A New Fermion Many-Body Theory Based on the SO(2N+1) Lie Algebra of the Fermion Operators

Abstract

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)

Nonlinear Bogolyubov–Valatin transformations and quaternions

“…(6)- (8)) are generators of the group SU (2) and they obey the Lie algebra of SO(3), SU (2). This has been observed earlier (in a more general context) in [61] (also see [19]). Related observations can be found in [62], Appendix A.1, p. 919, [63] and [64] (11)).…”

“…To apply the powerful tool of canonical transformations to the physically interesting class of nonquadratic Hamiltonians, however, requires to go beyond linear Bogolyubov-Valatin transformations. Certain aspects of nonlinear Bogolyubov-Valatin transformations have received some attention over time [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] (We disregard here work done within the framework of the coupled-cluster method (CCM) [4] which is nonunitary.). However, a systematic analytic study of general (nonlinear) Bogolyubov-Valatin transformations has not been undertaken so far.…”

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

“…Important special cases motivating these results are fermionic linear optics quantum computation (and equivalent matchgate models introduced by Valiant), which is efficiently classically simulatable [5,17,18], and models that also include linear fermionic operators (so (2N + 1)) for which an extension of the canonical Bogoliubov mapping exists [19]. Natural bosonic analogues of the fermionic results also exist.…”

Time Dependent Hartree–Bogoliubov Equation on the Coset Space SO(2N + 2)/U(N + 1) and Quasi Anti-Commutation Relation Approximation

“…The U(G) is the extended version of the well-known generalized Bogoliubov transformation 1 and is specified by the following SO(2N + 1) matrix G 3 The symbol * denotes the complex conjugation. The above canonical transformation is not strictly linear but an inhomogeneous linear transformation with the g-number gauge factor z -p where p = Xa$ a -x* a c a and p 2 --x* a x a = z 2 -l. 3 When z = 1, the above G becomes essentially an SO(2N) matrix g. The HB (SO(2N)) wave function \g) is generated as | <7) = U(g)\0) and |0) is the vacuum satisfying c a |0) = 0. The \g) is expressed as (2.3b)…”

“…In order to obtain a general microscopic means for achieving a unified self-consistent description for bose and fermi type collective excitations in such fermion systems, a new fermion many-body theory has been proposed by Fukutome, Yamamura and the present author in 1977 based on the SO(2N + 1) Lie algebra of the fermion operators. 3 An induced representation of SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. We start with the fact that the set of the fermion operators consisting of the annihilation-creation operators and the pair operators forms a larger Lie algebra, that of the SO(2N + 1) group.…”