A new many-body theory for fermions is proposed which is based on the S0(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A new cannonical transformation, which is the extension of the Bogoliubov transformation to the SO (2N + 1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the S0(2N+ 1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the S0(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the S0(2N+1) canonical transformation. The physical fermion space is shown to be the irreducible spinor representation of the SO (2N + 1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N + 1 dimensional rotator. § 1. IntroductionThe conventional standard approach to fermion many-body problems starts with the independent particle approximation (IPA), either the Hartree-Fock (HF) or the Hartree-Bogoliubov (HB) approximation, for the ground state. Excited states are then treated with the random phase approximation (RP A). The RPA treatment of excited states, however, meets a serious difficulty when an instability in the IPA ground state takes place. The lowest excitation energy evaluated by the RP A becomes zero at the instability boundary of the IP A ground state due to the equivalence of the instabilities of the IPA ground state and the RPA excited states.n In the region near the phase transition of the IP A ground state, the amplitudes of collective excitations become large and couplings between collective modes, which are neglected in the RPA, become of essential importance.Since the RPA is the approximation based on an approximate bose quantization for the fermion pair operators,"> attempts were made to take into account the effect of mode couplings with the use of the boson expansion satisfying exactly the commutation relation of the pair operators. 3 > However, the boson expansion theory treats mode couplings in a perturbational way and the convergence of the expansion becomes bad when the amplitudes of collective excitations become large. 4 > Furthermore, it is unable to describe the processes involving unpaired fermions because the bosons introduced can represent only the pair operators of fermions. at Bibliotheque de l'Universite Laval on July 6, 2015 http://ptp.oxfordjournals.org/ Downloaded from E a fJ == Cat C/J ---t (} afJ , The pair operators (2·1)

Exact solutions for the Lipkin-Meshkov-Glick (LMG) model Hamiltonian are obtained by solving the Bethe ansatz equation (BAE) which is derived from the variation equation based on the Bethe ansatz. Unlike Pan and Draayer, we do not use bosonization and infinite-dimensional algebra techniques. Consequently there are no restrictions on parameters specifying strengths of the interactions included in the LMG Hamiltonian. Thus, for all the regimes of the interaction parameters, we get the exact solutions for the LMG Hamiltonian by numerically solving the BAEs and give the numerical behaviour of an order parameter J 2x .

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