Nonlinear conformally invariant generalization of the Poisson equation to D{gt}2 dimensions
- Department of Condensed-Matter Physics, Weizmann Institute, Rehovot 76100 (Israel)
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)
- OSTI ID:
- 547357
- Journal Information:
- Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 56, Issue 1; Other Information: PBD: Jul 1997
- Country of Publication:
- United States
- Language:
- English
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