Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions

Milgrom, M

doi:10.1103/PhysRevE.56.1148

Title: Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions

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Abstract

I propound a nonlinear generalization of the scalar-field Poisson equation [({var_phi}{sub ,}{sup i}{var_phi}{sub ,i}){sup D/2{minus}1}{var_phi}{sub ;}{sup k}]{sub ;k}{proportional_to}{rho}, in curved D-dimensional space. It is derivable from the Lagrangian density L{sup D}=L{sub f}{sup D}{minus}A{rho}{var_phi}, with L{sub f}{sup D}{proportional_to}{minus}({var_phi}{sub ,}{sup i}{var_phi}{sub ,i}){sup D/2}, and {rho} the distribution of sources. Specializing to Euclidean spaces, where the field equation is {bold {del}}{center_dot}({vert_bar}{bold {del}}{var_phi}{vert_bar}{sup D{minus}2}{bold {del}}{var_phi}){proportional_to}{rho}, I find that L{sub f}{sup D} is the only conformally invariant (CI) Lagrangian in D dimensions, containing only first derivatives of {var_phi}, beside the free Lagrangian ({bold {del}}{var_phi}){sup 2}, which underlies the Laplace equation. When {var_phi} is coupled to the sources in the above manner, L{sup D} is left as the only CI Lagrangian. The symmetry is one`s only recourse in solving this nonlinear theory for some nontrivial configurations. Systems comprising N point charges are special and afford further application of the symmetry. In spite of the CI, the energy function for such a system is not invariant under conformal transformations of the charges` positions. The anomalous transformation properties of the energy stem from effects of the self-energies of the charges. It follows from these that the forces {bold F}{sub i} on the charges q{sub i} at positions {boldmore »« less

Authors:

Milgrom, M ^[1]

Department of Condensed-Matter Physics, Weizmann Institute, Rehovot 76100 (Israel)

Publication Date:: Tue Jul 01 00:00:00 EDT 1997

OSTI Identifier:: 547357

Resource Type:: Journal Article

Journal Name:: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Additional Journal Information:: Journal Volume: 56; Journal Issue: 1; Other Information: PBD: Jul 1997

Country of Publication:: United States

Language:: English

Subject:: 66 PHYSICS; POISSON EQUATION; CONFORMAL INVARIANCE; WAVE EQUATIONS; ELECTROSTATICS; NONLINEAR PROBLEMS; LAGRANGIAN FUNCTION; FIELD EQUATIONS; SCALAR FIELDS; KINETIC ENERGY; SELF-ENERGY

Citation Formats


                    Milgrom, M. Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions.  United States: N. p., 1997. 
        Web.  doi:10.1103/PhysRevE.56.1148.

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                    Milgrom, M. Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions.  United States.  https://doi.org/10.1103/PhysRevE.56.1148

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                    Milgrom, M. 1997.  
        "Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions".  United States.  https://doi.org/10.1103/PhysRevE.56.1148.

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@article{osti_547357,

  title        = {Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions},

  author       = {Milgrom, M},

  abstractNote = {I propound a nonlinear generalization of the scalar-field Poisson equation [({var_phi}{sub ,}{sup i}{var_phi}{sub ,i}){sup D/2{minus}1}{var_phi}{sub ;}{sup k}]{sub ;k}{proportional_to}{rho}, in curved D-dimensional space. It is derivable from the Lagrangian density L{sup D}=L{sub f}{sup D}{minus}A{rho}{var_phi}, with L{sub f}{sup D}{proportional_to}{minus}({var_phi}{sub ,}{sup i}{var_phi}{sub ,i}){sup D/2}, and {rho} the distribution of sources. Specializing to Euclidean spaces, where the field equation is {bold {del}}{center_dot}({vert_bar}{bold {del}}{var_phi}{vert_bar}{sup D{minus}2}{bold {del}}{var_phi}){proportional_to}{rho}, I find that L{sub f}{sup D} is the only conformally invariant (CI) Lagrangian in D dimensions, containing only first derivatives of {var_phi}, beside the free Lagrangian ({bold {del}}{var_phi}){sup 2}, which underlies the Laplace equation. When {var_phi} is coupled to the sources in the above manner, L{sup D} is left as the only CI Lagrangian. The symmetry is one`s only recourse in solving this nonlinear theory for some nontrivial configurations. Systems comprising N point charges are special and afford further application of the symmetry. In spite of the CI, the energy function for such a system is not invariant under conformal transformations of the charges` positions. The anomalous transformation properties of the energy stem from effects of the self-energies of the charges. It follows from these that the forces {bold F}{sub i} on the charges q{sub i} at positions {bold r}{sub i} must satisfy certain constraints beside the vanishing of the net force and net moment: e.g., {summation}{sub i}{bold r}{sub i}{center_dot}{bold F}{sub i} must equal some given function of the charges. The constraints total (D+1)(D+2)/2, which tallies with the dimension of the conformal group in D dimensions. Among other things I use all these to derive exact expressions for the following quantities: (1) The general two-point-charge force. (Abstract Truncated)},

  doi          = {10.1103/PhysRevE.56.1148},

  url          = {https://www.osti.gov/biblio/547357},
  journal      = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
number       = 1,

  volume       = 56,

  place        = {United States},

  year         = {Tue Jul 01 00:00:00 EDT 1997},

  month        = {Tue Jul 01 00:00:00 EDT 1997}

}

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