Communications in Mathematical Physics

, Volume 172, Issue 1, pp 187–220

The quantum structure of spacetime at the Planck scale and quantum fields

  • Sergio Doplicher
  • Klaus Fredenhagen
  • John E. Roberts
Article

Abstract

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.

We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper.

In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundleTS2 of the 2-sphere. The relations with Conne's theory of the standard model will be studied elsewhere.

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References

  1. 1.
    Amati, D., Ciafaloni, M., Veneziano, G.: Nucl. Phys.B347, 551 (1990); Amati, D.: On Spacetime at Small Distances. In: Sakharov Memorial Lectures, Kladysh, L.V., Fainberg, V.YA., eds., Nova Science P. Inc., 1992CrossRefGoogle Scholar
  2. 2.
    Ellis, J., Mavromatos, N.E., Nanopoulos, D.V.: A Liouville String Approach to Microscopic Time in Cosmology. Preprint CERN-TH. 7000/93Google Scholar
  3. 3.
    Ashtekar, A.: Quantum Gravity: A Mathematical Physics Perspective. Preprint CGPG-93/12-2Google Scholar
  4. 4.
    Kempf, A.: Uncertainty Relations in Quantum Mechanics with Quantum Group Symmetry. Preprint DAMT/93-65Google Scholar
  5. 5.
    Woronowicz, S.L.: Compact Matrix Pseudogroups. Commun. Math. Phys.111, 613–665 (1987)CrossRefGoogle Scholar
  6. 6.
    Wheeler, J.A.: Geometrodynamics and the Issue of the Final State. In: Relativity, Groups and Topology. De Witt, C., De Witt, B., (eds.) Gordon and Breach 1965. Hawking, S.W., Spacetime Foam. Nucl. Phys.B144, 349, (1978)Google Scholar
  7. 7.
    Madore, J.: Fuzzy Physics. Ann. Phys.219, 187–198 (1992)CrossRefGoogle Scholar
  8. 8.
    Connes, A.: Non-Commutative Geometry. Academic Press, 1994Google Scholar
  9. 9.
    Connes, A., Lott, J.: Particle Models and Non Commutative Geometry. Nucl. Phys. B. (Proc. Suppl.)11B, 19–47 (1990); Kastler, D.: A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I, II. Rev. Math. Phys.5, 477–532 (1993); III, Preprint CPT-92/P. 2814; IV (with Th. Schücker), Preprint CPT-94/P.3092Google Scholar
  10. 10.
    Schrader, R.: The Maxwell Group and the Quantum Theory of Particles in Classical Homogeneous Electric Fields. Forts. der Phys.20, 701–734 (1972)Google Scholar
  11. 11.
    Filk, T.: Field Theory on the Quantum Plane. Preprint, University of Freiburg, August 1990, THEP 90/12Google Scholar
  12. 12.
    Grosse, H., Madore, J.: A Noncommutative Version of the Schwinger Model. Phys. Lett.B283, 218 (1992)CrossRefGoogle Scholar
  13. 13.
    Mack, G., Schomerus, V.: Models of Quantum Space Time: Quantum Field Planes. Preprint, Harvard University HUTMP 93-B335. Lukierski, J., Ruegg, H.: Quantum κ-Poincaré in any Dimension. Phys. Lett. B, to appearGoogle Scholar
  14. 14.
    Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime Quantization Induced by Classical Gravity. Phys. Lett. B331, 33–44 (1994)Google Scholar
  15. 15.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of spacetime. Cambridge: Cambridge U.P., 1973Google Scholar
  16. 16.
    Knight, J.M.: Strict localization in Quantum Field Theory. J. Math. Phys.2, 439–471 (1961); Licht, A.L.: Strict localization. J. Math. Phys.4, 1443 (1963)CrossRefGoogle Scholar
  17. 17.
    Haag, R.: Local Quantum Physics. Berlin, Heidelberg, New York: Springer TMP, 1993Google Scholar
  18. 18.
    Straumann, N.: General Relativity and Relativistic Astrophysics. Berlin-Heidelberg-New York: Springer, 1984Google Scholar
  19. 19.
    Woronowicz, S.L.: Unbounded elements affiliated withC *-algebras and non-compact Quantum Groups. Commun. Math. Phys.136, 399–432 (1991). Woronowicz, S.L.:C *-Algebras Generated by Unbounded Elements. Preprint 1994; Baaj, S.: Multiplicateur non borné. Thèse, Université Paris VII 1980CrossRefGoogle Scholar
  20. 20.
    Pedersen, G.K.:C *-Algebras and their Automorphism Groups. New York: Acad. Press, 1979Google Scholar
  21. 21.
    Dixmier, J.:C *-Algebras. Amsterdam: North Holland, 1980Google Scholar
  22. 22.
    Von Neumann, J.: Über die Eindeutigkeit der Schrödingerschen Operatoren. Math. Annalen104, 570–578 (1931)CrossRefGoogle Scholar
  23. 23.
    Filk, T.: Renormalizability of Field Theories on Quantum Spaces, Preprint University of Freiburg, THEP 94/15, 1994Google Scholar
  24. 24.
    Didonato P. et al. eds.: Sympletic Geometry and Mathematical Physics. Actes du Colloque en Honneur de J.M. Souriau, Basel, Boston: Birkhäuser, 1991Google Scholar
  25. 25.
    Rieffel, M.A.: Deformation Quantization of Heisenberg Manifolds. Commun. Math. Phys.122, 531–562 (1989)CrossRefGoogle Scholar
  26. 26.
    Landsman, N.P.: Strict Deformation Quantization of a Particle in External Gravitational and Yang Mills Fields. J. Geom. Phys.12, 93–132 (1993)CrossRefGoogle Scholar
  27. 27.
    Doplicher, S., Fredenhagen, K.: in preparationGoogle Scholar
  28. 28.
    Maggiore, M.: Quantum Groups, Gravity, and the generalized uncertainty principle, Phys. Rev.D49, 5182–5187 (1994); Phys. Lett. B304, 65–69 and319, 83–86 (1993)CrossRefGoogle Scholar
  29. 29.
    Geray, L.J.: Quantum Gravity and minimum Length. Preprint Imperial/TP/93-94/20, gr-qc/9403008Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Sergio Doplicher
    • 1
  • Klaus Fredenhagen
    • 2
  • John E. Roberts
    • 3
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.II Institut für Theoretische Physik der Universität HamburgHamburgGermany
  3. 3.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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