Advertisement

Annales Henri Poincaré

, Volume 16, Issue 10, pp 2303–2365 | Cite as

Locally Covariant Quantum Field Theory with External Sources

  • Christopher J. Fewster
  • Alexander SchenkelEmail author
Article

Abstract

We provide a detailed analysis of the classical and quantized theory of a multiplet of inhomogeneous Klein–Gordon fields, which couple to the spacetime metric and also to an external source term; thus the solutions form an affine space. Following the formulation of affine field theories in terms of presymplectic vector spaces as proposed in Benini et al. (Ann. Henri Poincaré 15:171–211, 2014), we determine the relative Cauchy evolution induced by metric as well as source term perturbations and compute the automorphism group of natural isomorphisms of the presymplectic vector space functor. Two pathological features of this formulation are revealed: the automorphism group contains elements that cannot be interpreted as global gauge transformations of the theory; moreover, the presymplectic formulation does not respect a natural requirement on composition of subsystems. We therefore propose a systematic strategy to improve the original description of affine field theories at the classical and quantized level, first passing to a Poisson algebra description in the classical case. The idea is to consider state spaces on the classical and quantum algebras suggested by the physics of the theory (in the classical case, we use the affine solution space). The state spaces are not separating for the algebras, indicating a redundancy in the description. Removing this redundancy by a quotient, a functorial theory is obtained that is free of the above-mentioned pathologies. These techniques are applicable to general affine field theories and Abelian gauge theories. The resulting quantized theory is shown to be dynamically local.

Keywords

Automorphism Group Covariant Functor Poisson Algebra Canonical Commutation Relation Canonical Connection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Baez J.C., Segal I.E., Zhou Z.: Introduction to algebraic and constructive quantum field theory. In: Princeton Series in Physics. Princeton University Press, Princeton (1992)CrossRefGoogle Scholar
  2. 2.
    Brunetti, R., Duetsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13, 1541 (2009). arXiv:0901.2038 [math-ph]
  3. 3.
    Benini, M., Dappiaggi, C., Hack, T.-P., Schenkel, A.: A C *-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds. Commun. Math. Phys. 332, 477 (2014). arXiv:1307.3052 [math-ph]
  4. 4.
    Benini, M., Dappiaggi, C., Schenkel, A.: Quantum field theory on affine bundles. Ann. Henri Poincaré 15, 171–211 (2014). arXiv:1210.3457 [math-ph]
  5. 5.
    Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123 (2014). arXiv:1303.2515 [math-ph]
  6. 6.
    Borchers H.-J.: On structure of the algebra of field operators. Nuovo Cimento 24, 214–236 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Brunetti, R., Fredenhagen, K., Rejzner, K.: Quantum gravity from the point of view of locally covariant quantum field theory. arXiv:1306.1058 [math-ph]
  8. 8.
    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). math-ph/0112041
  9. 9.
    Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ferguson, M.: Dynamical locality of the nonminimally coupled scalar field and enlarged algebra of Wick polynomials. Ann. Henri Poincaré 14, 853 (2013). math-ph/1203.2151
  11. 11.
    Fewster, C.J.: Quantum energy inequalities and local covariance. II. Categorical formulation. Gen. Relativ. Gravit. 39, 1855 (2007). math-ph/0611058
  12. 12.
    Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25, 1350008 (2013). arXiv:1201.3295 [math-ph]
  13. 13.
    Fewster, C.J., Pfenning, M.J.: Quantum energy inequalities and local covariance. I. Globally hyperbolic spacetimes. J. Math. Phys. 47, 082303 (2006). math-ph/0602042
  14. 14.
    Fredenhagen, K., Rejzner, K.: Batalin–Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697 (2013). arXiv:1110.5232 [math-ph]
  15. 15.
    Fewster, C.J., Verch, R.: Dynamical locality and covariance: What makes a physical theory the same in all spacetimes? Ann. Henri Poincaré 13, 1613 (2012). arXiv:1106.4785 [math-ph]
  16. 16.
    Fewster, C.J., Verch, R.: Dynamical locality of the free scalar field. Ann. Henri Poincaré 13, 1675 (2012). arXiv:1109.6732 [math-ph]
  17. 17.
    Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 223, 289 (2001). gr-qc/0103074
  18. 18.
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 231, 309 (2002). gr-qc/0111108
  19. 19.
    Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005). gr-qc/0404074
  20. 20.
    Itzykson C., Zuber J.B.: Quantum Field Theory. McGraw-Hill, NewYork (1980)Google Scholar
  21. 21.
    Meisters G.H.: Translation-invariant linear forms and a formula for the Dirac measure. J. Funct. Anal. 8, 173–188 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Peierls R.E.: The commutation laws of relativistic field theory. Proc. R. Soc. A 214, 143–157 (1952)CrossRefADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Pinamonti, N., Siemssen, D.: Scale-invariant curvature fluctuations from an extended semiclassical gravity. arXiv:1303.3241 [gr-qc]
  24. 24.
    Pinamonti, N., Siemssen, D.: Global existence of solutions of the semiclassical einstein equation. Commun. Math. Phys. arXiv:1309.6303 [math-ph]
  25. 25.
    Sanders, K., Dappiaggi, C., Hack, T.-P.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625 (2014). arXiv:1211.6420 [math-ph]
  26. 26.
    Uhlmann A.:Über die Definition der Quantenfelder nach Wightman und Haag. Wiss. Zeit. Karl Marx Univ. 11, 213–217 (1962)zbMATHGoogle Scholar
  27. 27.
    Verch, R.: A spin statistics theorem for quantum fields on curved space-time manifolds in a generally covariant framework. Commun. Math. Phys. 223, 261 (2001). math-ph/0102035
  28. 28.
    Verch, R.: Local covariance, renormalization ambiguity, and local thermal equilibrium in cosmology. In: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds.) Quantum Field Theory and Gravity. Conceptual and Mathematical Advances in the Search for a Unified Framework, pp. 229–256. Birkhäuser, Boston (2012). arXiv:1105.6249 [gr-qc]

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Christopher J. Fewster
    • 1
  • Alexander Schenkel
    • 2
    • 3
    • 4
    • 5
    Email author
  1. 1.Department of MathematicsUniversity of YorkHeslington, YorkUK
  2. 2.Fachgruppe MathematikBergische Universität WuppertalWuppertalGermany
  3. 3.Department of MathematicsHeriot-Watt UniversityEdinburghUK
  4. 4.Maxwell Institute for Mathematical SciencesEdinburghUK
  5. 5.The Tait InstituteEdinburghUK

Personalised recommendations