Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 105, 13 pages      arXiv:0711.1671      http://dx.doi.org/10.3842/SIGMA.2007.105
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Wavelet-Based Quantum Field Theory

Mikhail V. Altaisky a, b
a) Joint Institute for Nuclear Research, Dubna, 141980, Russia
b) Space Research Institute RAS, 84/32 Profsoyuznaya Str., Moscow, 117997, Russia

Received August 15, 2007, in final form November 03, 2007; Published online November 11, 2007

Abstract
The Euclidean quantum field theory for the fields φΔx(x), which depend on both the position x and the resolution Δx, constructed in SIGMA 2 (2006), 046, on the base of the continuous wavelet transform, is considered. The Feynman diagrams in such a theory become finite under the assumption there should be no scales in internal lines smaller than the minimal of scales of external lines. This regularisation agrees with the existing calculations of radiative corrections to the electron magnetic moment. The transition from the newly constructed theory to a standard Euclidean field theory is achieved by integration over the scale arguments.

Key words: wavelets; quantum field theory; regularisation.

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References

  1. Zinn-Justin J., Quantum field theory and critical phenomena, Oxford University Press, 1989.
  2. Kadanoff L.P., Scaling laws for Ising models near Tc, Physics 2 (1966), 263-272.
  3. Vasiliev A.N., The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Boca Raton, Chapman and Hall/CRC, 2004.
  4. Altaisky M.V., Scale-dependent functions, stochastic quantization and renormalization, SIGMA 2 (2006), 046, 17 pages, hep-th/0604170.
  5. Isham C.J., Klauder J.R., Coherent states for n-dimensional Euclidean groups E(n) and their applications, J. Math. Phys. 32 (1991), 607-620.
  6. Battle G., Wavelets and renormalization, World Scientific, 1989.
  7. Federbush P.G., A mass zero cluster expansion, Comm. Math. Phys. 81 (1981), 327-340.
  8. Altaisky M.V., Wavelet based regularization for Euclidean field theory, in Proc. of the 24th Int. Coll. "Group Theoretical Methods in Physics", Editors J-P. Gazeau et al., IOP Conference Series, Vol. 173, Bristol, 2003, 893-897, hep-th/0305167.
  9. Carey A.L., Square-integrable representations of non-unimodular groups, Bull. Austr. Math. Soc. 15 (1976), 1-12.
  10. Duflo M., Moore C.C., On regular representations of nonunimodular locally compact group, J. Func. Anal. 21 (1976), 209-243.
  11. Chui C.K., An introduction to wavelets, Academic Press Inc., 1992.
  12. Handy C.M., Murenzi R., Continuous wavelet transform analysis of one-dimensional quantum bound states from first principles, Phys. Rev. A 54 (1996), 3754-3763.
  13. Federbush P., A new formulation and regularization of gauge theories using a non-linear wavelet expansion, Progr. Theor. Phys. 94 (1995), 1135-1146, hep-ph/9505368.
  14. Halliday I.G., Suranyi P., Simulation of field theories in wavelet representation, Nucl. Phys. B 436 (1995), 414-427, hep-lat/9407010.
  15. Best C., Wavelet-induced renormalization group for Landau-Ginzburg model, Nucl. Phys. B (Proc. Suppl.) 83-84 (2000), 848-850, hep-lat/9909151.
  16. Altaisky M.V., Wavelets: theory, applications, implementation, Universities Press Ltd., India, 2005.
  17. Christensen J.C., Crane L., Causal sites as quantum geometry, J. Math. Phys 46 (2005), 122502, 17 pages, gr-qc/0410104.
  18. Altaisky M.V., Causality and multiscale expansions in quantum field theory, Phys. Part. Nuclei Lett. 2 (2005), 337-339.
  19. Alebastrov V.A., Efimov G.V., Causality in quantum field theory with nonlocal interaction, Comm. Math. Phys. 38 (1974), 11-28.
  20. Efimov G.V., Problems in quantum theory of nonlocal interactions, Nauka, Moscow, 1985 (in Russian).
  21. Wilson K.G., Kogut J., Renormalization group and e-expansion, Phys. Rep. C 12 (1974), 77-200.
  22. Wilson K.G., The renormalization group and critical phenomena, Rev. Modern Phys. 55 (1983), 583-600.
  23. Berestetskii V.B., Lifshitz E.M., Pitaevskii L.P., Course of theoretical physics: quantum electrodynamics, 2nd ed., Pergamon, London, 1982.

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