Skip to main content
SpringerLink
Log in

Classical Theory

  • Chapter
  • First Online:
Perturbative Algebraic Quantum Field Theory

Part of the book series: Mathematical Physics Studies ((MPST))

  • 2057 Accesses

Abstract

Having defined the essential kinematical structure we are now ready to introduce the dynamics. To this end we use a generalization of the Lagrange formalism. The precise relation to notions known from classical mechanics will be explained in Sect. 4.5.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The name “dual” is meant as the indication that this notion behaves better under dualities of the type mentioned in Proposition 4.6 than the usual WF set.

  2. 2.

    Hypocontinuity of a bilinear map is a notion stronger than sequential continuity on the product space, but is weaker than the joint continuity.

References

  1. Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, Zurich (2007)

    Book  MATH  Google Scholar 

  2. Brennecke, F., Dütsch, M.: Removal of violations of the master ward identity in perturbative QFT. Rev. Math. Phys. 20(2), 119–151 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brouder, C., Dang, N., Laurent-Gengoux, C., Rejzner, K.: Functionals and their derivatives in quantum field theory, work in progress (2016)

    Google Scholar 

  4. Brouder, C., Dang, N.V., Hélein, F.: Boundedness and continuity of the fundamental operations on distributions having a specified wave front set. (with a counter example by Semyon Alesker) (2014). arXiv:1409.7662

  5. Brunetti, R., Dütsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brunetti, R., Fredenhagen, K., Rejzner, K.: Quantum gravity from the point of view of locally covariant quantum field theory (2013). arXiv:math-ph/1306.1058

  7. Brunetti, R., Fredenhagen, K., Ribeiro, P.L.: Algebraic structure of classical field theory I: kinematics and linearized dynamics for real scalar fields. arXiv:math-ph/1209.2148v2

  8. Chazarain, J., Piriou, A.: Introduction à la théorie des équations aux dérivées partielles linéaires, vol. 3. Gauthier-Villars (1981)

    Google Scholar 

  9. Dabrowski, Y.: Functional properties of generalized Hörmander spaces of distributions I: Duality theory, completions and bornologifications (2014). arXiv:1411.3012

  10. Dabrowski, Y.: Functional properties of generalized Hörmander spaces of distributions II: multilinear maps and applications to spaces of functionals with wave front set conditions (2014). arXiv:math-ph/1412.1749

  11. Dabrowski, Y., Brouder, C.: Functional properties of Hörmander’s space of distributions having a specified wavefront set. Commun. Math. Phys. 332(3), 1345–1380 (2014)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. Dütsch, M., Fredenhagen, K.: The master Ward identity and generalized Schwinger-Dyson equation in classical field theory 2, 275–314 (2002). arXiv.org hep-th

  13. Duistermaat, J.J.: Fourier Integral Operators, vol. 130. Springer Science and Business Media (1996)

    Google Scholar 

  14. Fewster, C.J., Verch, R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes? 1–57

    Google Scholar 

  15. Fewster, C.J., Verch, R.: Dynamical locality of the free scalar field (2011). arXiv:1109.6732

  16. Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314(1), 93–127 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Fredenhagen, K., Rejzner, K.: QFT on curved spacetimes: axiomatic framework and examples (2014). arXiv:math-ph/1412.5125

  18. Fredenhagen, K., Rejzner, K.: Perturbative construction of models of algebraic quantum field theory (2015). arXiv:math-ph/1503.07814

    Google Scholar 

  19. Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1), 79–183 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hörmander, L.: The analysis of the linear partial differential operators I: distribution theory and fourier analysis. Classics in Mathematics. Springer, Berlin (2003)

    Google Scholar 

  21. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. Courier Corporation (2006)

    Google Scholar 

  22. Jakobs, S.: Eichbrücken in der klassischen Feldtheorie. Thesis (2009)

    Google Scholar 

  23. Kriegl, A., Michor, P.W.: The convenient setting of global analysis. Mathematical Surveys and Monographs, vol. 53. AMS (1997)

    Google Scholar 

  24. Meyer, R.: Bornological versus topological analysis in metrizable spaces. Contemp. Math. 363, 249–278 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peierls, R.E.: The commutation laws of relativistic field theory. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 214(1117), 143–157 (1952)

    Article  MathSciNet  MATH  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Schwartz, L.: Théorie des distributions à valeurs vectorielles. I. Annales de l’Institut Fourier 7, 1–141 (1957)

    Article  MATH  Google Scholar 

  28. Trèves, F.: Topological Vector Spaces, Distributions and Kernels, vol. 25. Courier Corporation (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of York, York, UK

    Kasia Rejzner

Authors
  1. Kasia Rejzner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kasia Rejzner .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 The Author(s)

About this chapter

Cite this chapter

Rejzner, K. (2016). Classical Theory. In: Perturbative Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-25901-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation