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We study the $\nu=1/2$ Chern-Simons system and consider a self-consistentfield theory of the Singwi-Sj\"olander type which goes beyond the random phaseapproximation (RPA). By considering the Heisenberg equation of motion for thelongitudinal momentum operator, we are able to show that the zero-frequencydensity-density response function vanishes linearly in long wavelength limitindependent of any approximation. From this analysis, we derive a consistencycondition for a decoupling of the equal time density-density anddensity-momentum correlation functions. By using the Heisenberg equation ofmotion of the Wigner distribution function with a decoupling of the correlationfunctions which respects this consistency condition, we calculate the responsefunctions of the $\nu=1/2$ system. In our scheme, we get a density-densityresponse function which vanishes linearly in the Coulomb case forzero-frequency in the long wavelength limit. Furthermore, we derive thecompressibility, and the Landau energy as well as the Coulomb energy. Theseenergies are in better agreement to numerical and exact results, respectively,than the energies calculated in the RPA.Comment: 9 Revtex pages, 4 eps figures, typos correcte

Beyond the random phase approximation in the Singwi-Sjölander theory of the half-filled Landau level