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Gaussian as Test Functions in Operator Valued Distribution Formulation of QED

Received: 9 November 2018     Accepted: 3 December 2018     Published: 25 January 2019
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Abstract

As shown by Epstein and Glaser, the operator valued distribution (OPVD) formalism permits to obtain a non-standard regularization scheme which leads to a divergences-free quantum field theory. The present formulation gives a tool to calculate finite scattering matrix without adding counter-terms. After a short recall about the OPVD formalism in 3+1-dimensions, Gaussian functions and Harmonic Hermite-Gaussian functions are used as test functions. The field resulting from this approach obeys the Klein-Gordon equation. The vacuum fluctuation calculation then gives a result regularized by the Gaussian factor. The example of scalar quantum electrodynamics theory shows that Gaussian functions may be used as test functions in this approach. The result shows that the scale of the theory is described by a factor arising from Heisenberg uncertainties. The Feynman propagators and a study on loop convergence with the example of the tadpole diagram are given. The formulation is extended to Quantum Electrodynamics. Triangle diagrams anomaly are calculated efficiently. In the same approach, using Lagrange formulae to regularize the singular distribution involved in the scattering amplitude gives a good way to avoid infinities. After applying Lagrange formulae, the test function could be reduced to unity since the amplitude is regular. Ward-Takahashi identity is calculated with this method too and this shows that the symmetries of the theory are unbroken.

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 4)
DOI 10.11648/j.ijamtp.20180404.12
Page(s) 98-104
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Quantum Electrodynamics, Triangle Anomaly, Ward-Takahashi Identity, Partition of Unity, Operator Valued Distribution, Gaussian Functions, Tadpole Diagram

References
[1] Matthew Schwartz “Introduction to Quantum electrodynamics”, Harvard University, Fall 2008.
[2] Lewis H. Ryder “Quantum electrodynamics”, Cambridge University, 1985.
[3] Mikhaïl Shaposhnikov, Champs Quantiques Relativistes, Sven Bachmann, 2005.
[4] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of scalar theories November 22, 2016.
[5] Jean-François Mathiot, Toward finite field theory: the Taylor–Lagrange regularization scheme, 2012.
[6] Bruno Mutet, Pierre Grangé and Ernst Werner, Taylor Lagrange Renormalization Scheme: electron and gauge-boson self-energies, (PoS) Proceeding of Science (2010).
[7] Pierre Grange, Ernst Werner. Quantum Fields as Operator Valued Distributions and Causality. 20 pages, 2 figures - Contrat-IN2P3-CNRS., 2007.
[8] Laurent Schwartz, Théorie des distributions, Hermann, Paris 1966.
[9] Günther Hormann& Roland Steinbauer, Theory of distributions, Fakultat fur Mathematik, Universitat Wien Summer Term 2009.
[10] Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo. Time-Frequency analysis and harmonic Gaussian functions. Pure and Applied Mathematics Journal. Vol. 2, No. 2, 2013, pp. 71-78. doi: 10.11648/j.pamj.20130202.14, 2013.
[11] Ravo Tokiniaina Ranaivoson: Raoelina Andriambololona, Rakotoson Hanitriarivo, Roland Raboanary: Study on a Phase Space Representation of Quantum Theory, arXiv: 1304.1034 [quant-ph], International Journal of Latest Research in Science and Technology, ISSN (Online): 2278-5299, Volume 2, Issue 2: pp. 26-35, March-April 2013.
[12] Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Hanitriarivo Rakotoson, Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator, arXiv: 1711.07308, International Journal of Applied Mathematics and Theoretical Physics. Vol. 4, No. 1, 2018, pp. 8-14, 2018.
[13] Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Randriamisy Hasimbola Damo Emile, Hanitriarivo Rakotoson, Dispersion Operators Algebra and Linear Canonical Transformations, International Journal of Theoretical Physics, Volume 56, No. 4, pp 1258-1273, Springer, 2017.
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    Hasimbola Damo Emile Randriamisy, Raoelina Andriambololona, Hanitriarivo Rakotoson, Ravo Tokiniaina Ranaivoson, Roland Raboanary. (2019). Gaussian as Test Functions in Operator Valued Distribution Formulation of QED. International Journal of Applied Mathematics and Theoretical Physics, 4(4), 98-104. https://doi.org/10.11648/j.ijamtp.20180404.12

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    Hasimbola Damo Emile Randriamisy; Raoelina Andriambololona; Hanitriarivo Rakotoson; Ravo Tokiniaina Ranaivoson; Roland Raboanary. Gaussian as Test Functions in Operator Valued Distribution Formulation of QED. Int. J. Appl. Math. Theor. Phys. 2019, 4(4), 98-104. doi: 10.11648/j.ijamtp.20180404.12

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    AMA Style

    Hasimbola Damo Emile Randriamisy, Raoelina Andriambololona, Hanitriarivo Rakotoson, Ravo Tokiniaina Ranaivoson, Roland Raboanary. Gaussian as Test Functions in Operator Valued Distribution Formulation of QED. Int J Appl Math Theor Phys. 2019;4(4):98-104. doi: 10.11648/j.ijamtp.20180404.12

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  • @article{10.11648/j.ijamtp.20180404.12,
      author = {Hasimbola Damo Emile Randriamisy and Raoelina Andriambololona and Hanitriarivo Rakotoson and Ravo Tokiniaina Ranaivoson and Roland Raboanary},
      title = {Gaussian as Test Functions in Operator Valued Distribution Formulation of QED},
      journal = {International Journal of Applied Mathematics and Theoretical Physics},
      volume = {4},
      number = {4},
      pages = {98-104},
      doi = {10.11648/j.ijamtp.20180404.12},
      url = {https://doi.org/10.11648/j.ijamtp.20180404.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180404.12},
      abstract = {As shown by Epstein and Glaser, the operator valued distribution (OPVD) formalism permits to obtain a non-standard regularization scheme which leads to a divergences-free quantum field theory. The present formulation gives a tool to calculate finite scattering matrix without adding counter-terms. After a short recall about the OPVD formalism in 3+1-dimensions, Gaussian functions and Harmonic Hermite-Gaussian functions are used as test functions. The field resulting from this approach obeys the Klein-Gordon equation. The vacuum fluctuation calculation then gives a result regularized by the Gaussian factor. The example of scalar quantum electrodynamics theory shows that Gaussian functions may be used as test functions in this approach. The result shows that the scale of the theory is described by a factor arising from Heisenberg uncertainties. The Feynman propagators and a study on loop convergence with the example of the tadpole diagram are given. The formulation is extended to Quantum Electrodynamics. Triangle diagrams anomaly are calculated efficiently. In the same approach, using Lagrange formulae to regularize the singular distribution involved in the scattering amplitude gives a good way to avoid infinities. After applying Lagrange formulae, the test function could be reduced to unity since the amplitude is regular. Ward-Takahashi identity is calculated with this method too and this shows that the symmetries of the theory are unbroken.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Gaussian as Test Functions in Operator Valued Distribution Formulation of QED
    AU  - Hasimbola Damo Emile Randriamisy
    AU  - Raoelina Andriambololona
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    DO  - 10.11648/j.ijamtp.20180404.12
    T2  - International Journal of Applied Mathematics and Theoretical Physics
    JF  - International Journal of Applied Mathematics and Theoretical Physics
    JO  - International Journal of Applied Mathematics and Theoretical Physics
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    PB  - Science Publishing Group
    SN  - 2575-5927
    UR  - https://doi.org/10.11648/j.ijamtp.20180404.12
    AB  - As shown by Epstein and Glaser, the operator valued distribution (OPVD) formalism permits to obtain a non-standard regularization scheme which leads to a divergences-free quantum field theory. The present formulation gives a tool to calculate finite scattering matrix without adding counter-terms. After a short recall about the OPVD formalism in 3+1-dimensions, Gaussian functions and Harmonic Hermite-Gaussian functions are used as test functions. The field resulting from this approach obeys the Klein-Gordon equation. The vacuum fluctuation calculation then gives a result regularized by the Gaussian factor. The example of scalar quantum electrodynamics theory shows that Gaussian functions may be used as test functions in this approach. The result shows that the scale of the theory is described by a factor arising from Heisenberg uncertainties. The Feynman propagators and a study on loop convergence with the example of the tadpole diagram are given. The formulation is extended to Quantum Electrodynamics. Triangle diagrams anomaly are calculated efficiently. In the same approach, using Lagrange formulae to regularize the singular distribution involved in the scattering amplitude gives a good way to avoid infinities. After applying Lagrange formulae, the test function could be reduced to unity since the amplitude is regular. Ward-Takahashi identity is calculated with this method too and this shows that the symmetries of the theory are unbroken.
    VL  - 4
    IS  - 4
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Author Information
  • Theoretical Physics Department, National Institute for Nuclear Sciences and Technology, Antananarivo, Madagascar

  • Theoretical Physics Department, National Institute for Nuclear Sciences and Technology, Antananarivo, Madagascar

  • Theoretical Physics Department, National Institute for Nuclear Sciences and Technology, Antananarivo, Madagascar

  • Theoretical Physics Department, National Institute for Nuclear Sciences and Technology, Antananarivo, Madagascar

  • Physics and Applications Department, Faculty of Sciences, University of Antananarivo, Antananarivo, Madagascar

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