The many-body problem in the electron gas requires a balanced treatment of ring (long-range correlation), ladder (short-range correlation), and exchange (Pauli principle) terms. The requirement is fulfilled by the employment of the effective-potential expansion (EPX) method which Takada has invented and developed by combining the field-theoretic and variational methods [1,2]. With use of the EPX, the ground-state energy is calculated as a function of the valley degeneracy in the multivalley electron gas [3-5] and a novel concept of fermion-boson conversion is obtained as the valley degeneracy increases [6]. Based on a new exact relation derived from the Feynman's theorem [7], we have proven that the momentum distribution function obtained in EPX is very accurate [8]. 
  The EPX is extended to treat low-lying excited states so as to evaluate the Landau's fermi-liquid theory from first principles [9]. Although it has never been done in usual variational methods using the Jastrow function, the  extention of EPX to nonzero temperatures has also been done [10,11]. 
  The usual band theories based on the density-functional theory with the local-density approximation suffer from the absence of a prescription to determine the exchange-correlation potential, but EPX can provide the prescription. After preliminary consideration [12,13], we begin with the calculation on the He atom [14,15] and alkali metals [16,17]. By the analysis of both the compressibility and the spin susceptibility with the change of the electron density, Takada concludes that the alkali metals should not be regarded as the electron gas but as the three-dimensional antidot lattice