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Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

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Rigorous Quantum Field Theory

Part of the book series: Progress in Mathematics ((PM,volume 251))

Summary

Quantum Energy Inequalities (QEIs) are constraints on the extent to which quantum fields can violate the energy conditions of classical general relativity. As such they are closely related to the gravitational stability of quantised matter. In this contribution we discuss links between QEIs and other stability conditions in quantum field theory: the microlocal spectrum condition, passivity and nuclearity. The fist two links suggest an interconnection between stability conditions at three different length scales, while the third hints at a deeper origin of QEIs.

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References

  1. R. Brunetti and K. Fredenhagen: Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Commun. Math. Phys. 208:623–661 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. R. Brunetti, K. Fredenhagen and M. Köhler: The microlocal spectrum condition and Wick polynomials in curved spacetime. Commun. Math. Phys. 180:633–652 (1996).

    Article  ADS  MATH  Google Scholar 

  3. D. Buchholz and P. Junglas: Local properties of equilibrium states and the particle spectrum in quantum field theory. Lett. Math. Phys. 11:51–58 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. D. Buchholz and M. Porrmann: How small is the phase space in quantum field theory? Ann. Inst. H. Poincaré 52:237–257 (1990).

    MathSciNet  MATH  Google Scholar 

  5. D. Buchholz and E.H. Wichmann: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106:321–344 (1986).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. H. Epstein, V. Glaser and A. Jaffe: Nonpositivity of the energy density in quantized field theories. Nuovo Cimento 36:1016–1022 (1965).

    Article  MathSciNet  Google Scholar 

  7. S.P. Eveson, C.J. Fewster and R. Verch: Quantum Inequalities in Quantum Mechanics. To appear in Ann. H. Poincaré. Preprint arXiv:math-ph/0312046, 2003.

    Google Scholar 

  8. C.J. Fewster: A general worldline quantum inequality. Class. Quantum Grav. 17:1897–1911 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. C.J. Fewster: Energy Inequalities in Quantum Field Theory (2003). Available at http://www-users.york.ac.uk/~cjf3/QEIs.ps

    Google Scholar 

  10. C.J. Fewster and S.P. Eveson: Bounds on negative energy densities in flat spacetime. Phys. Rev. D 58:084010 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  11. C.J. Fewster and S. Hollands: Quantum Energy Inequalities in two-dimensional conformal field theory. Preprint arXiv:math-ph/0412028., 2004.

    Google Scholar 

  12. C.J. Fewster and B. Mistry: Quantum Weak Energy Inequalities for the Dirac field in Flat Spacetime. Phys. Rev. D 68:105010 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  13. C.J. Fewster, I. Ojima and M. Porrmann: p-Nuclearity in a New Perspective. Preprint arXiv:math-ph/0412027, 2004.

    Google Scholar 

  14. C.J. Fewster and M.J. Pfenning: A Quantum Weak Energy Inequality for spin-one fields in curved spacetime. J. Math. Phys. 44:4480–4513 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. C.J. Fewster and T.A. Roman: Null energy conditions in quantum field theory. Phys. Rev. D. 67:044003 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  16. C.J. Fewster and E. Teo: Bounds on negative energy densities in static space-times. Phys. Rev. D 59:104016 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  17. C.J. Fewster and R. Verch: A quantum weak energy inequality for Dirac fields in curved spacetime. Commun. Math. Phys. 225:331–359 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. C.J. Fewster and R. Verch: Stability of quantum systems at three scales: passivity, quantum weak energy inequalities and the microlcal spectrum condition. Commun. Math. Phys. 240:329–375 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. É.É. Flanagan: Quantum inequalities in two-dimensional Minkowski spacetime. Phys. Rev. D 56:4922–4926 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  20. É.É. Flanagan: Quantum inequalities in two dimensional curved spacetimes. Phys. Rev. D 66:104007 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  21. L.H. Ford: Quantum coherence effects and the second law of thermodynamics. Proc. Roy. Soc. Lond. A 364:227–236 (1978).

    Article  ADS  Google Scholar 

  22. L.H. Ford: Constraints on negative-energy fluxes. Phys. Rev. D 43:3972–3978 (1991).

    Article  ADS  Google Scholar 

  23. L.H. Ford, A. Helfer and T.A. Roman: Spatially averaged quantum inequalities do not exist in four-dimensional spacetime. Phys. Rev. D 66:124012 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  24. L.H. Ford and T.A. Roman: Averaged energy conditions and quantum inequalities. Phys. Rev. D 51:4277–4286 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  25. L.H. Ford and T.A. Roman: Quantum field theory constrains traversable wormhole geometries. Phys. Rev. D 53:5496–5507 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  26. L.H. Ford and T.A. Roman: Restrictions on negative energy density in flat spacetime. Phys. Rev. D 55:2082–2089 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  27. R. Haag: Local quantum physics: Fields, particles, algebras. Springer Verlag, Berlin, 1992.

    Book  MATH  Google Scholar 

  28. S.W. Hawking: Chronology protection conjecture. Phys. Rev. D 46:603–611 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  29. S.W. Hawking and G.F.R. Ellis: The large scale structure of space-time. Cambridge University Press, Cambridge, 1973.

    Book  MATH  Google Scholar 

  30. S. Hollands: The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes. Commun. Math. Phys. 216:635–661 (2001).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. L. Hörmander: The analysis of linear partial differential operators I. Springer Verlag, Berlin, 1983.

    MATH  Google Scholar 

  32. W. Junker and E. Schrohe: Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties. Ann. H. Poincaré 3:1113–1181 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  33. B.S. Kay and R.M. Wald: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207:49–136 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. K. Kratzert: Singularity structure of the free Dirac field on a globally hyperbolic spacetime. Annalen Phys. 9:475–498 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. M. Ludvigsen and J.A.G. Vickers: A simple proof of the positivity of the Bondi mass. J. Phys. A Math. Gen. 15:L67–L70 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  36. P. Marecki: Application of quantum inequalities to quantum optics. Phys. Rev. A 66:053801 (2002).

    Article  ADS  Google Scholar 

  37. M.J. Pfenning: Quantum inequalities for the electromagnetic field. Phys. Rev. D 65:024009 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  38. M.J. Pfenning and L.H. Ford: The unphysical nature of ‘warp drive’. Class. Quantum Grav. 14:1743–1751 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. M.J. Pfenning and L.H. Ford: Scalar field quantum inequalities in static spacetimes. Phys. Rev. D 57:3489–3502 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  40. W. Pusz and S.L. Woronowicz: Passive states and KMS states for general quantum systems. Commun. Math. Phys. 58:273–290 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  41. M.J. Radzikowski: Micro-local approach to the Hadamard condition in quantum field theory in curved spacetime. Commun. Math. Phys. 179:529–553 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. T.A. Roman: Some thoughts on energy conditions and wormholes. Preprint arXiv:gr-qc/0409090, 2004. To appear in: Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity and Gravitation.

    Google Scholar 

  43. H. Sahlmann and R. Verch: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214:705–731 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. H. Sahlmann and R. Verch: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13:1203–1246 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  45. R. Schumann: Operator ideals and the statistical independence in quantum field theory. Lett. Math. Phys. 37:249–271 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. A. Strohmaier, R. Verch and M. Wollenberg: Microlocal analysis of quantum fields on curved spacetimes: Analytic Wavefront sets and Reeh-Schlieder theorems. J. Math. Phys. 43:5514–5530 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. D.N. Vollick: Quantum inequalities in curved two-dimensional spacetimes. Phys. Rev. D 61:084022 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  48. E. Witten: A new proof of the positive energy theorem. Commun. Math. Phys. 80:381–402 (1981).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. H. Yu and P. Wu: Quantum inequalities for the free Rarita-Schwinger fields in flat spacetime. Phys. Rev. D 69:064008 (2004).

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

    Christopher J. Fewster

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  1. Christopher J. Fewster
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Editor information

Editors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Anne Boutet de Monvel

  2. Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, 37077, Göttingen, Germany

    Detlef Buchholz

  3. Service de Physique Théorique, CEA/Saclay, Orme des Merisiers, 91191, Gif-sur-Yvette cedex, France

    Daniel Iagolnitzer

  4. Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio, 11, 22100, Como, Italy

    Ugo Moschella

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© 2007 Birkhäuser Verlag

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Fewster, C.J. (2007). Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory. In: de Monvel, A.B., Buchholz, D., Iagolnitzer, D., Moschella, U. (eds) Rigorous Quantum Field Theory. Progress in Mathematics, vol 251. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7434-1_8

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