commun. Math. Phys. 243, 275 (2003). arXiv:hep-th/0211242 CrossRefADSMathSciNetzbMATHGoogle Scholar
  44. 44.
    Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. H. Poincaré A19, 211 (1973)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Bernal, A.N., Sánchez, M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77, 183 (2006). arXiv:gr-qc/0512095 CrossRefADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Ward, J.C.: An identity in quantum electrodynamics. Phys. Rev. 78, 182 (1950)CrossRefADSMathSciNetzbMATHGoogle Scholar
  47. 47.
    Rohrlich, F.: Quantum electrodynamics of charged particles without spin. Phys. Rev. 80, 666 (1950)CrossRefADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Takahashi, Y.: On the generalized Ward identity. Nuovo Cim. 6, 371 (1957)CrossRefADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    Buchholz, D., D’Antoni, C., Fredenhagen, K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)CrossRefADSMathSciNetzbMATHGoogle Scholar
  50. 50.
    Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195 (1995). arXiv:hep-th/9501063 CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Yngvason, J.: The role of type III factors in quantum field theory. Rept. Math. Phys. 55, 135 (2005). arXiv:math-ph/0411058 CrossRefADSMathSciNetzbMATHGoogle Scholar
  52. 52.
    Sakai, S.: Derivations of \(W^*\)-algebras. Ann. Math. 83, 273 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Kadison, R.V.: Derivations of operator algebras. Ann. Math. 83, 280 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Becchi, C., Rouet, A., Stora, R.: Renormalization of Gauge theories. Ann. Phys. 98, 287 (1976)CrossRefADSMathSciNetGoogle Scholar
  55. 55.
    Batalin, I., Vilkovisky, G.: Gauge algebra and quantization. Phys. Lett. B 102, 27 (1981)CrossRefADSMathSciNetGoogle Scholar
  56. 56.
    Batalin, I., Vilkovisky, G.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567 (1983)CrossRefADSMathSciNetGoogle Scholar
  57. 57.
    Batalin, I., Vilkovisky, G.: Erratum: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 30, 508 (1984)CrossRefADSMathSciNetGoogle Scholar
  58. 58.
    Henneaux, M.: Lectures on the antifield-BRST formalism for gauge theories. Nucl. Phys. B Proc. Suppl. 18A, 47 (1990)CrossRefADSMathSciNetzbMATHGoogle Scholar
  59. 59.
    Gomis, J., Paris, J., Samuel, S.: Antibracket, antifields and gauge-theory quantization. Phys. Rep. 259, 1 (1995). arXiv:hep-th/9412228 CrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Barnich, G., Brandt, F., Henneaux, M.: Local BRST cohomology in gauge theories. Phys. Rep. 338, 439 (2000). arXiv:hep-th/0002245 CrossRefADSMathSciNetzbMATHGoogle Scholar
  61. 61.
    Nakanishi, N.: Covariant quantization of the electromagnetic field in the Landau gauge. Prog. Theor. Phys. 35, 1111 (1966)CrossRefADSGoogle Scholar
  62. 62.
    Lautrup, B.: Canonical quantum electrodynamics in covariant gauges. Mat. Fys. Medd. Dan. Vid. Selsk. 35, 11 (1967)Google Scholar
  63. 63.
    Brandt, F., Henneaux, M., Wilch, A.: Global symmetries in the antifield formalism. Phys. Lett. B 387, 320 (1996). arXiv:hep-th/9606172 CrossRefADSMathSciNetGoogle Scholar
  64. 64.
    Brandt, F., Henneaux, M., Wilch, A.: Extended antifield formalism. Nucl. Phys. B 510, 640 (1998). arXiv:hep-th/9705007 CrossRefADSMathSciNetzbMATHGoogle Scholar
  65. 65.
    Townsend, P.K.: Covariant quantization of antisymmetric tensor gauge fields. Phys. Lett. B 88, 97 (1979)CrossRefADSMathSciNetGoogle Scholar
  66. 66.
    Namazie, M.A., Storey, D.: On secondary and higher-generation ghosts. J. Phys. A 13, L161 (1980)CrossRefADSGoogle Scholar
  67. 67.
    Thierry-Mieg, J.: BRS structure of the antisymmetric tensor gauge theories. Nucl. Phys. B 335, 334 (1990)CrossRefADSMathSciNetGoogle Scholar
  68. 68.
    Siegel, W.: Hidden ghosts. Phys. Lett. B 93, 170 (1980)CrossRefADSMathSciNetGoogle Scholar
  69. 69.
    Kimura, T.: Counting of ghosts in quantized antisymmetric tensor gauge field of third rank. J. Phys. A 13, L353 (1980)CrossRefADSGoogle Scholar
  70. 70.
    Kimura, T.: Quantum theory of antisymmetric higher rank tensor gauge field in higher dimensional space–time. Prog. Theor. Phys. 65, 338 (1981)CrossRefADSGoogle Scholar
  71. 71.
    Piguet, O., Sorella, S.P.: Algebraic Renormalization: Perturbative Renormalization, Symmetries and Anomalies. Springer, Berlin (1995)zbMATHGoogle Scholar
  72. 72.
    Batalin, I.A., Vilkovisky, G.A.: Closure of the gauge algebra, generalized Lie equations and Feynman rules. Nucl. Phys. B 234, 106 (1984)CrossRefADSMathSciNetGoogle Scholar
  73. 73.
    Batalin, I.A., Vilkovisky, G.A.: Existence theorem for gauge algebra. J. Math. Phys. 26, 172 (1985)CrossRefADSMathSciNetGoogle Scholar
  74. 74.
    Tyutin, I.V.: Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism. arXiv:0812.0580
  75. 75.
    Taslimi Tehrani, M.: Quantum BRST charge in gauge theories in curved space-time. J. Math. Phys. 60, 012304 (2019). arXiv:1703.04148 CrossRefADSMathSciNetzbMATHGoogle Scholar
  76. 76.
    Yang, C.-N., Mills, R.L.: Conservation of Isotopic Spin and Isotopic Gauge Invariance. Phys. Rev. 96, 191 (1954)CrossRefADSMathSciNetzbMATHGoogle Scholar
  77. 77.
    Ferrara, S., Zumino, B.: Supergauge invariant Yang-Mills theories. Nucl. Phys. B 79, 413 (1974)CrossRefADSGoogle Scholar
  78. 78.
    Salam, A., Strathdee, J.: Super-symmetry and non-Abelian gauges. Phys. Lett. B 51, 353 (1974)CrossRefADSMathSciNetGoogle Scholar
  79. 79.
    de Wit, B., Freedman, D.Z.: Combined supersymmetric and gauge-invariant field theories. Phys. Rev. D 12, 2286 (1975)CrossRefADSMathSciNetGoogle Scholar
  80. 80.
    Fierz, M.: Zur Fermischen Theorie des \(\beta \)-Zerfalls. Z. Physik 104, 553 (1937)CrossRefADSzbMATHGoogle Scholar
  81. 81.
    Freedman, D.Z., Van Proeyen, A.: Supergravity. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  82. 82.
    Wess, J., Zumino, B.: Consequences of anomalous Ward identities. Phys. Lett. B 37, 95 (1971)CrossRefADSMathSciNetGoogle Scholar
  83. 83.
    Drago, N., Hack, T.-P., Pinamonti, N.: The generalised principle of perturbative agreement and the thermal mass. Ann. H. Poincaré 18, 807 (2017). arXiv:1502.02705 CrossRefMathSciNetzbMATHGoogle Scholar
  84. 84.
    Adler, S.L.: Axial-Vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426 (1969)CrossRefADSGoogle Scholar
  85. 85.
    Bell, J.S., Jackiw, R.: A PCAC puzzle: \(\pi ^0\rightarrow \gamma \gamma \) in the \(\sigma \)-model. Nuovo Cim. A 60, 47 (1969)CrossRefADSGoogle Scholar
  86. 86.
    Fujikawa, K.: Path-Integral measure for gauge-invariant fermion theories. Phys. Rev. Lett. 42, 1195 (1979)CrossRefADSGoogle Scholar
  87. 87.
    Geng, C.Q., Marshak, R.E.: Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint. Phys. Rev. D 39, 693 (1989)CrossRefADSGoogle Scholar
  88. 88.
    Minahan, J.A., Ramond, P., Warner, R.C.: Comment on anomaly cancellation in the standard model. Phys. Rev. D 41, 715 (1990)CrossRefADSGoogle Scholar
  89. 89.
    Dütsch, M., Fredenhagen, K.: A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203, 71 (1999). arXiv:hep-th/9807078 CrossRefADSMathSciNetzbMATHGoogle Scholar
  90. 90.
    Lada, T., Stasheff, J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32, 1087 (1993). arXiv:hep-th/9209099 CrossRefMathSciNetzbMATHGoogle Scholar
  91. 91.
    Hohm, O., Zwiebach, B.: \(L_{\infty }\) algebras and field theory. Fortsch. Phys. 65, 1700014 (2017). arXiv:1701.08824 CrossRefADSMathSciNetzbMATHGoogle Scholar
  92. 92.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge systems. Princeton University Press, Princeton (1992)zbMATHGoogle Scholar
  93. 93.
    Zahn, J.: Private communication (2018)Google Scholar
  94. 94.
    Piguet, O., Sibold, K.: The anomaly in the Slavnov identity for \(N=1\) supersymmetric Yang-Mills theories. Nucl. Phys. B 247, 484 (1984)CrossRefADSGoogle Scholar
  95. 95.
    Brandt, F.: Extended BRST cohomology, consistent deformations and anomalies of four-dimensional supersymmetric gauge theories. JHEP 04, 035 (2003). arXiv:hep-th/0212070 CrossRefADSMathSciNetGoogle Scholar
  96. 96.
    Junker, W., Schrohe, E.: Adiabatic vacuum states on general spacetime manifolds: definition, construction, and physical properties. Ann. H. Poincaré 3, 1113 (2002). arXiv:math-ph/0109010 CrossRefMathSciNetzbMATHGoogle Scholar
  97. 97.
    Fewster, C.J., Pfenning, M.J.: A quantum weak energy inequality for spin-one fields in curved space-time. J. Math. Phys. 44, 4480 (2003). arXiv:gr-qc/0303106 CrossRefADSMathSciNetzbMATHGoogle Scholar
  98. 98.
    Duch, P.: Weak adiabatic limit in quantum field theories with massless particles. Ann. H. Poincaré 19, 875 (2018). arXiv:1801.10147 CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

Personalised recommendations