Self-Consistent-Field Method and $\tau$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C=0. Our method is constructed manifesting itself the structure of the group under consideration.>...


Historical background on microscopic study of nuclear collective motions
A standard description of fermion many-body systems starts with the most basic approximation that is based on an independent-particle (IP) picture, i.e., self-consistent-field (SCF) approximation for the fermions. The Hartree-Fock theory (HFT) is typical one of such an approximation for ground states of the systems. Excited states are treated with the random phase approximation (RPA). The time dependent Hartree-Fock (TDHF) equation and time dependent Hartree-Bogoliubov (TDHB) equation are nonlinear equations owing to their SCF characters and may have no unique solution. The HFT and HBT are given by variational method to optimize energy expectation value by a Slater determinant (S-det) and an HB wave function, respectively [2]. Particle-hole (p-h) operators of the fermions with N single-particle states form a Lie algebra u(N ) [3] and generate a Thouless transformation [4] which induces a representation of the corresponding Lie group U (N ). The U (N ) canonical transformation transforms an S-det with M particles to another S-det. Any S-det is obtained by such a transformation of a given reference S-det, i.e., Thouless theorem provides an exact wave function of fermion state vector which is the generalized coherent state representation (CS rep) of U (N ) Lie group [5]. Following Yamamura and Kuriyama [6], we give a brief history of methods extracting collective motions out of fully parametrized TDHF/TDHB manifolds in SCF. Arvieu and Veneroni, and Baranger and independently Marumori have proposed a theory for spherical even nuclei [7] called quasi-particle RPA (QRPA) and it has been a standard approximation for the excited states of the systems. In nuclei, there exist a short-range correlation and a long-range one [8]. The former is induced by a pairing interaction and generates a superconducting state. The excited state is classified by a seniority-scheme and described in terms of quasi-particles given by the BCS-Bogoliubov theory [9]. The latter is occured by p-h interactions and gives rise to collective motions related to a density fluctuation around equilibrium states. The p-h RPA (RPA) describes such collective motions like vibrational and rotational motions. It, however, stands on a harmonic approximation and should be extended to take some nonlinear effects into account. To solve such a problem, the boson expansion HB theory (BEHBT) has been developed by Belyaev and Zelevinsky [10], and Marumori, Yamamura and Tokunaga [11]. The essence of the BEHBT is to express the fermion-pairs in terms of boson operators keeping a pure boson-character. The boson representation is constructed to reproduce the Lie algebra of the fermion-pairs. The state vector in the fermion Fock space corresponds to the one in the boson Fock space by one-to-one mapping. Such a boson representation makes any transition-matrixvalued quantity for the boson-state vectors coincide with that for the fermion-state ones. The algebra of fermion-pairs and the boson representation have been extensively investigated. The fermion-pairs form an algebra so(2N ). As for the boson representation, e.g., da Providência and Weneser and Marshalek [12] have proposed boson operators basing on p-h pairs forming an algebra su(N ). By Fukutome, Yamamura and Nishiyama [13,14], the fermions were found to span the algebras so(2N + 1) and so(2N + 2) accompanying with u(N + 1). The BET expressed by Schwinger-type and Dyson-type bosons has been intensively studied by Fukutome and Nishiyama [15,16,17,18]. However, the above BET's themselves do not contain any scheme under which collective degree of freedom can be selected from the whole degrees of freedom.
On the contrary, we have a traditional approach to the microscopic theory of collective motion, the TDHF theory (TDHFT) and TDHB theory (TDHBT), e.g., [19,20]. The pioneering idea of the TDHFT was suggested by Marumori [7] for the case of small amplitude vibrational motions. Using this idea, one can determine the time dependence of any physical quantity, e.g., frequency of the small fluctuation around a static HF/HB field. The equation for the frequency has the same form as that given by RPA. A quantum energy given by this method means an excitation energy of the first excited state. Then, the RPA is a possible quantization of the TDHFT/TDHBT in the small amplitude limit. In fact, as was proved by Marshalek and Horzwarth [21], the BEHBT is reduced to the TDHBT under the replacement of boson operators with classical canonical variables. Using a canonical transformation in a classical mechanics, it is expected to obtain a scheme for choosing the collective degree of freedom in the SCF. Historically, there was another stream, i.e., an adiabatic perturbation approach. This approach starts from an assumption that the speed of collective motion is much slower than that of any other noncollective motion. At an early stage of the study of this stream, the adiabatic treatment of the TDHFT (ATDHFT) was presented by Thouless and Valatin [22]. Such a theory has a feature common to the one of the theory for large-amplitude collective motion. Later the ATDHFT was developed mainly by Baranger and Veneroni, Brink, Villars, Goeke and Reinhard, and Mukherjee and Pal [23]. The most important point of the ATDHFT by Villars is in introducing a "collective path" into a phase space. A collective motion corresponds to a trajectory in the phase space which moves along the collective path. Standing on the same spirit, Holtzwarth and Yukawa, Rowe and Bassermann [24], gave the TDHFT and Marumori, Maskawa, Sakata and Kuriyama so-called "maximally decoupled" method in a canonical form [25]. So, various techniques of classical mechanics are useful and then canonical quantization is expected. By solving the equation of collective path, one can obtain some corrections to the TDHF result. The TDHFT has a possibility to illustrate not only collective modes but also intrinsic modes. However, the following three points remain to be solved yet: (i) to determine a microscopic structure of collective motion, which may be a superposition of each particle motion, in relation to dynamics under consideration (ii) to determine IP motion which should be orthogonal to collective motion and (iii) to give a coupling between both the motions. The canonical-formed TDHFT enables us to select the collective motion in relation to the dynamics, though it makes no role to take IP motion into account, because the TD S-det contains only canonical variables to represent the collective motion. Along the same way as the TDHFT, Yamamura and Kuriyama have extended the TDHFT to that on a fermion CS constructed on the TD S-det. The CS rep contains not only the usual canonical variables but also the Grassmann variables. A classical image of fermions can be obtained by regarding the Grassmann variables as canonical ones [26]. The constraints governing the variables to remove the overcounted degrees of freedom were decided under the physical consideration. Owing to the Dirac's canonical theory for a constrained system, the TDHFT was successfully developed for a unified description of collective and IP motions in the classical mechanics [27].

Viewpoint of symmetry of evolution equations
The TDHF/TDHB can be summarized to find optimal coordinate-systems on a group manifold basing on Lie algebras of the finite-dimensional fermion-pairs and to describe dynamics on the manifold. The boson operators in BET are generators occurring in the coordinate system of tangent space on the manifold in the fermion Fock space. But the BET's themselves do not contain any scheme under which collective degrees of freedom can be selected from the whole degrees of freedom. Approaches to collective motions by the TDHFT suggest that the coordinate system on which collective motions is describable deeply relates not only to the global symmetry of the finite-dimensional group manifold itself but also to hidden local symmetries, besides the Hamiltonian. Various collective motions may be well understood by taking the local symmetries into account. The local symmetries may be closely connected with infinite-dimensional Lie algebras. However, there has been little attempts to manifestly understand collective motions in relation to the local symmetries. From the viewpoint of symmetry of evolution equations, we will study the algebro-geometric structures toward a unified understanding of both the collective and IP motions.
The first issue is to investigate fundamental "curvature equations" to extract collective submanifolds out of the full TDHF/TDHB manifold. We show that the expression in a quasi-particle frame (QPF) of the zero-curvature equations described later becomes the nonlinear RPA which is the natural extension of the usual RPA. We abbreviate RPA and QRPA to only RPA. We had at first started from a question whether soliton equations exist in the TDHF/TDHB manifold or do not, in spite of the difference that the solitons are described in terms of infinite degrees of freedom and the RPA in terms of finite ones. We had met with the inverse-scattering-transform method by AKNS [28] and the differential geometrical approaches on group manifolds [29]. An integrable system is explained by the zero-curvature, i.e., integrability condition of connection on the corresponding Lie group. Approaches to collective motions had been little from the viewpoint of the curvature. If a collective submanifold is a collection of collective paths, an infinitesimal condition to transfer a path to another may be nothing but the integrability condition for the submanifold with respect to a parameter time t describing a trajectory of an SCF Hamiltonian and to other parameters specifying any point on the submanifold. However the trajectory of the SCF Hamiltonian is unable to remain on the manifold. Then the curvature may be able to work as a criterion of effectiveness of the collective submanifold. From a wide viewpoint of symmetry the RPA is extended to any point on the manifold because an equilibrium state which we select as a starting point must be equipotent with any other point on the manifold. The well-known RPA had been introduced as a linear approximation to treat excited states around a ground state (the equilibrium state), which is essentially a harmonic approximation. When an amplitude of oscillation becomes larger and then an anharmonicity appears, then we have to treat the anharmonicity by taking nonlinear effects in the equation of motion into account. It is shown that equations defining the curvature of the collective submanifold becomes fundamental equations to treat the anharmonicity. We call them "the formal RPA equation". It will be useful to understand algebro-geometric meanings of large-amplitude collective motions.
The second issue is to go beyond the perturbative method with respect to the collective variables [25]. For this aim, we investigate an interrelation between the SCF method (SCFM) extracting collective motions and τ -functional method (τ -FM) [30] constructing integrable equations in solitons. In a soliton theory on a group manifold, transformation groups governing solutions for soliton equations become infinite-dimensional Lie groups whose generators of the corresponding Lie algebras are expressed as infinite-order differential operators of affine Kac-Moody algebras. An infinite-dimensional fermion Fock space F ∞ is realized in terms of a space of complex polynomial algebra. The infinite-dimensional fermions are given in terms of the infiniteorder differential operators and the soliton equation is nothing but the differential equation to determine the group orbit of the highest weight vector in the F ∞ [30]. The generalized CS rep gives a key to elucidate relationship of a HF wave function to a τ -function in the soliton theory. This has been pointed out first by D'Ariano and Rasetti [31] for an infinite-dimensional harmonic electron gas. Standing on their observation, for the SCFM one can give a theoretical frame for an integrable sub-dynamics on an abstract F ∞ . The relation between SCFM in finite-dimensional fermions and τ -FM in infinite ones, however, has not been investigated because dynamical descriptions of fermion systems by them have looked very different manners. In the papers [32,33,34,35,1], we have first tried to clarify it using SCFM on U (N ) group and τ -FM on that group. To attain this object we will have to solve the following main problems: first, how we embed the finite-dimensional fermion system into a certain infinite one and how we rebuilt the TDHFT on it; second, how any algebraic mechanism works behind particle and collective motions and how any relation between collective variables and a spectral parameter in soliton theory is there; last, how the SCF Hamiltonian selects various subgroup-orbits and how a collective submanifold is made from them and further how the submanifold relates to the formal RPA. To understand microscopically cooperative phenomena, the concept of collective motion is introduced in relation to a TD variation of SC mean-field. IP motion is described in terms of particles referring to a stationary mean-field. The variation of a TD mean-field gives rise to couplings between collective and IP motions and couplings among quantum fluctuations of the TD mean-field itself [6], while in τ -FM a soliton equation is derived as follows: Consider an infinitedimensional Lie algebra and its representation on a functional space. The group-orbit of the highest weight vector becomes an infinite-dimensional Grassmannian G ∞ . The bilinear equation (Plücker relation) is nothing else than the soliton equation. This means that a solution space of the soliton equation corresponds to a group-orbit of the vacuum state. The SCFM does not use the Plücker relation in the context of a bilinear differential equation defining finite-dimensional Grassmannian G M but seems to use implicitly such a relation. In the SCFM a physical concept of quasi-particle and vacuum and a coset space is used instead. If we develop a perturbative theory for large-amplitude collective motion [25], an infinite-dimensional Lie algebra might been necessarily used. The sub-group orbits consisting of several loop-group paths [36] classified by the Plücker relation exist innumerably in G M so that the SCFM is related to the soliton theory in G ∞ . The Plücker relation in a coset space [37] becomes analogous with the Hirota's bilinear form [38,39]. Toward an ultimate goal we aim to reconstruct a theoretical frame for a υ (external parameter)-dependent SCFM to describe more precisely the dynamics on the F ∞ . In the abstract fermion Fock space, we find common features in both SCFM and τ -FM. (i) Each solution space is described as Grassmannian that is group orbit of the corresponding vacuum state. (ii) The former may implicitly explain the Plücker relation not in terms of bilinear differential equations defining G M but in terms of the physical concept of quasi-particle and vacuum and mathematical language of coset space and coset variable. The various BETs are built on the Plücker relation to hold the Grassmannian. The latter asserts that the soliton equations are nothing but the bilinear differential equations giving a boson representation of the Plücker relation. The relation, however, has been unsatisfactorily investigated yet within the framework of the usual SCFM. We study it and show that both the methods stand on the common features, Plücker relation or bilinear differential equation defining the Grassmannian. On the contrary, we observe different points: (i) The former is built on a finite-dimensional Lie algebra but the latter on an infinite-dimensional one. (ii) The former has an SCF Hamiltonian consisting of a fermion one-body operator, which is derived from a functional derivative of an expectation value of a fermion Hamiltonian by a ground-state wave function. The latter introduces artificially a fermion Hamiltonian of one-body type operator as a boson mapping operator from states on fermion Fock space to corresponding ones on τ -functional space (τ -FS).
The last issue is, despite a difference due to the dimension of fermions, to aim at obtaining a close connection between concept of mean-field potential and gauge of fermions inherent in the SCFM and at making a role of a loop group [36] to be clear. Through the observation, we construct infinite-dimensional fermion operators from the finite-dimensional ones by Laurent expansion with respect to a circle S 1 . Then with the use of an affine Kac-Moody (KM) algebra according to the idea of Dirac's positron theory [40], we rebuilt a TDHFT in F ∞ . The TDHFT results in a gauge theory of fermions and the collective motion, fluctuation of the mean-field potential, appears as the motion of fermion gauges with a common factor. The physical concept of the quasi-particle and vacuum in the SCFM on the S 1 connects to the "Plücker relations" due to the Dirac theory, in other words, the algebraic mechanism extracting various sub-group orbits consisting of loop path out of the full TDHF manifold is just the "Hirota's bilinear form" [39] which is an su(N )(∈ sl(N )) reduction of gl(N ) in the τ -FM. As a result, it is shown that an infinite-dimensional fermion many-body system is also realizable in a finite-dimensional one and that roles of the soliton equation (Plücker relation) and the TDHF equation are made clear. We also understand an SCF dynamics through gauge of interacting infinite-dimensional fermions. A bilinear equation for the υ-HFT has been transcribed onto the corresponding τfunction using the regular representation for the group and the Schur polynomials. The υ-HF SCFM on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ). We have given explicit expressions for Laurent coefficients of soliton solutions for sl(N ) and su(N ) on the Gr ∞ using Chevalley bases for sl(N ) and su(N ) [41]. As an illustration we will attempt to make a υ-HFT approach to an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick (LMG) model [42]. For this aim, we give an affine KM algebra sl(2, C) (complexification of su (2)) to which the LMG generators subject, and their τ representations and the σ K mappings for them. We can represent an infinite-dimensional matrix of the LMG Hamiltonian and its HF Hamiltonian in terms of the Schur polynomials. Its infinite-dimensional HF operator is also given through the mapping σ M for ψ i ψ * j of infinite-dimensional fermions ψ i and ψ * i , which is expressed by the Schur polynomials S k (x) and S k (∂ x ). Further its τ -function for a simple case is provided by the Plücker coordinates and Schur polynomials.
In Section 2, we propose curvature equations as fundamental equations to extract a collective submanifold out of the full TDHB manifold. Basing on these ideas, we construct the curvature equations and study the relation between the maximal decoupled method and the curvature equations. We further investigate the role of the non-zero curvature arising from the residual Hamiltonian. Making use of the expression of the zero-curvature equations in the QPF, we find the formal RPA equation. In Section 3, we present a simply unified aspect for the SCFM and the τ -FM and show a simple idea connecting both the methods. We study the algebraic relation between coset coordinate and Plücker coordinate. Basing on the above idea, we attempt to rebuilt the TDHFT in τ -FS. We introduce υ-dependent infinite-dimensional fermion operators and a F ∞ through Laurent expansion with respect to the degrees of freedom of the original fermions. The algebraic relation between both the methods is manifestly described. We embed a HF u(N ) Lie algebra into a gl(∞) by means of infinite-dimensional fermions. The υ-SCFM in τ -FS is developed. The role of the shift operators in the τ -FM is studied. As an illustration, explicit expressions for Laurent coefficients of soliton solutions for sl(N ) and su(N ) are presented. A problem related to a nonlinear Schrödinger equation is also discussed. In Section 4, we construct a formal RPA equation on F ∞ and also argue about the relation between a loop collective path and a formal RPAEQ. Consequently, it can be proved that the usual perturbative method with respect to periodic collective variables in the TDHFT is involved in the present method which aims for constructing the TDHFT on the affine KM algebra. In Section 5, we introduce infinite-dimensional "particle" and "hole" operators and operators K 0 and K ± defined by infinite-dimensional "particle-hole" pair operators. Using these operators, we construct an infinite-dimensional Heisenberg subalgebra of the affine KM algebra sl(2, C). The LMG Hamiltonian and its HF Hamiltonian are expressed in terms of the Heisenberg basic-elements whose representations are isomorphic to those in the corresponding boson space. They are given in terms of infinite numbers of variables x k and derivatives ∂ x k through the Schur polynomials S k (x). We give also an infinite-dimensional representation of SU (2N ) ∞ transformation of the particle and hole operators. Finally, in Section 6, we summarize and discuss the results and future problems. We give Appendices A-J. Especially in Appendix J, we show an explicit expression for Plücker coordinate for the LMG model and calculate a quantity, det(1 N + p † p) (p : coset variable) for the LMG model, in terms of the Schur polynomials.
2 Integrability conditions and collective submanifolds

Introduction
Let us consider an abstract evolution equation ∂ t u(t) = K(u(t)) for u, which is dependent only on a parameter, time t. If there exists a symmetry operation to transfer a solution for u to another one, then introducing another parameter s specifying various solutions, we can derive another form of evolution equation with respect to s, ∂ s u(t, s) = K(u(t, s)) for which we should want to search. The infinitesimal condition for the existence of such a symmetry appears as the well-known integrability condition ∂ s K(u(t, s)) = ∂ t K(u(t, s)). The "maximally decoupled" method proposed by Marumori et al. [25], invariance principle of Schrödinger equation and canonicity condition, can be considered as a theory to apply the above basic idea to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe the symmetries of the collective submanifold in terms of t and collective variables, in which the canonicity conditions make the collective variables to be orthogonal coordinate-systems.
Therefore we adopt a concept of curvature unfamiliar in the conventional TDHBT. The reason why we take such a thing is the following: let us consider a description of motions of systems on a group manifold. An arbitrary state of the system induced by a transitive group action corresponds to any point of the full group parameter space and therefore its time evolution is represented by an integral curve in this space. In the whole representation space adopted, we assume the existence of 2m parameters specifying the proper subspace in which the original motion of the system can be approximated well, the existence of the well-defined symmetries. Suppose we start from a given point on a space, which consists of t and the 2m parameters, and end at the same point again along the closed curve. Then we have the value of the group parameter different from the one at an initial point on the proper subspace. We search for some quantities characterizing the difference of the value. For our aim, we introduce a differential geometrical viewpoint. The our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe collective coordinate systems. This manner enables us to take a one-form which is linearly composed of TDHB Hamiltonian and infinitesimal generators induced by collective variable differentials of an SO(2N ) canonical transformation. The integrability conditions of the system read the curvature C = 0. Our methods are constructed manifesting themselves the structure of the group under consideration to make easy to understand physical characters at any point on the group manifold.

Integrability conditions
We consider many fermion systems with pair correlations. Let c α and c † α (α = 1, . . . , N ) be the annihilation-creation operators of the fermion. Owing to the anti-commutation relations among them, some sets of fermion operators with simple construction become the basis of a Lie algebra. The operators in the fermion so(2N ) Lie algebra, generate a canonical transformation U (g) (the Bogoliubov transformation [9]) which is specified by an SO(2N ) matrix g: where (c, c † ) = ((c α ), (c † α )) and (d, d † ) = ((d i ), (d † i )) are 2N -dimensional row vectors and a = (a α i ) and b = (b α i ) (i = 1, . . . , N ) are N × N matrices. 1 2N is a 2N -dimensional unit matrix. The symbols †, ⋆ and T mean the hermitian conjugate, the complex conjugation and the transposition, respectively. The explicit expressions of the canonical transformations are given by Fukutome for the various types of the fermion Lie algebra [16]. The fermion Lie operators of the quasiparticles (E i j , E ij , E ij ) are constructed from the operators d and d † in (2.1) by the same way as the one to define the set (Ē α β ,Ē αβ ,Ē αβ ). The set E in the quasi-particle frame is transformed into the setĒ in the particle frame as follows: A TDHB Hamiltonian of the system is given by where the HB matrices F = (F αβ ) and D = (D αβ ) are related to the quasi-particle vacuum expectation values of the Lie operators Ē as The quantities h αβ and [αβ|γδ] are the matrix element of the single-particle Hamiltonian and the antisymmetrized one of the interaction potential, respectively. Here and hereafter we use the dummy index convention to take summation over the repeated index. Let |0 be the free-particle vacuum satisfying c α |0 = 0. The SO(2N )(HB) wave function |φ(ǧ) is constructed by a transitive action of the SO(2N ) canonical transformation U (ǧ) on |0 : |φ(ǧ) = U −1 (ǧ)|0 ,ǧ ∈ SO(2N ). In the conventional TDHBT, the TD wave function |φ(ǧ) is given through that of the TD group parameters a (a ⋆ ) and b (b ⋆ ). They characterize the TD self-consistent mean HB fields F and D whose dynamical changes induce the collective motions of the many fermion systems. As was made in the TDHF case [25] and [27], we introduce a TD SO(2N ) canonical transformation U (ǧ) = U [ǧ(Λ(t),Λ ⋆ (t))]. A set of TD complex variables (Λ(t),Λ ⋆ (t)) = (Λ n (t),Λ ⋆ n (t); n = 1, . . . , m) associated with the collective motions specifies the group parameters. The number m is assumed to be much smaller than the order of the SO(2N ) Lie algebra, which means there exist only a few "collective degrees of freedom". The above is the natural extension of the method in TDHF case to the TDHB case.
However, differing from the above usual manner, we have another way, may be called a Lagrange-like manner, to introduce a set of complex variables. This is realized if we regard the above-mentioned variables (Λ(t),Λ ⋆ (t)) as functions of independent variables (Λ, Λ ⋆ ) = (Λ n , Λ ⋆ n ) and t, where time-independent variables (Λ, Λ ⋆ ) are introduced as local coordinates to specify any point of a 2m-dimensional collective submanifold. A collective motion in the 2m-dimensional manifold is possibly determined in the usual manner if we could know the explicit forms ofΛ andΛ ⋆ in terms of (Λ, Λ ⋆ ) and t. The above manner seems to be very analogous to the Lagrange manner in the fluid dynamics. The pair of variables (Λ, Λ ⋆ ) specifies variations of the SCF associated with the collective motion described by a pair of collective coordinates α and their conjugate π in the Lagrange-like manner, α = 1 √ 2 (Λ ⋆ + Λ) and π = i 1 √ 2 (Λ ⋆ − Λ) [25]. Thus, the SO(2N ) canonical transformation is rewritten as U (ǧ) = U [g(Λ, Λ ⋆ , t)] ∈ SO(2N ). Notice that a functional formǧ(Λ(t),Λ ⋆ (t)) changes into another form g(Λ, Λ ⋆ , t) due to an adoption of the Lagrange-like manner. This manner enables us to take a one-form Ω which is linearly composed of the infinitesimal generators induced by the time differential and the collective variable ones (∂ t , ∂ Λ , ∂ Λ ⋆ ) of the SO(2N ) canonical transformation U [g(Λ, Λ ⋆ , t)]. By introducing the oneform Ω, it is possible to search for the collective path and the collective hamiltonian almost separated from other remaining degrees of freedom of the systems. It may be achieved to study the integrability conditions of our systems which are expressed as the set of the Lie-algebravalued equations.
We define the Lie-algebra-valued infinitesimal generators of collective submanifolds as follows: We regard these equations (2.7) as partial differential equations for |φ(g) . In order to discuss the conditions under which the differential equations (2.7) can be solved, the mathematical method well known as integrability conditions is useful. For this aim, we take a one-form Ω linearly composed of the infinitesimal generators (2.4): . With the aid of the Ω, the integrability conditions of the system read C d = dΩ − Ω ∧ Ω = 0, where d and ∧ denote the exterior differentiation and the exterior product, respectively. From the differential geometrical viewpoint, the quantity C means the curvature of a connection. Then the integrability conditions may be interpreted as the vanishing of the curvature of the connection (D t , D Λn , D ⋆ Λn ). The detailed structure of the curvature is calculated to be The vanishing of the curvature C means C •,• = 0.
Finally with the use of the explicit forms of (2.5) and (2.6), we can get the set of Lie-algebravalued equations as the integrability conditions of partial differential equations (2.7) Here the quantities F c , θ † n , θ n are defined through partial differential equations, (2.10) The quantity C •,• may be naturally regarded as the curvature of the connection on the group manifold. The reason becomes clear if we take the following procedure quite parallel with the above: Starting from (2.10), we are led to a set of partial differential equations on the SO(2N ) Lie group, is easily shown to be equivalent to the quantity C •,• in (2.9). The above set of the Lie-algebra-valued equations (2.9) evidently leads us to putting all the curvatures C •,• in (2.9) equal to zero. On the other hand, the TDHB Hamiltonian (2.3), being the full Hamiltonian on the full SO(2N ) wave function space, can be represented in the same form as (2.4), H HB / = (i∂ t U −1 (g ′ ))U (g ′ ), where g ′ is any point on the SO(2N ) group manifold. This Hamiltonian is also transformed into the same form as (2.5). It is self-evident that the above fact leads us to the well-known TDHBEQ, i ∂ t g ′ = Fg ′ . The full TDHB Hamiltonian can be decomposed into two components at the reference point g ′ = g: where the second part H res (F res ) means a residual component out of a well-defined collective submanifold for which we should search now. For our purpose, let us introduce another curvature C ′ t,Λn and C ′ t,Λ ⋆ n with the same forms as those in (2.9), except that the Hamiltonian F c is replaced by F| g ′ =g (= F c + F res ). The quasiparticle vacuum expectation values of the Lie-algebra-valued curvatures are easily calculated as where we have used C t,Λn = 0 and C t,Λ ⋆ n = 0. The above equations (2.12) and (2.13) are interpreted that the values of C ′ t,Λn g and C ′ t,Λ ⋆ n g represent the gradient of energy of the residual Hamiltonian in the 2m-dimensional manifold. Suppose there exists the well-defined collective submanifold. Then it will be not so wrong to deduce the following remarks: the energy value of the residual Hamiltonian becomes almost constant on the collective submanifold, i.e., δ g H res g ∼ = 0 and ∂ Λn H res g ∼ = 0, where δ g means g-variation, regarding g as function of (Λ, Λ ⋆ ) and t. It may be achieved if we should determine g (collective path) and F c (collective Hamiltonian) through auxiliary quantity (θ, θ † ) so as to satisfy H c + const = H HB as far as possible. Putting F c = F in (2.9), we seek for g and F c satisfying 14) The set of the equations C •,• = 0 makes an essential role to determine the collective submanifold in the TDHBT. The set of the equations (2.14) and (2.11) becomes our fundamental equation for describing the collective motions, under the restrictions (2.21).
If we want to describe the collective motions through the TD complex variables (Λ(t),Λ ⋆ (t)) in the usual manner, we must inevitably knowΛ andΛ ⋆ as functions of (Λ, Λ ⋆ ) and t. For this aim, it is necessary to discuss the correspondence of the Lagrange-like manner to the usual one.
First let us define the Lie-algebra-valued infinitesimal generator of collective submanifolds aš whose form is the same as the one in (2.4). To guaranteeΛ n (t) andΛ ⋆ n (t) to be canonical, according to [25,27], we set up the following expectation values with use of the SO(2N ) (HB) wave function |φ(ǧ) : The above relation leads us to the weak canonical commutation relation the proof of which was shown in [25] and [27]. Using (2.7), the collective Hamiltonian H c / and the infinitesimal generators O † n and O n in the Lagrange-like manner are expressed in terms of infinitesimal onesǑ † n andǑ n in the usual way as follows: where an SO(2N ) (HB) density matrix R(g) is defined as 20) in which g becomes function of the complex variables (Λ, Λ ⋆ ) and t. We here have used the transformation property (2.2), the trace formulae equations (2.5) and (2.6) and the differential formulae, i.e., which owe to the canonicity condition (2.15) and weak canonical commutation relation (2.16).
Through the above procedure, as a final goal, we get the correspondence of the Lagrangelike manner to the usual one. We have no unknown quantities in the r.h.s. of equations (2.18) and (2.19), if we could completely solve our fundamental equations to describe the collective motion. Then we come up to be able to know in principle the explicit forms of (Λ,Λ ⋆ ) in terms of (Λ, Λ ⋆ ) and t by solving the partial differential equations (2.18) and (2.19). However we should take enough notice of roles different from each other made by equations (2.18) and (2.19), respectively, to construct the solutions. Especially, it turns out that the l.h.s. in (2.19) has a close connection with Lagrange bracket. From the outset we have set up the canonicity condition to guarantee the complex variables (Λ,Λ ⋆ ) in the usual manner to be canonical. Thus the variables (Λ,Λ ⋆ ) are interpreted as functions giving a canonical transformation from (Λ,Λ ⋆ ) to another complex variables (Λ, Λ ⋆ ) in the Lagrange-like manner. From this interpretation, we see that the canonical invariance requirements impose the following restrictions on the r.h.s. of (2.19): Using (2.19) and (2.21), we get Lagrange brackets for canonical transformation of (Λ,Λ ⋆ ) to (Λ, Λ ⋆ ).

Validity of maximally-decoupled theory
First we transform the set of the fundamental equations in the particle frame into the one in the quasi-particle frame. The SO(2N ) (TDHB) Hamiltonian of the system is expressed as the relation of which to the original TDHB Hamiltonian F is given by The infinitesimal generators of collective submanifolds and their integrability conditions expressed as the Lie-algebra-valued equations are also rewritten into the ones in the quasi-particle frame as follows: The quantities F o−c , θ † o−n (= g † θ † n g) and θ o−n (= g † θ n g) are defined through partial differential equations on the SO(2N ) Lie group manifold, In the above set of (2.23), all the curvatures C o−•,• should be made equal to zero. The full TDHB Hamiltonian is decomposed into the collective one and the residual one as Then the curvature C ′ t,Λn and C ′ t,Λ ⋆ n can be regarded as the gradients of quantum-mechanical potentials due to the existence of the residual Hamiltonian H res on the collective submanifolds. The potentials become almost flat on the collective submanifolds, i.e., H HB = H c + const, if the proper subspace determined is an almost invariant subspace of the full TDHB Hamiltonian. This collective subspace is an almost degenerate eigenspace of the residual Hamiltonian. Therefore it is naturally deduced that, provided there exists the well-defined collective subspace, the residual curvatures at a point on the subspace are extremely small. Thus, the way of extracting the collective submanifolds out of the full TDHB manifold is made possible by the minimization of the residual curvature, for which a deep insight into (2.29) becomes necessary.
Finally, the restrictions to assure the Lagrange bracket for the usual collective variables and Lagrange-like ones are transformed into the following forms represented in the QPF: We discuss here how the Lagrange-like manner picture is transformed into the usual one. First let us regard any point on the collective submanifold as a set of initial points (initial value) in the usual manner. Suppose we observe the time evolution of the system with various initial values. Then we have the following relations which make a connection between the Lagrange-like manner and the usual one in which the transformation functions are set up by the initial conditions, in order to guarantee both pictures to coincide at time t = 0. On the other hand, our collective Hamiltonian F o−c can also be expressed in the form Finally we impose the canonicity conditions in the usual manner, which leads us to the weak canonical commutation relation with the aid of (2.28) and (2.21). The TDHBT for maximally-decoupled collective motions can be formulated parallel with TD-HFT [25]. The basic concept of the theory lies in an introduction of the invariance principle of the Schrödinger equation, and the TDHBEQ is solved under the canonicity condition and the vanishing of non-collective dangerous terms. However, as we have no justification on the validity of the maximally-decoupled method, we must give a criterion how it extracts the collective submanifold effectively out of the full TDHB manifold. We are now in a position to derive some quantities by which the criterion is established. For this aim, we express the collective Hamiltonian F o−c and the residual one F o−res in the same form as the one of the TDHB Hamiltonian F o given in (2.22). We also represent quantities θ † o−n , C res o−t,Λn and C res o−t,Λ ⋆ n which consist of N × N block matrices as follows: together with their complex conjugate and where we have used (2.25) and the explicit forms of the Hamiltonian. As was mentioned, the way of extracting collective submanifolds out of the full TDHB manifold is made possible by minimization of the residual curvature. This is achieved if we require at least expectation values of the residual curvatures to be minimized as much as possible, i.e., We here adopt a condition similar to one of the stationary HB method as was done in the TDHF [27]: The so-called dangerous terms in the residual Hamiltonian F o−res are made to vanish, With aid of equations (2.36), (2.37) and (2.38), equations (2.40) and (2.39) are rewritten as First, we will discuss how equation (2.41) leads us to the equation of path for the collective motion. Notice that the quantitiesθ † o−n andθ o−n are subjected to satisfy the same type of partial differential equation as that of (2.24). Remember the explicit representation of an SO(2N ) matrix g given in the previous section. Then we have partial differential equationš together with its complex conjugate. Putting the relation F o = g † Fg and (2.43) into (2.41), we getǎ Let H be an exact Hamiltonian of the system with certain two-body interaction and let us denote the expectation value of H by |φ(g) as H g . It can be easily proved that the relations and their complex conjugate relations do hold, through which the well-known TDHBEQ is converted into a matrix form as The quantity H g/ As one way of satisfying (2.47), we may adopt the following type of partial differential equations: Here we notice the invariance principle of the Schrödinger equation and the canonicity condition which leads us necessarily to the equation of collective motion expressed in the canonical forms  [25,27] to the one in the TDHB case. The set of (2.48) and (2.49) is expected to determine the behaviour of the maximally decoupled collective motions in the TDHB case. However, it means nothing else than the rewriting of the TDHBEQ with the use of canonicity condition, if we are able to assume only the existence of invariant subspace in the full TDHB solution space. The above interpretation is due to the natural consequence of the maximally decoupled theory because there exists, as a matter of case, the invariant subspace, if the invariance principle of the Schrödinger equation does hold true.
The maximally decoupled equation can be solved with the additional RPA boundary condition, though its solution is, strictly speaking, different from the true motion of the system on the full SO(2N ) group manifold. But how can we convince that the solution describes the well-defined maximally decoupled collective motions from the other remaining degrees of freedom of motion? Therefore, in order to answer such a question, we must establish a criterion how we extract the collective submanifolds effectively out of the full TDHB manifold. Up to the present stage, equation (2.42) remains unused yet and makes no role for approaching to our aim. Finally with the aid of (2.42), we will derive some quantity by which the range of the validity of the maximally decoupled theory can be evaluated. As was mentioned previously, we demanded that the expectation values of the residual curvatures are minimized as far as possible and adopted the canonical equation in place of our fundamental equation (2.27). Then, by combining both the above propositions, it may be expected that we can reach our final goal of the present task. Further substitution of the equation of motion (2.49 Here we have used the transformation property of the differential ∂ Λn = ∂ ΛnΛn ′ ∂Λ n ′ +∂ ΛnΛ ⋆ n ′ ∂Λ⋆ n ′ and the differential formulae for the expectation values of the Hamiltonians H and H HB Then we must show that the fundamental equation has necessarily the RPA solution at the lowest power of the collective variables which approach in the small amplitude limit. A paired mode amplitude g(Λ, Λ ⋆ , t) is separated into stationary and fluctuating components as g = g (o)g . This means that the SO(2N ) matrix g is decomposed into a product of stationary matrix g (o)
Using the above decomposition of g, an original SO(2N )(HB) density matrix R(Λ, Λ ⋆ , t) and a HB matrix The fluctuating R and the HB matrix F in fluctuating QPF are given in the following forms: in which all the quantities are redefined in [14]. Quasi-particle energies ε include a chemical potential.
Introducing fluctuating auxiliary quantitiesθ where the quantities F c ,θ † n andθ n satisfy partial differential equations, Putting F c = F (2.52) in (2.53), we are able to look for a collective path (g) and a collective Hamiltonian ( F c ) under the minimization of the residual curvature arising from a residual Hamiltonian ( F res ). Next, for convenience of further discussion, we introduce modified fluctuating auxiliary quantitiesθ † o−n =g †θ † ng andθ o−n =g †θ ng . Then we can rewrite our fundamental equations (2.53), (2.54) and (2.55) in terms of the above quantities as follows: In the derivation of equations (2.57) and (2.58), we have used (2.56). The equation (2.59) is easily obtained with the aid of another expression for the fluctuating density matrix R (2.51), In order to investigate the set of the matrix-valued nonlinear time evolution equation (2.57) arising from the zero curvature equation, we give here the ∂ Λn and ∂ Λ ⋆ n differential forms of the TDHB density matrix and collective hamiltonian. First, using (2.60), we have where we have used the relationg †g =gg † = 1. Using (2.56), the above equation is written as On the other hand, from (2.51), we easily obtain into the r.h.s. of (2.62) and combine it with (2.63). Then, we obtain the final ∂ Λn differential form of the TDHB (SO(2N )) density matrix as follows: The ∂ Λ ⋆ n differentiation of the SO(2N ) density matrix is also made analogously to the above. As shown in [14], the fluctuating components of the HB matrix F (2.52) are linear functionals of R(g) and K(g). We can easily calculate the ∂ Λn differential as follows: We here derive a new equation formally analogous to the SO(2N ) RPA equation. To achieve this, we first further decompose the fluctuating pair mode amplitudeg into a product of a fluctuating SO(2N ) matrix and a 2N -dimensional diagonal matrix with an exponential time dependence as follows: where we redenote a new fluctuating pair mode asg and ε i is the Λ and Λ ⋆ dependent quasiparticle energy including the chemical potential. Next, using (2.56),θ † o−n =g †θ † ng andθ o−n = g †θ ng , the modified fluctuating auxiliary quantitiesθ † o−n can be written as where we again redenote the new fluctuating auxiliary quantities asθ † o−n . Accompanying the above change, ∂ tθ † o−n are modified to the following forms by using the explicit expression forθ † o−n : In the above, hereafter we adopt (Λ, Λ ⋆ )-independent ε (o) given in (2.52) as the quasi-particle energy ε. If we substitute equations (2.65), (2.66) and (2.67) into the set of the matrix-valued nonlinear time evolution equation, i.e., the equation of (2.57), we finally obtain the following set of matrix-valued equations: with the modified new matrices {D} etc. defined through in which summation is made over indices i ′ and j ′ running from 1 to N . In (2.68) by making block off-diagonal matrices vanish, we get a TD equation with respect to ψ o−n and ϕ o−n which is formally analogous to that of the SO(2N ) RPA, though our TD amplitude and matrices {D} etc. have (Λ, Λ ⋆ , t)-dependence.

Summary and discussions
We have studied integrability conditions of the TDHBEQ to determine collective submanifolds from the group theoretical viewpoint. As we have seen above, the basic idea lies in the introduction of the Lagrange-like manner to describe the collective coordinates. It should be noted that the variables are nothing but the parameters to describe the symmetry of TDHBEQ. By introducing the one-form, we gave the integrability conditions, the vanishing of the curvatures of the connection, expressed as the Lie-algebra-valued equations. The full TDHB Hamiltonian H HB is decomposed into the collective Hamiltonian H c and the residual one H res . To search for the welldefined collective submanifold, we have demanded that the expectation value of the curvature is minimized so as to satisfy H res ∼ = const or H c + const = H HB as far as possible. Further we have imposed the restriction to assure the Lagrange bracket for the usual variables and Lagrange-like ones. Our fundamental equation together with the restricted condition describes the collective motion of the system. We have proposed the minimization of the residual curvature arising from the residual part of the full TDHB Hamiltonian to determine the collective submanifold. With our theory it is also possible to investigate the range of the validity of the maximally decoupled theory of the TDHBT with use of the condition to satisfy (2.50). This condition makes an essential role to give the criterion how we extract well the collective submanifold out of the full TDHB manifold. The reason why the condition occurs in our theory which did not appear in the maximally decoupled theory lies in the consideration of the d † d-type in the residual Hamiltonian to calculate the residual curvature and in the adoption of the canonical equation. Since the maximally decoupled theory has no consideration of such type from the outset, the condition is trivially fulfilled. This is the essential difference between the maximally decoupled theory and ours.
We have investigated the nonlinear time-evolution equation arising from zero-curvature equation on TDHB (SO(2N ) Lie group) manifold. It is self-evident that the new equation has an SO(2N ) RPA solution as a small-amplitude limit. The new equation depends on the collective variables (Λ, Λ ⋆ ) defined in a Lagrange-like manner. It works well in the large scale beyond the SO(2N ) RPA under appropriate boundary and initial conditions. The integrability condition is just the infinitesimal condition to transfer a solution to another solution for the evolution equation under consideration. The usual treatment of the RPA for small amplitude around ground state is nothing but a method of determining an infinitesimal transformation of symmetry under the assumption that fluctuating fields are composed of only normal-modes. We conclude that the set of equations defining the symmetry of the SCF equation and the weak boson commutation relations on the QPF becomes the nonlinear RPA theory.
Finally, following Rajeev [43], we also show the existence of the homogeneous symplectic 2-form ω. From (2.20), using the SO(2N ) U (N ) coset variable q (= ba −1 ) = −q T , the SO(2N ) (HB) density matrix R(g) is expressed as which has quite the same form as the one given by Rajeev [43]. The two-form ω is given as which is a symplectic form and makes it possible to discuss geometric quantization on a finite/infinite-dimensional Grassmannian [44,45].
3 SCF method and τ -functional method on group manifolds

Introduction
Despite the difference due to the dimension of fermions mentioned in Section 1, we ask the following: How is a collective submanifold, truncated through the SCF equation, related to a subgroup orbit in the infinite-dimensional Grassmannian by the τ -FM? To get a microscopic understanding of cooperative phenomena, the concept of collective motion is introduced in relation to TD variation of a SCF. Independent-particle (IP) motion is described in terms of particles referring to a stationary MF. The TD variation of the TD SCF is attributed to couplings between the collective and the IP motions and couplings among quantal fluctuations of the TD SCF [6].
There is a one-to-one correspondence between MF potentials and vacuum states of the system. Decoupling of collective motion out of full-parameterized TDHF dynamics corresponds to truncation of the integrable sub-dynamics from a full-parameterized TDHF manifold. The collective submanifold is a collection of collective paths developed by the SCF equation. The collectivity of each path reflects the geometrical attribute of the Grassmannian, which is independent of the characteristic of the SCF Hamiltonian. Then the collective submanifold should be understood in relation to the collectivity of various subgroup orbits in the Grassmannian. The collectivity arises through interference among interacting fermions and links with the concept of the MF potential. The perturbative method has been considered to be useful to describe the periodic collective motion with large amplitude [25,6]. If we do not break the group structure of the Grassmannian in the perturbative method, the loop group may work under that treatment. Thus we notice the following point in both methods: Various subgroup orbits consisting of loop path may infinitely exist in the full-parameterized TDHF manifold. They must satisfy an infinite set of Plücker relations to hold the Grassmannian. As a result, the finite-dimensional Grassmannian on the circle S 1 is identified with an infinite-dimensional one. Namely the τ -FM works as an algebraic tool to classify the subgroup orbits. The SCF Hamiltonian is able to exist in the infinite-dimensional Grassmannian. Then the SCFT can be rebuilt on the infinite-dimensional fermion Fock space and also on the τ -functional space. The infinite-dimensional fermions are introduced through Laurent expansion of the finite-dimensional fermions with respect to the degrees of freedom of the fermions related to the MF potential. Inversely, the collectivity of the MF potential is attributed to gauges of interacting infinite-dimensional fermions and interference among fermions is elucidated via the Laurent parameter. These are described with the use of affine KM algebra according to the Dirac theory [40]. Algebro-geometric structure of infinite-dimensional fermion many-body systems is realized in the finite-dimensional ones.

Bilinear differential equation in SCF method
Owing to the anti-commutation relations Let |0 be a free vacuum and |φ M be an M particle S-det 2) shows that the M particle S-det is an exterior product of M single-particle states and that U (g) transforms |φ M to another S-det (Thouless transformation) [4] under (3.1). Such states are called "simple" states. The set of all the "simple" states of unit modulus together with the equivalence relation, identifying distinct states only in phases with the same state, constitutes a manifold known as Grassmannian Gr M . The Gr M is an orbit of the group given through (3.2). Any simple state |φ M ∈ Gr M defines a decomposition of single-particle Hilbert spaces into sub-Hilbert spaces of occupied and unoccupied states [46]. Thus, the Gr M corresponds to a coset space . Using a variable p of the coset space, following [15,16] and [37], we express the third equation of (3.2) as where we have used the relations and the definition In In the Gr M we can introduce an expression called the Plücker coordinate which has played important roles for an algebraic construction of a soliton theory in its early stage [38], From elementary determinantal calculus, we prove easily the Plücker coordinate has a relation where the indices denote the distinct sets The Plücker relation is equivalent to a bilinear identity equation The bilinear equation has a more general form where |φ k and |φ l denote k-particle simple state and l-one, respectively. It is noted that the Gr M is essentially an SU (N ) group manifold since the phase equivalence theorem does hold. Now we study the relation between the coset coordinate appeared in (3.3) and the Plücker coordinates in (3.5). Both the well-known coordinates make a crucial role to clarify the algebraic relation between the SCFT, i.e. TDHFT, and the soliton theory.
Using the expressions for unoccupied and occupied states in (3.3), we can rewrite (3.5) as The last line of the above is recast again into the form of (3.5) after many time exchanges between c a 1 · · · c aρ and all creation operators so that all the annihilation operators are ordered in such a way that they are to the right of all the creation operators including the ones in |φ M . Then we have the relation Here matrix elements in the a 1 -th, . . . and a ρ -th rows, C(ζ) a 1 ,1∼M , . . . and C(ζ) aρ,1∼M are replaced with S(ζ) i 1 ,1∼M , . . . and S(ζ) iρ,1∼M to describe ρ (1 < ρ < M ) times particle-hole excitations from hole state a 1 to particle state i 1 , . . . and those of hole state a ρ to particle state i ρ , respectively. Equating equations (3.3) and (3.5) with equations (3.6) and (3.7), respectively, we obtain the anti-symmetrized A(· · · ) and the coset variable expressed in terms of Plücker coordinates as , (3.8) in the second Plücker coordinate of which, only one row matrix elements of its determinantal form (3.7) C(ζ) a,1∼M are replaced with S(ζ) i,1∼M . Expanding the anti-symmetrized A(· · · ) in the left-hand side of the first equation of (3.8) with respect to, for example, the first column, we have a decomposition rule  in which hole state a 1 in the last Plücker coordinate make no changes (a 1 → a 1 ) since in the second one particle-hole excitation already occurred from hole state a 1 to particle state i j [1].
It is well-known that the Plücker relation is equivalent to a bilinear identity equation which have made an important role to construct many kinds of solitons on various group manifolds [30]. Parallel to the regular representation method by Fukutome [15,16], we can prove that the Lie commutation relation is also satisfied by the differential operators for particle-hole pairs in Appendix B: From the calculations in Appendix B, these differential operators are also proved to satisfy relations Further we can introduce higher order differential operators obeying the relation which show that by operating the differential operator D on the vacuum function Φ, we obtain the Plücker coordinate A. The Plücker relation (3.9) becomes a finite set of partial differential equations satisfying Thus, in both the SCFT and the soliton theory on a group, we can find the common feature that the Grassmannian is just identical with the solution space of the bilinear differential equation. The solution space of each differential equation becomes an integral surface [32, 34, 1].

SCF method in F ∞
We will give here a brief sketch of the SCF equation, i.e., the TDHFM. According to Rowe et al. [46], we start with a geometrical aspect of the method in the following way: Let us consider the time dependent Schrödinger equation i ∂ t Ψ = HΨ with a Hamiltonian where γα|δβ denotes a matrix element of an interaction potential. The starting point for the TDHFT lies in an extremal condition of an action integral (3.14) To get an explicit expression for the TDHFEQ, we calculate an expectation value of one-and two-body operators for the S-det (3.2). Using the canonical transformation (3.1), we have (3.16) Introducing triangular matrices C(ζ) and S(ζ) in Gr M [37,16] and using an isometric matrix u T The Lagrange function L(g(t)) in (3.14) is computed as using ∂ t U (g † (t))U (g(t)) + U (g † (t)) · ∂ t U (g(t)) = 0. The condition (3.14) gives the TDHFEQ d dt and then we obtain a compact form of the TDHFEQ i ∂ t g(t) = F[W {g(t)}]g(t). The time evolution of the S-det (3.2) is given by On the other hand, using a υ-dependent fermion operator given soon later, from (3.14) we also obtain a compact form of υ-dependent HF equation, instead of time t, as Following the observation by D'Ariano and Rasetti's [31] for the relation between "soliton equations and coherent states", we may assert that the SCFM presents the theoretical scheme for an integrable sub-dynamics on a certain infinite-dimensional fermion Fock space, by identifying |φ M with the highest weight vector and by regarding the TDHF-manifold Gr M as the projection onto a subspace of the τ -function.
We reconstruct a υ-dependent SCFM in a F ∞ and study a relation between soliton equation and υ-dependent HF equation [41]. We start from a single-particle Schrödinger equation with a υ-dependent and a Υ-periodic potential V (r , υ) = V (r , υ + Υ) Here we have supposed that an eigen-spectrum ǫ α is υ independent, though the potential is dependent on υ. It holds an iso-spectrum under a υ-evolution of the potential. An eigenfunction ψ α (r , υ) constitutes an orthonormal complete set and satisfies the same periodicity, ψ α (r , υ + Υ) = ψ α (r , υ) (Floquet's theorem). This picture is very different from that in [1] and [32]. According to Goddard and Olive [47], we can make Laurent expansion of a fermionfield creation-operator ψ † (r , υ) with a parameter υ as where z = exp i2π υ Υ given on a unit circle. Thus, the ψ N r+α can be regarded as a new fermion creation-operator. We obtain also a new fermion annihilation-operator in the same way. The anti-commutation relations can be rewritten as Through Laurent expansion of the fermion-field operators, infinite-dimensional fermion operators with particle spectra and Laurent spectra can be obtained as where Z means the set of the integers. The indices α and r are called the label on particle spectra and that on Laurent spectra, respectively. Substitution of (3.23) into (3.22) leads to the anti-commutation relations If the canonical transformation (3.1) has the υ-dependence and generates the υ-evolution of the potential, it is possible to embed a U (N ) group induced from (3.1) into a group which can be induced from a canonical transformation of the infinite-dimensional fermion operators (3.24).
Then, for one-body and two-body type operators we obtain The W is just the so-called density matrix since it is easily proved to satisfy the idempotency relation W 2 = W which has not been given explicitly in [1] and [32]. It provides a strong tool to develop our SCF scenario on the F ∞ . Taking summation over infinite integers, inevitably we have an anomaly in the above expectation value. To avoid this anomaly, the one-body operator (3.33) must be changed to a normal-ordered product as where we have used the correspondence relation between basic elements. The W k is identical with a coefficient of the Laurent expansion of the density matrix W (3.15) We compute formal expectation value of (3.32) for the state U (ĝ)|M . Introducing a new integer K, due to the equivalence conditions for H F∞ , from (3.31) we get Changing W in (3.35) into its normal-ordered product and using (3.34), we obtain This result coincides with formal Laurent polynomials of H[W ] (3.17) in the sense of the expansion To get a υ-independent Hamiltonian, in (3.36) it is enough for us to pick up only the term with Laurent spectrum l = 0. Then, we may select a density-functional Hamiltonian We here adopt (3.37) as an energy functional for the u(N ) HF system on the F ∞ . Through the variation For the υ-dependent HF equation on the F ∞ , the state vector U (ĝ)|M is required to satisfy the variational principle where we use = 1 here and hereafter. First by using U (ĝ) = e Xγ we get the following relations:
Demand the extremal condition of (3.40) leads to D υĝ = F(ĝ)ĝ where F(ĝ) has an infinite N -periodic sequence of block form {. . . , which suggest symmetry breaking and arising of collective motion due to recovery of symmetry. Suppose thatĝ to diagonalize F p in H p F∞;HF and U (ĝ)|M to do F c in H c F∞;HF are determined spontaneously whenĝ ≃ĝ 0 e −iǫ υ and ∂ υĝ 0 = 0. Using the definition of F c we have ω c Γ(ĝ 0 ) = F(ĝ 0 )ĝ 0 −ĝ 0ǫ where Γ(ĝ 0 ) has an infinite N -periodic sequence of block form {. . . , −g 0 −1 , 0, g 0 1 , . . . } like (C.5) andǫ = diag{. . . , ǫ, . . . }. We also obtain g r z r ∝ e −i(ǫ +ωcI N )υ . Thus the quasi-particle energy ǫ(ǫ αβ = ǫ α δ αβ ) and the boson energy ω c are unified into gauge phase. The static υ-HFT on Gr M has obviously no collective term and leads inevitably to ω c Γ(ĝ 0 ) = 0.ĝ 0 should compose of only a block-diagonal g 0 0 = e γ 0 , γ 0 being a block-diagonal su(N ) matrix. Equation (3.48) brings a υ-evolution of particle degrees of freedom and a common language, infinite-dimensional Gr ∞ and affine KM algebra, to discuss the relation between SCFT and soliton theory. The SCFT on F ∞ is nothing else than the zero-th order of the Laurent expansion on Gr M . Through the construction of the SCFT an explicit algebraic structure of the SCFT on F ∞ is made clear since it is just the gauge theory inherent in the SCFT. The mean-field potential degrees of freedom occur from the gauge degrees of freedom of fermions and the fermions make pairs among them absorbing a change of gauges. The sub-Hamiltonian (3.38) exhibits such a phenomenon in u(N ) algebra, which allows us to interpret absorption of gauge as a coherent property of fermion pairs. Thus the SCFTM is regarded as a method to determine self-consistently both quasi-particle energy ǫ α (ĝ) and boson energy ω c , to the υ-evolution of the "fermion gauge". Then we can say that both the energies have been unif ied into the gauge phase.

SCF method in τ -functional space
Along the soliton theory in the infinite-dimensional fermion Fock space [30,48,49,54], we transcribe the υ-dependent HFT in F ∞ to the one in τ -functional space. We restrict ourselves mainly to the cases of sl(N ) and su(N ) and the group orbit of the fundamental highest weight vector |M . Let us consider infinite-dimensional charged fermions ψ i and ψ * i (i ∈ Z) satisfying the canonical anti-commutation relation {ψ * i , ψ j } = δ ij and {ψ * i , ψ * j } = {ψ i , ψ j } = 0. The perfect vacuum |Vac and the simple state |M given by (3.25) are represented in terms of another basis ν i (i ∈ Z) and the present fermions ψ i and ψ * i as |Vac ≃ ν 0 ∧ ν −1 ∧ ν −2 ∧ · · · (∧ : exterior product), Vac|Vac = 1, The basis {ν i |i ∈ Z} is given by the column vector with 1 as the i-th row and 0 elsewhere. The number M is called the charge number. The fermions ψ i and ψ * i (i ∈ Z) generate an algebra We consider further a bigger Lie algebra than gl(∞) so as to include a Heisenberg subalgebra (bosons). Following Appendix C, it is defined as the vector space a ∞ = { i,j∈Z a ij : ψ i ψ * j : +C · c, a ij = 0, for |i − j| > N} and : For detail see Appendix C. A Heisenberg subalgebra S [49] is defined as From the definition of the normal-ordered product, the boson algebra is obtained as [Λ k , Λ l ] = kδ k+l,0 · c. Λ k is called the shift operator and Λ 0 belongs to the center. Suppose level-one c = 1. Let F (M ) be a linear span of semi-infinite monomials with charge number M . For the representation of Λ k on F (M ) , Λ k |M = 0 holds for k > 0. All the elements Λ −ks · · · Λ −k 1 |M (0 < k 1 ≤ k 2 ≤ · · · ≤ k s ) are linear independent with each other. Thus, we have obtained an irreducible representation of the algebra S in the fermion space F (M ) . This is isomorphic to the representation of S in the corresponding boson space B (M ) below. Let σ M denote this isomorphism The contravariant hermitian form on the B (M ) is given as where the P * means the complex conjugation of all the coefficients of the polynomial P and x = (x 1 , x 2 , . . . ). We construct a representation in B (M ) in reduction to sl(N ). Let the generating series be and introduce Schur polynomials S k (x) given in Appendix D. It should be emphasized that in [1] and [32], we already have obtained explicit expressions for the basic elements which makes a crucial role to construct a υ-dependent HFT on U (ĝ)|M as shown later. For any element of the gl(∞) and the a ∞ in the B (M ) , we have got Using exp{H(x)} = exp j≥1 x j Λ j 1 = j≥ 0 Λ j S j (x) and H(x)|M = 0 due to Λ j |M = 0, x-evolution of the infinite-dimensional fermion operator is given in terms of the Schur polynomials as By using (3.5) and (3.55), the derivation of the above is made as follows: . This equation reads (3.56), the generalization of which to infinite-dimension is given in [49].
To see that the affine Kac-Moody algebra associated with the Lie algebra gl ∞ is contained as a subalgebra, we give a reduction of gl ∞ to sl n . A subalgebra X a = i,j∈Z a ij : ψ i ψ * j : +C · 1 of a ∞ is called n-reduced if and only if the following two conditions are satisfied: (1) U (ĝ)|M U (ĝ) = e Xγ ; X γ ∈ sl(N ) → N -reduced KP τ -function,
We can generalize the above n-soliton solutions to the cases of sl(N ) and su(N ). Using the Chevalley bases for sl(N ) and for su(N ) [48,51], the Laurent coefficients can be derived for each case as Suppose the external parameter υ to be a time t and solve the TDHFEQ by restricting a solution space to the above spaces. We get a soliton solution, i.e., a solitary wave propagating on a surface rather than a colliding soliton [55]. This is in contrast with a two-dimensional soliton [56], i.e., dromion [57], of the Davey-Stewartson equation (DSE) [58] which provides a two-dimensional generalization of the celebrated nonlinear Schrödinger equation (NLSE) [57]. The dromion for gl(2∞) was derived from the standpoint of a Clifford algebra with generators of infinite free fermions, in terms of a reduction of the two-component KP hierarchy [59,60,61]. As Kac and van der Leur pointed out [61], the dromion solution of the DSE was first studied from the point of view of the spinor formalism by Heredero et al. [62].

Summary and discussions
We have transcribed a bilinear equation for υ-HFT into the corresponding τ -function using regular representations for groups [3] and Schur polynomials. The concept of quasi-particle and vacuum in SCFT is connected with bilinear differential equations. So far SCFM has focused mainly on construction of various types of boson expansions for quantum fluctuations of meanfield(MF) rather than taking the bilinear differential equations(Plücker relations) into account. These methods turn out to be essentially equivalent with each other. Various subgroup-manifolds consisting of several loop-group paths [36] exist innumerably in Gr M relating to collective motions. To go beyond the perturbative method in terms of the collective variables, we have aimed to construct υ-HFT on affine Kac-Moody algebras along soliton theory, using infinite-dimensional fermions. These fermions have been introduced through Laurent expansion of finite-dimensional fermions with respect to the degrees of freedom of fermions related to MF. Consequently υ-SCFT on F ∞ leads to dynamics on infinite-dimensional Grassmannian Gr ∞ . Gr M is identified with Gr ∞ affiliated with a manifold obtained by reduction of gl ∞ to sl(N ) and su(N )(reduction of KP hierarchy to DS, NLS and KdV hierarchies). We have given explicit expressions for Laurent coefficients of soliton solutions for sl(N ) and su(N ) using Chevalley bases for sl(N ) and su(N ). In this sence the algebraic treatment of extracting subgroup-orbits with z(|z| = 1) from Gr M exactly forms the differential equation (Hirota's bilinear equation). The υ-SCFT on F ∞ results in gauge theory of fermions and collective motion due to quantal fluctuations of υ-dependent SCMF potential is attributed to motion of the gauge of fermions in which common gauge factor causes interference among fermions. The concept of particle and collective motions is regarded as the compatible condition for particle and collective modes. The collective variables may have close relation with a spectral parameter in soliton theory. These show that υ-SCFT on F ∞ presents us new algebraic method on S 1 for microscopic understanding of fermion many-body systems.
We have studied the relation between υ-SCFT and soliton theory on group manifold and shown that both the theories describe dynamics on each Grassmannian Gr which is the group orbit of highest weight vector. The former stands on the finite-dimensional fermion operators but the latter does on the infinite-dimensional ones. Each Gr is just identical with the solution space for respective finite and infinite set of bilinear differential equations on the boson space mapped from those on the fermion space. We have investigated the dynamics on υ-dependent HF manifold using regular representations for groups [16]. A picture of quasi particle and vacuum in υ-SCFT is connected with the bilinear differential equation. υ-HFT on finite-dimensional Fock space is embedded into υ-HFT on infinite one. The wave function in an SC Υ-periodic MF potential becomes dependent on the Laurent parameter z on a unit circle S 1 . This owes to the introduction of affine Kac-Moody algebra by infinite-dimensional fermion operators, Laurent expansion of finite-dimensional fermions with respect to z. The Plücker relation on coset variables becomes analogous to Hirota's bilinear form. The υ-SCFM has been mainly devoted to the construction of boson-coordinate systems rather than that of soliton solution by τ -FM. It turns out that both the methods are equivalent with each other due to the Plücker relation defining Gr. From loop group viewpoint and with clearer physical picture we have proposed description of particle and collective motions in υ-SCFT on F ∞ in relation to iso-spectral equation in soliton theory. Then the υ-SCFT on F ∞ may be regarded as soliton theory in the sense that it bases on Gr ∞ and may describe dynamics on infinite set of real fermion-harmonic oscillators though the soliton theory describes dynamics on complex ones. The soliton equation is nothing but the bilinear equation and the boson coordinate x k with highest degree plays a role of an evolutional variable on τ -functional space (FS) on which in υ-HFT, the bilinear equation provides algebraic means to extract subgroup orbits parametrized with z from Gr M . The infinite set of x k becomes coordinates on τ -FS and their υ-evolution yield trajectories of the SCF Hamiltonian H p F∞ . Though we have started with a periodic potential to introduce infinite-dimensional fermions, it is easy to see that υ-dependence with periodicity Υ is by no means a necessary condition. The fact that Schrödinger function is dependent on an unit circle S 1 , however, makes a crucial role for construction of infinite-dimensional fermions. As pointed out in [32], it turns out that the fully parametrized υ-dependent SCF Hamiltonian is made up of only the υ-dependent Hamiltonian H F∞;HF . Then, we have a very important question why infinite-dimensional Lie algebras work well in fermion systems. As concerns this problem, Pan and Draayer (PD) [63] have developed an infinite-dimensional algebraic approach using affine Lie algebras su(2) and su(1, 1). They have introduced fermion pair operators with two parameters for the general pairing Hamiltonian and boson operators through Jordan-Schwinger fermion-boson mapping for an exactly solvable su(2) Lipkin-Meshkov-Glick (LMG) model [42]. They have obtained analytical expressions for exact eigenvalues and eigenfunctions of this Hamiltonian based on the Bethe anzatz (BA), from which BA equation [64] or Richardson equation [65] is derived.
It is interesting to study a relationship between various subgroup-manifolds of Gr ∞ and collective sub-manifolds of υ-SCF Hamiltonian by using a simple and exactly solvable LMG model. Notwithstanding, it is possible to provide a theoretical frame of formal RPA [34,35] as a tool of truncating a collective motion with only one normal mode, i.e., a collective submanifold out of Gr ∞ . As mentioned in [34,35], the collective submanifold may be interpreted as a rotator on curved surface in Gr ∞ . It is stressed that the υ-HFT on F ∞ describes a dynamics on real fermion-harmonic oscillators while soliton theory does the same but on complex oscillators. This remark gives us an attractive task to extend the υ-HFT on real space su(N ) to the theory on complex space sl(N, C) removing the restriction |z| = 1. We have discussed a close connection between υ-SCFM and τ -FM on an abstract fermion Fock space and denoted them independently on S 1 . It means that algebro-geometric structures of infinite-dimensional fermion many-body systems is also realisable in finite-dimensional ones. The υ-dependent HF equation on τ M (x,ĝ), however, should lead to multi-circles, relating closely to a problem of construction of multi-dimensional soliton theory [66,67,61]. It is also a very exciting problem to investigate such new motions on the multi-circles (d: Number of circles) in finite fermion many-body systems. As suggested by the referee, the motions, on the other hand, may be related to the coupled (d + 1)D systems [61] or the linear flows on the Birkhoff strata of the universal Sato Grassmannian [68].

RPA equation embedded into infinite-dimensional Fock space 4.1 Introduction
The purpose of this section is to give a geometrical aspect of RPA equation (RPAEQ) [50,69] and an explicit expression for the RPAEQ with a normal mode on F ∞ . We also argue about the relation between a loop collective path and a formal RPAEQ (FRPAEQ). Consequently, it can be proved that the usual perturbative method with respect to periodic collective variables η and η * in TDHFT [25], is involved in the present method which aims for constructing TDHFT on the affine KM algebra. It turns out that the collective submanifold is exactly a rotator on a curved surface in the Gr ∞ . If we could arrive successfully at our final goal of clarifying relation between the SCFT and the soliton theory on a group, the present work may give us important clues for description of large-amplitude collective motions in nuclei and molecules and for construction of multi-dimensional soliton equations [61,66] since the collective motions usually occur in multi-dimensional loop space.

Construction of formal RPA equation on F ∞
We construct the FRPAEQ on F ∞ . We put the following canonicity conditions which guarantee the variables (ǫ, ǫ * ) to be an orthogonally canonical coordinate system [6,25,32]: Previously we define the infinitesimal generators of the collective submanifold X θ and X θ † (3.51) in which the term C(ĝ −1 ∂ ǫĝ ) is proved to vanish. From these infinitesimal generators and ∂ ǫ * ĝ|∂ ǫ |ĝ − ∂ ǫ ĝ|∂ ǫ * |ĝ , we obtain the weak orthogonality condition where we have used (3.26) and (3.34). As shown in [32], using Lax's ideas [70] we recast (3.47) and D tĝ = F(ĝ)ĝ, and (3.51) into From (4.4), the weak orthogonality condition (4.2) is expressed as To satisfy integrability conditions for ǫ, ǫ * and t, curvatures obtained from (4.3) should vanish; and ∂ tĝ 0 = 0. Here D t θ and D t θ † are defined as The expressions for the curvatures on the quasi-particle frame (QPF) have the same form as those of RPAEQs in the finite Fock space [50]. As mentioned before, the TDHFEQ on the F ∞ leads to the RPAEQ if we take into account only a small fluctuation around a stationary ground-state solution. The form of RPAEQ on the QPF has a following simple geometrical interpretation: Relative vector fields made of the SCF Hamiltonian around each point on loop pathes also take the form of RPAEQ around the same point which is in turn a fixed point in the QPF. Thus, the curvature equation in the QPF is regarded as the FRPAEQ on the Gr ∞ . Using (3.28), the canonical transformation forĝ is given by together with its hermitian conjugate. Owing to [50], (4.3) is rewritten on the above QPF as The subscript "qpf" means the quasi-particle frame (QPF). For (4.5) we obtain also another expression on this QPF as Further, using (4.6) and the relation i∂ ǫ F| qpf = i∂ ǫ (ĝ † F(ĝ)ĝ) = −[θ † , F]| qpf +ĝ † i∂ ǫ Fĝ, one can rewrite equations in the first line of (4.7) as From (4.6) and (4.3), the infinitesimal operators are expressed as together with the same relation for θ(ĝ † )| qpf . We have also θ † (ĝ 0 † )| qpf = −i∂ ǫĝ 0 † ·ĝ 0 and θ(ĝ 0 † )| qpf = −i∂ ǫ * ĝ 0 † ·ĝ 0 . Then, from (4.8) we can derive the FRPAEQ on the Gr ∞ in the form To obtain an explicit expression for the last term of the l.h.s. of (4.10), we introduce an auxi- The R is related to density matrix W as R = I − 2 W ( I: infinite-dimensional unit matrix). Then, we obtain Substituting the above result into (4.10), we can derive the FRPAEQ on F ∞ . Finally we show the following equations to determine the collective submanifold and motion: The canonicity condition (4.1): The FRPAEQ (4.10): Through constructions of the TDHFT and the FRPAEQ on F ∞ , the following become apparent: The ordinary perturbative method for collective variables η and η * [25] is involved in the way of construction of the TDHFT on the affine KM algebra if we restrict ourselves to su(N ). When the η and η * are represented as η = √ Ωe iϕ , we can always express γ(η, η * ) = r,s∈Zγ r,s η * r η s = r γ r z r on the Lie algebra if we put z = e iϕ . This means that the infinite-dimensional Lie algebra in the SCFT is introduced in a natural way and is useful to study various motions of fermion many-body systems.

Summary and discussions
FRPAEQ has been provided as a tool for truncating a collective submanifold with only one normal mode out of an Gr ∞ . We have given a simple geometrical interpretation for FRPAEQ. The collective submanifold is interpreted as a rotator on a curved surface in the Gr ∞ . In F ∞ , to study motions of finite fermion systems, it is manifestly natural and useful to introduce an infinite-dimensional Lie algebra arising from anti-commutation relations among fermions. In order to discuss the relation between TDHFT and soliton theory, we have given expressions for TDHFT on τ -FS along soliton theory. From the loop group viewpoint and with a clearer physical picture, we have proposed a way of describing particle and collective motions in SCFT on F ∞ in relation to an iso-spectral equation in soliton theory. Then, SCFT on F ∞ may be regarded as soliton theory in the sense that it is based on the Gr ∞ and may describe dynamics on an infinite set of real fermion-harmonic oscillators. On the other hand, soliton theory describes dynamics on complex fermion-harmonic oscillators. It is one of the most challenging problem to extend real space su(N ) to complex space sl(N ) in TDHFT on F ∞ together with removal of the restriction |z| = 1. Concerning the construction of soliton theory on multi-dimensional space [61,66], we have an interesting future problem that is to extend the Plücker relation (Hirota's form) with only one circle to the case of multi-circles such that SCFM on F ∞ can describe dynamics of fermion systems in terms of multi-RPA bosons.
5 Infinite-dimensional KM algebraic approach to LMG model

Introduction
To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ-HFT. In constructing the υ-HF theory we must observe, however, the following two different points between the maximaly-decoupled method and the υ-HF SCFM (i) The former is built on the finite-dimensional Lie algebra but the latter on the infinite-dimensional one.
(ii) The former has an SCF Hamiltonian consisting of a fermion one-body operator, which is derived from a functional derivative of an expectaion value of a fermion Hamiltonian by a ground-state wave function. The latter has a fermion Hamiltonian with a one-body type operator brought artificially as an operator which maps states on a fermion Fock space into corresponding ones on a τ -FS. Toward such an ultimate goal, the υ-HFT has been reconstructed on an affine KM algebra along the soliton theory, using infinite-dimensional fermion. An infinitedimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HFT has been transcribed onto the corresponding τ -function using the regular representation for the group and the Schurpolynomials. The υ-HF SCFM on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ). We have given explicit expressions for Laurent coefficients of soliton solutions for sl(N ) and su(N ) on the Gr ∞ using Chevalley bases for sl(N ) and su(N ). As an illustration we make the υ-HFT approach to an infinite-dimensional matrix model extended from the finite-dimensional su(2) LMG model and represent an infinite-dimensional matrix LMG model in terms of the Schur polynomials.

Application to Lipkin-Meshkov-Glick model
To show the usefulness of the infinite KM algebra and to avoid an unnecessary complication, we apply it to a simple model, the LMG model consisting of N (= M ) particles. Let us introduce the LMG Hamiltonian which has two N -fold degenerate levels with energies 1 2 ε and − 1 2 ε, respectively The operators K 0 , K + and K − are defined by where the indices i and a stand for particle-state and hole-state, respectively and satisfy the SU (2) quasi-spin algebra Then the S-det, |S N , in which the N particles fill the lower level, satisfies We here use a notation [c † ] denoting a 2N -dimensional row vector [c † a , c † i ] (i = 1, 2, . . . , N ; a = 1, 2, . . . , N ). We introduce the following SU (2N ) Thouless transformation: The above SU (2N ) matrix is essentially the direct sum of the SU (2) matrix. Any N -particle S-det is constructed by the Thouless transformation of a reference S-det, |S N (the Thouless theorem) as which is the CS rep of fermion state vector on the SU (2N ) group [5].
The HF density matrix is given as whereÎ 2N is a 2N -dimensional unit matrix andĥ 2N ,ê 2N andf 2N are defined in the next subsection. The usual HF energy g|H|g (= H[W ]) is obtained as The Fock operator F [W ] (= δH[W ]/δW T ), in the HF approximation is represented as
Then the K 0 and K ± (5.1) are expressed in terms of the above infinite-dimensional operators as follows: : ψ N (s−r)+α ψ * N s+β : : ψ N (s−r)+a ψ * N s+a : , : ψ N (s−r)+i ψ * N s+a :, : ψ N (s−r)+a ψ * N s+i :, and Let us introduce the following 2N -dimensional dual elements of the direct sum of the algebra sl(2, C) multiplied by z r : Using the formulas in Appendix C, the τ reps of the operators K 0 and K ± are given, respectively, in the following forms: from which we have the KM brackets among the operators K 0 (r) and K ± (r). For detailed calculations see Appendix F.
For details see Appendix F. Then we have the following KM brackets and the map σ K : √ N X k (k = 2r + 1) and 1 √ N Y k (k = 2r + 1) are clearly an infinite-dimensional Heisenberg subalgebra of the KM algebra sl(2, C). We also introduce the element in the form ofĥ 2N as

Representation of infinite-dimensional LMG model in terms of Schur polynomials
The expressions for the operators K 0 and K ± (5.1) in terms of the operators 1 √ N X 2r+1 , 1 √ N Y 2r+1 and 1 √ N Y 2r and the expressions for the map σ K for the operators 1 √ N X 2r+1 and 1 √ N X −(2r+1) are given as follows: Consequently we can obtain important sum-rules for the operators Y 2r+1 and Y 2r as where we have used the relation which is derived from the definition of the Schur polynomial S k (x) in Appendix D. Further we have an expression for a quadratic operator K 2 + + K 2 − as Finally from (5.5), (5.6), (5.7) and (5.8) we get an expression for the LMG Hamiltonian as . (5.9)

Summary and discussions
In the preceding section, for the LMG model we have an approximate HF operator and a τfunction up to the first order in S k (− ∂ x ), S k ( ∂ x ), S k (x) and S k (−x) which should satisfy the υ-dependent HF equation. Then, we meet inevitably with an interesting and exciting problem of solving the υ-dependent HF equation on the τ N (x, g(υ)). After determining HF parameters θ and ψ self-consistently, a further study should be made to obtain a soliton solution derived from a υ-dependent Hirota's bilinear equation regarding υ as time t and relationship between a collective motion and the soliton solution. Such attractive problems have not been treated yet and just begin to open.
Here, we will recur to the representation of the infinite-dimensional LMG model. In the previous section, we already have obtained the expression for the LMG Hamiltonian (5.9) in which, however, the commutator term in the first term in the forth line of the equation brings an anomaly (an infinitely divergent result) for us, as shown below, For the present, we ought to discard this anomalous term to construct the anomaly-free infinitedimensional LMG model.
Finally, we will point out the possibility of the extension of the present algebra sl(2, C) to the affine Lie algebra A (1) 1 . Corresponding to an extension of the adopted simple LMG model to the so-called coupled LMG model [73,74], we have a very interesting problem of constructing an A (1) 1 LMG model for which the idea given in the paper [51] is considered to be very suggestive and useful. According to [51], we can choose the Chevalley basis for A (1) 1 as follows: in which the total Hamiltonian of the coupled-LMG system is composed of two LMG model Hamitonians H 1 = H 1 K 1;0 , K 1;± and H 2 = H 2 K 2;0 , K 2;± and an interaction term K 1;+ K 2;− + K 2;+ K 1;− . We will discuss the above A 1 LMG model elsewhere using the present infinitedimensional SCF method in τ -functional space on F ∞ .

Summary and future problems
In Section 1, from algebro-geometric viewpoint, we have given a brief history of microscopic understanding of theoretical nuclear physics. It is summarized that we seek for an optimal coordinate-system describing dynamics on a group manifold based on a Lie algebra of fermion pairs. The TDHF/TDHB are nonlinear dynamics owing to their SCF characters. Seeking for collective coordinates in a fully parametrized dynamical system is exactly finding a symmetry of an evolution equation in nonlinear dynamics. In differential geometrical approaches for nonlinear problems, the integrability conditions are stated as the zero curvature of connection on the corresponding Lie groups of systems. Nonlinear evolution equations, e.g., the famous KdV and sine/sinh-Gordan equations and etc., come from the well-known Lax equation [70] which arises as the zero curvature [29]. These soliton equations describe motions of the tangent space of local gauge fields on a time t and a space x, which are Lie group/algebra-valued-equations arising from the integrability condition of gauge field with respect to t and x. In the TDHFT/TDHBT, the corresponding Lie groups are unitary transformation groups of their ortho-normal bases dependent on t but not on x.
In Section 2, along the Lax form for integrable systems, we have studied essential curvature equations to extract collective submanifolds out of the full TDHF/TDHB manifold and shown the following: (i) Expectation values of the zero curvatures for a state function become a set of equations of motion, imposing weak orthogonal conditions among infinitesimal generators, i.e., equations for tangent vector fields on the group submanifold. Those of non-zero curvatures become gradients of a potential arising from a residual Hamiltonian along collective variables. These quantities are expected to give a criterion how the collective submanifold is truncated well.
(ii) The zero-curvature equation in QPF is nothing but the FRPAEQ imposed by the weak orthogonal conditions and has a simple geometrical interpretation: Relative vector fields made of the SCF Hamiltonian around each point on an integral curve constitute solutions for the FRPA around the same point which is in turn a fixed point in QPF. It means the FRPA is a natural extension of the usual RPA for small-amplitude fluctuations around a ground state to RPA at any point on the collective submanifold. The enveloping curve, made of a solution of the FRPA at each point on an integral curve, becomes another integral curve. The integrability condition is the infinitesimal condition to transfer a solution to another solution for the evolution equation. Then the usual RPAEQ becomes nothing but a method of determining an infinitesimal transformation of symmetry if fluctuating fields are composed of only normal modes.
In Section 3, to go beyond a perturbative method with respect to collective variables to extract large-amplitude collective motions, we have studied an algebro-geometric relation between SCFM and τ -FM, method of constructing integrable equations (Hirota's equations) in soliton theory. At the beginning, descriptions of dynamical fermion systems in both the methods had looked very different manners at first glance. In abstract fermion Fock spaces, each solution space of dynamics in both the methods is the corresponding Grassmannian. There is, however, a difference between finite-dimensional and infinite-dimensional fermion systems. In spite of such a difference, we have aimed at closely connecting the concept of mean-field potential with gauge of fermions and at making a role of loop group clear and consequently we have shown the relation between both the methods: (i) The Plücker relation on the coset variable becomes analogous to the Hirota's bilinear form. The SCFM has been mainly devoted to the construction of boson-coordinate systems rather than to the construction of soliton solution by the τ -FM. It turns out that both the methods are equivalent with each other from the viewpoint of the Plücker relation or the bilinear identity equation defining Grassmannian.
(ii) The infinite-dimensional fermion operators are introduced through Laurent expansion of the finite-dimensional fermions with respect to degrees of freedom of fermions related to a υ dependent mean-field potential. Inversely, the mean-field potential is attributed to gauges of cooperating infinite-dimensional fermions. The construction of fermion operators can be contained in that of a Clifford algebra. This fact permits us to introduce an affine KM algebra. It means that the usual perturbative method with respect to collective variables with time periodicity has implicitly stood on a Gr ∞ . Then we rebuilt the υ-HFT with the use of the affine KM algebra and map it to the corresponding τ -functional space. As a result, the υ-HFT becomes a gauge theory of fermions and the collective motion appears as the motion of fermion gauges with a common factor. The physical concept of quasi-particle and vacuum in the SCFM on S 1 is connected with the Plücker relations. Extracting sub-group orbits consisting of loop paths out of the Gr M is just the formation of the Hirota's bilinear equation for the reduced KP hierarchy to su(N ) (⊂ sl(N )). The present theory gives the manifest structure of gauge theory of fermions inherent in SCFM and provides a new algebraic tool for microscopic understanding of the fermion many-body system.
(iii) Through the investigation of physical meanings for the infinite-dimensional shift operators and the conditions of reduction to sl(N ) in τ -FM from the loop group viewpoint, it is induced that there is the close connection between collective variables and spectral parameter in soliton theory and that the algebraic mechanism bringing the physical concept of particle and collective motions arises from the reduction from u(N ) to su(N ) for the υ-dependent HF Hamiltonian.
(iv) It must be stressed that though the υ-HFT describes a dynamics on real fermionharmonic oscillators, the soliton theory does on complex fermion-harmonic oscillators. This suggests us an important task to extend the υ-HFT on real space su(N ) to that on complex space sl(N ). It gives us a deeper understanding of the concept of quasi-particle energies and boson ones, in other words, independent particles and mean-field potential, in a microscopic treatment. Recently Wiegmann et al. have developed an approach in which the theory of classical integrable systems is applied to studies of 1D-fermion systems and the so-called orthogonality catastrophe in a Fermi gas. They have introduced a boundary condition changing operator [75] but have made no map σ M : : ψ i ψ * j : → z ij (x, ∂ x ) (3.53) contrary to the present υ-HFT. In Section 4, we have given a geometrical aspect of RPAEQ [50,69] and an explicit expression for the RPAEQ with a normal mode on F ∞ . We also have argued about the relation between a loop collective path and a FRPAEQ. Consequently, the usual perturbative method is shown to be involved in the present method which aims for constructing TDHFT on the affine KM algebra. It turns out that the collective submanifold is interpreted as a rotator on a curved surface in the Gr ∞ . The present theory may lead to multi-circles occurring multiple parameterized collective motions. If we could arrive successfully at such a final goal, the present work may give us important clues for description of large-amplitude collective motions in nuclei and molecules and for construction of multi-dimensional soliton equations [61,66] since the collective motions usually occur in multi-dimensional loop space.
In Section 5, as an illustration we have attempted to make a υ-HFT approach to an infinitedimensional matrix model extended from the finite-dimensional su(2) LMG model [42]. For this aim, we have given an affine KM algebra sl(2, C) (complexification of su (2)) to which the LMG generators subject and their τ representations and the σ K mappings for them. Further we have introduced infinite-dimensional "particle" and "hole" operators and operators K 0 and K ± defined by the infinite-dimensional "particle-hole" pair operators. Using these operators, we have constructed the infinite-dimensional Heisenberg subalgebra of the affine KM algebra sl(2, C). Thus the LMG Hamiltonian and its HF Hamiltonian have been represented in terms of the Heisenberg basic-elements whose representations are isomorphic to those in the corresponding boson space. They have been expressed in terms of infinite numbers of the variables x k and the derivatives ∂ x k through the Schur polynomials S k (x). Further we have obtained an approximate HF operator and a τ -function up to the first order in S k (− ∂ x ), S k ( ∂ x ), S k (x) and S k (−x).
In Appendices, we have given the infinite-dimensional representation of SU (2N ) ∞ transformation of the "particle" and "hole" operators. The expression for τ rep of (Y −(2i+1) + Y (2i+1) ), i.e., g Y −(2i+1) +Y (2i+1) (z) has been first given in terms of the Bessel functions. We have also shown an explicit expression for Plüker coordinate and calculated a quantity, det(1 N + p † p), in terms of the Schur polynomials.
Finally intimate relation of SCFT to soliton theory has been shown to come from ways of constructing a closed system of solution spaces. The ordinary SCFM has been almost devoted to approach cooperative phenomena in finite fermion systems. We must contrive construction of the optimal coordinate-system on the group manifold. For this purpose the relation between the boson expansion method for finite fermion systems and the τ -FM for infinite ones should be intensively investigated to clarify algebro-geometric structures of integrable systems. Such algebro-geometric approach will make a bridge between finite fermion systems and infinite ones. Various physical concepts and mathematical methods will work well also in the infinite ones. The SCFM based on global symmetry should be much improved noticing local symmetry of the infinite ones and then may open a new area in vigorous pursuit of wider fields of physics.
We have many future problems in connection with the above discussions, which are itemized as follows: (i) To study the relation between the quantity of non-zero curvature and the collectivity: It is interesting to study the relation using the simple LMG model, which leads to an investigation of the effective condition for the collective submanifold extracted by the zero-curvature equation. Temporarily digressing from the integrability condition, adopting the Bethe anzatz (BA) we have obtained exact solutions for the LMG model solving the BA equation [76]. Contrary to Pan and Draayer's work [63] and our previous works [77], we do not use any bosonization nor infinite-dimensional techniques and hence have no restrictions on interaction-strengths of LMG Hamiltonian. Considering the advantage of the integrability condition, the famous Gaudin model plays an important role to solve effectively the BA equation [78]. From the loop group viewpoint, as shown by Sklyanin, with the use of the exactly-solvable Gaudin model obeying the Gaudin algebra, an exponential generating function of correlators is obtained from the Gauss decomposition for sl(2, C) loop algebra, which gives correlators including the Richardson-Gaudin determinant formula for the Bethe eigen-function [79]. A generalization of the Gaudin algebra is given by Ortiz et al. [80]. These works may have an intimate relation with our recent work [35] and the present work.
(ii) To clarify the explicit relation between spectral parameter and collective variable and the physical concept of the geometrical connection: The spectral parameter of the iso-spectral equation in soliton theory and the collective variable in SCFM, though showing different aspects at a glance, work as scaling parameters on S 1 . The former relates to a scaling by analytical continuation of S 1 , i.e., z. The latter makes roles of deformation parameters of loop paths in Gr M .
(iii) To study the relation between weak boson operators and boson mapping operators, i.e., the shift operators in τ -FM: The generators for collective variables in F ∞ can never take exact boson commutation relations because of the finite-dimensional matrices.
(iv) To study a relation hidden behind gauge of state functions and construction of fermion pairs: In the usual algebraic treatment of fermion many-body systems, we assign an abstract number to each of the set of quantal numbers and let their fermions make the Lie algebras (u(N ), so(2N ), so(2N + 1) and etc.). For the pair-constructions we have an interpretation as classifications of Laurent spectra in the infinite-dimensional fermions, although we did not manifestly state. On the other hand, as well known, electron spin can be described as a geometrical phase of gauge with the help of Möbius band. Then we inversely start from fermions with the abstract numbers and through any way we could return to the original fermions with the physical quantal numbers. We think it not so wrong to attempt to understand quantal numbers as the geometrical attribute of the Grassmannian made of the abstract fermions.
(v) To establish mathematical tools to obtain subgroup-orbits of loop paths in Gr M , basing on the Plücker relations on S 1 , i.e., soliton equations. This problem is most fundamental to solve the new theory: For this aim we must know sub-group orbits or corresponding sub-Lie algebras and establish mathematical tools to extract them out of Gr M . In this concern, we are intensively interested in the algebraic mechanism for spontaneous decision of a fixed point and a collective submanifold around the point.
(vi) To study a relation between nonlinear superposition principle in soliton theory and generator coordinate method (GCM) in SCFM: The GCM may provide a superposition principle on a nonlinear space [81,16]. Standing on the viewpoint of local symmetry of infinite fermion systems behind global symmetry of finite ones, we might reconstructed the GCM and nonlinear superposition methods using the infinite-dimensional shift operators. What relation does exist between the construction of exact solutions based on the idea of the imbricate series in soliton equations [82] and resonating mean-field theories [83]?
(vii) To study why soliton solutions for classical wave equations show fermion-like behaviours in quantum dynamics and about what symmetries are hidden in soliton equations. To both the questions suggested by Tajiri et al., we cannot give a satisfactory answer yet within the present framework. Because both the methods are a priori based on the fermion system from the outset. That is to say: SCFM describes a quasi classical dynamics on Grassmannian (Sdet orbit) which is induced owing to the anti-commutative property of fermions. On the other hand, τ -FM also uses the fermions to explain Grassmannian of solution space which reflects the fermion-like behaviours of soliton solutions. Therefore we should study further why extraction of soliton equations out of classical wave equations brings out Grassmannian. In the reductive perturbation method [84], soliton equations appear as the symmetry space in a classical wave equation with respect to a transformation and a scaling transformation with a common parameter for independent and dependent variables. We can see a relation between the power exponents of the parameter and the degree of shift operators. At any point on extracting way of soliton equations from classical wave equations did we introduce the "anti-commutative character", in other words, did we introduce the structure of Grassmannian?

C Affine Kac-Moody algebra
According to Kac and Raina [49], let gl(N ) be the Lie algebra of all N ×N matrices with complex entries acting in C N and let C[z, z −1 ] be the ring of Laurent polynomials in indeterminate z and z −1 . The loop algebra [36] gl(N )(⊃ u(N )) is defined as gl(N )(C[z, z −1 ]) (⊃u(N )(C[z, z −1 ])), i.e., as the complex Lie algebra of N × N matrices with Laurent polynomials as entries. An element of gl(N ) is given in the form a(z) = r∈Z z r a r (a r ∈ gl(N )). (C.1) As was pointed out by Kac and Raina, we regard the fermion pair-and single-operators as (C. 2) The vectors defined as also form a basis of the vector space C[z, z −1 ] N indexed by Z and its dual space. The {ν i | i = N r + α ∈ Z} is given by the column vector with 1 as the i-th row and 0 elsewhere. Thus it is possible to identify C[z, z −1 ] N with C ∞ . The relation e αβ (r)ν N s+β = ν N (s−r)+α is easily derived. For a(z) ∈ gl(N ) we denote the corresponding matrix in a ∞ by τ{a(z)}. Then we deduce a matrix representation for τ{e αβ (r)} in a ∞ as E ij (i, j ∈ Z) have 1 as the (i, j) entry and 0 elsewhere and form a gl ∞ . Suppose a bigger algebra a ∞ There exists such an N satisfying the above condition. The corresponding matrix of the a(z) in (C.1) in a ∞ has an infinite N periodic sequence of block form We regard (C.5) as a representation of the matrix a ∞ in which elements on each diagonal parallel to the principal diagonal form a periodic sequence with period N . Let X(k) = z k X (X ∈ gl(N )) be an element of gl(N ). Define an antilinear anti-involution ω on gl(N ) by ω[X(k)] = z −k X † . Then in the a ∞ we get τ{ω[X(k)]} = τ † {X(k)}.
Using the fundamental idea of the Dirac theory [40], we define the vacuum state in which the state labeled by the "Laurent spectrum" with positive energy is empty but all the negative energy states labeled by the "Laurent spectra" are occupied. Denoting an exterior product of vectors as ∧, a perfect vacuum Ψ 0 and a reference vacuum Ψ M are expressed as Let the space V = ⊕ i∈Z Cν i and its dual V * = ⊕ j∈Z Cν * j be an infinite-dimensional complex space with a basis {ν i , ν * j | i, j ∈ Z} giving a linear functional ν * j on V by ν * j (ν i ) = δ ij (i, j ∈ Z). Each ν ∈ V and ν * ∈ V * defines a wedging operatorν and a contracting operatorν * on the F ∞ asν (C.6) Then the operators {ν i ,ν * j | i, j ∈ Z} generate a Clifford algebra Thus the anti-commutation relations lead us to the identification of the new fermion annhilationcreation operators (3.23) with the present operators at a pointwise z with |z| = 1 as ψ * N r+α →ν * N r+α , ψ N r+α →ν N r+α . (C.8) Using the above identification, the corresponding perfect vacuum and reference vacuum, it can be shown that Ψ 0 → |Vac , Ψ M → |M = ψ M · · · ψ 1 |Vac , ψ N r+α |Vac = 0, Vac|ψ * N r+α = 0 (r ≤ −1), ψ * N r+α |Vac = 0, Vac|ψ N r+α = 0 (r ≥ 0).
For any elements a(z) and b(z) in the gl(N ) the formula (C.12) is written as α(τ{a(z)}, τ{b(z)}) = Res 0 Tr a ′ (z)b(z), where a ′ (z) is the derivative of a with respect to z and Res 0 is the residue at z = 0, i.e., the coefficient of z −1 . Investigation of the highest weight representation of gl(N ) leads to its central extension gl(N ) = gl N + C · c in which general elements a(z) and b(z) and center C · c satisfy the KM brackets The Lie algebra gl(N ) is called the affine KM algebra associated with the Lie algebra gl(N ). For simplicity consider the level one case, c|M = 1 · |M . Using the one-level formula it is possible to rewrite (C.13) as X a = X a + C · c,
A group orbit of the highest weight vector |M under the action U (g) is mapped to a space of τ -function τ M (x, g) = M |e H(x) U (g)|M . Using (D.2), the Schur-polynomial expression for τ -function is given by (3.56). Noting the relation S λ |S µ = S λ ( ∂ x )S µ (x)| x=0 = δ λµ , It is described by the Hirota's bilinear differential operator as If we expand (E.2) into a multiple Taylor series of variables y 1 , y 2 , . . . and make each coefficient of this series vanishing, we get an infinite set of nonlinear partial differential equation for the KP hierarchy.