Algebro-geometric approach to a fermion self-consistent field theory on coset space SU(m+n)/S(U(m) x U(n))
aa r X i v : . [ h e p - t h ] J a n Toward an infinite-dimensional Kac-Moodyalgebraic approach to a fermion SCF theory ∗ Seiya NISHIYAMA † , Jo˜ao da PROVID ˆENCIA ‡ CFisUC, Department of Physics, University of Coimbra.3004-516 Coimbra, PortugalJanuary 14, 2021
To go beyond a perturbative method with respect to collective variables, we aim at con-structing a fermion self consistent field (SCF) theory, i.e., time dependent Hartree-Fock the-ory on associative affine Kac-Moody algebras along a soliton theory on infinite-dimensional(ID) fermions. The ID fermion operators are introduced through the Laurent expansion offinite-dimensional fermion operators with respect to the degrees of freedom of the fermionsrelated to a mean-field potential. The mean-field potential degrees of freedom of fermionsarise from the gauge degrees of freedom of the fermions. We study a relation betweenthe SCF theory and the soliton theory from the viewpoint of dynamics on an infinite-dimensional Grassmannian ( Gr ) with Lie algebras. Behind the Gr , the associative affineKac-Moody algebras play a crucial role to determine a dynamics of the fermion systems. According to Yamamura and Kuriyama (YK) [1], we view simply a history: In the Bohr-Mottelson (BM) model [2], a liquid drop model is taken for the collective motion andthe independent particle motion is described by the shell model. However, microscopically,the constituents of the liquid drop are nucleons themselves, which move in a nucleus in-dependently. In the year 1960, Arvieu and Veneroni, Baranger and Marumori proposed,independently, a theory for spherical even -mass nuclei [3] called the quasi-particle randomphase approximation (QRPA). There exist two types of fundamental correlations, that is,a short-range correlation and a long-range one [4]. The former is expressed in terms ofthe pairing interaction and generates a superconducting mode. Excited states are classi-fied by a seniority coupling scheme and well described in terms of quasi-particles given bythe Hartree-Bogoliubov (HB) theory [5]. The latter is expressed in terms of particle andhole operators and give rise to collective motions related to density fluctuations around theequilibrium state. The particle-hole random phase approximation (phRPA) is a method forthe collective motions such as the rotational motion and the vibrational motion around theequilibrium state. The Q/phRPA theories are systematic methods for treating phenomenaof coexistence of both correlations. However, these theories are essentially a harmonicoscillator approximation. They can be extended to take into account of nonlinear terms ∗ Using the particle–hole representation this work is made based on a part of the paper: S.Nishiyama,J.daProvidˆencia, C.Proidˆencia, F.Cordeiro and T.Komatsu,
Self-Consistent-Field Method and τ -FunctionalMethod on Group Manifold in Soliton Theory: a Review and New Results , SIGMA (2009) 009-1-76. † Corresponding author. E-mail address: [email protected] ‡ E-mail address: providencia@teor.fis.uc.pt SO (2 N ) algebra with N single-particle statesas functions of the boson operators, to estimate the deviation of the fermion-pairs fromthe pure boson character. Keeping the original idea, up to now, various BEHBs have beenproposed. These methods are classified into mainly the two classes: the first class: theboson representation is constructed to reproduce the Lie algebra which the fermion-pairsobey; the second class: in so-called the Marumori-type, let the state vectors in the fermionFock space correspond to the state vectors in the boson Fock space by one-to-one mappingand the boson representation is constructed to achieve the coincidence of the transitionmatrix-values of any physical quantity for the boson-state vectors with those in the originalfermion states. In the former, there are Holstein-Primakoff (HP)-type, Schwinger-type andDyson-type. The HP expresses the fermion-pairs in terms of an infinite series of bosonoperators, the other two do that in terms of the finite series ones. The algebraic struc-ture governing fermion-pairs and the BEHB theories have been investigated. The bosonoperators proposed by Providencia and Weneser and Marshalek [8] based on the bosonrepresentation of particle-hole pairs forming SU ( N ) algebra. Fukutome, Yamamura andone of the present authors, Nishiyama (S. N.) [9, 10, 11], found the fermions to obey the SO (2 N +1), SO (2 N +2) and U ( N +1) Lie algebras, respectively. The BEHB theory formulatedby the Schwinger type boson representation has been greatly developed by Fukutome andS. N. However, BEHB theories themselves do not contain any scheme under which thecollective degree of freedom is selected from the whole degrees of freedom.Meanwhile, there exists another approach to the microscopic theory of collective motion,that is, the time dependent Hartree-Fock (TDHF) or time dependent Hartree-Bogoliubov(TDHB) theory. At an early stage, idea of the TDHF theory was proposed by Nogami[12] and soon later by Marumori [3] for small amplitude vibrational motions. With thehelp of this method, we determine the time dependence of any physical quantity, functionof density matrix. The frequency of the small fluctuation around static HF field and theequation for the frequency are the same forms as those given by the RPA theory. The RPAtheory is quantal and the frequency given by this method means the excitation energyof the first excited state. Then, the RPA theory is a possible quantization of the TDHFtheory in the small amplitude limit.In fact, as proved by Malshalek and Horzwarth [13], theBEHB theory reduces to the TDHB theory at any order, under the replacement of bosonoperators by classical canonical variables. Using a technique analogous to the canonicaltransformation in the classical mechanics, it is expected to obtain a scheme for choosingthe collective degree of freedom in the TDHF theory. Historically, there is a stream, whosesource is the cranking model given by Inglis [14]. The basic standpoint of this model is akind of adiabatic perturbation theories. It starts from the assumption that the speed of thecollective motion is much slower than that of any other non-collective motions. The TDHFtheory with the adiabatic treatment (ATDHF) was presented by Shono and Tanaka, andThouless and Valatin at the early stage of the study of this field [15, 16].At the middle of 1970, TDHF theory was revived not only for the study of anharmonicvibration but also for studies of heavy ion-reaction and nuclear fission. These problems havea common feature from the viewpoint of large amplitude collective motions. In this newsituation, ATDHF theory was developed mainly by Baranger and Veneroni [17], Brink [18]2nd Goeke and Reinhard and Mukherjee and Pal [19]. Especially, the most important point ofthe ATDHF theory by Villars [20] is the introduction of the concept of collective path intothe phase space. A collective motion corresponds to a trajectory in the phase space whichmoves along the collective path . Along the same spirit, Holtzwarth and Yukawa [21], Roweand Bassermann [22], and Marumori, Maskawa, Sakata and Kuriyama gave the TDHF theoryso called the maximal decoupling method [23]. This theory is formulated in a canonical formframework. Therefore, various techniques of classical mechanics are useful and canonicalquantization is expected. By solving the equation, we obtain corrections whose order ishigher than the RPA order. The collective submanifold is, in a certain sense, a possibleextension from the collective path. This theory has a potentiality to give not only collectivemodes but also intrinsic modes.In the BM model, the degrees of freedom of collective motion and independent-particleone are overestimated. This problem has been inquired from the microscopical theoreticalviewpoint. See also the exact canonical momenta approach by the present authors [24]. Atan early stage of the study, Marumori, Yukawa and Tanaka [25], and Tomonaga [26, 27]proposed independently remarkable theories which, however, are, in certain sense, kine-matical because the collective motion is given a priori . In this concern, there exist threepoints to be solved dynamically: i) to determine the microscopic structure of the collectivemotion, which is the ensemble of the individual particles motion, in relation to dynamicsunder consideration, ii) to determine the independent-particle motions which should be orthogonal to the collective motion and iii) to give a coupling between these two types ofmotions. The TDHF theory in the canonical form enables us to select the collective motionin relation to the dynamics. However, it gives us no scheme to take into account the effectof the quasi-fermion, because the TD Slater determinant (S-det) contains only variablesto represent the collective motion. Then, for example, an odd-particle system cannot bedescribed. Along the same spirit as TDHF, YK extended the TDHF theory to that on afermion coherent state constructed on the TD S-det. The coherent state contains not onlythe usual canonical variables but also Grassmann (Gr) variables. Candlin [28], Berezin[29] and Casalbuoni obtained a classical image of fermions by regarding the Gr variablesas canonical [30]. The constraints governing the variables to remove the overestimated de-grees of freedom were decided under a physical consideration. With the use of the Dirac’scanonical theory [31] for a constrained system, the TDHF theory was successfully solved byYK, for a unified description of collective and independent-particle motions in the classicalmechanics [32]. The historical evolution is summarized in order to present the optimal coordinate systemto describe the group manifold itself, based on the Lie algebra in which the pairs of thefinite-dimensional fermion obey, or to do dynamics on the manifold. The boson operatorin the BEHB is the operator arising from the coordinate system of the tangent spaces onthe manifold in the fermion Fock space. The BEHB itself does not contain any schemeunder which the collective degree of freedom can be selected from the whole degrees offreedom. On the other hand, approaches to collective motions by the TDHF theory suggestthat the coordinate system on which the collective motions can be described is deeplyrelated not only to the global symmetry of the finite-dimensional group manifold itself butis also behind the local symmetry , besides the Hamiltonian [33]. Therefore, the various3ollective motions are better understood by taking the local symmetry into account. Thelocal symmetry may have close connection with an infinite-dimensional (ID) Lie algebra.The TDHF leads to nonlinear dynamics owing to the SCF characters. However, there havenot been enough attempts to manifestly understand collective motions in relation to thelocal symmetry. Therefore, from the viewpoint of symmetry of the evolution equations innonlinear problems, we study the algebro-geometric structures toward unified understand-ing of both motions. The first theme of our study is to investigate the curvature equation as a fundamental equation to extract collective submanifolds out of the TDHF manifold.We show that the zero-curvature equation in a quasi particle-hole frame (PHF) leads tothe nonlinear RPA theory which is the natural extension of the usual RPA. We denoteRPA and QRPA as RPA. We had started from a question whether soliton equations existor not in the TDHF manifold, in spite of the difference that the former is described interms of the continuously infinite degrees of freedom and the latter uses finite ones. Wehad met with AKNS formulation in the Inverse Scattering Transform (IST) method [34]and differential geometrical approaches on group manifolds developed by Sattinger [35].The essential points in the geometrical viewpoint have attracted us much:AKNS stands onthe group manifold SL (2) and integrable system can be explained by the zero-curvature(integrability condition) of connection on the corresponding Lie groups of systems. Theimage of both methods is illustrated below asFigure 1: IST method method in SL (2) group and SCF method in U ( n ) groupIn various approaches to the collective motions, those from the viewpoint of curvature werescarce. In studying the maximal decoupling method by Marumori and YK, we had an imagefor the large amplitude collective motions: If a collective submanifold is the collection ofcollective paths, the infinitesimal condition to switch from a path to another is nothingbut an integrability condition for the submanifold with respect to time t describing thetrajectory under the SCF Hamiltonian and the parameters specifying any point on thesubmanifold. However, the trajectory is unable to exactly remain on the manifold. Thecurvature is able to work as a certain criterion of effectiveness of the collective submanifold.The RPA is considered as an integrability condition under a linear approximation so thatthe idea existing behind the theory can be contained in that of curvature. From the wideviewpoint of symmetry, RPA must be extended to any point on the manifold because anequilibrium state which we select as a starting point must be equivalent to other points onthe manifold among one another. The RPA was introduced as a linear approximation totreat excited states around the equilibrium ground state which is essentially a harmonicoscillator approximation. The amplitude of oscillation becomes larger and then anhar-monicity appears so that we treat the anharmonicity by taking into account nonlinearterms in the equation of motion. We present a set of equations defining the curvature ofthe collective submanifold which become a fundamental equation to treat anharmonicity .We call it the formal RPA . It is useful to understand the algebro-geometric meanings ofthe large amplitude collective motions. The solution procedure, due to the perturbativemethod [36], suggests that it is important to study an ID Lie algebra working behind it.4o go beyond perturbative method, we study the relation between the method extracting collective motions in the SCF method and τ -functional method ( τ -FM) [37](a) constructing integrable equations in solitons , Dickey [38]. This is the second theme. The relation betweenthe τ -function and coherent state has been pointed out first by D’Ariano and Rasetti [39] foranID harmonic charged Fermi gas[40].If we stand on the observation,weassert that the SCFmethod presents the theoretical scheme for integrable subdynamics on a certain ID fermionFock space. Up to now, however, it has been insufficiently investigated the relation betweenthe SCF method in finite-dimensional fermion system and the τ -FM in ID fermion system because the description of dynamical fermion systems by them have looked very differently.We investigate the relation between the collective submanifold and various subgroup orbitsin theSCFmanifold.Makinguseof theTDHFtheoryonthe U ( N ) group, we study the relationbetween both methods, according to an essential point of the picture in the conventionalSCF method. To tackle this problem we have to solve the following main problems: first,how to imbed the finite-dimensional fermion system into a certain infinite one and torebuilt the TDHF theory on it; second, to make sure that any algebraic mechanism workingbehind the particle and collective motions, and any relation between the collective variableand a spectral parameter in soliton theory remains present; last, to select from the SCFhamiltonian various subgroup orbits and to generate a collective submanifold of them, andhow to relate the above to the formal RPA in the first theme.We obtain a unified aspect for both methods. The HF theory is made by a variationalmethod to optimize energy expectation value by S-det and to obtain a variational equationfor orbitals in S-det [41]. The particle-hole pair operators of fermion with N single-particlestates are closed under Lie multiplication and forms basis of Lie algebra u N [42]. It generatesthe Thouless transformation [43] which induces a representation (rep) of the corresponding U ( N ) group. The U ( N ) canonical transformation changes the S-det with m particles intoanother S-det. Any S-det is obtained by such a transformation of a given reference S-det(Thouless theorem). The Thouless transform provides an exact generator coordinate (GC)rep of fermion state vectors. The GC is a U ( N ) group and generating wave function (WF)is an independent-particle one. This is the generalized coherent state rep (GCS rep) [44].In soliton theory on a group manifold, transformation group to cover the solution forthe soliton equation is ID Lie group whose infinitesimal generator of the correspondingLie algebra is expressed as infinite-order differential operator of the associative affine Kac-Moody algebra. The operator is represented in terms of ID fermion. A space of complexpolynomial algebra is realized in terms of a Fock space of ID fermion . The soliton equationbecomes nothing else than the differential equation defining the group orbit of the highestweight vector in the ID Fock space F ∞ . The generating WF, GCS rep is just S-det byThouless theorem and provides a key to elucidate the interrelation of HF WF to the τ -function in soliton theory [37](a).In the abstract fermion Fock spaces, we find common features in both methods, i.e., SCFmethod and τ -FM as follows: (1) Each solution space is described as ID Grassmannian ( Gr )that is the group orbit of corresponding vacuum state.(2) The former may implicitly explainthe Pl¨ucker relation not in terms of bilinear differential equations defining finite-dimensional Gr but of a physical concept of quasi-particle and vacuum and of the mathematical languageof a coset space and coset variable.Various boson expansion methods are builtonthe Pl¨uckerrelation to hold the Gr . The latter asserts that the soliton equations are else nothing butthe bilinear differential equations. This fact gives a boson rep of the Pl¨ucker relation . We5tudy them and show both the methods stand on the common feature to be the Pl¨uckerrelation or the bilinear differential equation defining the Gr .Meanwhile, we observe the two different points between both methods: (1) The formeris built on finite -dimensional Lie algebra but the latter on ID one. (2) The former has SCF Hamiltonian consisting of fermion one-body operator, derived from functional derivativeof expectation value of fermion Hamiltonian by ground-state WF but the latter introducesartificially one-body type fermion Hamiltonian as boson mapping operator from states onfermion Fock space to corresponding ones on τ -functional space ( τ -FS).Overcoming difference due to the dimension of fermion, we aim at having close con-nection between concept of mean-field (MF) potential and gauge of fermions inherent inthe SCF method and to making a role of loop group [45] clear. Through the observation,we make ID fermion operators with finite-dimension by Laurent expansion with respectto circle S . Then with the use of an affine Kac-Moody algebra owing to the idea of theDirac’s electron-positron theory, we rebuilt a TDHF theory in F ∞ . The TDHF result ina gauge theory of fermions and the collective motion (motion of MF potential) appears asthe motion of fermion gauges with common factor ( S ). The physical concept of particle-hole and vacuum in the SCF method on S connects to the Pl¨ucker relations accordingto the idea of Dirac, saying, the algebraic mechanism extracting various sub-group orbitsconsisting of loop path out of the TDHF manifold is just
Hirota’s bilinear form [46] whichis su N ( ∈ sl N ) reduction to gl N in the τ -FM. For the sl case, see Lepowsky and Wilson [47].As a result, it is shown that the algebraic structure of ID fermion system is also realizable inthe finite-dimension. In such a constructive way, the roles of the soliton equation (Pl¨uckerrelation) and the TDHF equation are made clear and we also understand non-dispersiveproperty of the Gr and the SCF dynamics through gauge of interacting ID fermions. Thuswe have a simple unified aspect for both methods .Studying the manifest roles of the ID shift operators (bosons) in τ -FM, we derive analgebraic mechanism arising the concept of the particle and collective motions and inducea close connection between collective variable and spectral parameter, but we clear therelation more explicitly. For the last problem, further research must be made.As the last theme, basing on the above viewpoint of TDHF theory on circle S , weshow the algebraic mechanism bringing both motions. The mechanism can be elucidatedfrom the following points: first, the su N -condition for HF Hamiltonian; second, the vacuumstate (highest weight vector) according to the idea of Dirac’s electron-positron theory; andlast, the phase of fermion gauge is separated into the particle mode and the collective one .We propose a new theory for unified description of both the motions, beyond the staticHF equation and the RPA equation in the usual manner. This theory simply and clearlyelucidates not only the collective motion as that of MF potential but also the symmetrybreaking and occurrence of collective motion due to recovery of symmetry.Tajiri has suggested significant problems to be inquired about why soliton solution forclassical wave equation shows fermion-like behaviors in quantum dynamics and about whatsymmetry is hidden in soliton equation [48]. This is a very interesting problem. As we startfrom a quasi-classical dynamics in TDHF and do not notice the fermion systems behindthem, we inquire why their solution space is Grassmannian. However, we have not beenconcerned with this problem yet but gave a few observations to future problems. The anti-commutation relations of fermions are regarded as quantal orthogonal conditions amongthe canonical coordinate variables described by the Gr number [32]. The tangent space has6o norm as like as that of bosons, since the Gr number has only anti-commutative propertybut no measure as a classical number. On the contrary, the SO (2 N +1) theory [49, 50, 9]describes all degrees of freedom with respect to pair and unpaired fermions with the useof a classical number. Then we inquire how both systems of boson and fermion to relatewith each other from algebro-geometric viewpoint. We, further, inquire whether the abovequestions are concerned with the fermion-like and boson-like behaviors of solitons suggestedby Tajiri and Watanabe [48]. In which the idea of nonlinear superposition principle so calledimbricate series [51] is similar to that of GC method in the SCF method. The SCF methodand soliton theory have some similar features. However, the relation between them hasbeen insufficiently investigated yet. The explicit approaches from the viewpoint of localsymmetry are not sufficient in the SCF method. Let c α and c † α ( α = 1 , · · · ,N ) be the annihilation-creation operators of the fermion. Ow-ing to the anti-commutation relations among the fermion operators, the fermion operators E αβ = c † α c β forming the U ( N ) Lie algebra [ E αβ , E γδ ] = δ γβ E αδ − δ αδ E γβ generates a canonicaltransformation U ( G ) (Thouless transformation [43]) specified by a matrix g belonging toa U ( N ) unitary group. We decompose the creation operator [ c † α ] as [ c † α ] = [ˆ c † a , ˇ c † i ] and theannihilation operator [ c α ] as [ c α ]=[ˆ c a , ˇ c i ]. Then, the canonical transformation is expressedas[ ˆ d † , ˇ d † ] = U ( g )[ˆ c † , ˇ c † ] U − ( g ) = [ˆ c † , ˇ c † ] g, g = (cid:20) ˆ a ´ b ` b ˇ a (cid:21) , gg † = g † g = 1 N ,U − ( g )= U † ( g ) , U † ( g )= U ( g † ) , U ( g ) U ( g ′ )= U ( gg ′ ) , (1 N : N -dimensional unit matrix) (3.1)where [ˆ c † , ˇ c † ]=[(ˆ c † a ) , (ˇ c † i )] and [ ˆ d † , ˇ d † ]=[( ˆ d † a ) , ( ˇ d † i )] are row vectors and ˆ a =(ˆ a ab ) m × m matrix, a, b = 1 , · · · m (occupied states), ´ b =(´ b ai ) m × ( N − m ) matrix, i = m +1 , · · · N , ` b =(` b ia )( N − m ) × m matrix and ˇ a = (ˇ a ij ) ( N − m ) × ( N − m ) matrix i, j = m +1 , · · · N (unoccupied states) and c α | i = 0(vacuum), ˇ c i | i = 0 and ˆ c † a | i = 0. The symbols † , ⋆ and T mean hermitian conjugate, com-plex conjugation and transposition, respectively. The set of fermion operators ( ¯ E ij ) areconstructed from the operators ˆ d † and ˇ d † in (3.1) by the same way as the one to define theset ( E αβ ). Using the second equation of (A. 6), the set ( ¯ E ) in the quasi PHF is transformedinto the set ( E ) in the original particle frame as, (cid:20) ˆ d ˇ d (cid:21) [ ˆ d † , ˇ d † ] = " ¯ E ˆ •† ˆ • ¯ E ˆ •† ˇ • ¯ E ˇ •† ˆ • ¯ E ˇ •† ˇ • = g † (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] g = g † " E ˆ •† ˆ • E ˆ •† ˇ • E ˇ •† ˆ • E ˇ •† ˇ • g PHF= ⇒ (cid:20) ( hh ) ( hp )( ph ) ( pp ) (cid:21) . (3.2)¯ E ˆ • ˆ • =( ¯ E ab ) , ¯ E ˆ • ˇ • =( ¯ E ai ) , ¯ E ˇ • ˆ • =( ¯ E ia ) , ¯ E ˇ • ˇ • =( ¯ E ij ) : E ˆ • ˆ • =( E ab ) , E ˆ • ˇ • =( E ai ) , E ˇ • ˆ • =( E ia ) , E ˇ • ˇ • =( E ij )are the m × m, m × ( N − m ) , ( N − m ) × m, ( N − m ) × ( N − m ) matrices, respectively.The HF energy functional is defined in (A. 15) and the Fock operator H HF is given as E HF d = h βα Q αβ + 12 [ γα | δβ ] Q αγ Q βδ , Q≡ (cid:20) ˆ a ` b (cid:21)h ˆ a † , ` b † i = (cid:20) ˆ a ˆ a † ˆ a ` b † ` b ˆ a † ` b ` b † (cid:21) ≡ (cid:20) ˆ Q ´ Q ` Q ˇ Q (cid:21) ,H HF ≡ [ˆ c † , ˇ c † ] F (cid:20) ˆ c ˇ c (cid:21) = ( h αβ +[ αβ | γδ ] h E γδ i g ) c † α c β , F = (cid:20) ˆ F ´ F ` F ˇ F (cid:21) , F † = F , (3.3)where Q is the HF density matrix given in detail in (A. 14) and F is the HF matrix.The set ( F ) = ( F αβ ) is a particle-hole vacuum expectation value of the Lie operator, ex-pressed, using the dummy index convention to take summation over the repeated index, as, F αβ = " ˆ F ab = h ab +[ ab | cd ] h E cd i g , ´ F ai = h ai +[ ai | bj ] h E bj i g ` F ia = h ia +[ ia | jb ] h E jb i g , ˇ F ij = h ij +[ ij | kl ] h E kl i g , " h E ab i g =(` b ⋆ ´ b † ) ab , h E ai i g =(` b ⋆ ´ b † ) ai h E ia i g =(` b ⋆ ´ b † ) ia , h E ij i g =(` b ⋆ ´ b † ) ij . (3.4)7he quantities h αβ and [ αβ | γδ ] are the matrix element of the one-body Hamiltonian andthe antisymmterized one of the interaction potential. Here, it should be emphasized thatthe TDHB equation has also been derived using a path integral on the coset space of SO (2 N ) group, SO (2 N ) U ( N ) , by one of the present authors (S. N.) [52, 53]. We consider an evolution equation ∂ t u = K ( u ) for u ( x,t ). The K ( u ) is an operator acting ona function of u dependent on x and is expressed as a polynomial of differential to u withrespect to x . Regarding ∂ t u = K ( u ) as an equation to give an infinitesimal transformation offunction u , we search for a symmetry included in the evolution equation. For our aim, weintroduce another evolution equation, ∂ s u ( x,t,s )= b K ( u ( x,t,s )) for which we should search.The b K also consists of the polynomial operator of differential. This does not necessarilymean [ K, b K ] = 0 to seek for the required symmetry of the present subject, e.g,, rotationalsymmetry. The infinitesimal condition for the existence of the symmetry appears as thewell-known integrability condition ∂ s K ( u ( x,t,s )) = ∂ t b K ( u ( x,t,s )) [37](b). The maximal de-coupling method by Marumori based on the invariance principle of Schr¨odinger equationand the canonicity condition [23] are considered to adopt the integrability condition intoparametrized symmetries. This method is seen as a description of the symmetries of thecollective submanifold with respect to time and collective variables, in which the canonicitycondition makes the collective variables roles of the orthogonal coordinate of a system.In a differential geometrical approach to nonlinear problem, integrability conditionsare regarded as zero curvature of connection on the corresponding Lie groups of systems.The nonlinear evolution equations, e.g., famous KdV and sine/sinh-Gordan equations etc.,come from the well-known Lax equation [54] which arises as the zero curvature [35]. Thesesoliton equations describe motions of tangent space of local gauge fields dependent on time t and space x :i.e., equations of Lie-valued-function arising from the integrability conditionof the gauge field with respect to t and x . We also get a two-dimensional soliton solution[55], i.e., dromion of the Davey-Stewartson equation [56, 57, 58, 59]. On the other hand, inthe SCF method, TDHF etc., the corresponding Lie groups are the unitary transformationgroups of their orthonormal bases dependent on t but not on x . Although at this pointthe construction of means for both the dynamical systems are different from each other,the RPA theory also describes a motion of tangent space on the group manifold.We aim at the concept of curvature unfamiliar in the usual way in the nuclear physics.The reason why is the following: let us consider a description of motions of systems ona group manifold. An arbitrary state of the system induced by a transitive group actioncorresponds to any one point in the full group parameter space and its time evolution isgiven by an integral curve. In the whole rep space adopted, assume the existence of 2 µ parameters specifying a proper subspace in which the original motion of the system is welldefined; saying, the existence of certain symmetry. Suppose we start from a given point on aspace consisting of time and 2 µ parameters, and end at the same point again along a closedcurve. Then we have in general value of the group parameter different from the one at aninitial point on the subspace. We search for some quantity characterizing such a difference.To achieve our aim, standing on the differential geometrical viewpoint, we introduce a sortof Lagrange manner familiar to fluid dynamics to describe collective coordinate systemsand take a one-form Ω which is linearly composed of the MF Hamiltonian and infinitesimalgenerators induced by collective variable differentials of certain canonical transformation.8he integrability conditions of our system read the zero curvature.We study curvature equations of the TDHF equation to determine collective submani-folds from group theoretical viewpoint. The non-zero curvature with respect to time andcollective variable are shown to be gradient of the expectation value of the residual Hamil-tonian along the direction of the collective coordinate. The set of expectation value of thezero-curvature equation for vacuum state function is shown to be nothing but the set ofequations of motion, imposing restrictions of weak canonical commutation relations. Westudy the relation between Marumori’s ≪ maximal decoupled ≫ theory and YK and ours.We discuss the role of the non zero-curvature equation not appearing in the former theory.We investigate the nonlinear time evolution equation arising from the zero-curvature equa-tion. We find that the expression for equation in a quasi PHF is nothing but the formal RPA equation and show it to be regarded as the nonlinear RPA theory. We give a solutionprocedure by the power expansion with respect to the collective variables. Our methodsare constructed manifesting itself the structure of group under consideration in order tomake easy to understand physical characters at any point on the group manifold.
The U ( N ) WF | φ (˘ g ) i is constructed by a transitive action of U ( N ) canonical transformation U − (˘ g ) on | i ; | φ (˘ g ) i = U − (˘ g ) | i , ˘ g ∈ U ( N ). In the conventional TDHF, TD WF | φ (˘ g ) i is giventhrough the group U ( N ) parameters ˆ a, ` b, ´ b and ˇ a . They characterize the TD SC mean HFfield F whose dynamical change induces the collective motions of many fermion systems.As in the TDHF by Marumori et al. [23] and YK [32], we introduce the following TD U ( N )canonical transformation to derive collective motions within the TDHF framework: U (˘ g ) = U [˘ g ( ˘Λ( t ) , ˘Λ ⋆ ( t ))] , ˘ g ( ˘Λ( t ) , ˘Λ ⋆ ( t )) = (cid:20) ˆ a ( ˘Λ( t ) , ˘Λ ⋆ ( t )) ´ b ( ˘Λ( t ) , ˘Λ ⋆ ( t ))` b ( ˘Λ( t ) , ˘Λ ⋆ ( t )) ˇ a ( ˘Λ( t ) , ˘Λ ⋆ ( t )) (cid:21) , (4.1)where a set of TD complex functions (˘Λ( t ) , ˘Λ ⋆ ( t ))=( ˘Λ ν ( t ) , ˘Λ ⋆ν ( t ); ν = 1 , · · · ,µ ) associated withthe collective motions specifies the group parameters ˘ g . The number µ is assumed to bemuch smaller than the degree of the U ( N ) Lie algebra, which means there exist only a few collective degrees of freedom . The U (˘ g ) is a natural extension of the method in the simpleTDHF case to the TDHF on the 2 µ -dimensional (2 µ D) collective submanifold [60]. Forour aim, taking a
Lagrange-like manner , it is convenient to introduce complex parameters { Λ , Λ ⋆ } which are regarded as local coordinates to specify any point of 2 µ D collective sub-manifold. Instead of ( ˘Λ( t ) , ˘Λ ⋆ ( t )), we use the functions of { Λ , Λ ⋆ } and t ,˘Λ ν ( t ) = ˘ Λ ν (Λ , Λ ⋆ , t ) , ˘Λ ⋆ν ( t ) = ˘ Λ ⋆ν (Λ , Λ ⋆ , t ) . (4.2)This means that the set { Λ , Λ ⋆ ,t }∈ ǫ is mapped into the set { Λ , Λ ⋆ }∈ ǫ ,i.e., { ˘Λ , ˘Λ ⋆ ,t }→{ ˘Λ , ˘Λ ⋆ } through the functions { ˘ Λ , ˘ Λ ⋆ } . This means that one is considering at t a coordinate set S t thatdepends on t and is labeled by the set of pairs { (Λ , Λ ⋆ ) } . At t ′ the set S t ′ is still labeled by thesame set of pairs { (Λ , Λ ⋆ ) } . However, the state which is labeled by the pair (Λ , Λ ⋆ ) at t isdifferent from the state which is labeled by the same pair (Λ , Λ ⋆ ) at t ′ . This manner seemsvery analogous to the one founded by Lagrange in the fluid dynamics.An invariant subspacelabeled by the parameters { ˘Λ , ˘Λ ⋆ } is being assumed.The collective subspace becomes definedonly when such a transformation as the one which has been formulated is considered,but this is not enough. Now comes the definition of collective subspace. The invariantsubspace is the collective subspace if the evolution of the physical system determined bythe TDHF theory is such that its coordinates in S t , namely (Λ , Λ ⋆ ), do not change with t .9sing (4.2), U ( N ) canonical transformation is rewritten as U (˘ g )= U [ g (Λ , Λ ⋆ ,t )] ∈ U ( N ). Noticethat the functional form ˘ g (˘Λ( t ) , ˘Λ ⋆ ( t )) in (4.1) changes into another functional form g (Λ , Λ ⋆ ,t )due to the Lagrange-like manner. This enables us to take a one-form Ω, composed of theinfinitesimal generators induced by time and collective variable differentials ( ∂ t ,∂ Λ ,∂ Λ ⋆ ) ofcanonical transformation U [ g ]. Introducing the one-form Ω, we search for collective pathand collective Hamiltonian separated from other remaining degrees of freedom of systems. Following YK [32], we define Lie-algebra-valued infinitesimal generators for collective Hamil-tonian H c and collective coordinates ( O † ν , O ν ) in collective submanifolds as follows: H c d = ( i ~ ∂ t U ( g )) U − ( g ) , (4.3) O † ν d = ( i∂ Λ ν U ( g )) U − ( g ) , O ν d = ( i∂ Λ ⋆ν U ( g )) U − ( g ) , ( ν = 1 , · · · , µ ) . (4.4)We abbreviate g (Λ , Λ ⋆ ,t ) simply as g . In the TDHF theory, the Lie-algebra-valued infinites-imal generators are expressed by the trace form as H c = − Tr ( ( i ~ ∂ t g.g † ) " E ˆ •† ˆ • E ˆ •† ˇ • E ˇ •† ˆ • E ˇ •† ˇ • = [ˆ c † , ˇ c † ]( i ~ ∂ t g.g † ) (cid:20) ˆ c ˇ c (cid:21) , (4.5) O † ν = − Tr ( ( i∂ Λ n g.g † ) " E ˆ •† ˆ • E ˆ •† ˇ • E ˇ •† ˆ • E ˇ •† ˇ • = [ˆ c † , ˇ c † ]( i∂ Λ ν g.g † ) (cid:20) ˆ c ˇ c (cid:21) ,O ν = − Tr ( ( i∂ Λ ⋆n g.g † ) " E ˆ •† ˆ • E ˆ •† ˇ • E ˇ •† ˆ • E ˇ •† ˇ • = [ˆ c † , ˇ c † ]( i∂ Λ ⋆ν g.g † ) (cid:20) ˆ c ˇ c (cid:21) . (4.6)The image of a construction of collective subspace is illustrated below asFigure 2: Collective subspace in the case of single pair of collective variables ( η, η ⋆ )The direct derivation of (4.6) using (4.3) and (4.4) is given in detail in Appendix A. We,however, for the moment, remove the factor e Ξ in (4.5) and (4.6). Multiplying | φ ( g ) i onthe both sides of (4.3) and (4.4), we get a set of equations on the U ( N ) Lie algebra: D t | φ ( g ) i d = ( i ~ ∂ t − H c ) | φ ( g ) i = 0 ,D Λ ν | φ ( g ) i d = ( ∂ Λ ν + iO † ν ) | φ ( g ) i = 0 , D Λ ⋆ν | φ ( g ) i d = ( ∂ Λ ⋆ν + iO ν ) | φ ( g ) i = 0 . ) (4.7)We regard these equations (4.7) as partial differential equations for | φ ( g ) i . In order to dis-cuss the conditions for that the differential equation (4.7) can be solved, the mathematicalmethod well known as integrability conditions is useful. For this aim, we take the followingone-form Ω linearly composed of the infinitesimal generators (4.3) and (4.4):Ω = H c · dt + ~ O † ν · d Λ ν + ~ O ν · d Λ ⋆ν . (4.8)With the aid of the one-form Ω, the integrability conditions of the system read C d = d Ω − Ω ∧ Ω = 0 , (4.9)where d and ∧ denote the exterior differentiation and the exterior product, respectively.10rom the differential geometrical viewpoint, the quantity C defined in the above meansthe curvature of a connection. Then the integrability conditions may be interpreted as thevanishing of the curvature of the connection ( D t , D Λ ν , D Λ ⋆ν ; ν = 1 , · · · , µ ). The detailedstructure of the curvature is calculated to be C = C t, Λ ν d Λ ν ∧ dt + C t, Λ ⋆ν d Λ ⋆ν ∧ dt + C Λ ν ′ , Λ ⋆ν d Λ ⋆ν ∧ d Λ ν ′ + C Λ ν ′ , Λ ν d Λ ν ∧ d Λ ν ′ + C Λ ⋆ν ′ , Λ ⋆ν d Λ ⋆ν ∧ d Λ ⋆ν ′ , where C t, Λ ν d = [ D t , D Λ ν ] = i ~ ∂ t O † ν − i∂ Λ ν H c + [ O † ν , H c ] ,C t, Λ ⋆ν d = [ D t , D Λ ⋆ν ] = i ~ ∂ t O n − i∂ Λ ⋆ν H c + [ O ν , H c ] ,C Λ ν ′ , Λ ⋆ν d = [ D Λ ν ′ , D Λ ⋆ν ] = i∂ Λ ν ′ O ν − i∂ Λ ⋆ν O † ν ′ + [ O ν , O † ν ′ ] ,C Λ ν ′ , Λ ν d = [ D Λ ν ′ , D Λ ν ] = i∂ Λ ν ′ O † ν − i∂ Λ ν O † ν ′ + [ O † ν , O † ν ′ ] ,C Λ ⋆ν ′ , Λ ⋆ν d = [ D Λ ⋆ν ′ , D Λ ⋆ν ] = i∂ Λ ⋆ν ′ O ν − i∂ Λ ⋆ν O ν ′ + [ O ν , O ν ′ ] . (4.10)The vanishing of curvature C means C • , • = 0. For basic study of differential geometry, seethe famous textbooks [61] and [62].Finally using the expressions of (4.5) and (4.6), we get the following set of Lie-algebra-valued equations as the integrability conditions of the partial differential equations (4.7). C t, Λ ν = [ˆ c † , ˇ c † ] C t, Λ ν (cid:20) ˆ c ˇ c (cid:21) , C t, Λ ⋆ν = [ˆ c † , ˇ c † ] C t, Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) ,C Λ ν ′ , Λ ⋆ν =[ˆ c † , ˇ c † ] C Λ ν ′ , Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) , C Λ ν ′ , Λ ν =[ˆ c † , ˇ c † ] C Λ ν ′ , Λ ν (cid:20) ˆ c ˇ c (cid:21) , C Λ ⋆ν ′ , Λ ⋆ν =[ˆ c † , ˇ c † ] C Λ ⋆ν ′ , Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) , (4.11)where C t, Λ ν = i ~ ∂ t θ † ν − i∂ Λ ν F c +[ θ † ν , F c ] , C t, Λ ⋆ν = i ~ ∂ t θ ν − i∂ Λ ⋆ν F c +[ θ ν , F c ] , C Λ ν , Λ ⋆ν = i∂ Λ ν ′ θ ν − i∂ Λ ⋆ν θ † ν ′ +[ θ ν , θ † ν ′ ] , C Λ ν ′ , Λ ν = i∂ Λ ν θ † ν − i∂ Λ ν θ † ν ′ +[ θ † ν , θ † ν ′ ] , C Λ ⋆ν ′ , Λ ⋆ν = i∂ Λ ⋆ν θ ν − i∂ Λ ⋆ν θ ν ′ +[ θ ν , θ ν ′ ] . (4.12)The quantities F c , θ † ν and θ ν are defined through partial differential equations, i ~ ∂ t g = F c g, (4.13) i∂ Λ ν g = θ † ν g, i∂ Λ ⋆ν g = θ ν g, ( ν = 1 , · · · , µ ) . (4.14)The quantities C • , • are naturally regarded as the curvature of the connection on the groupmanifold. The reason becomes clear if we take the following procedure quite parallel withthe above one: Starting from (4.13) and (4.14), we are led to a set of partial differentialequations on the U ( N ) Lie group g for ( ν = 1 , · · · , µ ) as D t g d = ( i ~ ∂ t − F c ) g = 0 , D Λ ν g d = ( ∂ Λ ν + iθ † ν ) g = 0 , D Λ ⋆ν g d = ( ∂ Λ ⋆ν + iθ ν ) g = 0 . ) (4.15)The curvature C • , • ( d = [ D • , D • ]) of the connection ( D t , D Λ ν , D Λ ⋆ν ; ) is shown to be equivalentto the quantity C • , • in (4.12). The above set of the Lie-algebra-valued equations (4.11)evidently leads us to putting all the curvatures C • , • equal to be zero.On the other hand, TDHF Hamiltonian (3.3), being Hamiltonian on U ( N ) WF space,is represented in the same form as (4.3), H HF = ( i∂ t U ( g ′ )) U − ( g ′ ) , (4.16)where g ′ is any point on the U ( N ) group manifold. The RHS of (4.16) is transformed intothe same form as (4.5). This fact leads us to the well-known TDHF equation, i ~ ∂ t g ′ = F g ′ . (4.17)The TDHF Hamiltonian is decomposed into two components at the reference point g ′ = g
11n the group manifold as H HF | U ( g ′ )= U ( g ) = H c + H res , F | g ′ = g = F c + F res , (4.18)where the second parts H res and F res mean residual components extracted out of a well-defined collective submanifold for which we must search now. For our aim, let us introduceanother curvatures C ′ t, Λ ν and C ′ t, Λ ν ⋆ with the same forms as those in (4.11) and (4.12), inwhich instead of F c , it is replaced by F | g ′ = g (= F c + F res ). The vacuum expectation valuesof another Lie-algebra-valued curvatures are easily calculated for ν = 1 , · · · , µ as h C t, Λ ν i = h C t, Λ ν ⋆ i = 0 , hC ′ t, Λ i g = − i∂ Λ h H res i g and h C ′ t, Λ ⋆ν i g = − i∂ Λ ⋆ν h H res i g , (4.19) h H res i g = − Tr (cid:26) g (cid:20) m − N − m (cid:21) g † ( F g − i ~ ∂ t g.g † ) (cid:27) , (4.20)The first Eq. of (4.19) is derived through the relations i ~ ˙ θ ν = i∂ Λ ν F c and i ~ ˙ θ † ν = i∂ Λ ν ⋆ F c which are derived with the aid of (4.13) and (4.14). Eqs.(4.19) and (4.20) are interpretedthat the values of C ′ t, Λ ν and C ′ t, Λ ν ⋆ represent the gradient of energy of the residual Hamil-tonian in 2 µ -dimensional manifold. Suppose the existence of a well-defined collective sub-manifold. Then it is not so wrong to deduce the following result: The energy value of theresidual Hamiltonian becomes almost constant on the collective submanifold, δ g h H res i g ∼ = 0 , ∂ Λ ν h H res i g ∼ = 0 and ∂ Λ ⋆ν h H res i g ∼ = 0 , (4.21)where δ g means g -variation, regarding g as functions of (Λ , Λ ⋆ ) and t . It may be achievedif we should determine g (collective path) and F c (collective Hamiltonian) through theauxiliary quantity ( θ, θ † ) so as to satisfy H c + const = H HF as far as possible. Then putting F c = F in (4.12), we seek for g and F c satisfying C t, Λ ν ∼ = 0 and C t, Λ ⋆ν ∼ = 0 , C Λ ν ′ , Λ ⋆ν = 0 , C Λ ν ′ , Λ ν = 0 and C Λ ⋆ν ′ , Λ ⋆ν = 0 . (4.22)The set of the equations C • , • = 0 makes an essential role to determine the collective sub-manifold in the TDHF method. The set of the Eqs. (4.22) and (4.15) becomes an ourfundamental equation for describing the collective motions, under the restrictions (4.32). If we hope to describe collective motions through TD complex variables ( ˘Λ( t ) , ˘Λ ⋆ ( t )), wemust know explicit forms of ˘Λ and ˘Λ ⋆ in (4.2) in terms of (Λ , Λ ⋆ ) and t . For this aim, it isnecessary to discuss the correspondence of Lagrange-like manner to the usual one.We define the Lie-algebra-valued infinitesimal generators of collective submanifolds as˘ O † ν d = ( i∂ ˘Λ n U (˘ g )) U − (˘ g ) , ˘ O ν d = ( i∂ ˘Λ ⋆ν U (˘ g )) U − (˘ g ) , ( ν = 1 , · · · , µ ) , (4.23)whose form is the same as the one in (4.4). In order to guarantee the variables ˘Λ ν (= ˘Λ ν ( t ))and ˘Λ ⋆ν (= ˘Λ ⋆ν ( t )) to be canonical, following Marumori [23] and YK [32], we set up thefollowing expectation values with use of the U ( N ) WF | φ (˘ g ) i : h φ (˘ g ) | i∂ ˘Λ ν | φ (˘ g ) i = h φ (˘ g ) | ˘ O † ν | φ (˘ g ) i = i ˘Λ ⋆ν , h φ (˘ g ) | i∂ ˘Λ ⋆ν | φ (˘ g ) i = h φ (˘ g ) | ˘ O ν | φ (˘ g ) i = − i ˘Λ ν . ) (4.24)The above relation leads us to the week canonical commutation relation h φ (˘ g ) | [ ˘ O ν , ˘ O † ν ′ ] | φ (˘ g ) i = δ νν ′ , ( ν, ν ′ = 1 , · · · , µ ) , h φ (˘ g ) | [ ˘ O † ν , ˘ O † ν ′ ] | φ (˘ g ) i = 0 , h φ (˘ g ) | [ ˘ O ν , ˘ O ν ′ ] | φ (˘ g ) i = 0 , (4.25)proof of which is, shown by Marumori and YK [23, 32], due to the integrability conditions.Using (4.7), the collective Hamiltonian H c and the infinitesimal generators O † ν and O ν in the Lagrange-like manner are expressed by ˘ O † ν and ˘ O ν in the usual manner as follows:12 c = ~ ∂ t ˘ Λ ν ˘ O † ν + ~ ∂ t ˘Λ ⋆ν ˘ O ν ,O † ν = ∂ Λ ν ˘ Λ ν ′ ˘ O † ν ′ + ∂ Λ ν ˘ Λ ⋆ν ′ ˘ O ν ′ , O ν = ∂ Λ ⋆ν ˘ Λ ν ′ ˘ O † ν ′ + ∂ Λ ⋆ν ˘ Λ ⋆ν ′ ˘ O ν ′ . ) (4.26)Substituting (4.26) into (4.10), it is carried out to evaluate the expectation values of theLie-algebra-valued curvatures C • , • by the U ( N ) WF | φ [˘ g ( ˘Λ( t ) , ˘Λ ⋆ ( t ))] i (= | φ [ g (Λ , Λ ⋆ , t )] i ).A weak integrability condition requiring the expectation values h φ (˘ g ) | C • , • | φ (˘ g ) i = 0 yieldspartial differential equations with aid of quasi particle-hole vacuum property, d | φ ( g ) i = 0: ∂ Λ ν ˘ Λ ν ′ ∂ t ˘ Λ ⋆ν ′ − ∂ Λ ν ˘ Λ ⋆ν ′ ∂ t ˘ Λ ν ′ = − Tr { / Q ( g )[ θ † ν , F c / ~ ] } ,∂ Λ ⋆ν ˘ Λ ν ′ ∂ t ˘ Λ ⋆ν ′ − ∂ Λ ⋆ν ˘ Λ ⋆ν ′ ∂ t ˘ Λ ν ′ = − Tr { / Q ( g )[ θ † ν , F c / ~ ] } , ) (4.27) ∂ Λ ⋆ν ˘ Λ ν ′′ ∂ Λ ν ′ ˘ Λ ⋆ν ′′ − ∂ Λ ⋆ν ˘ Λ ⋆ν ′′ ∂ Λ ν ′ ˘ Λ ν ′′ = Tr { / Q ( g )[ θ ν , θ † ν ′ ] } ,∂ Λ ν ˘ Λ ν ′′ ∂ Λ ν ′ ˘ Λ ⋆ν ′′ − ∂ Λ ν ˘ Λ ⋆ν ′′ ∂ Λ ν ′ ˘ Λ ν ′′ = Tr { / Q ( g )[ θ † ν , θ † ν ′ ] } ,∂ Λ ⋆ν ˘ Λ ν ′′ ∂ Λ ⋆ν ′ ˘ Λ ⋆ν ′′ − ∂ Λ ⋆ν ˘ Λ ⋆ν ′′ ∂ Λ ⋆ν ′ ˘ Λ ν ′′ = Tr { / Q ( g )[ θ ν , θ ν ′ ] } , (4.28)where another U ( N ) HF density matrix / Q ( g ) is defined as / Q ( g ) d = g " m − N − m g † ≡ " ˆ / Q ´ / Q ` / Q ˇ / Q , / Q † = / Q , / Q = 1 N , (4.29)using the expression for g , (A. 4), the density matrix / Q is explicitly expressed as / Q = (cid:20) C ( ξ ) e υ − S † ( ξ ) e υ ⋆ S ( ξ ) e υ ˜ C ( ξ ) e υ ⋆ (cid:21)(cid:20) m − N − m (cid:21)(cid:20) e − υ C ( ξ ) e − υ S † ( ξ ) − e − υ ⋆ S ( ξ ) e − υ ⋆ ˜ C ( ξ ) (cid:21) = 2 (cid:20) C ( ξ ) C ( ξ ) S † ( ξ ) S ( ξ ) C ( ξ ) S ( ξ ) S † ( ξ ) (cid:21) − N = 2 Q− N , (due to ( A. , i.e. , " ˆ / Q ´ / Q ` / Q ˇ / Q = (cid:20) Q− m Q Q Q− N − m (cid:21) . (4.30)The parameters involved in g are functions given in terms of the complex variables (Λ , Λ ⋆ )and t . In derivation of (4.27) and (4.28) , we have used the transformation property (3.2),the trace formulae appeared in (4.5) and (4.6) and the differential formulae given as h φ (˘ g ) | i∂ ˘Λ ν ′ ˘ O ν | φ (˘ g ) i = − δ νν ′ , h φ (˘ g ) | i∂ ˘Λ ⋆ν ′ ˘ O † n | φ (˘ g ) i = δ νν ′ , h φ (˘ g ) | i∂ ˘Λ ν ′ ˘ O † ν | φ (˘ g ) i = 0 , h φ (˘ g ) | i∂ ˘Λ ⋆ν ′ ˘ O ν | φ (˘ g ) i = 0 , (due to (4.24) - (4.25)) . (4.31)This is the consequence of canonicity condition and weak canonical commutation relation.Through the above procedure, as a final goal, we get the correspondence of Lagrange-like manner to the usual one. We have no unknown quantities in the RHS of (4.27) and(4.28), if we could completely solve the fundamental equations. Then we become able toknow in principle the explicit forms of the functions ˘ Λ and ˘ Λ ⋆ in terms of (Λ , Λ ⋆ ) and t bysolving the set of partial differential equations (4.27) and (4.28). However we should takeenough notice of the roles different from each other made by (4.27) and (4.28), respectively,to construct the solutions. It turns out that the LHS in (4.28) has a close connection withLagrange bracket often appeared in analytical dynamics. Since we have set up from theoutset the canonicity conditions to guarantee the complex variables ( ˘Λ , ˘Λ ⋆ ) in the usualmanner being canonical, the functions ˘ Λ and ˘ Λ ⋆ in (4.2) can be interpreted as functionsgiving a canonical transformation from ( ˘Λ , ˘Λ ⋆ ) to another complex variables (Λ , Λ ⋆ ) in theLagrange-like manner. From this interpretation, we see that requirement of the canonicalinvariance imposes the following restrictions on the RHS of (4.28) for ν, ν ′ = 1 , · · · , µ :Tr { / Q ( g )[ θ ν , θ † ν ′ ] } = δ νν ′ , Tr { / Q ( g )[ θ † ν , θ † ν ′ ] } = 0 , Tr { / Q ( g )[ θ ν , θ ν ′ ] } = 0 . (4.32)13t is quite self-evident that, combining (4.28) with (4.32), we get the Lagrange bracket forthe canonical transformation from ( ˘Λ , ˘Λ ⋆ ) to (Λ , Λ ⋆ ). From the correspondence arguments,it is reasonable to add the restrictions (4.32) to our fundamental equations in order todescribe the collective motion in terms of the canonical coordinate variables.We have studied the integrability conditions of the TDHF equation to determine thecollective submanifolds from the group theoretical viewpoint. Our idea lies in the adoptionof the Lagrange-like manner to describe the collective coordinates. It should be notedthat the variables are nothing but the parameters to describe the symmetry of TDHFequation. Introducing the one-form, we gave the integrability conditions, the vanishingof the curvatures of the connection, expressed as the Lie-algebra-valued equations. TheTDHF Hamiltonian H HF is decomposed into a collective Hamiltonian H c and a residualone H res . To search for well-defined collective submanifold, we demand that the expectationvalue of the curvature is minimized so as to satisfy H res ∼ =const or H c +const= H HF as faras possible. Further, we impose the restriction to assure the Lagrange bracket for theusual and Lagrange-like variables. Our fundamental equation together with the restrictedcondition describes the collective motion of a system. It is expected to work well in a largescale beyond U ( N ) RPA as the small amplitude limit, with certain boundary condition. To extract the collective submanifolds, we demand the zero curvature of the connection onthe TDHF manifold. We transform the fundamental equation into the equation in quasiPHF. The TDHF Hamiltonian in the quasi PHF is expressed as U − ( g ) H HF U ( g ) = [ ˆ d † , ˇ d † ] F (cid:20) ˆ d ˇ d (cid:21) = [ˆ c † , ˇ c † ] g † F g (cid:20) ˆ c ˇ c (cid:21) , g † F g = F o . (4.33)The infinitesimal generators of collective submanifolds and integrability conditions for ν = 1 , · · · , µ , expressed as Lie-algebra-valued equation, are rewritten in the quasi PHFas, H c = [ˆ c † , ˇ c † ] F o − c (cid:20) ˆ c ˇ c (cid:21) , F o − c = (cid:20) ˆ F o − c ´ F o − c ` F o − c ˇ F o − c (cid:21) , (4.34) O † ν = [ˆ c † , ˇ c † ] θ † o − ν (cid:20) ˆ c ˇ c (cid:21) , ( θ † o − ν ≡ g † θ † ν g ) , O ν = [ˆ c † , ˇ c † ] θ o − ν (cid:20) ˆ c ˇ c (cid:21) , ( θ o − ν ≡ g † θ ν g ) , (4.35) U − ( g ) C t, Λ ν U ( g ) = [ˆ c † , ˇ c † ] C o − t, Λ ν (cid:20) ˆ c ˇ c (cid:21) = U − ( g ) C t, Λ ⋆ν U ( g ) = [ˆ c † , ˇ c † ] C o − t, Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) = 0 ,U − ( g ) C Λ ν ′ , Λ ⋆ν U ( g ) = [ˆ c † , ˇ c † ] C o − Λ ν ′ , Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) = 0 ,U − ( g ) C Λ ν ′ , Λ ν U ( g )=[ˆ c † , ˇ c † ] C o − Λ ν ′ , Λ ν (cid:20) ˆ c ˇ c (cid:21) = U − ( g ) C Λ ⋆ν ′ , Λ ⋆ν U ( g )=[ˆ c † , ˇ c † ] C o − Λ ⋆ν ′ , Λ ⋆ν (cid:20) ˆ c ˇ c (cid:21) = 0 , (4.36) U − ( g ) C t, Λ ν U ( g ) = i ~ ∂ t θ † o − ν − i∂ Λ ν F o − c − [ θ † o − ν , F o − c ] ,U − ( g ) C t, Λ ⋆ν U ( g ) = i ~ ∂ t θ o − ν − i∂ Λ ⋆ν F o − c − [ θ o − ν , F o − c ] ,U − ( g ) C Λ ν ′ , Λ ⋆ν U − ( g ) = i∂ Λ ν ′ θ o − ν − i∂ Λ ⋆ν θ † o − ν ′ − [ θ o − ν , θ † o − ν ′ ] , ( ν, ν ′ = 1 , · · · , µ ) ,U − ( g ) C Λ ν ′ , Λ ν U ( g ) = i∂ Λ ν θ † o − ν − i∂ Λ ν θ † o − ν ′ − [ θ † o − ν , θ † o − ν ′ ] ,U − ( g ) C Λ ⋆ν ′ , Λ ⋆ν U ( g ) = i∂ Λ ⋆ν θ o − ν − i∂ Λ ⋆ν θ o − ν ′ − [ θ o − ν , θ o − ν ′ ] , (4.37)where all the curvatures C o −• , • should be vanished. The F o − c , θ † o − ν and θ o − ν satisfy thepartial differential equations on the U ( N ) Lie group manifold g , − i ~ ∂ t g † = F o − c g † , (4.38) − i∂ Λ ν g † = θ † o − ν g † , − i∂ Λ ⋆ν g † = θ o − ν g † . (4.39)14he TDHF Hamiltonian in a quasi PHF is decomposed into collective and residual ones as U − ( g ) H HF U ( g ) = H c + H res , F o = F o − c + F o − res , (4.40)at a reference point g on U ( N ) group. The curvatures C ′ t, Λ ν and C ′ t, Λ ⋆ν introduced previouslyhave the same forms as those in (4.36) and (4.37) except that F o − c is replaced by F o . Thenthe corresponding curvatures C ′ o − t, Λ ν and C ′ o − t, Λ ⋆ν are also decomposed into as, C ′ o − t, Λ ν = C co − t, Λ ν + C reso − t, Λ ν , C ′ o − t, Λ ⋆ν = C co − t, Λ ⋆ν + C reso − t, Λ ⋆ν . (4.41)The collective curvatures C co − t, Λ ν and C co − t, Λ ⋆ν arising from F o − c are given in the same formsas the ones in (4.37). The residual ones C reso − t, Λ ν and C reso − t, Λ ⋆ν arising from F o − res are defined as C reso − t, Λ ν = − i∂ Λ ν F o − res − [ θ † o − ν , F o − res ] , C reso − t, Λ ⋆ν = − i∂ Λ ⋆ν F o − res − [ θ o − ν , F o − res ] . (4.42)Using (4.39) and (4.40), Lie-algebra-valued forms of curvatures are calculated to be C rest, Λ ν = − i∂ Λ ν
6) and (4 . . (5. 4)The way of extracting the collective submanifolds out of the full TDHF manifold ismade possible by the minimization of the residual curvature. This is achieved if we requireat least the expectation values of the residual curvatures be minimized as far as possible, h φ (˘ g ) | C rest, Λ ν | φ (˘ g ) i = Tr C resξ,ν ∼ = 0 , h φ (˘ g ) | C rest, Λ ⋆ν | φ (˘ g ) i = Tr C resξ ⋆ ,ν ∼ = 0 . (5. 5) C rest, Λ ν and C rest, Λ ⋆ν are given in the same way as the one in (4.36). We adopt the condition ofthe stationary HF method (YK) [32]. The so-called dangerous term in F o − res must vanish,` F o − res = 0 , ´ F o − res = 0 . (5. 6)With the aid of (5. 3) and (5. 4), (5. 6) and (5. 5) are rewritten as` F o = ~ ˙˘ Λ ν ˘ ψ o,ν + ~ ˙˘ Λ ⋆ν ˘ ϕ † o,ν , ´ F o = ~ ˙˘ Λ ν ˘ ϕ o,ν + ~ ˙˘ Λ ⋆ν ˘ ψ † o,ν , (5. 7) i∂ Λ ν Tr ˆ F o − res = i∂ Λ ν Tr ˆ F o + ∂ Λ ν ( ~ ˙˘ Λ ν ′ ˘Λ ⋆ν ′ − ~ ˙˘ Λ ⋆ν ′ ˘Λ ν ′ ) ∼ = 0 ,i∂ Λ ⋆ν Tr ˆ F o − res = i∂ Λ ⋆ν Tr ˆ F o + ∂ Λ ⋆ν ( ~ ˙˘ Λ ν ′ ˘Λ ⋆ν ′ − ~ ˙˘ Λ ⋆ν ′ ˘Λ ν ′ ) ∼ = 0 . (5. 8)16 .1 Fundamental equation in fluctuating quasi PHF A paired-mode amplitude g (Λ , Λ ⋆ ,t ) is devided into stationary and fluctuating components, g = g ˜ g , i.e., the product of stationary g and fluctuating ˜ g (Λ , Λ ⋆ ,t ). The g satisfies theHF eigenvalue equation. A density matrix / Q (Λ , Λ ⋆ ,t ) is decomposed as / Q = g e / Q g † . Thefluctuating ˜ θ ν and ˜ θ † ν are given as ˜ θ ν = g † θ ν g and ˜ θ † ν = g † θ † ν g , ( ν = 1 , · · · , µ ) . Under thedecomposition g = g ˜ g , the set of fundamental equation (4.44), is transformed to i ~ ∂ t ˜ θ † ν − i∂ Λ ν e F c +[˜ θ † ν , e F c ] = 0 , i ~ ∂ t ˜ θ ν − i∂ Λ ⋆ν e F c +[˜ θ ν , e F c ] = 0 , (5. 9) i∂ Λ ν ′ ˜ θ ν − i∂ Λ ⋆ν ˜ θ † ν ′ +[˜ θ ν , ˜ θ † ν ′ ] = 0 ,i∂ Λ ν ′ ˜ θ † ν − i∂ Λ ν ˜ θ † ν ′ +[˜ θ † ν , ˜ θ † ν ′ ] = 0 , i∂ Λ ⋆ν ′ ˜ θ ν − i∂ Λ ⋆ν ˜ θ ν ′ +[˜ θ ν , ˜ θ ν ′ ] = 0 , (5. 10)Tr ne / Q (˜ g )[˜ θ ν , ˜ θ † ν ′ ] o = δ νν ′ , Tr ne / Q (˜ g )[˜ θ † ν , ˜ θ † ν ′ ] o = 0 , Tr ne / Q (˜ g )[˜ θ ν , ˜ θ ν ′ ] o = 0 , (5. 11)where e F c , ˜ θ † ν and ˜ θ ν satisfy partial differential equations given below, i ~ ∂ t ˜ g = e F c ˜ g, (5. 12) i∂ Λ ν ˜ g = ˜ θ † ν ˜ g, i∂ Λ ⋆ν ˜ g = ˜ θ ν ˜ g. (5. 13)Putting e F c = e F of ( 6. 1) in (5. 9), we look for collective-path ˜ g and collective-Hamiltonian e F c under the minimization of the residual curvature arising from residual Hamiltonian e F res .Next, for our convenience of further discussion, we also introduce modified fluctuatingauxiliary quantities with the same forms as those in the previous section, through,˜ θ † o − ν = ˜ g † ˜ θ † ν ˜ g, ˜ θ o − ν = ˜ g † ˜ θ ν ˜ g. (5. 14)We rewrite a set of equations (5. 9), (5. 10) and (5. 11) in terms of the above quantitiesas i ~ ∂ t ˜ θ † o − ν − i ˜ g † ( ∂ Λ ν ˜ F c )˜ g = 0 , i ~ ∂ t ˜ θ o − ν − i ˜ g † ( ∂ Λ ⋆ν ˜ F c )˜ g = 0 , (5. 15) i∂ Λ ν ′ ˜ θ o − ν − i∂ Λ ⋆ν ˜ θ † o − ν ′ − [˜ θ o − ν , ˜ θ † o − ν ′ ] = 0 ,i∂ Λ ν ˜ θ † o − ν − i∂ Λ ν ˜ θ † o − ν ′ − [˜ θ † o − ν , ˜ θ † o − ν ′ ] = 0 , i∂ Λ ⋆ν ˜ θ o − ν − i∂ Λ ⋆ν ˜ θ o − ν ′ − [˜ θ o − ν , ˜ θ o − ν ′ ] = 0 . (5. 16)In the derivation of Eqs. (5. 15) and (5. 16), we have used (5. 12) and (5. 13), respectively.While, ˘ θ † o − ν (cid:18) = (cid:20) ˘ ξ o ˘ ϕ o ˘ ψ o ˘ ζ o (cid:21) ν (cid:19) obeys − i∂ ˘Λ ν ˘ g † = ˘ θ † o − ν ˘ g † by which (5. 7) is changed to the equationof path for collective motion. Using the rep. (3.1) of g , we get partial differential equations˘ ξ ,ν = − i ( ∂ ˘Λ ν ˘ˆ a † ˘ˆ a + ∂ ˇΛ ν ˘` b † ˘` b ) , ˘ ϕ ,ν = − i ( ∂ ˘Λ ν ˘ˆ a † ˘´ b + ∂ ˇΛ ν ˘` b † ˘ˇ a ) , ˘ ψ ,ν = − i ( ∂ ˘Λ ν ˘´ b † ˘ˆ a + ∂ ˘Λ ν ˘ˇ a † ˘` b ) , ˘ ζ ,ν = − i ( ∂ ˘Λ ν ˘´ b † ˘´ b + ∂ ˇΛ ν ˘ˇ a † ˘ˇ a ) . (5. 17)Substituting ˘ F o = ˘ g † ˘ F ˘ g and (5. 17) into (5. 7), we obtain for ˘` F o and ˘´ F o approximately as˘ˇ a † ( ˘` F ˘ˆ a + ˘ˆ F ˘` b )+˘´ b † ( ˘ˆ F ˘ˆ a + ˘´ F ˘` b )+ ~ i ˙˘Λ ν ∂ ˘Λ ν ˘´ b † ˘ˆ a − ~ i ˙˘Λ ⋆ν ˘ˇ a † ∂ ˘Λ ⋆ν ˘` b + ~ i ˙ˇΛ ν ∂ ˇΛ n ˘ˇ a † ˘` b − ~ i ˙˘Λ ⋆ν ˘´ b † ∂ ˘Λ ⋆ν ˘ˆ a = 0 , = ˘ˇ a † n (˘` F ˘ˆ a + ˘ˆ F ˘` b ) − ~ i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘` b o +˘´ b † n (˘ˆ F ˘ˆ a + ˘´ F ˘` b ) − ~ i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘ˆ a o + ~ i ˙˘Λ ν ∂ ˘Λ ν ˘´ b † ˘ˆ a + ~ i ˙ˇΛ ν ∂ ˇΛ n ˘ˇ a † ˘` b, = ˘ˇ a † n(cid:16) ˘` F ˘ˆ a + ˘ˆ F ˘` b (cid:17) − ~ (cid:16) i ˙ˇΛ ν ∂ ˇΛ n ˘` b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘` b (cid:17)o +˘´ b † n(cid:16) ˘ˆ F ˘ˆ a + ˘´ F ˘` b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˆ a + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘ˆ a (cid:17)o , (5. 18)17ˆ a † ( ˘´ F ˘ˇ a + ˘ˆ F ˘´ b )+˘` b † ( ˘ˇ F ˘ˇ a + ˘` F ˘´ b )+ ~ i ˙˘Λ ν ∂ ˘Λ ν ˘ˆ a † ˘´ b − ~ i ˙˘Λ ⋆ν ˘ˆ a † ∂ ˘Λ ⋆ν ˘` b + ~ i ˙˘Λ ⋆ν ∂ ˘Λ ν ˘` b † ˘ˇ a − ~ i ˙ˇΛ ν ˘ˆ a † ∂ ˇΛ ⋆ν ˘´ b =0 , = ˘ˆ a † n ( ˘´ F ˘ˇ a + ˘ˆ F ˘´ b ) − ~ i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘´ b o +˘` b † n ( ˘ˇ F ˘ˇ a + ˘` F ˘´ b ) − ~ i ˙ˇΛ ⋆ν ∂ ˇΛ ⋆ν ˘ˇ a o + ~ i ˙˘Λ ν ∂ ˘Λ ν ˘ˆ a † ˘´ b + ~ i ˙˘Λ ν ∂ ˘Λ ν ˘` b † ˘ˇ a = ˘ˆ a † n(cid:16) ˘´ F ˘ˇ a + ˘ˆ F ˘´ b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘´ b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘´ b (cid:17)o +˘` b † n(cid:16) ˘ˇ F ˘ˇ a + ˘` F ˘´ b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˇ a + i ˙ˇΛ ⋆ν ∂ ˇΛ ⋆ν ˘ˇ a (cid:17)o , (5. 19)where we have assumed the anti-commutatibity ∂ ˘Λ ν ˘ˆ a † = − ˘ˆ a † ∂ ˘Λ ν , and ∂ ˘Λ ν ˘` b † = − ˘` b † ∂ ˘Λ ν . From (3.3), we have the differential formulas for Q as follows: ∂ Q ∂ ˆ a ≡ (cid:20) ˆ a ` b (cid:21)h ˆ a † , ` b † i = (cid:20) ˆ a † ` b † (cid:21) , ∂ Q ∂ ˇ a ≡ (cid:20) ´ b ˇ a (cid:21)h ´ b † , ˇ a † i = (cid:20) b † , ˇ a † (cid:21) , (5. 20)Let us denote HF energy functional (3.3) simply as h H i g . Then we prove that the relations ∂ ˘ˆ a h H i ˘ g = ˘ˆ F ˘ˆ a † + ˘´ F ˘` b † , ∂ ˘ˇ a h H i ˘ g = ˘ˇ F ˘ˇ a † + ˘` F ˘´ b † , (5. 21)and that show, through which the TDHF equation is rewritten as i ~ ˙˘ g = " ∂ ˘ˆ a ∂ ˘´ b ∂ ˘` b ∂ ˘ˇ a h H i ˘ g , where ∂ ˘´ b h H i ˘ g = ˘´ F ˘ˇ a + ˘ˆ F ˘´ b, ∂ ˘` b h H i ˘ g = ˘´ F ˘ˇ a + ˘ˆ F ˘´ b. (5. 22)Using the relation similar to (5. 21), Eqs. (5. 18) and (5. 19) are reduced respectively to˘ˇ a † n(cid:16) ˘` F ˘ˆ a + ˘ˆ F ˘` b (cid:17) − ~ (cid:16) i ˙ˇΛ ν ∂ ˇΛ n ˘` b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘` b (cid:17)o +˘´ b † n(cid:16) ˘ˆ F ˘ˆ a + ˘´ F ˘` b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˆ a + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘ˆ a (cid:17)o = 0 , (5. 23)˘ˆ a † n(cid:16) ˘´ F ˘ˇ a + ˘ˆ F ˘´ b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘´ b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘´ b (cid:17)o +˘` b † n(cid:16) ˘ˇ F ˘ˇ a + ˘` F ˘´ b (cid:17) − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˇ a + i ˙ˇΛ ⋆ν ∂ ˇΛ ⋆ν ˘ˇ a (cid:17)o = 0 . (5. 24)As a way of satisfying (5. 23), we adopt the following partial differential equations: ∂ ˘ˆ a h H i ˘ g − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˆ a + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘ˆ a (cid:17) = 0 , ∂ ` b h H i ˇ g − ~ (cid:16) i ˙ˇΛ ν ∂ ˇΛ n ˘` b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘` b (cid:17) = 0 and c.c. (5. 25) ∂ ˘ˇ a h H i ˘ g − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘´ b + i ˙˘Λ ⋆ν ∂ ˘Λ ⋆ν ˘´ b (cid:17) = 0 , ∂ ´ b h H i ˇ g − ~ (cid:16) i ˙˘Λ ν ∂ ˘Λ ν ˘ˇ a + i ˙ˇΛ ⋆ν ∂ ˇΛ ⋆ν ˘ˇ a (cid:17) = 0 and c.c. (5. 26)From (4.36) and (4.44), we have h φ (˘ g ) | C t, Λ ν | φ (˘ g ) i = 0 and h φ (˘ g ) | C t, Λ ⋆ν | φ (˘ g ) i = 0 in which F c is replaced by F o . Using the transformation property of the differentials ∂ Λ ν = ∂ Λ ν ˇΛ ν ′ ∂ ˇΛ ν ′ + ∂ Λ ν ˇΛ ⋆ν ′ ∂ ˇΛ ⋆ν ′ , ∂ Λ ⋆ν = ∂ Λ ⋆ν ˇΛ ν ′ ∂ ˇΛ ν ′ + ∂ Λ ⋆ν ˇΛ ⋆ν ′ ∂ ˇΛ ⋆ν ′ , (5. 27)and differential formulas for expectation values of Hamiltonian H and HF one H HF ∂ ˇΛ ν h H i ˇ g = − Tr[ ∂ ˇΛ ν ˇ Q (ˇ g ) ˇ F ] ,∂ ˇΛ ν h H HF i ˇ g = − ∂ ˇΛ ν Tr ˇ F o = ∂ ˇΛ ν h H i ˇ g − Tr[ ˇ Q (ˇ g ) ∂ ˇΛ ν ˇ F ] . ) (5. 28)Then, due to (5. 21), (5. 22) and (5. 4) and the approximate relation ˘Λ ν Tr F o ≈ − ∂ ˘Λ ν h H i ˇ g , the invariance principle of Schr¨odinger equation and canonicity condition leads to thecanonical forms of equations of collective motion i ~ ˙˘Λ ⋆ν = − ∂ ˘Λ ν h H i ˇ g , i ~ ˙˘Λ ν = ∂ ˘Λ ⋆ν h H i ˇ g . (5. 29)The structure of (5. 25) shows that it becomes the equation of path for collective motionunder substitution of (5. 29). In this sense, it realizes a construction of the equation of pathin the TDHF [63], [23], [32]. (5. 25) and (5. 29) determine behavior of maximally decoupled collective motions in the TDHF. It is a renewal of TDHF equation by using the canonicitycondition under the existence of invariant subspace in the TDHF. This is due to a naturalconsequence of the maximally decoupled theory because there exists an invariant subspace,if the invariance principle of Schr¨odinger equation is realized. Then we can investigate avalidity of maximally decoupled theory with the use of the condition. The reason why thecondition occurs in our theory, which did not appear in the maximally decoupled theory,owes to ˆ d † ˆ d -terms. Since the maximally decoupled theory has no such terms, the condition istrivially fulfilled. This is the essential difference between maximally decoupled theory andours. We stress that our theory has been formulated by manifesting the group structurewhich makes the present work applicable to the SO (2 N +1) group [64].18 Nonlinear RPA arising from zero-curvature equation
The integrability condition of TDHF equation determining a collective-submanifold hasbeen studied based on a differential geometrical viewpoint. A fundamental equation workswell in large scale beyond random phase approximation (RPA). A linearly approximatedsolution of TDHF equation becomes RPA. Suppose we solved the fundamental equationby expanding it in power series of collective variables (Λ ν , Λ ⋆ν ) ( ν =1 , · · · , µ ; µ ≪ N ). Thefundamental equation has a RPA solution at the lowest power of collective variables. Thefluctuating density matrix e / Q and HF matrix e F in quasi PHF are expressed as follows: e / Q (˜ g ) = g † / Q ( g ) g , e / Q ≡ e ˆ / Q e ´ / Q e ` / Q e ˇ / Q , e F = g † ˜ F g , ˜ F ≡ ˆ ε +ˆ f ´ f ` f ˇ ε +ˇ f , g † F g = ˆ ε
00 ˇ ε , (6. 1)where F is a stationary HF matrix and ˆ ε and ˆ ε denote hole-energy and particle-energy.The ˆ f , ´ f , ` f and ˇ f are given later. A part of the fundamental equation (5. 11) is rewritten asTr (cid:26)(cid:20) m
00 1 N − m (cid:21) [˜ θ o − ν , ˜ θ † o − ν ′ ] (cid:27) = δ νν ′ , Tr (cid:26)(cid:20) m − N − m (cid:21) [˜ θ † o − ν , ˜ θ † o − ν ′ ] (cid:27) = 0 , Tr (cid:26)(cid:20) m − N − m (cid:21) [˜ θ o − ν , ˜ θ o − ν ′ ] (cid:27) = 0 . (6. 2)( 6. 2) is obtained with the aid of the fluctuating density matrix e / Q defined in ( 6. 1). Using(4.29) and the first of ( 6. 1), we have a relation between / Q ( g ) and e / Q (˜ g ) used later as / Q ( g ) ≡ ˆ / Q ( g ) ´ / Q ( g )` / Q ( g ) ˇ / Q ( g ) = / Q +2 g e ˆ / Q (˜ g ) e ´ / Q (˜ g ) e ` / Q (˜ g ) e ˇ / Q (˜ g ) g † , / Q ≡ g (cid:20) m − N − m (cid:21) g † = ˆ / Q ´ / Q ` / Q ˇ / Q . (6. 3)In order to investigate the matrix-valued nonlinear time evolution equation (5. 15) arisingfrom the zero curvature one, we give here the ∂ Λ ν and ∂ Λ ⋆ν differential forms of the TDHFdensity matrix and collective Hamiltonian. First, using ( 6. 3) and ˜ g † ˜ g =˜ g ˜ g † =1 N , we have ∂ Λ ν e / Q (˜ g ) = ∂ Λ ν ˜ g (cid:20) m − N − m (cid:21) ˜ g † +˜ g (cid:20) m − N − m (cid:21) ∂ Λ ν ˜ g † = ∂ Λ ν ˜ g (˜ g † ˜ g ) (cid:20) m − N − m (cid:21) ˜ g † +˜ g (cid:20) m − N − m (cid:21) ∂ Λ ν ˜ g † (˜ g ˜ g † ) , = − i ˜ θ † ν ∂ Λ ν ˜ g (cid:20) m − N − m (cid:21) ˜ g † + i ˜ g (cid:20) m − N − m (cid:21) ˜ g † ˜ θ † ν = − i (˜ g ˜ g † )˜ θ † ν ˜ g (cid:20) m − N − m (cid:21) ˜ g † + i ˜ g (cid:20) m − N − m (cid:21) ˜ g † ˜ θ † ν (˜ g ˜ g † ) , (6. 4)where we have used i∂ Λ ν ˜ g = ˜ θ † ν ˜ g . Next, by using (5. 14), ( 6. 4) is transformed into ∂ Λ ν e / Q (˜ g ) = − i ˜ g (cid:20) ˜ θ † o − ν , (cid:20) m − N − m (cid:21)(cid:21) ˜ g † . (6. 5)From ( 6. 1) and ( 6. 3), we have ∂ Λ ν e / Q (˜ g ) = 2 ∂ Λ ν e Q (˜ g ) = 2 g ∂ Λ ν e ˆ Q (˜ g ) , ∂ Λ ν e ´ Q (˜ g ) ∂ Λ ν e ` Q (˜ g ) , ∂ Λ ν e ˇ Q (˜ g ) g † , g = (cid:20) ˆ a ´ b ` b ˇ a (cid:21) . (6. 6)Let us substitute explicit reps for ˜ g (cid:18)(cid:20) ˜ˆ a ˜´ b ˜` b ˜ˇ a (cid:21)(cid:19) and ˜ θ † o − ν (cid:18)(cid:20) ξ o − ν ϕ o − ν ψ o − ν ζ o − ν (cid:21)(cid:19) into the RHS of( 6. 5) and combine it with ( 6. 6). Then, we obtain the final ∂ Λ n differential form of theTDHF ( U ( N )) density matrix as follows: ∂ Λ ν e ˆ Q (˜ g ) , ∂ Λ ν e ´ Q (˜ g ) ∂ Λ ν e ` Q (˜ g ) , ∂ Λ ν e ˇ Q (˜ g ) = g † i (cid:16) ˜´ bψ o − ν ˜ˆ a † − ˜ˆ aϕ o − ν ˜´ b † (cid:17) , − i (cid:16) ˜ˆ aψ o − ν ˜ˇ a † − ˜´ bϕ o − ν ˜` b † (cid:17) i (cid:16) ˜ˇ aψ o − ν ˜ˆ a † − ˜` bϕ o − ν ˜´ b † (cid:17) , − i (cid:16) ˜` bψ o − ν ˜ˇ a † − ˜ˇ aϕ o − ν ˜` b † (cid:17) g . (6. 7)19 Λ ⋆ν differentiation of the U ( N ) density matrix is also made analogously to the above case.The explicit form of ( 6. 3) without constant matrices is given as h E ˆ • ˆ • i g h E ˆ • ˇ • i g h E ˇ • ˆ • i g h E ˇ • ˇ • i g = g e ˆ / Q e ´ / Q e ` / Q e ˇ / Q g † = ˆ a ( e ˆ / Q ˆ a † + e ´ / Q ´ b † )+´ b ( e ` / Q ˆ a † + e ˇ / Q ´ b † ) , ˆ a ( e ˆ / Q ` b † + e ´ / Q ˇ a † )+´ b ( e ` / Q ` b † + e ˇ / Q ˇ a † )` b ( e ˆ / Q ˆ a † + e ´ / Q ´ b † )+ˇ a ( e ` / Q ˆ a † + e ˇ / Q ´ b † ) , ` b ( e ˆ / Q ` b † + e ´ / Q ˇ a † )+ˇ a ( e ` / Q ` b † + e ˇ / Q ˇ a † ) . (6. 8)From ( 6. 1), we have ˜ F = (cid:20) ˆ f ´ f ` f ˇ f (cid:21) (except diagonal ˆ ε and ˇ ε terms) and ˜ F = g e F g † .Substituting ( 6. 8) without constant matrices into (3.4), we have ˆ f ab =[ ab | cd ] h E cd i g , ´ f ai =[ ai | bj ] h E bj i g ` f ia =[ ia | jb ] h E jb i g , ˇ f ij =[ ij | kl ] h E kl i g = [ ab | cd ] { ˆ a ( e ˆ / Q ˆ a † + e ´ / Q ´ b † )+´ b ( e ` / Q ˆ a † + e ˇ / Q ´ b † ) } cd , [ ai | bj ] { ˆ a ( e ˆ / Q ` b † + e ´ / Q ˇ a † )+´ b ( e ` / Q ` b † + e ˇ / Q ˇ a † ) } bj [ ia | jb ] { ` b ( e ˆ / Q ˆ a † + e ´ / Q ´ b † )+ˇ a ( e ` / Q ˆ a † + e ˇ / Q ´ b † ) } jb , [ ij | kl ] { ` b ( e ˆ / Q ` b † + e ´ / Q ˇ a † )+ˇ a ( e ` / Q ` b † + e ˇ / Q ˇ a † ) } kl . (6. 9)Then the components of g e F g † become linear functionals of e ˆ Q (˜ g ) , e ´ Q (˜ g ) , e ` Q (˜ g ) and e ˇ Q (˜ g ).Using ( 6. 7), we easily calculate the ∂ Λ n differential of, for example, ˆ f and ` f as follows: ∂ Λ ν ˆ f ab =[ ab | cd ] { ˆ a ( ∂ Λ ν e ˆ / Q ˆ a † + ∂ Λ ν e ´ / Q ´ b † )+´ b ( ∂ Λ ν e ` / Q ˆ a † + ∂ Λ ν e ˇ / Q ´ b † ) } cd = i ( F } ψ o − ν + i ( F } ϕ o − ν ,∂ Λ ν ` f ia =[ ia | jb ] { ` b ( ∂ Λ ν e ˆ / Q ˆ a † + ∂ Λ ν e ´ / Q ´ b † )+ˇ a ( ∂ Λ ν e ` / Q ˆ a † + ∂ Λ ν e ˇ / Q ´ b † ) } jb = i ( D } ψ o − ν + i ( D } ϕ o − ν . (6. 10)The ( D ) etc. have the same form as the one in [64] and new ( D } etc. are also defined by( D } ≡ k ( ij | D | kl }k , ( D } ≡ k ( ij | D | kl }k , ( F } ≡ k ( ij | F | kl }k , ( F } ≡ k ( ij | F | kl }k . (6. 11)Using the simplified notations a for (ˆ a , ˇ a ) and b for (´ b , ` b ), we define ( ij | D | kl } etc. as( ij | D | kl } = ( ij | D | k ′ l ′ )˜ a ⋆k ′ k ˜ a ⋆l ′ l − ( ij | D | k ′ l ′ )˜ b ⋆k ′ k ˜ b ⋆l ′ l + ( ij | d | k ′ l ′ )˜ b ⋆k ′ k ˜ a ⋆l ′ l , − ( ij | D | kl } = ( ij | D | k ′ l ′ )˜ b k ′ k ˜ b l ′ l − ( ij | D | k ′ l ′ )˜ a k ′ k ˜ a l ′ l + ( ij | d | k ′ l ′ )˜ a k ′ k ˜ b l ′ l , ( ij | F | kl } = ( ij | F | k ′ l ′ )˜ a ⋆k ′ k ˜ a ⋆l ′ l − ( ij | F | k ′ l ′ )˜ b ⋆k ′ k ˜ b ⋆l ′ l + ( ij | f | k ′ l ′ )˜ b ⋆k ′ k ˜ a ⋆l ′ l , − ( ij | F | kl } = ( ij | F | k ′ l ′ )˜ b k ′ k ˜ b l ′ l − ( ij | F | k ′ l ′ )˜ a k ′ k ˜ a l ′ l + ( ij | f | k ′ l ′ )˜ a k ′ k ˜ b l ′ l , (6. 12)( ij | D | kl ) = − [ a i a ⋆k | a j a ⋆l ] − [ b ⋆k b i | b ⋆l b j ] + [ a i b j − a j b i | b ⋆k a ⋆l − b ⋆l a ⋆k ] , etc . (6. 13)We have used the notation [ a i a ⋆k | a j a ⋆l ] = [ αβ | γδ ] a αi a ⋆ βk a γj a ⋆ δl . Under the minimizationof the residual curvature, putting e F = e F c , we get ∂ Λ ν e F c = i " − ( F } ψ o − ν − ( F } ϕ o − ν − ( D } ⋆ ψ o − ν − ( D } ⋆ ϕ o − ν ( D } ψ o − ν +( D } ϕ o − ν T ( F } ⋆ ψ o − ν + T ( F } ⋆ ϕ o − ν . (6. 14)We derive a new equation formally analogous to the U ( N ) RPA equation. To do this,we decompose the fluctuating pair mode amplitude ˜ g → ˜ g ˜ g (ˆ ε , ˇ ε ): We finally obtain thefollowing matrix-valued equations:˜ g † (ˆ ε , ˇ ε ) · i ~ ∂ t ξ o − ν +[ˆ ε , ξ o − ν ] i ~ ∂ t ϕ o − ν +[ˆ ε , ϕ o − ν ] + −{ F } ψ o − ν − { F } ϕ o − ν −{ D } ⋆ ψ o − ν − { D } ⋆ ϕ o − ν i ~ ∂ t ψ o − ν − [ˇ ε , ψ o − ν ] + − i ~ ∂ t ξ T o − ν +[ˇ ε , ξ T o − ν ]+ { D } ψ o − ν + { D } ϕ o − ν + T { F } ⋆ ψ o − ν + T { F } ⋆ ϕ o − ν · ˜ g (ˆ ε , ˇ ε ) = 0 , (6. 15)with the modified new matrices { F } etc. defined through { F } = k{ ij | F | kl }k , { F } = k{ ij | F | kl }k , { D } = k{ ij | D | kl }k , { D } = k{ ij | D | kl }k , (6. 16)20hose matrix elements are given by − { ij | F | kl }k = k ˜ b ⋆i ′ i ˜ a j ′ j ( i ′ j ′ | D | kl }k − k ˜ a ⋆i ′ i ˜ b j ′ j ( i ′ j ′ | D | kl } ⋆ − ˜ a ⋆i ′ i ˜ a j ′ j ( i ′ j ′ | F | kl }k + k ˜ b ⋆i ′ i ˜ b j ′ j ( i ′ j ′ | F | kl } ⋆ , { ij | F | kl }k = k ˜ a ⋆i ′ i ˜ b j ′ j ( i ′ j ′ | D | kl } ⋆ k − k ˜ b ⋆i ′ i ˜ a j ′ j ( i ′ j ′ | D | kl }− ˜ b ⋆i ′ i ˜ b j ′ j ( i ′ j ′ | F | kl } ⋆ k + k ˜ a ⋆i ′ i ˜ a j ′ j ( i ′ j ′ | F | kl } , (6. 17) { ij | D | kl }k = k ˜ a i ′ i ˜ a j ′ j ( i ′ j ′ | D | kl }k − k ˜ b i ′ i ˜ b j ′ j ( i ′ j ′ | D | kl } ⋆ − ˜ b i ′ i ˜ a j ′ j ( i ′ j ′ | F | kl }k + k ˜ a i ′ i ˜ b j ′ j ( i ′ j ′ | F | kl } ⋆ , −{ ij | D | kl }k = k ˜ b i ′ i ˜ b j ′ j ( i ′ j ′ | D | kl } ⋆ k − k ˜ a i ′ i ˜ a j ′ j ( i ′ j ′ | D | kl }− ˜ a i ′ i ˜ b j ′ j ( i ′ j ′ | F | kl } ⋆ k + k ˜ b i ′ i ˜ a j ′ j ( i ′ j ′ | F | kl } . (6. 18)By making the block off-diagonal matrices of the LHS of the equation vanish, we get i ~ ∂ t ψ o − ν = [ˇ ε , ψ o − ν ] + −{ D } ψ o − ν −{ D } ϕ o − ν ,i ~ ∂ t ϕ o − ν = − [ˆ ε , ϕ o − ν ] + + { D } ⋆ ψ o − ν + { D } ⋆ ϕ o − ν . ) (6. 19)Due to the vanishing of the block diagonal matrices in the LHS of ( 6. 15) , we obtain i ~ ∂ t ξ o − ν = − [ˆ ε , ξ o − ν ]+ { F } ψ o − ν + { F } ϕ o − ν . (6. 20)( 6. 19) and ( 6. 20) are very similar to TD equation [64] and have matrices { D } etc. with(Λ , Λ ⋆ ,t )-dependence. Starting from the lowest solution of ψ o − ν , ϕ o − ν and ξ o − ν , we proceedto the next leading solution of ψ o − ν and ϕ o − ν . Using these, ξ o − ν with the same power isobtained. Then, we determine time dependence of the amplitudes corresponding to eachpower iteratively. It works well in a large scale beyond the SO (2 N ) RPA.Finally, following Rajeev [65], we show the existence of symplectic 2-form ω . Using SO (2 N ) U ( N ) coset variable q (= ba − )= − q T , SO (2 N ) (HB) density matrix R ( G ) is expressed as R ( G ) = G " − N
00 1 N G † = " R ( G ) − N − K ⋆ ( G )2 K ( G ) − R ⋆ ( G )+1 N , R ( G ) = q † q (cid:0) N + q † q (cid:1) − ,K ( G ) = − q (cid:0) N + q † q (cid:1) − . ) (6. 21)The two-form ω is given as ω = − i Tr (cid:8) ( d R ( G )) (cid:9) = − i Tr (cid:8) ( d R ( G )) R ( G ) (cid:9) , dω = − dω = 0 , (closed form) . (6. 22) ω ( U, V ) = − i Tr (cid:26)(cid:20) N − N (cid:21) [ U, V ] (cid:27) = i Tr (cid:8) u † v − v † u (cid:9) , U ≡ (cid:20) uu † (cid:21) and V ≡ (cid:20) vv † (cid:21) , (6. 23)which is a symplectic form and makes it possible to practice a geometric quantization ona finite/infinite-dimensional Grassmannian [66, 67]. SU m + n S ( U m × U n ) structure In this section, we construct another fundamental equation, noticing the special structureof the coset space SU m + n S ( U m × U n ) . Composing linearly of the same kind as the infinitesimal gen-erators (4.5) and (4.6), we adopt the following one-form Ω o and the integrability condition:Ω o = ~ F o · dt + θ † o − ν · d Λ ν + θ o − ν · d Λ ⋆ν , d Ω o − Ω o ∧ Ω o = 0 , (7. 1)where the expressions for F o , θ † o − ν and θ o − ν are given as F o = (cid:20) ˆ F o ´ F o ` F o ˇ F o (cid:21) , θ † o − ν = (cid:20) ξ o ϕ o ψ o ζ o (cid:21) ν , θ o − ν = (cid:20) ξ † o ψ † o ϕ † o ζ † o (cid:21) ν . (7. 2)Under the conditions ´ F o = − ` F † o , ϕ o − ν = − ψ † o − ν and n = N − m , Ω o is divided into as follows:Ω o = (cid:20) ˆΩ o − m
00 ˇΩ o − n (cid:21) + (cid:20) − `Ω † o `Ω o (cid:21) , TrΩ o = 0 , using indices m and n insteadof µ and ν ! . (7. 3)21here the ˆΩ o − m and ˇΩ o − n are expressed asˆΩ o − m = ~ ˆ F o · dt + ξ o − ν · d Λ ν + ξ † o − ν · d Λ ⋆ν , ˇΩ o − n = ~ ˇ F o · dt + ζ o − ν · d Λ ν + ζ † o − ν · d Λ ⋆ν . (7. 4)and according to Caseller–Megna-Sciuto [68], we adopt the λ -modified `Ω o given as`Ω o = λ ~ ` F o · dt + λ ψ o − ν · d Λ ν + λ ϕ † o − ν · d Λ ⋆ν . (7. 5)Further, the integrability condition is calculated as " d ˆΩ o − m − d `Ω † o d `Ω o d ˇΩ o − n = " ˆΩ o − m − `Ω † o `Ω o ˇΩ o − n ∧ " ˆΩ o − m − `Ω † o `Ω o ˇΩ o − n = " ˆΩ o − m ∧ ˆΩ o − m − `Ω † o ∧ `Ω o − ˆΩ o − m ∧ `Ω † o − `Ω † o ∧ ˇΩ o − n `Ω o ∧ ˆΩ o − m + ˇΩ o − n ∧ `Ω o ˇΩ o − n ∧ ˇΩ o − n − `Ω o ∧ `Ω † o , (7. 6)in which d `Ω o is given as d `Ω o = − ∂ t `Ω o ∧ dt − ∂ Λ ν `Ω o ∧ d Λ ν − ∂ Λ ⋆ν `Ω o ∧ d Λ ⋆ν = − (cid:16) λ ~ ∂ t ` F o · dt + λ ∂ t ψ o − ν · d Λ ν + λ ∂ t ϕ † o − ν · d Λ ⋆ν (cid:17) ∧ dt − (cid:16) λ ~ ∂ Λ ν ` F o · dt + λ ∂ Λ ν ψ o − ν ′ · d Λ ν ′ + λ ∂ Λ ν ϕ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ν − (cid:16) λ ~ ∂ Λ ⋆ν ` F o · dt + λ ∂ Λ ⋆ν ψ o − ν ′ · d Λ ν ′ + λ ∂ Λ ⋆ν ϕ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ⋆ν = − λ (cid:16) ~ ∂ Λ ν ` F o dt ∧ d Λ ν + ~ ∂ Λ ⋆ν ` F o dt ∧ d Λ ⋆ν (cid:17) + λ (cid:16) ∂ t ψ o − ν dt ∧ d Λ ν + ∂ t ϕ † o − ν dt ∧ d Λ ⋆ν + ∂ Λ ν ψ o − ν ′ d Λ ν ∧ d Λ ν ′ + ∂ Λ ⋆ν ϕ † o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ + ∂ Λ ν ϕ † o − ν ′ d Λ ν ∧ d Λ ⋆ν ′ + ∂ Λ ⋆ν ψ o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ (cid:17) . (7. 7)On the other hand, the term to equalize with d `Ω o is computed as`Ω o ∧ ˆΩ o − m + ˇΩ o − n ∧ `Ω o = (cid:16) λ ~ ` F o · dt + λ ψ o − ν · d Λ ν + λ ϕ † o − ν · d Λ ⋆ν (cid:17) ∧ (cid:16) ~ ˆ F o · dt + ξ o − ν ′ · d Λ ν ′ + ξ † o − ν ′ · d Λ ⋆ν ′ (cid:17) + (cid:16) ~ ˇ F o · dt + ζ o − ν · d Λ ν + ζ † o − ν · d Λ ⋆ν (cid:17) ∧ (cid:16) λ ~ ` F o · dt + λ ψ o − ν ′ · d Λ ν ′ + λ ϕ † o − ν ′ · d Λ ⋆ν ′ (cid:17) = λ n(cid:16) ~ ` F o ξ o − ν − ζ o − ν ~ ` F o (cid:17) dt ∧ d Λ ν + (cid:16) ~ ` F o ξ † o − ν − ζ † o − ν ~ ` F o (cid:17) dt ∧ d Λ ⋆ν o + λ n(cid:16) ~ ˇ F o ψ o − ν − ψ o − ν ~ ˆ F o (cid:17) dt ∧ d Λ ν + (cid:16) ~ ˇ F o ϕ † o − ν − ϕ † o − ν ~ ˆ F o (cid:17) dt ∧ d Λ ⋆ν + ( ψ o − ν ξ o − ν ′ + ζ o − ν ψ o − ν ′ ) d Λ ν ∧ d Λ ν ′ + (cid:16) ϕ † o − ν ′ ξ † o − ν ′ + ζ † o − ν ϕ † o − ν ′ (cid:17) d Λ ⋆ν ∧ d Λ ⋆ν ′ + (cid:16) ψ o − ν ξ † o − ν ′ + ζ o − ν ϕ † o − ν ′ (cid:17) d Λ ν ∧ d Λ ⋆ν ′ + (cid:16) ϕ † o − ν ′ ξ o − ν ′ + ζ † o − ν ϕ o − ν ′ (cid:17) d Λ ⋆ν ∧ d Λ ⋆ν ′ o . (7. 8)Equating the λ term and the λ term in both sides of ( 7. 7) and ( 7. 8), we get the relations λ -term : ∂ Λ ν ` F o = ζ o − ν ` F o − ` F o ξ o − ν , ∂ Λ ⋆ν ` F o = ζ † o − ν ` F o − ` F o ξ † o − ν , (7. 9) λ -term : ~ ∂ t ψ o − ν = ˇ F o ψ o − ν − ψ o − ν ˆ F o , ~ ∂ t ϕ † o − ν = ˇ F o ϕ † o − ν − ϕ † o − ν ˆ F o ,∂ Λ ν ψ o − ν ′ = ψ o − ν ξ o − ν ′ + ζ o − ν ψ o − ν ′ , ∂ Λ ⋆ν ϕ † o − ν ′ = ϕ † o − ν ′ ξ † o − ν ′ + ζ † o − ν ϕ † o − ν ′ ,∂ Λ ν ϕ † o − ν ′ = ψ o − ν ξ † o − ν ′ + ζ o − ν ϕ † o − ν ′ , ∂ Λ ⋆ν ψ o − ν ′ = ϕ † o − ν ′ ξ o − ν ′ + ζ † o − ν ϕ o − ν ′ , (7. 10)using ϕ o − n = − ψ † o − n , from Eqs. of second and third lines in ( 7. 10), we acquire ξ † o − ν = − ξ o − ν .In ( 7. 6), we have d ˇΩ o − n = − ∂ t ˇΩ o − n ∧ dt − ∂ Λ ν ˇΩ o − n ∧ d Λ ν − ∂ Λ ⋆ν ˇΩ o − n ∧ d Λ ⋆ν = − (cid:16) ∂ t ~ ˇ F o · dt + ∂ t ζ o − ν · d Λ ν + ∂ t ζ † o − ν · d Λ ⋆ν (cid:17) ∧ dt − (cid:16) ∂ Λ ν ~ ˇ F o · dt + ∂ Λ ν ζ o − ν ′ · d Λ ν ′ + ∂ Λ ν ζ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ν − (cid:16) ∂ Λ ⋆ν ~ ˇ F o · dt + ∂ Λ ⋆ν ζ o − ν ′ · d Λ ν ′ + ∂ Λ ⋆ν ζ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ⋆ν = − ∂ Λ ν ~ ˇ F o dt ∧ d Λ ν − ∂ Λ ⋆ν ~ ˇ F o dt ∧ d Λ ⋆ν + ∂ t ζ o − ν dt ∧ d Λ ν + ∂ t ζ † o − ν dt ∧ d Λ ⋆ν + ∂ Λ ν ζ o − ν ′ d Λ ν ∧ d Λ ν ′ + ∂ Λ ⋆ν ζ † o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ + ∂ Λ ν ζ † o − ν ′ d Λ ν ∧ d Λ ⋆ν ′ + ∂ Λ ⋆ν ζ o − ν ′ d Λ ⋆ν ∧ d Λ ν ′ . (7. 11)22he terms to equalize with d `Ω o − n are computed asˇΩ o − n ∧ ˇΩ o − n − `Ω o ∧ `Ω † o = (cid:16) ~ ˇ F o · dt + ζ o − ν · d Λ ν + ζ † o − ν · d Λ ⋆ν (cid:17) ∧ (cid:16) ~ ˇ F o · dt + ζ o − ν ′ · d Λ ν ′ + ζ † o − ν ′ · d Λ ⋆ν ′ (cid:17) − (cid:16) λ ~ ` F o · dt + λ ψ o − ν · d Λ ν + λ ϕ † o − ν · d Λ ⋆ν (cid:17) ∧ (cid:16) λ ~ ` F † o · dt + λ ψ † o − ν ′ · d Λ ν ′ + λ ϕ o − ν ′ · d Λ ⋆ν ′ (cid:17) = − (cid:16) ~ ` F o ψ † o − ν − ψ o − ν ~ ` F † o (cid:17) dt ∧ d Λ ν − (cid:16) ~ ` F o ϕ o − ν − ϕ † o − ν ~ ` F † o (cid:17) dt ∧ d Λ ⋆ν + (cid:0) ~ ˇ F o ζ o − ν − ζ o − ν ~ ˇ F o (cid:1) dt ∧ d Λ ν + (cid:16) ~ ˇ F o ζ † o − ν − ζ † o − ν ~ ˇ F o (cid:17) dt ∧ d Λ ⋆ν + ζ o − ν ζ o − ν ′ d Λ ν ∧ d Λ ν ′ + ζ † o − ν ζ † o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ + ζ o − ν ζ † o − ν ′ d Λ ν ∧ d Λ ⋆ν ′ + ζ † o − ν ζ o − ν ′ d Λ ⋆ν ∧ d Λ ν ′ . (7. 12)Here we should discard the λ term. Equating both sides of ( 7. 11) and ( 7. 12), we get ∂ Λ ν ˇ F o = ` F o ψ † o − ν − ψ o − ν ` F † o , ∂ Λ ⋆ν ˇ F o = ` F o ϕ o − ν − ϕ † o − ν ` F † o , (7. 13) ~ ∂ t ζ o − ν = ˇ F o ζ o − ν − ζ o − ν ˇ F o , ~ ∂ t ζ † o − ν = ˇ F o ζ † o − ν − ζ † o − ν ˇ F o ,∂ Λ ν ζ o − ν ′ = ζ o − ν ζ o − ν ′ , ∂ Λ ⋆ν ζ † o − ν ′ = ζ † o − ν ζ † o − ν ′ , ∂ Λ ν ζ † o − ν ′ = ζ o − ν ζ † o − ν ′ , ∂ Λ ⋆ν ζ o − ν ′ = ζ † o − ν ζ o − ν ′ . ) (7. 14)Following Caseller–Megna-Sciuto [68], under h m , k m ∈ U m and h n , k n ∈ U n , we assume` F o = h n ` F h m , ` F = · · · ` F ′ · ··· · · · ·· | {z } m } m } n − m , ψ o − ν = k n ψ o − ν k m , ( ` F ′ ) ij = δ ij ` f ,i , ∂ Λ ν ` f ,i =0 , ( ψ o − ν ) ij = δ ij f i , ~ ∂ t f i =0 . (7. 15)As also suggested by Caseller–Megna-Sciuto [68], Eqs. ( 7. 9) and ( 7. 10) imply ∂ Λ ν Tr (cid:16) ` F † o ` F o (cid:17) T =0 , ` F † o ` F o = c F I m , ~ ∂ t Tr (cid:16) ψ † o − ν ψ o − ν (cid:17) T =0 , ψ † o − ν ψ o − ν ψ o − ν ψ † o − ν = − c ψ I m c ψ I m , (7. 16)which is proved as0 = ∂ Λ ν Tr (cid:16) ` F † o ` F o (cid:17) T =Tr (cid:16) ∂ Λ ν ` F T o ` F ⋆o + ` F T o ∂ Λ ν ` F ⋆o (cid:17) =Tr n(cid:16) ` F T o ζ T o − ν − ξ T o − ν ` F T o (cid:17) ` F ⋆o + ` F T o (cid:16) ζ ⋆o − ν ` F ⋆o − ` F ⋆o ξ ⋆o − ν (cid:17)o =Tr n(cid:16) ` F o ` F † o − ` F † o ` F o (cid:17)(cid:16) ζ o − ν + ξ † o − ν (cid:17)o , ~ ∂ t Tr (cid:16) ψ † o − ν ψ o − ν (cid:17) T = Tr (cid:0) ~ ∂ t ψ T o − ν ψ ⋆o − ν + ψ T o − ν ~ ∂ t ψ ⋆o − ν (cid:1) =Tr n ψ T o − ν (cid:16) ˆ F ⋆o ψ ⋆o − ν − ψ ⋆o − ν ˇ F ⋆o (cid:17) − (cid:16) ˇ F T o ψ T o − ν − ψ T o − ν ˆ F T o (cid:17) ψ ⋆o − ν o =Tr n ψ o − ν ψ † o − ν (cid:16) ˆ F o + ˆ F † o (cid:17)o − Tr n ψ † o − ν ψ o − ν (cid:0) ˇ F o + ˇ F † o (cid:1)o , (Due to Tr( ˆ F o + ˇ F o )=0) . (7. 17)Further putting ζ o − ν = 0 and substituting it into ( 7. 9) and ( 7. 10), we have the relations ∂ Λ ν ` F o = − ` F o ξ o − ν , i.e. , ξ o − ν = − h − m ∂ Λ ν h m and ∂ Λ ν ψ o − ν ′ = ψ o − ν ξ o − ν ′ . (7. 18)Lastly, in ( 7. 6), we have d ˆΩ o − m = − ∂ t ˆΩ o − m ∧ dt − ∂ Λ ν ˆΩ o − m ∧ d Λ ν − ∂ Λ ⋆ν ˆΩ o − m ∧ d Λ ⋆ν = − (cid:16) ∂ t ~ ˆ F o · dt + ∂ t ξ o − ν · d Λ ν + ∂ t ξ † o − ν · d Λ ⋆ν (cid:17) ∧ dt − (cid:16) ∂ Λ ν ~ ˆ F o · dt + ∂ Λ ν ξ o − ν ′ · d Λ ν ′ + ∂ Λ ν ξ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ν − (cid:16) ∂ Λ ⋆ν ~ ˆ F o · dt + ∂ Λ ⋆ν ξ o − ν ′ · d Λ ν ′ + ∂ Λ ⋆ν ξ † o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ d Λ ⋆ν = − ∂ Λ ν ~ ˆ F o dt ∧ d Λ ν − ∂ Λ ⋆ν ~ ˆ F o dt ∧ d Λ ⋆ν + ∂ t ξ o − ν dt ∧ d Λ ν + ∂ t ξ † o − ν dt ∧ d Λ ⋆ν + ∂ Λ ν ξ o − ν ′ d Λ ν ∧ d Λ ν ′ + ∂ Λ ⋆ν ξ † o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ + ∂ Λ ν ξ † o − ν ′ d Λ ν ∧ d Λ ⋆ν ′ + ∂ Λ ⋆ν ξ o − ν ′ d Λ ⋆ν ∧ d Λ ν ′ . (7. 19)23hile, the term to equalize with d ˆΩ o − m is computed asˆΩ o − m ∧ ˆΩ o − m − `Ω † o ∧ `Ω o = (cid:16) ~ ˆ F o · dt + ξ o − ν · d Λ ν + ξ † o − ν · d Λ ⋆ν (cid:17) ∧ (cid:16) ~ ˆ F o · dt + ξ o − ν ′ · d Λ ν ′ + ξ † o − ν ′ · d Λ ⋆ν ′ (cid:17) − (cid:16) λ ~ ` F † o · dt + λ ψ † o − ν ′ · d Λ ν ′ + λ ϕ o − ν ′ · d Λ ⋆ν ′ (cid:17) ∧ (cid:16) λ ~ ` F o · dt + λ ψ o − ν · d Λ ν + λ ϕ † o − ν · d Λ ⋆ν (cid:17) = (cid:16) ~ ˆ F o ξ o − ν − ξ o − ν ~ ˆ F o (cid:17) dt ∧ d Λ ν + (cid:16) ~ ˆ F o ξ † o − ν − ξ † o − ν ~ ˆ F o (cid:17) dt ∧ d Λ ⋆ν − (cid:16) ~ ` F † o ψ o − ν − ψ † o − ν ~ ` F o (cid:17) dt ∧ d Λ ν − (cid:16) ~ ` F † o ϕ † o − ν − ϕ o − ν ~ ` F o (cid:17) dt ∧ d Λ ⋆ν + ξ o − ν ξ o − ν ′ d Λ ν ∧ d Λ ν ′ + ξ † o − ν ξ † o − ν ′ d Λ ⋆ν ∧ d Λ ⋆ν ′ + ξ o − ν ξ † o − ν ′ d Λ ν ∧ d Λ ⋆ν ′ + ξ † o − ν ξ o − ν ′ d Λ ⋆ν ∧ d Λ ν ′ . (7. 20)We should discard the λ term. Equating both sides of ( 7. 19) and ( 7. 20), we get ∂ Λ ν ˆ F o − ~ ∂ t ξ o − ν = [ ξ o − ν , ˆ F o ]+ ` F † o ψ o − ν − ψ † o − ν ` F o ,∂ Λ ⋆ν ˆ F o − ~ ∂ t ξ † o − ν = [ ξ † o − ν , ˆ F o ]+ ` F † o ψ o − ν − ψ † o − ν ` F o , ) (7. 21) ∂ Λ ν ξ o − ν ′ = ξ o − ν ξ o − ν ′ , ∂ Λ ⋆ν ξ † o − ν ′ = ξ † o − ν ξ † o − ν ′ , ∂ Λ ν ξ † o − ν ′ = ξ o − ν ξ † o − ν ′ , ∂ Λ ⋆ν ξ o − ν ′ = ξ † o − ν ξ o − ν ′ . (7. 22)By exploiting the invariance of the integrability condition ( 7. 6), especially by noticingEqs. ( 7. 13) and ( 7. 14), under the transformation U = U (Λ , Λ ⋆ ,t ) ∈ S ( U m × U n ), (cid:20) ˆΩ o − m
00 ˇΩ o − n (cid:21) ⇒ U − (cid:20) ˆΩ o − m
00 ˇΩ o − n (cid:21) U − U dU, `Ω o ⇒ U − `Ω o U, (7. 23)we have reasonably put ` F o = ` F h m , ζ o − ν = 0 , ψ o − ν = k n ψ o − ν . (7. 24)According to Caseller–Megna-Sciuto [68], we take ψ o − ν ≡ ψ o − ν, ψ o − ν, | {z } m } m } n − m , ˆ F o ≡ iα · I m , ˇ F o ≡ S − T † T |{z} m |{z} n − m } m } n − m , (7. 25)where ψ o − ν, is an invertible m × m matrix. Then, Eq. of the first line in ( 7. 10) becomes ~ ∂ t (cid:20) ψ o − ν, ψ o − ν, (cid:21) = (cid:20) S − T † T (cid:21)(cid:20) ψ o − ν, ψ o − ν, (cid:21) − (cid:20) ψ o − ν, ψ o − ν, (cid:21) iα · I n = (cid:20) Sψ o − ν, − T † ψ o − ν, − iαψ o − ν, T ψ o − ν, − iαψ o − ν, (cid:21) , (7. 26)from which we have the explicit expressions for T and S as T = ( ~ ∂ t + iα ) ψ o − ν, · ψ − o − ν, , S = (cid:8) ( ~ ∂ t + iα ) ψ o − ν, + T † ψ o − ν, (cid:9) ψ − o − ν, = ( ψ † o − ν, ) − n iα (cid:16) ψ † o − ν, ψ o − ν, − ψ † o − ν, ψ o − ν, (cid:17) ψ o − ν, + ψ † o − ν, ~ ∂ t ψ o − ν, + ~ ∂ t ψ † o − ν, · ψ o − ν, o ψ − o − ν, . (7. 27)The unknown field α can be determined through Eq. ( 7. 27) by imposing the conditionTrΩ o = Tr( ˆΩ o − m + ˇΩ o − n ) = 0 given in ( 7. 3) which readsTrΩ o = iαm +Tr S = 0 , Tr S =Tr h iα n I m − ( ψ † o − ν, ψ o − ν, ) − ψ † o − ν, ψ o − ν, o + ψ − o − ν, ~ ∂ t ψ o − ν, − ( ψ † o − ν, ψ o − ν, ) − ~ ∂ t ψ † o − ν, · ψ o − ν, i , (7. 28)and one can determine α as α = i Tr (cid:0) ψ − o − ν, ~ ∂ t ψ o − ν, (cid:1) − Tr h ( ψ † o − ν, ψ o − ν, ) − ~ ∂ t ψ † o − ν, · ψ o − ν, i m − Tr h ( ψ † o − ν, ψ o − ν, ) − ψ † o − ν, ψ o − ν, i . (7. 29)24sing ξ o − ν = − h − m ∂ Λ ν h m , ` F o =` F h m ≡ ` F , , ψ o − ν = k n ψ o − ν and ˆ F o = iα · I m , ( 7. 21) is changed as iα · I m + ~ ∂ t ( h − n ∂ Λ ν h n ) = h † m ` F † k n ψ o − ν − ψ ⋆o − ν k † n ` F h m = ` F † , ψ o − ν, − ψ † o − ν, ` F , . (7. 30)With the aid of the relation ( 7. 13) and the expression for ˇ F o ( 7. 25), we obtain the dif-ferential operation ∂ Λ ν on ˇ F o in the following forms: " ∂ Λ ν S − ∂ Λ ν T † ∂ Λ ν T = ` F o ψ † o − ν − ψ o − ν ` F † o = " ` F ′ , [ ψ † o − ν, , ψ † o − ν, ] − (cid:20) ψ o − ν, ψ o − ν, (cid:21) [ ` F ′† , , " ` F ′ , ψ † o − ν, ` F ′ , ψ † o − ν, − " ψ o − ν, ` F ′† , ψ o − ν, ` F ′† , = " ` F ′ , ψ † o − ν, − ψ o − ν, ` F ′† , ` F ′ , ψ † o − ν, − ψ o − ν, ` F ′† , , (7. 31)from which, thus we have ∂ Λ ν S = ` F ′ , ψ † o − ν, − ψ o − ν, ` F ′† , , ∂ Λ ν T = − ψ o − ν, ` F ′† , . (7. 32)Standing on the basic idea of construction of (1+1)-dimensional model with SU m + n S ( U m × U n ) structure by Caseller–Megna-Sciuto [68], we have extended the model to the (1 + 2 µ )-dimensional model. We have successfully derived the fundamental equation with SU m + n S ( U m × U n ) structure to get the classical equation of motion for an integrable system. The presentparticle-hole formalism for the fundamental equation is also applicable to the supercon-ducting formalism on the coset space SO (2 N ) U ( N ) for the paired mode [52] and on the cosetspace its extension SO (2 N +2) U ( N +1) for both the paired and unpaired modes [9, 11, 50]. This workwill be possible in the near future. τ -FM on group manifold Getting over the difference due to dimension of fermions, we ask the followings: How is a collective submanifold which is truncated through SCF equation related to a subgroup orbit in ID Grassmannian ( Gr ) by τ -FM? To get a microscopic understanding of cooperativephenomena, the concept of collective motion is introduced in relation to the TD variationof SC MF which brings couplings between collective and independent-particle motions andcouplings among quantum fluctuations of the TD MF [1]. The independent-particle motionis described in terms of particles referring to a stationary MF. There is the one-to-onecorrespondence between MF potential and vacuum state of system. Decoupling of collectivemotion out of fully-parameterized TDHF dynamics means a truncation of integrable sub-dynamics from the fully-parameterized TDHF manifold. The collective submanifold is acollection of collective paths developed by the SCF equation. The collectivity of each pathreflects a geometrical contribution in the Gr , independent from the SCF Hamiltonian. Thenthe collective submanifold is understood in relation to the collectivity of various subgrouporbits in the Gr . The collectivity arises through interferences among interacting fermionsand links with the concept of MF potential. The perturbative method has been consideredto be useful for description of periodic collective motion with large amplitude [1, 23]. If wemight not break the group structure of the Gr in the perturbative method, the loop group may work behind that treatment.Thus in both the methods, the following points are noticed: Various subgroup orbitsconsisting of loop path may infinitely exist in the fully-parameterized TDHF manifold.Theysatisfy an infinite set of the Pl¨ucker relations to hold Gr . Thus, the finite-dimensional Gr on a circle S is identified with an ID Gr . Namely, the τ -FM works as an algebraic tool toclassify the subgroup orbits. The SCF Hamiltonian is able to exist in the ID Gr . Then,25he SCF theory is rebuilt on the ID fermion Fock space on τ -FS. The ID fermions areintroduced through Laurent expansion of the finite-dimensional fermions with respect todegrees of freedom of fermions related to the MF potential. Inversely, the collectivity ofthe MF potential is attributed to gauges of interacting ID fermions and the interferencesamong fermions are elucidated via Laurent parameter. They are described with the use ofaffine Kac-Moody algebra according to the idea of the Dirac’s positron theory [31]. Thealgebro-geometric structure of ID fermion systems is realized in finite -dimensional ones. Following the same way as the one in Appendix A, fermion pair operators E αβ = c † α c β spanthe U ( N ) Lie algebra [ E αβ , E γδ ]= δ γβ E αδ − δ αδ E γβ and generate a canonical transformation U ( g )(= e γ αβ c † α c β ; γ † = − γ ) which is given by a U ( N ) matrix g as U ( g ) c † α U − ( g ) = c † β g βα , U ( g ) c α U − ( g ) = c β g ∗ βα , ( g = e γ , g † g = gg † = 1 N ) . (9. 1)Let | i be a free particle vacuum c α | i = 0 ( α = 1 , · · · , N ) and a m particle S-det be | φ i = c † m · · · c † | i . Due to the Thouless theorem [43], a different S-det is produced as U ( g ) | φ i = ( c † g ) m · · · ( c † g ) | i d = | g i , U ( g ) | i = | i . (9. 2)The m particle S-det is an exterior product of m single-particle states called simple states.All the simple states together with the equivalence relation identifying state different fromeach other in phase with the same state, constitutes a manifold Gr m , the group orbit. Anysimple state | φ i defines a decomposition of single-particle Hilbert space into sub-Hilbertones of occupied and unoccupied states [69]. Thus the Gr m corresponds to a coset space Gr m ∼ U ( m + n ) / ( U ( m ) × U ( n )) , ( g m + n / ( g m × g n ) , N = m + n ) . (9. 3)Following Fukutome [11, 42], for the coset variable p defined later, we express it as U ( g ) | φ i = h φ | U ( g ξ g υ ) | φ i e p ia c † i c a | φ i , using the relations g = g ξ g υ , h φ | U ( g ξ g υ ) | φ i = [det(1+ p † p )] − det υ and P Mρ =0 P ≤ a < ··· ··· >α ≥ v m, ··· , α m ,...,α c † α m · · · c † α | i ,v m, ··· , α m , ··· ,α = det g α , · · · g α ,m ... ... g α m , · · · g α m ,m , (Pl¨ucker coordinate) . (9. 6)We easily find that the Pl¨ucker coordinate has a relation P m +1 i =1 ( − i − v m, ··· , β i ,α m − , ··· ,α · v m, ··· , β m +1 , ··· ,β i +1 ,β i − , ··· ,β = 0 , (Pl¨ucker relation) (9. 7)where the indices denote the distinct sets 1 ≤ α , · · · , α m − ≤ N and 1 ≤ β , · · · , β m +1 ≤ N .The Pl¨ucker relation is equivalent to a bilinear identity equation P Nα =1 c † α U ( g ) | φ i ⊗ c α U ( g ) | φ i = P Nα =1 U ( g ) c † α | φ i ⊗ U ( g ) c α | φ i = 0 . (9. 8)The Gr m is an SU ( N ) group manifold since the phase equivalence theorem does hold.26e regard the above Pl¨ucker relation as analogous relation of Hirota’s form. Due to theregular rep of SO (2 N +1) group by Fukutome [72], it is shown that under the introductionof a phase variable τ = i ln det υ , Lie commutation relation which is given by [ E αβ ,E γδ ] = δ γβ E αδ − δ αδ E γβ is satisfied by the following differential operators for particle-hole pairs: e ∗ ia d = − ( p ib p ja ∂∂p jb + ∂∂p ∗ ia + i p ia ∂∂τ ) , e ∗ ai d = p ∗ ja p ∗ ib ∂∂p ∗ jb + ∂∂p ia − i p ∗ ia ∂∂τ ,e ∗ ab d = p ∗ ia ∂∂p ∗ ib − p ib ∂∂p ia − iδ ab ∂∂τ , e ∗ ij d = p ia ∂∂p ja − p ∗ ja ∂∂p ∗ ia . ) (9. 9)These differential operators are also proved to satisfy e ∗ ia Φ m,m ( p, p ∗ , τ ) = p ia Φ m,m ( p, p ∗ , τ ) , e ∗ ai Φ m,m ( p, p ∗ , τ ) = 0 ,e ∗ ij Φ m,m ( p, p ∗ , τ ) = 0 , e ∗ ab Φ m,m ( p, p ∗ , τ ) = δ ab Φ m,m ( p, p ∗ , τ ) , ) (9. 10)and [ e ∗ ia , p jb ] = − p ib p ja . The free particle vacuum function Φ m,m ( p, p ∗ , τ ) is defiend asΦ m,m ( p, p ∗ , τ ) d = v m ··· m ··· ( g ξ g υ ) = [det(1 + p † p )] − e iτ . (9. 11)The present formulation becomes the dual of regular rep by Fukutome. In both the SCFtheory and the soliton theory on a group, we find the common features that the Gr isjust identical with solution space of the bilinear differential equation. The solution spaceof each differential equation becomes an integral surface, subspace in the Gr m on whichthe residual coset variables are remained to be constant. From the viewpoint of bosonexpression in the SCF method, the above bilinear equation is regarded as that whichextracts the Slater determinantal orbit which contains in the U ( N ) boson Fock space [42].
10 SCF method in F ∞ We give here a brief sketch of the SCF equation, i.e., the TDHF method. According toRowe et al. [69], we start with a geometrical aspect of the method in the following way:Let us consider the TD Schr¨odinger equation i ~ ∂ t Ψ = H Ψ with a Hamiltonian H = h βα c † α c β + h γα | δβ i c † γ c † δ c β c α , (10. 1)where h βα and h γα | δβ i denote a single-particle Hamiltonian and a matrix element of aninteraction potential, respectively. This equation is linear but has generally shows disper-sive behavior. A TDHF equation gives a dynamics constrained to a nonlinear space on the Gr m . Suppression of the dispersion of a WF results in the nonlinear TDHF equation inwhich a non-dispersive solution is a path on the Gr m . The starting point for the TDHFtheory lies in an extremal condition of an action integral δ Z t t dt L ( g ( t )) = 0 , L ( g ( t )) d = h φ | U † ( g ( t ))( i ~ ∂ t − H ) U ( g ( t )) | φ i . (10. 2)To obtain an expression for the TDHF equation, we calculate an expectation value of one-body operators for S-det. Using the canonical transformation ( 9. 1), we have W αβ d = h φ | U † ( g ) c † β c α U ( g ) | φ i = ( g † ) β ′ β ( g T ) α ′ α h φ | c † β ′ c α ′ | φ i = P mα ′ =1 g αα ′ g † α ′ β , (10. 3)which is an element of density matrix and satisfies the hermite and idempotent relations W † αβ = W αβ , W αβ = W αβ , (10. 4)and an expectation value of two-body operators for S-det as h φ | U † ( g ) c † γ c † δ c β c α U ( g ) | φ i = ( g † ) γ ′ γ ( g † ) δ ′ δ ( g T ) β ′ β ( g T ) α ′ α h φ | c † γ ′ c † δ ′ c β ′ c α ′ | φ i = W αγ W βδ − W αδ W βγ . (10. 5)From ( 10. 3) and ( 10. 5), we get an energy functional, expectation value of Hamiltonian H [ W ] d = h φ | U † ( g ) HU ( g ) | φ i = h βα W αβ + [ γα | δβ ] W αγ W βδ , [ γα | δβ ]= h γα | δβ i−h γβ | δα i . (10. 6)27e also obtain the HF Hamiltonian H HF [ W ], the projected Hamiltonian onto the Gr m , H HF [ W ] = F αβ [ W ] E βα , F αβ [ W ] = δH [ W ] δW βα = h αβ + [ αβ | γδ ] W δγ . (10. 7)The Lagrange function L ( g ( t )) in ( 10. 2) is computed as L ( g ( t )) = i ~ (cid:16) g † ab ˙ g ba + g † ai ˙ g ia − ˙ g † ab g ba − ˙ g † ai g ia (cid:17) − H [ W ] , (10. 8)using ∂ t U † ( g ( t )) U ( g ( t ))+ U † ( g ( t )) · ∂ t U ( g ( t ))=0.The condition ( 10. 2) gives the TDHF equation ddt (cid:18) ∂ L ∂ ˙ g † (cid:19) − ∂ L ∂g † = 0 , ddt (cid:18) ∂ L ∂ ˙ g (cid:19) − ∂ L ∂g = 0 , (10. 9)and then we obtain a compact form of the TDHF equation i ~ ∂ t g ( t ) = F [ W { g ( t ) } ] g ( t ). Thetime evolution of the S-det ( 9. 2) is given by i ~ ∂ t U ( g ( t )) | φ i = H HF [ W ( g ( t ))] U ( g ( t )) | φ i .Standing on the same observation as the D’Ariano-Rasetti’s [39] for the relation be-tween soliton equations and coherent states , we may insist that the SCF method presentsthe theoretical scheme for an integrable sub-dynamics on an ID fermion Fock space, byidentifying the simple state | φ i with the highest weight vector and by regarding the TDHF-manifold Gr m as the projection onto a subspace (local chart) of the τ -function. Furtherit is noticed that the bilinear equation ( 9. 8) becomes just the expression for the Pl¨uckerrelation if we introduce a coordinate on the local chart. F ∞ We construct the SCF method, TDHF theory on ID fermion Fock space F ∞ basing on thephysical spirit of SCF method. Although it is different from the usual standpoint of space-coordinate construction of ID fermion [73], we start from that the canonical transformation( 9. 1) preserves a unitary equivalence for the original single-particle Schr¨odinger equationand induces an iso-spectral deformation. The Schr¨odinger equation with a TD potentialholds iso-spectrum under the time evolution of the potential. Taking Pauli principle and time-energy indeterminacy into account, we have the anti-commutation relations { c α ( t ) , c † β ( t ′ ) } = δ αβ δ ( t − t ′ ) , { c α ( t ) , c β ( t ′ ) } = { c † α ( t ) , c † β ( t ′ ) } = 0 . (10. 10)If the TD potential has periodicity in time T , the eigen-function has the same periodicity.Through Laurent expansion of the fermion operators, ID fermion operators are obtained as c α ( t )= q ~ T P r ∈ Z ψ ∗ Nr + α z r , c † α ( t )= q ~ T P r ∈ Z ψ Nr + α z − r , δ ( t − t ′ )= ~ T P r ∈ Z e − i ~ πT r ( t − t ′ ) . (10. 11) Z means the integer and the indices α and r are called particle spectra and Laurent spectra,respectively. Substituting ( 10. 11) into ( 10. 10), we get the anti-commutation relations { ψ ∗ Nr + α , ψ Ns + β } = δ αβ δ rs , { ψ ∗ Nr + α , ψ ∗ Ns + β } = { ψ Nr + α , ψ Ns + β } = 0 . (10. 12)Suppose that collective motion is dependent only on the collective variables η and η ∗ with aperiod of time T , η ( t )= η ( t + T ) and η ∗ ( t )= η ∗ ( t + T ). The matrix γ ( ∈ u N ) given in ( 9. 1) alsohas the same periodicity via collective variables satisfying γ ( t + T )= γ ( η ( t + T ) , η ∗ ( t + T ))= γ ( η ( t ) , η ∗ ( t )) = γ ( t ). As for the ordinary perturbative method with respect to η and η ∗ with a periodicity [23], we represent the collective variables η and η ∗ as η = √ Ω e iφ and η ∗ = √ Ω e − iφ . Then we always express the matrix γ as γ ( η, η ∗ )= P r,s ∈ Z ¯ γ r,s η ∗ r η s = P r γ r z r on the Lie algebra u N if we put z = e iφ . Regarding this expression as relation of Lie algebraof map from the unit circle S to the Lie algebra u N [73], we make Laurent expansion of γ as γ ( z ) = P r ∈ Z γ r z r , z = e − i ~ ω c t (cid:0) ω c = πT (cid:1) . (10. 13)For the anti-hermitian condition for the γ ( z ), we impose the constraints γ † ( z ) = − γ ( z ) γ † r = − γ − r and z − = z ∗ ( | z | = 1). We can consider these maps as loop groups .28ccording to Kac and Raina [74], we introduce ID Fock space F ∞ and associativeaffine Kac-Moody algebra restricting ourselves to u N algebra. Using the idea of Dirac’selectron-positron theory [31], perfect vacuum | Vac i and reference vacuum | m i are shown as ψ Nr + α | Vac i =0 , h Vac | ψ ∗ Nr + α =0 , ( r ≤ − ψ ∗ Nr + α | Vac i =0 , h Vac | ψ Nr + α =0 , ( r ≥ , | m i = ψ m · · · ψ | Vac i , h Vac | Vac i = 1 and h m | m i = 1 . ) (10. 14)Then we embed the free vacuum | i and simple state | φ i into the ID Fock space F ∞ as | i 7→ | Vac i , | φ i 7→ | m i , ( m = 1 , · · · , N ) . We assume state with the Laurent spectrumcorresponding to the | i to be stable state with minimal energy. This assumption meansfor us to make a choice of a gauge under which we adopt the correspondence | i 7→ | Vac i .We regard fermion pair c † α c β as e αβ (1 ≤ α,β ≤ N ). Using correspondence among basic ele-ments: c † α c β z r τ ( e αβ ( r )) d = P s ∈ Z ψ N ( s − r )+ α ψ ∗ Ns + β and normal-orderd product : ψ Nr + α ψ ⋆Nns + β : d = ψ Nr + α ψ ⋆Ns + β − δ αβ δ rs ( s < d su N ( ⊂ c sl N ) Lie algebra whose correspond-ing matrix rep has an infinite N periodic sequence of block form. See Appendix B: X γ = b X γ + C · c, C ∗ = − C , (pure imaginary , c : center) , b X γ = P Nr = − N P s ∈ Z ( γ r ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β : , Tr γ r = 0 , [ X γ , c ] KM = 0 , [ X γ , X γ ′ ] KM = b X [ γ,γ ′ ] + α ( γ, γ ′ ) · c, c | m i = 1 ·| m i . (10. 15) γ is divided into four blocks by specifying occupied state and unoccupied state for perfectvacuum | Vac i . Corresponding to this division, matrix in the 2- cocycle α [75](a) (AppendixB) is also divided into four blocks on the analogy of Dirac’s positron theory [31]. Note therelations α ∗ ( γ, γ ′ ) = − α ( γ, γ ′ ) and γ † = − γ , γ ′† = − γ ′ . Using ( 10. 12) and ( 10. 15) andidentity [ AB,C ]= A { B,C } − {
A,C } B , adjoint actions of X γ for ψ and ψ ∗ are computed as[ X γ , ψ Nr + α ] = P Ns = − N ψ N ( r − s )+ β ( γ s ) βα , [ X γ , ψ ∗ Nr + α ] = P Ns = − N ψ ∗ N ( r − s )+ β ( γ ∗ s ) αβ . (10. 16)Further using ( 10. 16) and the operator identity e Xγ Ae − Xγ = A + [ X γ , A ] + [ X γ , [ X γ , A ]] + · · · , (10. 17)ID fermion operator is transformed by canonical transformation U (ˆ g )(ˆ g = e γ ), satisfying U − (ˆ g ) = U (ˆ g − ) = U (ˆ g † ) and U (ˆ g ˆ g ′ ) = U (ˆ g ) U (ˆ g ′ ) with ˆ g † ˆ g = ˆ g ˆ g † = 1 ∞ (= I ), into the forms ψ Nr + α (ˆ g ) d = U (ˆ g ) ψ Nr + α U − (ˆ g ) = P s ∈ Z ψ N ( r − s )+ β ( g s ) βα ,ψ ∗ Nr + α (ˆ g ) d = U (ˆ g ) ψ Nr + α U − (ˆ g ) = P s ∈ Z ψ ∗ N ( r − s )+ β ( g ∗ s ) βα , ) (10. 18) δ rs δ αβ = (ˆ g ˆ g † ) Nr + α,Ns + β = P t ∈ Z ( g t g † t +( r − s ) ) αβ , ˆ g Nr + α,Ns + β = ( g s − r ) αβ ,δ rs δ αβ = (ˆ g † ˆ g ) Nr + α,Ns + β = P t ∈ Z ( g † t g t − ( r − s ) ) αβ , ˆ g † Nr + α,Ns + β = ( g † r − s ) αβ . ) (10. 19)ˆ g forms the periodic sequence with period N and s and t run over formally infinite set of Z .To built the TDHF theory in F ∞ , bilinear equation ( 9. 8) can be embedded into thatin F ∞ . Using the corresponding arguments U ( g ) U (ˆ g )(= e X γ ) and P Nα =1 c † α ⊗ c α P Nα =1 P r ∈ Z c † α z − r ⊗ c α z r ≃ P Nα =1 P r ∈ Z ψ Nr + α ⊗ ψ ∗ Nr + α , (10. 20)they are embedded into the bilinear equations on F ∞ for m = 1 , · · · , N as P Nα =1 P r ∈ Z ψ Nr + α U (ˆ g ) | m i⊗ ψ ∗ Nr + α U (ˆ g ) | m i = P Nα =1 P r ∈ Z U (ˆ g ) ψ Nr + α | m i⊗ U (ˆ g ) ψ ∗ Nr + α | m i =0 . (10. 21)Relieving from the restriction of su N and taking γ ∈ sl N with m , ( 10. 21) is regarded asthe bilinear equation of the reduced KP (Kadomtsev-Petviashvili) hierarchy and modifiedKP in the soliton theory [37, 75]. This suggests us the possibility of constructing theSCF theory for equation of collective motion governed by the condition ( 10. 21) on biggerspace sl N than su N . In the SCF theory the bilinear equation is considered to play roles ofconditions ensuring the existence of subgroup orbits on Gr m , different from soliton theory29here boson expressions become an infinite set of dynamical equations. The concept ofparticle-hole and vacuum in the SCF theory on S is connected to the Pl¨ucker relation.We embed the original Hamiltonian ( 10. 1) into F ∞ . By replacing the annihilation-creation operators as c α P r ∈ Z ψ ∗ Nr + α and c † α P r ∈ Z ψ Nr + α , we obtain H F ∞ = h βα P r,s ∈ Z ψ Nr + β ψ ∗ Ns + α + h γα | δβ i P k,l ∈ Z r,s ∈ Z ψ Nk + γ ψ Nl + δ ψ ∗ Ns + β ψ ∗ Nr + α . (10. 22)To embed the SCF Hamiltonian ( 10. 7), we introduce a general Hamiltonian on the F ∞ as H F ∞ = P r,s ∈ Z h Ns + β,Nr + α ψ Ns + β ψ ∗ Nr + α + P r,s ∈ Z k,l ∈ Z h N k + γ,N r + α | N l + δ,N s + β i ψ Nk + γ ψ Nl + δ ψ ∗ Ns + β ψ ∗ Nr + α , (10. 23)which is same as ( 10. 22) if h Ns + β,Nr + α = h βα and h N k + γ,N r + α | N l + δ,N s + β i = h γα | δβ i : equiv-alence condition for H F ∞ . For one- and two-body operators, we get formal expressions as c W Nr + α,Ns + β = h m | U † (ˆ g ) ψ Ns + β ψ ∗ Nr + α U (ˆ g ) | m i = P mγ =1 ˆ g Nr + α,γ ˆ g † γ,Ns + β + P t< P Nγ =1 ˆ g Nr + α,Nt + γ ˆ g † Nt + γ,Ns + β = P mγ =1 ( g − r ) αγ ( g †− s ) γβ + P t< P Nγ =1 ( g t − r ) αγ ( g † t − s ) γβ , (10. 24) h m | U † (ˆ g ) ψ Nk + γ ψ Nl + δ ψ ∗ Ns + β ψ ∗ Nr + α U (ˆ g ) | m i = c W Nr + α,Nk + γ c W Ns + β,Nl + δ − c W Nr + α,Nl + δ c W Ns + β,Nk + γ . (10. 25)The c W is just the so-called density matrix since it is easily proved to satify the idempo-tency relation c W = c W . Then we obtain the formal expectation value of ( 10. 23) as h H F ∞ i [ c W ] = P r,s ∈ Z h Ns + β,Nr + α c W Nr + α,Ns + β + P r,s ∈ Z [ N k + γ,N r + α | N l + δ,N s + β ] c W Nr + α,Nk + γ c W Ns + β,Nl + δ , [ N k + γ,N r + α | N l + δ,N s + β ] = h N k + γ,N r + α | N l + δ,N s + β i−h δ ←→ γ i . (10. 26)For Hamiltonian ( 10. 22), from ( 10. 26) and the equivalence condition for H F ∞ , we get h H F ∞ i [ c W ] = P k ∈ Z ( h k ) βα P s ∈ Z c W Ns + α,N ( s − k )+ β (( h k ) βα ≡ h βα )+ P k,L ∈ Z [( k,γ ) ,α | ( l,δ ) ,β ] P γ ∈ Z c W Nr + α,N ( r − k )+ γ P s ∈ Z c W Ns + β,N ( s − l )+ δ , [( k,γ ) ,α | ( l,δ ) ,β ] ≡ [ γα | δβ ] . (10. 27)Taking summation over infinite-number, we inevitably get an anomalous expectation value.To avoid the anomaly , we change the operator ( 10. 24) to its normal-ordered product,( W k ) αβ d = P r ∈ Z h m | U † (ˆ g ) : ψ N ( r + k )+ β ψ ∗ Nr + α : U (ˆ g ) | m i = P r ∈ Z c W Nr + α,N ( r + k )+ β − P r< δ k, δ βα = P r ∈ Z P mγ =1 ( g r ) αγ ( g † r − k ) γβ . (10. 28)The W k is identical to a coefficient of Laurent expansion of the density matrix W ( 10. 3) W αβ ( z ) = P k ∈ Z ( W k ) αβ z k = P k ∈ Z P s ∈ Z P mγ =1 ( g s ) αγ ( g † s − k ) γβ z k . (10. 29)Changing c W in ( 10. 27) into its normal-ordered product and using ( 10. 28), we obtain h H F ∞ i [ W ] = P k ∈ Z (cid:8) h βα ( W − k ) αβ + [ γα | δβ ] P l ∈ Z ( W − k + l ) αγ ( W − l ) βδ (cid:9) . (10. 30)The result coincides with the formal Laurent polynomial of H HF [ W ] ( 10. 6) in the sense of H [ W ( z )] = P l ∈ Z (cid:8) h βα ( W l ) αβ + [ γα | δβ ] P k ∈ Z ( W l − k ) αγ ( W k ) βδ (cid:9) z l . (10. 31)The time dependence of energy functional is brought by z ( t ).To preserve time independenceof ( 10. 31) we pick up only l =0 Laurent spectrum.Then we select the index of density W as h H F ∞ i [ W ] = h βα ( W ) αβ + [ γα | δβ ] P k ∈ Z ( W k ) αγ ( W − k ) βδ . (10. 32)This Hamiltonian is the sub-Hamiltonian H sub F ∞ extracted out of ( 10. 22) as H sub F ∞ = h βα P s ∈ Z ψ Ns + β ψ ∗ Ns + α + [ γα | δβ ] P k ∈ Z P r,s ∈ Z ψ N ( r − k )+ γ ψ N ( s + k )+ δ ψ ∗ Ns + β ψ ∗ Nr + α . (10. 33)30he above extraction permits us to interpret H F ∞ [ W ] as H [ W ( z )] | z = πi I H [ W ( z )] z dz . Wetake ( 10. 32) as an energy functional for the u N HF on the F ∞ . Through the variation δ h H F ∞ i [ W ] = P k ∈ Z ( F − k ) αβ δ ( W k ) βα , ( F k ) αβ d = h αβ δ k,o + [ αβ | γδ ]( W k ) δγ , (10. 34)we get an SCF Hamiltonian on the F ∞ similar to the formal Laurent expansion of H HF [ W ], H F ∞ ;HF = P k ∈ Z P s ∈ Z ( F k ) αβ : ψ N ( s − k )+ α ψ ∗ Ns + β : . (10. 35)This is the ID SCF Hamiltonian accompanying with the ID Fock operator F k which hasnever been seen in the usual TDHF Hamiltonian.For TDHF equation on the F ∞ , U (ˆ g ) | m i is required to satisfy the variational principle δS = δ ˆ g Z t t dtL (ˆ g ) = 0 , L (ˆ g ) = h m | U † (ˆ g )( i ~ ∂ t − H F ∞ ) U (ˆ g ) | m i . (10. 36)First, by using U (ˆ g ) = e X γ , we get the following relations: δ ˆ g Z dt h m | U † (ˆ g ) i ~ ∂ t U (ˆ g ) | m i = δ ˆ g Z dt h m | i ~ ∂ t | m i + δ ˆ g Z dt h m | i ~ ∂ t X γ − [ X γ ,i ~ ∂ t X γ ]+ · · · | m i , (10. 37) i ~ ∂ t X γ = P Nr = − N P s ∈ Z (cid:8) ( i ~ ∂ t γ r ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β :+( γ r ) αβ i ~ ∂ t : ψ N ( s − r )+ α ψ ∗ Ns + β : (cid:9) + i ~ ∂ t ( C · , (10. 38)Assuming a Laurent expansion parameter to be given by z = e − iω c t , ( 10. 38) is rewritten as i ~ ∂ t X γ = P Nr = − N P s ∈ Z D t ; r ( γ r ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β : + i ~ ∂ t ( C · , D t ; r d = i ~ ∂ t + rω c . (10. 39)A time differential i ~ ∂ t acting on γ r ( t ) and on ψ and ψ ∗ through z ( t ) is transformed to acovariant differential D t ; r ( D ) with a connection rω c which acts only on the γ r ( t ) from thegauge theoretical viewpoint. We put h m | i ~ ∂ t | m i =0 and C =0 since it has no influence onthe energy functional ( 10. 32). Then the time-differential term in ( 10. 36) is calculatedas U † (ˆ g ) i ~ ∂ t U (ˆ g ) = i ~ ∂ t X γ + [ i ~ ∂ t X γ , X γ ]+ [[ i ~ ∂ t X γ , X γ ] , X γ ] + · · · = b X Dγ + P k ≥ k ! [ · · · [ i ~ ∂ t X γ , X γ ] , · · · ] , X γ ]+ · · · . (10. 40)Using ( 10. 15) and D for the covariant differential, each commutator is calculated as[ i ~ ∂ t X γ , X γ ] = b X [ Dγ,γ ] + P r ∈ Z r Tr { ( Dγ ) r γ − r } , [[ i ~ ∂ t X γ , X γ ] , X γ ] = b X [[ Dγ,γ ] ,γ ] + P r ∈ Z r Tr { ([ Dγ, γ ]) r γ − r } , · · · · · · · · · [ · · · [ i ~ ∂ t X γ , X γ ] , · · · ] , X γ ] = b X [ ··· [ Dγ,γ ] , ··· ] ,γ ] + P r ∈ Z r Tr { ([ · · · [ Dγ, γ ] , · · · ]) r γ − r } . (10. 41)Substituting ( 10. 41) into ( 10. 40) and using D r d = D t ; r and ˆ g = e γ , we obtain U † (ˆ g ) i ~ ∂ t U (ˆ g ) = b X ˆ g † D ˆ g + C (ˆ g † D ˆ g ) , ˆ g † D ˆ g d = . . . . . . g † g † g †− g † g † g †− g † g † g †− . . . . . . . . . . . . D − g − D g D g D − g − D g D g D − g − D g D g . . . . . . , C (ˆ g † D ˆ g ) = − Tr (cid:20) − I I (cid:21)P k ≥ k ! [ · · · [ Dγ, γ ] , · · · ] , γ ]= P r ∈ Z r Tr n(cid:16)P k ≥ k ! [ · · · [ Dγ, γ ] , · · · ] , γ ] (cid:17) r γ − r o . (10. 42)The γ denotes an off-diagonal part of γ and an ID unit matrix I ∞ is written simply as I .The expectation value of ( 10. 40) for the reference vacuum is expressed as h m | U † (ˆ g ) i ~ ∂ t U (ˆ g ) | m i = P s ∈ Z P mα =1 P Nγ =1 ( g † s ) αγ ( D t ; s g s ) γα + C (ˆ g † D ˆ g ) . (10. 43)31sing C (ˆ g † D ˆ g ) − C ( D ˆ g † · ˆ g )=0, proved later, we obtain an explicit expression for the L (ˆ g ) as L (ˆ g ) = P s ∈ Z P mα =1 P Nγ =1 { ( g † s ) αγ ( D t ; s g s ) γα − ( D t ; − s g † s ) αγ ( g s ) γα }−h H F ∞ i [ W ] . (10. 44)Thus L (ˆ g ) is just the coefficient of z in Laurent expansion of L ( g ( z )). We give the TDHFequation identical with Laurent expansion of i ~ ∂ t g ( t )= F [ W { g ( t ) } ] g ( t ) and i ~ ∂ t U ( g ( t )) | φ i = H HF [ W ( g ( t ))] U ( g ( t )) | φ i . Demand of extremum of ( 10. 35) and definition of ( F cr ) αβ (ˆ g, ω c ) d = ω c P s ∈ Z s ( g s g † s − r ) αβ lead to the D operation on ˆ g and the “collective”-Fock operator F c as D ˆ g = F (ˆ g )ˆ g, F (ˆ g ) d = . . . . . . F − F F F − F F F − F F . . . . . . , Γ( g ) d = . . . . . . − g − g − g − g − g − g . . . . . . F c (ˆ g, ω c ) = ω c Γ( g )ˆ g † , ( c : “collective” part) . (10. 45)The set of equations ( 10. 45) including the definition of ( F cr ) αβ (ˆ g, ω c ) are transformed to i ~ ∂ t ˆ g = F p (ˆ g )ˆ g, F p (ˆ g ) d = F (ˆ g ) −F c (ˆ g ) , ( p : “particle” part) , ( F pr ) αβ d = ( F r −F cr ) αβ = h αβ δ r, +[ αβ | γδ ]( W r ) δγ − ω c P s ∈ Z s ( g s g † s − r ) αβ . (10. 46)Introducing b D t d = i ~ ∂ t − H cF ∞ ;HF , ( 10. 46) is cast into equations on the state vector U (ˆ g ) | m i as b D t U (ˆ g ) | m i = H F ∞ ;HF U (ˆ g ) | m i , H cF ∞ ;HF = P r,s ∈ Z ( F cr ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β : ,i ~ ∂ t U (ˆ g ) | m i = H pF ∞ ;HF U (ˆ g ) | m i , H pF ∞ ;HF d = P r,s ∈ Z ( F pr ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β : , ) (10. 47)which suggests symmetry breaking by the “particle” motion and occurrence of the “collec-tive” motion due to the recovery of the symmetry. This is a noticeable aspect of the presenttheory. Suppose that both ˆ g and U (ˆ g ) | m i diagonalizing F p in H pF ∞ ;HF and F c in H cF ∞ ;HF aredetermined spontaneously if ˆ g ≃ ˆ g e − i ˆ ǫt ~ and ∂ t ˆ g = 0. Then, using the expression for F c , thelast equation of ( 10. 45), we have a relation ω c Γ(ˆ g )= F (ˆ g )ˆ g − ˆ g ˆ ǫ , ˆ ǫ d =diag.[ · · · ǫ ǫ ǫ · · · ]and g r z r ∝ e − i ( ǫ + ωcrIN ) t ~ . Thus the particle-hole energy ǫ ( ǫ αβ = ǫ α δ αβ ) and boson one ω c are unifiedinto a gauge phase. The HF theory on the Gr m has no such the term Γ(ˆ g ) and leadsinevitably to ω c Γ(ˆ g ) = 0. Then ˆ g must be composed of only a block-diagonal g = e γ .Eqs. (10. 47) give a time evolution of particle degrees of freedom . Thus we get a commonlanguage, namely, ID Gr and affine Kac-Moody algebra , to discuss the relation betweenSCF and soliton theories. The SCF theory under level one on the F ∞ is nothing thanthe zeroth order of the Laurent expansion on the Gr m . Through construction of the SCFtheory on the F ∞ , the algebraic structure of SCF theory on the F ∞ is made clear as itis just the gauge theory inherent in the SCF theory. The MF potential degrees of free-dom occurs from the gauge degrees of freedom of fermions and the fermions make pairsamong them absorbing change of gauges. The sub-Hamiltonian ( 10. 33) exhibits such aphenomenon in the u N algebra, which allows us to interpret the absorption of the gaugeas a coherent property of fermion pairs. The SCF theory is regarded as a method to de-termine self-consistently both the particle-hole energy ǫ α (ˆ g ) and boson energy ω c whichowe to the time evolution of the fermion gauge . Then it becomes to be the first adven-turous attempt to possibly give a unified description of both the energies in terms of themotions of the gauge phases . Up to the present stage, we have made clear a unified aspectof SCF theory and soliton theory through the abstract fermion Fock space on the S . It32eans that an algebro-geometric structures of infinite -dimensional fermions many-bodysystems is realizable in finite -dimensional ones. We point out that in order to improvedrawbacks in the perturbative SCF method, it is useful to find a way how to construct theinfinite-dimensional boson variables from the parameters of the original group g .Let ǫ and ǫ ∗ be parameters specifying a continuous deformation of loop path on Gr m .Using notation in ( 10. 15) and calculating similarly to ( 10. 42), e − X γ ∂ ǫ e X γ is obtained as e − X γ ∂ ǫ e X γ = b X ˆ g − ∂ ǫ ˆ g + ∂ ǫ ( C · C (ˆ g − ∂ ǫ ˆ g ) : ˆ g = e γ , C (ˆ g − ∂ ǫ ˆ g ) = ∂ ǫ + ∂ ǫ γ + P k ≥ k ! [ · · · [ ∂ ǫ γ, γ ] , · · · ] , γ ] . ) (10. 48)To avoid the anomaly , b X γ must be read as P r ∈ Z ( γ r ) αβ (cid:8)P s ∈ Z ψ N ( s − r )+ α ψ ∗ Ns + β + δ r P s< δ αβ (cid:9) . Then e − X γ ∂ ǫ e X γ is computed as e − X γ ∂ ǫ e X γ = P r,s ∈ Z (ˆ g − ∂ ǫ ˆ g ) r : ψ N ( s − r )+ α ψ ∗ Ns + β : + P s< Tr(ˆ g − ∂ ǫ ˆ g ) . (10. 49)From ( 10. 49), we get C (ˆ g − ∂ ǫ ˆ g ) = P s< Tr(ˆ g − ∂ ǫ ˆ g ) = 0 , (ˆ g − ∂ ǫ ˆ g ) ∈ sl N and C (ˆ g † ∂ ǫ ∗ ˆ g ) = C ( ∂ ǫ ∗ ˆ g † · ˆ g ) = 0 . Similarly we obtain C (ˆ g † D ˆ g ) = C ( D ˆ g † · ˆ g ) = 0 in the same way as the above.For the last discussion it is convenient to define the infinitesimal generators of the col-lective submanifold as follows: X θ † d = i∂ ǫ U (ˆ g ) · U (ˆ g ) − = X θ † , θ † d = i∂ ǫ ˆ g · ˆ g † ,X θ d = i∂ ǫ ∗ U (ˆ g ) · U (ˆ g ) − = X θ , θ d = i∂ ǫ ∗ ˆ g · ˆ g † , ) (10. 50)which are functions of ǫ and ǫ ∗ and they are changed to the differential operators. Usingthese differential operators, we obtain the weak orthogonality condition [1, 23] as follows:1 = h ˆ g | [ X θ , X θ † ] | ˆ g i = P mα =1 P Nβ =1 P r ∈ Z ([ θ, θ † ] r ) αβ ( W − r ) βα + P r ∈ Z r Tr (cid:16) θ r , θ †− r (cid:17) . (10. 51)Thus we can treat the equation of collective motion with large amplitudes [60] and theollective submanifold along the idea of Lax pairs [54] for constructing integrable systems.
11 SCF method in τ -FS Along the soliton theory in the ID fermion Fock space [76, 77, 37, 74], we transcribe theTDHF theory in F ∞ to the one in τ -FS. A Heisenberg subalgebra S [74] is given by S = ⊕ k =0 Λ k + C · c , Λ k d = P i ∈ Z : ψ i ψ ∗ i + k : . ( k ∈ Z ) (11. 1)The boson algebra is obtained as [Λ k , Λ l ]= kδ k + l, . Λ k is called the shift operator and Λ belongs to the center. Take only c = 1 (level-one case). The boson mapping operator isintroduced as σ m d = h m | e H ( x ) , H ( x ) = P j ≥ x j Λ j is the Hamiltonian in τ -FM [37]. By whichthe following isomorphism is described as σ m ; F ( m ) B ( m ) = C ( x , x , · · · ) and | m i7→
1, thenΛ k ∂∂x k , Λ − k kx k , ( k >
0) Λ m. (11. 2) F ( m ) and B ( m ) denote m -charged fermion space and corresponding boson space and thedegree is defined by deg( x j )= j . The contravariant hermitian form on the B ( m ) is given as h | i = 1 , (cid:16) ∂∂x k (cid:17) † = kx k , h P | Q i = P ∗ (cid:16) ∂∂x , ∂∂x , · · · (cid:17) Q ( x ) | x =0 . (11. 3) P ∗ means the complex conjugation of all the coefficients of polynomial P and x = ( x ,x , · · · ).Then the group orbit of the highest weight vector | m i under the action U (ˆ g )(= e X a ) of GL ( ∞ ) is mapped to a space of the τ -function given as τ m ( x, ˆ g )= h m | e H ( x ) U (ˆ g ) | m i [78].We construct the rep in B ( m ) in reduction to affine c sl N . Let the generating series beΨ( p ) = P j ∈ Z p j ψ j , Ψ ∗ ( p ) = P j ∈ Z p − j ψ ∗ j . ( p ∈ C \ . (11. 4)33he action X a (= P i,j ∈ Z a ij : ψ i ψ ∗ j : + C · ∈ c sl N must satisfy (i) a i + N,j + N = a ij ( i, j ∈ Z ) and(ii) P Ni =1 a i,i + jN = 0 ( j ∈ Z ) . From (i) and Λ jN = P i ∈ Z : ψ i ψ ∗ i + jN : ( j ∈ Z ), [ X a , Λ jN ]=0, i.e., τ m ( x, ˆ g ) (ˆ g ∈ c sl N ) is independent of x jN . Using (i), the generating function is rewritten asΨ( p )Ψ ∗ ( q ) = P i,j ∈ Z ψ i ψ ∗ j p i q − j = P i,j ∈ Z ψ i + N ψ ∗ j + N p i q − j p N q − N . (11. 5)Using the Schur polynomials, exp P ∞ k ≥ x k p k = P k ≥ S k ( x ) p k [37, 76], X a is obtained as σ m ; X a = P i,j ∈ Z a ij : ψ i ψ ∗ j : + C · P i,j a ij e z ij ( x, e ∂ x ) + C · ,z ij ( x, e ∂ x ) ≡ P µ,ν ≥ ,k ≥ S i + k + µ − m ( x ) S − j − k + ν + m ( − x ) S µ ( − e ∂ x ) S ν ( e ∂ x ) , e z ij ≡ z ij − δ ij · j ≤ , ) (11. 6)being independent on x jN and e ∂ x d = (cid:16) ∂∂x , ∂∂x , · · · (cid:17) . We transcribe the fundamental equationfor TDHF theory on U (ˆ g ) | m i⊂ F ( m ) that on ⊂ B ( m ) by the corresponding τ -function:(1) U (ˆ g ) | m i ( U (ˆ g ) = e X γ ; X γ ∈ d su N ⊂ c sl N ) τ -function; τ m ( x, ˆ g ) = h m | e H ( x ) U (ˆ g ) | m i , ∂∂x jN τ m ( x, ˆ g ) = 0 . (11. 7)(2) Quasi-particle and vacuum states Hirota’s bilinear equation;A vector | τ M i ∈ F ( M ) belongs to the group orbit of the highest weight vector if and onlyif it satisfies the bilinear identity equation P i ∈ Z ψ i | τ M i ⊗ ψ ∗ i | τ M i = 0 , ⇐⇒ | τ M i = U (ˆ g ) | M i , ˆ g ∈ GL ( ∞ ) . (11. 8)This identity is cast into nonlinear differential equations for τ M ( x ) d = h M | e H ( x ) | τ M i πi I dpp h M +1 | e H ( x ′ ) Ψ( p ) | τ M i ⊗ h M − | e H ( x ′′ ) Ψ ∗ ( p ) | τ M i = 12 πi I dp exp nP j ≥ p j ( x ′ j − x ′′ j ) o exp n − P j ≥ p − j j (cid:16) ∂∂x ′ j − ∂∂x ′′ j (cid:17)o τ M ( x ′ ) · τ M ( x ′′ ) . (11. 9)It is described by the Hirota’s bilinear differential operator as P ( D ) f · g d = P (cid:16) ∂∂y , ∂∂y , · · · (cid:17) ( f ( x + y ) · g ( x − y )) | y =0 . (11. 10) P ( D ) is a polynomial in D =( D ,D , · · · ). Note that P f · f ≡ P ( D )= − P ( − D ).Defining new variables x and y by x ′ = x − y and x ′′ = x + y , with the help of the relation e P ∞ k ≥ x k p k = P k ≥ S k ( x ) p k and notation e D = ( D , D , · · · ), ( 11. 9) is brought to the form P j ≥ S j ( − y ) S j +1 ( e D ) exp (cid:0)P s ≥ y s D s (cid:1) τ M ( x ) · τ M ( x ) = 0 . (11. 11)If we expand ( 11. 11) into a multiple Taylor series of variables y , y , · · · and make eachcoefficient of this series vanishing, we get an infinite set of nonlinear partial differentialequation for the KP hierarchy. P Nα =1 P r ∈ Z ψ Nr + α U (ˆ g ) | m i ⊗ ψ ∗ Nr + α U (ˆ g ) | m i = 0 P j ≥ S j ( − y ) S j +1 ( ˜ D ) exp( P s ≥ y s D s ) τ m ( x, ˆ g ) τ m ( x, ˆ g ) = 0 . ) (11. 12) D = ( D ,D , · · · ) denotes the Hirota’s bilinear differential operator and e D = ( D , D , · · · ).(3) TDHF equation on U (ˆ g ) | m i 7→ TDHF equation on τ m ( x, ˆ g ); i ~ ∂ t U (ˆ g ( t )) | m i = H pF ∞ (ˆ g ( t )) U (ˆ g ( t )) | m i7→ i ~ ∂ t τ m ( x, ˆ g ( t ))= H pF ∞ ( x, e ∂ x , ˆ g ( t )) τ m ( x, ˆ g ( t )) , (11. 13)in which using ( 11. 6), H pF ∞ ( x, e ∂ x , ˆ g ), is given as H pF ∞ ( x, e ∂ x , ˆ g ) = P r,s ∈ Z ( F pr (ˆ g )) αβ e z N ( s − r )+ α,Ns + β ( x, e ∂ x ) , ( F pr (ˆ g )) αβ = h αβ δ r, +[ αβ | γδ ]( W r ) δγ − ω c P s ∈ Z s ( g s g † s − r ) αβ , ( W r ) αβ = P mγ =1 P s ∈ Z ( g s ) αγ ( g † s − r ) γβ . (11. 14)34igure 3: TDHF theory on τ -functional spaceThe image of the τ -functional space is illustrated above.The trajectories of the TDHF equation are running in their various subgroup orbits. Ifwe only have to know the form of X γ deciding the subgroup orbits by the soliton equations,we can construct a Hamiltonian made of only the element of X γ . Then the Hamiltonianbecomes integrable on the subgroup orbit.By K-N [79, 80, 81], we have clarified the relationof concept of particle-hole and collective motion in TDHF to that of soliton theory fromthe loop group viewpoint. The identification by Pressely and Segal [45] connects the space H ( N ) = L ( S ; C N ) with the standard Hilbert space H = H (1) = L ( S ; C ), square summable C N -valued function, by obvious lexicographic correspondence between orthonormal basis.
12 SCF method and τ -FM The subgroup orbits made of loop paths exist infinitely in Gr m and τ -FM is recognized asan algebraic tool to classify them. To go beyond the perturbative method [36] with respectto collective variables, using ID fermions, we have aimed at constructing the SCF theoryon affine Kac-Moody algebra along the soliton theory. The ID fermion operators have beenintroduced through the Laurent expansion of finite-dimensional fermion operators withrespect to degrees of freedom of the fermions related to the mean-field (MF) potential. The Gr m is identified with the infinite one which is affiliated with the manifold obtained bythe reduction of gl ∞ to su N (reduced KP hierarchy). In this sense, algebraic treatment ofextracting subgroup orbits with z ( | z | =1) from the Gr m is just the construction of thedifferential equation (Hirota’s bilinear equation) for su N ( ⊂ sl N ) reduced KP hierarchy [75].The SCF theory on F ∞ results in gauge theory of fermions . The collective motion due toquantal fluctuations of SC MF potential is attributed to the motion of gauge of fermions .A common factor explains interference among fermions. The concept of particle-hole andcollective motions is regarded as compatible condition for particle-hole and collective modes .The SCF theory on F ∞ gives us a new algebraic method for understanding of fermions.Assumption of periodic collective motion is not necessarily important.Prescribing fermionto form pair by absorbing change of gauge , SCF Hamiltonian made of only H F ∞ ; HF is in-duced. Through a compatible condition for particle and collective modes and a specialchoice of Laurent expansion, time dependence of the fermion gauge arises. In this concern,we have expressions for pair operators of ID fermions in terms of Laurent spectral numbers.35hough the TDHF theory on F ∞ describes dynamics on real fermion-harmonic oscillators,the soliton theory does on complex fermion-harmonic oscillators. This gives a problem onrelation between the present theory and resonating MF one [82, 83] and a task how toconstruct ID boson variables from group parameters of g ∈ U ( N ) [84].We start with su N algebra consisting of particle-hole (p-h) component and u N algebraincluding particle-particle (p-p) and hole-hole (h-h) ones for the state | φ i [42].Then we have U ( N ) group orbit. Further we must notice the equivalence relation which identifies statesdifferent from each other in phases dependent only on diagonal components in h-h types,with the same state [69]. Under this equivalence relation, we ought to treat SU ( N ) grouporbit. However the HF Hamiltonian has value on u N but not on su N . To describe dynamicson SU ( N ) group orbit, we must remove extra components not satisfying su N from the fullyparametrized HF Hamiltonian. With the help of the equivalence relation we ought to takediagonal components in p-p and h-h types into account to assign them.These quantities canplay a role of fermion gauge phase as the canonical transformation describes. The gauge-phase is separated into a term which comes from single pair of fermions and a term comingfrom the particle-number operator, on the Lie algebra through which the fermion pairsare governed. The former relates to the particle mode and the latter to the collective one.Removing the above superfluous components of the HF Hamiltonian, assignment of themto both the modes turns out to bring the concept of particle-hole and collective motions.The usual TDHF theory has not a complete scheme to treat separately both the motions.The TDHF equation on S , however, has such a scheme. It provides not only manifest andalgebraic understanding of the motions but also a scheme to describe a large amplitudecollective motion. As for particle-hole and collective motions, assuming a time-periodicityof motions, we can derive a new unified equation for both the motions beyond the HFand RPA ones [85, 86]. The new TDHF theory on S is constructed on a collection ofvarious subgroup orbits consisting of loop paths in the Gr m of finite-dimensional fermionFock space and is shown to be built on ID Gr of ID fermion Fock space F ∞ . If we choosea broken-symmetry vacuum on S , the vacuum state is able to deform by through theshift operators associated with collective modes so that τ -function in soliton theory is alsodeformed. We find an algebraic mechanism for appearance of collective motion induced byTD MF potential. This gives an algebraic understanding of physical concepts of symmetrybreaking and successive occurrence of collective motion due to recovery of the symmetry.A multi-circle TDHF theory is exciting but has a problem how the Pl¨ucker relation onmulti circle is constructed. It ais related to a multi-component soliton theory [87, 88, 89]. TheID algebraic approaches have been proposed using the Bethe ansatz (BA) WF [90], Lipkin-Meshkov-Glick (LMG)-model [91, 92] and pairing theory [93, 94]. Rowe has showed that anumber-projected WF satisfies some recursion relations and expressed it in a determinantalform [93] with completely anti-symmetric Schur function in the theory of group charactersby Littlewood [95] and MacDonald [96]. The WF is described by an ID Lie algebra [97, 98].The algebra is constructed by power series expansion of finite-dimensional Lie algebra withrespect to parameters involved in Hamiltonian. See [97](b) which has given an ID affine Liealgebra [ su (2) and an exactly solvable pairing Hamiltonian. It has also shown the conditionsfor solving the eigenvalue problem using the BA method. It is a very exciting problem tocompare the ID affine Lie algebra [ su (2) with the ID affine Lie algebra \ sl (2 , C ) of p-h LMG-model. They have the [ su (2) and \ sl (2 , C ) invariant Casimir operators which provide thealmost pairing and the LMG Hamiltonians. This will be solved and presented elsewhere36n the near future. Further one has possible generalization and initial value problem forsolution of Sine-Gordon equation, Mansfield [99]. This suggests an infinitesimal form ofthe corresponding nonlinear algebra generated by building blocks of the BA WFs [100]. Weobtain a collective submanifold decided by the SCF Hamiltonian of LMG-model. Then weacquire a clue to build a relation between the methods using Gaudin model [101, 100], Oritzexactly-solvable model [102] and present method, and further using the minimal matrixproduct states and the geminal WFs [103]. Finally we give a short remark: Daboul hasshown that the dynamical symmetry of the hydrogen atom leads in a natural way to theinfinite-dimensional algebra, twisted affine Kac-Moody algebras of [ so (4) and \ so (3 ,
1) [104]based on a suggestive paper by Goddard and Oliv [73].
13 Summary and concluding remarks
We have studied a curvature equation according to the idea of Lax unfamiliar to the con-ventional treatments of fermion system, as a fundamental equation to truncate collectivesubmanifolds out of the TDHF manifold. We show the following subjects to be studied:(I) The expectation values of the zero curvatures for the state function become the setof equations of motion in analytical mechanics, imposing the weak orthogonal conditionsamong the infinitesimal generators, i.e., equation for the tangent vector fields on the groupsubmanifold with respect to the collective variables. Those of the non-zero curvatures be-come the gradients of potential, which arises from the existence of residual Hamiltonian,along the collective variables. These quantities can be expected to give a criterion how thecollective submanifolds is effectively truncated, if we might understand the treatments.(II) The expression for the zero curvature conditions is nothing but the formal RPA equa-tion imposed by the weak orthogonal conditions. The formal RPA equation has a simplegeometrical interpretation: relative vector fields made of the SCF Hamiltonian around eachpoint on an integral curve also constitute solutions for the formal RPA equation aroundthe same point which is in turn a fixed point. It means that the formal RPA equation isa natural extension of the usual RPA equation for small-amplitude quantal fluctuationsaround the ground state to that at any point on the collective submanifold which shouldbe studied. Moreover, the enveloping curve, made of solutions of the formal RPA at eachpoint on an integral curve, becomes another integral curve. The integrability condition isjust the infinitesimal condition to transfer a solution to another one for the evolution equa-tion under consideration. Then the usual treatment of RPA equation for small amplitudearound the ground state becomes nothing but a method of determining an infinitesimaltransformation of symmetry under the assumption that fluctuating fields are composed ofonly normal-modes. See, e.g., Klein-Walet-Dang [105]. At the beginning, their descrip-tions of dynamical fermion systems in both methods had looked very different mannersat first glance. In the abstract fermion Fock spaces, each solution space of the mechanicsin both methods is the corresponding Gr . There is a difference of finite and ID fermionsystems. Overcoming the difference, we have aimed at clearing a close connection betweenthe concepts of MF potential and gauge of fermions making a role of loop group.(1) The Pl¨ucker relations on coset variables becomes analogous io
Hirota’s bilinear form .The SCF method has been devoted to a construction of boson-coordinate systems ratherthan soliton solution by the τ -FM. Both methods are equivalent to each other, if we standon the viewpoint of the Pl¨ucker relation or the bilinear identity equation defining the Gr .372) The ID fermion operators are introduced through the Laurent expansion of the finite-dimensional fermions with respect to the degrees of freedom of fermions related to theMF potential. Inversely, the collectivity of the MF potential is attributed to the gauges ofinteracting ID fermions. The construction of the fermion operators is contained in that ofClifford algebra according to the idea of Dirac by Kac and Raina [74]. This fact permits usto introduce an affine Kac-Moody algebra. It means that the usual perturbative methodwith respect to single collective variable having time periodicity has implicitly stood on τ -FS (reduction to su N ). Then we can rebuilt TDHF theory along the affine Kac-Moodyalgebra and map on the corresponding τ -FS. The TDHF theory becomes a gauge theory offermions and the collective motion appears as the motion of fermion gauges with commonfactor. The physical concept of particle-hole and vacuum in the SCF method dependent on S connects to the Pl¨ucker relations . The algebraic treatment of extracting sub-group orbitconsisting of loop paths out of Gr m is just the formation of Hirota’s bilinear equation forthe reduced KP hierarchy [75](b) to su N ( ⊂ sl N ) in the soliton theory. The present frameworkgives the manifest structure of gauge theory of fermions inherent in the SCF method andpresents a new algebraic tool for microscopic understandings of fermion system.(3) Through the investigation of physical meanings for the ID shift operators and the con-ditions of reduction to sl N in τ -FM from the loop group viewpoint [106], it is induced thatthere is the close connection between collective variables and spectral parameter in solitontheory and that the algebraic mechanism bringing the physical concept of particle andcollective motions arises from the reduction from u N to su N for the HF Hamiltonian.(4) Though TDHF theory describes a dynamics on real fermion-harmonic oscillator , solitontheory does on complex fermion-harmonic oscillator . These remarks give important tasksto extend the TDHF theory on real space affine d su N to complex space affine c sl N and tounderstand the concept of particle energy and boson energy. They are itemized as follows:(i) The algebraic mechanism can be derived from the three important elements: first, su N condition for HF Hamiltonian; second, the vacuum state according to the idea of Dirac;and last, the phase of fermion gauge can be separated into particle and collective modes .(ii) From them, a new theory for unified description of both the motions can be presented.In the theory, taking values with a time periodicity into both the modes, a unified de-scription of both the motions beyond the usual static HF equation and RPA one can beobtained. With the help of the affine Kac-Moody algebra, the theory provides the alge-braic mechanism in order to elucidate the physical concepts simply and clearly: collectivemotion induced by a TD MF potential and symmetry breaking of fermion systems andsuccessive occurrence of the collective motion due to the recovery of the symmetry. In thetheory, circle S takes an active role in causing resonance (interference) between fermions.(iii) The theory give a toy model to clear algebraic structure among original fermion field,vacuum field defined in SCF potential and bosonic field associated with Laurent spectra.(iv) To solve the new equation we must study how to extract various subgroup orbits sat-isfying the Pl¨ucker relation and how to determine a solution in the su N ( ∈ sl N ) hierarchy.We attempt construction of optimal coordinate systems on group manifolds. Then, therelation of boson expansion method for finite fermion systems and τ -FM for infinite onesis investigated to clarify algebro-geometric structures of integrable systems. Some physicalconcepts and mathematical methods work well in the ID one. The SCF method based onthe global symmetry is much improved, if we notice a local symmetry of the ID ones. TheGC method provides superposition principles on nonlinear space [51, 11]. Standing on the38ocal symmetry of ID fermion system behind a global symmetry in finite one, we havereconstructed the GC and nonlinear superposition methods using ID shift operator. Thereare many problems in connection with discussion. They are itemized as follows [80]:1. To study the relation between quantity of non zero curvature and collectivity:It is useful to investigate the relations in a schematic model of nuclei with paringplus quadruple interactions. Which may leads us to an investigation of the effectivecondition for the collective submanifold extracted by the zero curvature equations.2. To study path integral method (PIM) [52]-formal RPA from viewpoint of symmetry:InPIM, the RPA equation is described as the flactuational mode with time periodicityof Jacobi field around a classical path on the phase space. Then we intend to illus-trate the relation between both the methods from the viewpoint of symmetry.3. To extend the particle-hole formalism for (1 + 2 µ )-dimensional SU m + n S ( U m × U n ) model,Caseller–Megna-Sciuto [68] to a superconducting formalism on the SO (2 N ) U ( N ) coset space:We have successfully derived the SU m + n S ( U m × U n ) fundamental equation to get the classicalequation of motion for an integrable system. The particle-hole formalism is applica-ble to the superconducting formalism on the SO (2 N ) U ( N ) and SO (2 N +2) U ( N +1) coset spaces [52, 50].We will study the relation of the present model with the σ -model on the Gr [107].4. To study hiddenness behind gauge of state function and formation of fermion pairs:For forming pair we give classifications of Laurent spectra in ID fermions. The Lau-rent coefficient of n -soliton solution and τ -function for affine c su ( N ) have been givenin [81]. Is there any relation between soliton and gauge? While a spin of electron isdescribed as a geometrical phase of gauge using a M¨obius band.5. To clarify the relation between the spectral parameter and the collective variables:A spectral parameter of the iso-spectral equation [108] in soliton theory and collectivevariable in the SCF method, though being seen different aspect, work as scaling pa-rameters on S . The former relates to scaling parameters in description by analyticalcontinuation. The latter makes a role of deformation parameter of loop paths in Gr m .6. To study the relation between collective motions in SCF theory and Far Fields [109]:In the SCF method, we have not known approaches from the viewpoint of Far Fields.How do we bring in the idea of Far Fields to relate it to the adiabatic one which isintroduced to separate a collective motion from the others? The theory bases on thatthe speed of collective motion is much slower than that of any other non-collectivemotions. Therefore we may estimate a quantity on the assumption in what degreeto connect with a time scale transformation in the reductive perturbation method.7. To study the relation between the boson operators obeying the weak-orthogonalitycondition and the boson mapping operators i.e., the shift operators in τ -FM:The generators related to collective variables in finite-dimensional Fock space neversatisfy exact boson commutation relations because of their finite dimensionality.8. To study why soliton solution for classical wave equation shows fermion-like behaviorin quantum dynamics and what symmetry is hidden in soliton equation [48]:The nonlinear Schr¨odinger equation as a classical image of corresponding boson fields[1] has multi-soliton solutions and appears as solutions for non-scattering poten-tial [110]. The potential is derived with the use of τ -function. Nogami used a multi-component nonlinear Schr¨odinger equation instead of the TDHF equation [110].39. To study another expression of the TDHB equation using the K¨ahler coset space:To describe the classical motion on the coset manifold, we start from the local equa-tion of motion which becomes a Riccati-type equation. We get a general solution ofTD Riccati-HB equation for coset variables. We obtain the Harish-Chandra decom-position for the SO (2 N ) matrix based on the nonlinear M¨obius transformation [111].10. To study the relation between nonlinear superposition principle in soliton theory andresonating (Res) MF theory in SCF method and the algebraic (Alg) MF one:What relation does exist between the construction of exact solutions based on the ideaof imbricate series in soliton equation [48] and Res/Alg MF theories [112, 82]? Suchthe idea bases on the group integration on the solution space of soliton equation [113].The GC method in SCF method stands on the group integration [114, 115]. Furtherwe discuss on the ordinary MF theory related to the Alg MF theory based on thecoadjoint orbit leading to the non-degenerate symplectic form [116].40 ppendices A Derivation of (4.3) and (4.4) and density matrix
We give a decomposition of the generator U ( g ), i.e., U ( N ) canonical transformation (3.1), U − ( g (Λ , Λ ⋆ ,t )) = e b Ξ(Λ , Λ ⋆ ,t ) e b Υ(Λ , Λ ⋆ ,t ) (cid:0) U − ( g )= U † ( g ) , U † ( g )= U ( g † ) (cid:1) . We express g (Λ , Λ ⋆ ,t ) as g .For our aim, we set up variable ξ , ( N − m ) × m matrix ( ξ ia ), a =1 , · · · m (occupied state)and i = m + 1 , · · · N (unoccupied state) and variable υ (= − υ † ) , m × m matrix ( υ ab ) , a, b =1 , · · · m and variable υ ⋆ (= − υ T ) , ( N − m ) × ( N − m ) matrix ( υ ⋆ij ) , i, j = m +1 , · · · N , respectively.Further, we decompose the creation operator [ c † α ] as [ c † α ] = [ˆ c † a , ˇ c † i ] and the annihilation op-erator [ c α ] as [ c α ] = [ˆ c a , ˇ c i ] and prepare the following two operators b Ξ and b Υ: b Ξ ≡ [ˆ c † , ˇ c † ]Ξ (cid:20) ˆ c ˇ c (cid:21) , Ξ = (cid:20) − ξ † ξ (cid:21) , g ξ = e Ξ = (cid:20) C ( ξ ) − S † ( ξ ) S ( ξ ) ˜ C ( ξ ) (cid:21) , b Υ ≡ [ˆ c † , ˇ c † ]Υ (cid:20) ˆ c ˇ c (cid:21) , Υ ≡ (cid:20) υ υ ⋆ (cid:21) , g υ = e Υ = (cid:20) e υ e υ⋆ (cid:21) , (A. 1)where the triangular matrix functions ( S ( ξ ) , C ( ξ ) , ˜ C ( ξ )) are given by Fukutome [11, 42] as, S ( ξ ) = P ∞ k =0 ( − k k + 1)! ξ ( ξ † ξ ) k = sin p ξ † ξ ) p ξ † ξ ξ,C ( ξ ) = 1 m + P ∞ k =1 ( − k k )! ( ξ † ξ ) k = cos p ξ † ξ = C † ( ξ ) , ˜ C ( ξ ) = 1 n + P ∞ k =1 ( − k k )! ( ξ † ) k = cos p ξξ † = ˜ C ⋆ ( ξ ) , (A. 2)which hold the properties analogous to the usual triangular functions C ( ξ ) + S † ( ξ ) S ( ξ ) = 1 m , ˜ C ( ξ ) + S ( ξ ) S † ( ξ ) = 1 n (=1 N − m ) , S ( ξ ) C ( ξ ) = ˜ C ( ξ ) S ( ξ ) . (A. 3)Then using the matrices g ξ and g υ , the matrix g is given as g = g ξ g υ = e Ξ e Υ = (cid:20) C ( ξ ) − S † ( ξ ) S ( ξ ) ˜ C ( ξ ) (cid:21)(cid:20) e υ e υ ⋆ (cid:21) = (cid:20) C ( ξ ) e υ − S † ( ξ ) e υ ⋆ S ( ξ ) e υ ˜ C ( ξ ) e υ ⋆ (cid:21) = (cid:20) ˆ a ´ b ` b ˇ a (cid:21) . (A. 4)From (A. 1), we get, U ( g ξ )ˆ c † a U − ( g ξ )=ˆ c † b [ C ( ξ )] ba +ˇ c † i [ S ( ξ )] ia , U ( g ξ )ˇ c † i U − ( g ξ )=ˇ c † j h ˜ C ( ξ ) i ji − ˆ c † a (cid:2) S † ( ξ ) (cid:3) ai ,U ( g ξ )[ˆ c † , ˇ c † ] U − ( g ξ ) = [ˆ c † , ˇ c † ] g ξ , (cid:20) ˆ d ˇ d (cid:21) = g † ξ (cid:20) ˆ c ˇ c (cid:21) g ξ , g † ξ g ξ = g ξ g † ξ = 1 N ,U ( g υ )ˆ c † a U − ( g υ ) = ˆ c † b ( e υ ) ba , U ( g υ )ˇ c † i U − ( g υ ) = ˇ c † j ( e υ⋆ ) ji ,U ( g υ )[ˆ c † , ˇ c † ] U − ( g υ ) = (cid:2) ˆ c † , ˇ c † (cid:3)(cid:20) e υ e υ ⋆ (cid:21) . (A. 5)Finally, from (A. 5), we obtain the following formula:[ ˆ d † , ˇ d † ]= U ( g ξ ) U ( g υ )[ˆ c † , ˇ c † ] U − ( g υ ) U − ( g ξ )= U ( g ξ )[ˆ c † , ˇ c † ] U − ( g ξ ) g υ =[ˆ c † , ˇ c † ] g, (cid:20) ˆ d ˇ d (cid:21) [ ˆ d † , ˇ d † ] = g † (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] g. (A. 6)41he differential ∂ t e b Ξ is calculated as ∂ t e b Ξ = Z t dτ e τ b Ξ( t ) ∂ t b Ξ( t ) e − τ b Ξ( t ) e b Ξ( t ) = Z t dτ e τ ∆ b Ξ( t ) ∂ t b Ξ( t ) e b Ξ( t ) = e ∆ b Ξ − ∆ b Ξ ∂ t b Ξ( t ) e b Ξ( t ) = ∂ t b Ξ + 12! [ b Ξ , ∂ t b Ξ] + · · · + 1 n ! [ b Ξ , · · · [ b Ξ , ∂ t b Ξ ] · · · ] | {z } n − + · · · e b Ξ = [ˆ c † , ˇ c † ] ∂ t e Ξ e − Ξ (cid:20) ˆ c ˇ c (cid:21) e b Ξ = ∂ t b Ξ L e b Ξ , (due to [ b Ξ , ∂ t b Ξ] = 0) , (A. 7)Similarly, the differential ∂ t e b Υ is also computed as ∂ t e b Υ = [ˆ c † , ˇ c † ] ∂ t e Υ e − Υ (cid:20) ˆ c ˇ c (cid:21) e b Υ = ∂ t b Υ L e b Υ , (due to [ b Υ , ∂ t b Υ] = 0) . (A. 8)The proof of [ b Ξ , ∂ t b Ξ] = 0 is given as follows:[ b Ξ , ∂ t b Ξ] = [ˆ c † , ˇ c † ]Ξ (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] ∂ t Ξ (cid:20) ˆ c ˇ c (cid:21) − [ˆ c † , ˇ c † ] ∂ t Ξ (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ]Ξ (cid:20) ˆ c ˇ c (cid:21) = − Tr (cid:20)(cid:26) Ξ (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] ∂ t Ξ − ∂ t Ξ (cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] Ξ (cid:27)(cid:20) ˆ c ˇ c (cid:21) [ˆ c † , ˇ c † ] (cid:21) = − Tr (cid:20)(cid:26) Ξ (cid:20) E •• E •• E •• − E •†• (cid:21) ∂ t Ξ (cid:20) E •• E •• E •• − E •†• (cid:21) − ∂ t Ξ (cid:20) E •• E •• E •• − E •†• (cid:21) Ξ (cid:20) E •• E •• E •• − E •†• (cid:21)(cid:27)(cid:21) = 0 . (A. 9)Using (4.3) and (A. 7), collective Hamiltonian H c is rewritten as H c = i ~ ∂ t U ( g ) U − ( g ) = n i ~ ∂ t (cid:16) e b Ξ e b Υ (cid:17)o e − b Υ e − b Ξ = n i ~ ∂ t e b Ξ e b Υ + e b Ξ i ~ ∂ t e b Υ o e − b Υ e − b Ξ = n i ~ ∂ t b Ξ L e b Ξ e b Υ + e b Ξ i ~ ∂ t b Υ L e b Υ o e − b Υ e − b Ξ = i ~ ∂ t b Ξ L + e b Ξ i ~ ∂ t b Υ L e b Υ U − ( g ) = i ~ ∂ t b Ξ L + U ( g ) e − b Υ i ~ ∂ t b Υ L e b Υ U − ( g ) . (A. 10)We demand that the canonical transformation U ( g υ ) , g υ ∈ U ( N ) ( c α | i =0) leaves the free-particle vacuum | i invariant; U ( g υ ) | i = | i . Due to this demand, we set up the condition, i ~ ∂ t b Υ L = i ~ ∂ t e Υ = 0. From this condition and (A. 4) we get the final expression for H c (4.5) as H c =[ˆ c † , ˇ c † ] i ~ ∂ t e Ξ e − Ξ (cid:20) ˆ c ˇ c (cid:21) e b Ξ =[ˆ c † , ˇ c † ] i ~ ∂ t (cid:0) e Ξ e Υ (cid:1) e − Υ e − Ξ (cid:20) ˆ c ˇ c (cid:21) e b Ξ =[ˆ c † , ˇ c † ] i ~ ∂ t g · g † (cid:20) ˆ c ˇ c (cid:21) e b Ξ . (A. 11)For the Lie-algebra-valued coordinates ( O † ν , O ν ) (4.6), we also get the final expressions as O † ν =[ˆ c † , ˇ c † ] i ~ ∂ Λ ν g · g † (cid:20) ˆ c ˇ c (cid:21) e b Ξ , O ν =[ˆ c † , ˇ c † ] i ~ ∂ Λ ⋆ν g · g † (cid:20) ˆ c ˇ c (cid:21) e b Ξ , e b Ξ =exp (cid:26) [ˆ c † , ˇ c † ] Ξ (cid:20) ˆ c ˇ c (cid:21)(cid:27) . (A. 12)Following Fukutome [11, 42], let q Lξ,αa and q Rξ,αa be the N × m matrix and the N × n matrix( q Lξ,αa ) = (cid:20) ˆ a ` b (cid:21) , ( q Rξ,αi ) = (cid:20) ´ b ˇ a (cid:21) , (cid:18) g = (cid:20) ˆ a ´ b ` b ˇ a (cid:21) mode= ⇒ (cid:20) ( hh ) ( hp )( ph ) ( pp ) (cid:21)(cid:19) . (A. 13)The HF density matrix Q = ( Q αβ ) is expressed as Q ( g )= q Lξ q L † ξ = (cid:20) C ( ξ ) C ( ξ ) S † ( ξ ) S ( ξ ) C ( ξ ) S ( ξ ) S † ( ξ ) (cid:21) = (cid:20) C ( ξ ) C ( ξ ) S † ( ξ ) S ( ξ ) C ( ξ ) 1 N − m − ˜ C ( ξ ) (cid:21) , Q = Q † , Q = Q . (A. 14)We also have q Rξ q R † ξ = 1 N − Q ( g ). The HF energy functional is given in terms of Q as E HF d = h U ( g ) | H | U − ( g ) i = h φ ( g ) | H | φ ( g ) i = h βα Q αβ + [ γα | δβ ] Q αγ Q βδ = Tr { h Q} + [Tr {Q • •}| Tr {• • Q} ] . (A. 15)42 Affine Kac-Moody algebra and 2-cocycle
According to Kac and Raina [74], let gl ( N ) be the Lie algebra of N × N matrix with complexentries acting in C N and let C [ z,z − ] be the ring of Laurent polynomials in indeterminate z and z − . The loop algebra [45] e gl ( N )( ⊃ e u ( N )) is defined as gl ( N )( C [ z,z − ]) ( ⊃ u ( N )( C [ z, z − ])),i.e., as the complex Lie algebra of N × N matrix with Laurent polynomial as entries. Anelement of e gl ( N ) is given in the form a ( z ) = P r ∈ Z z r a r . ( a r ∈ gl ( N )) (B.1)We regard the fermion pair- and single-operators (OPs) as c † α c β e αβ , (1 ≤ α, β ≤ N ) ,c † α u α , (1 ≤ α ≤ N ) , c α u T α , (1 ≤ α ≤ N ) . ) (B.2)The e αβ has 1 as the ( α,β ) entry and 0 elsewhere and forms a basis of gl ( N ). The componentof N × u α is equal to 1 in the α -th row and 0 elsewhere.They span a vectorspace C N in which gl ( N ) acts. The e αβ ( r ) d = z r e αβ constitutes a basis of e gl ( N ). The loop algebra e gl ( N ) acts in the vector space C [ z,z − ] N consisting of N × z and z − as entries. The Lie bracket on e gl ( N ) is the commutator[ e αβ ( r ) , e γδ ( s )] = δ βγ e αδ ( r + s ) − δ αδ e γβ ( r + s ) . (B.3)The vectors ν Nr + α = z − r u α and ν ∗ Nr + α = u T α z r also form a basis of the vector space C [ z,z − ] N indexed by Z and its dual space. The { ν i | i = N r + α ∈ Z } is given by the column vector with1 as the i -th row and 0 elsewhere. The relation e αβ ( r ) ν Ns + β = ν N ( s − r )+ α is easily derived.For a ( z ) ∈ e gl ( N ) we denote the corresponding matrix in a ∞ by τ { a ( z ) } . We deduce matrixrep for τ { e αβ ( r ) } as τ { e αβ ( r ) } = P s ∈ Z E N ( s − r )+ α,Ns + β . (B.4) E ij ( i,j ∈ Z ) has 1 as ( i,j ) entry and 0 elsewhere and forms gl ∞ . Suppose a bigger algebra a ∞ a ∞ = { ( a ij ) | i, j ∈ Z , a ij = 0 for | i − j | ≫ N } . (B.5)There exists such an N satisfying the above condition. The corresponding matrix of the a ( z ) in (B.1) in a ∞ has the following block form with an infinite N periodic sequence: τ { a ( z ) } = . . . . . . . . . . . . a − a a . . .. . . a − a a . . .. . . a − a a . . . . . . . . . . . . . (B.6)We regard (B.6) as rep of a ∞ in which the elements on each diagonal parallel to the principaldiagonal form a periodic sequence with period N .It is well known that the OP of (B.4) acting in the F ∞ has in general an anomaly . Toavoid the anomaly , we had better use either of the normal-ordered products given below: E Nr + α,Ns + β : d = E Nr + α,Ns + β − δ αβ δ rs , ( s < , : (cid:0) ˆ νψ (cid:1) Nr + α (cid:0) ˇ ν ∗ ψ ∗ (cid:1) Ns + β : d = (cid:0) ˆ νψ (cid:1) Nr + α (cid:0) ˇ ν ∗ ψ ∗ (cid:1) Ns + β − δ αβ δ rs , ( s < . ) (B.7)We define important shift operators Λ j and Λ − j ( j ∈ Z + ) in the soliton theory on a group asΛ j = τ · · · · · · z · · · · · · · · · j , Λ − j = τ · · · · · · · · · z -1 · · · · · · j = Λ † j . (B.8)43y (B.7), 2- cocycle α on ¯ a ∞ makes 2- cocycle on pair of basis elements of e gl N as α ( τ { e αβ ( k ) } , τ { e γδ ( l ) } ) = δ αδ δ βγ δ k + l, · k, (B.9)where the Kac-Peterson 2- cocycle α [77, 76] is defined as α ( E ij , E kl ) = δ jk δ il ( i ≤ − δ li δ kj ( j ≤
0) = , for j = k, j ≥ , i = l, i ≤ , − , for i = l, i ≥ , j = k, j ≤ , , otherwise . (B.10)Then we have for the shift OPs (B.8) α (Λ k , Λ l ) = δ k + l, · k. (B.11)The Lie algebra b gl ( N ) is called the affine KM algebra associated with Lie algebra gl ( N ).For simplicity consider the level one case, c | m i = 1 ·| m i . Using the one-level formula we get X a = b X a + C · c, b X a d = P Nr = − N P s ∈ Z ( a r ) αβ : ψ N ( s − r )+ α ψ ∗ Ns + β : , ( a r ) αβ ≡ ( a r ) N ( s − r )+ α,Ns + β , [ X a , c ] KM = 0 , [ X a ,X b ] KM = b X [ a,b ] + α ( a, b ) , α ( a, b ) ≡ − Tr (cid:20) − I I (cid:21) [ a, b ] · c. (B.12) a d = a − N, ... . . . a , · · · a ,N a , − N · · · a , . . . ... a N, , b d = b − N, ... . . . b , · · · b ,N b , − N · · · b , . . . ... b N, . (B.13)The matrices a and b are elements of (B.6) and ¯ a and ¯ b are non-digonal elements of a and b [79]. [ a, b ] denotes the Lie bracket. We give a simple affine KM algebra [117, 77]. Wecall matrix A = ( a ij ) Ni,j =1 as generalized Cartan matrix if it satisfies the conditions: (i) a ii =2 and a ij are non-positive integers for i = j ; (ii) a ij = 0 impliees a ji = 0. We introduceKM algebra with Chevalley generators e i , f i , h i , (1 ≤ i ≤ N ) satisfying the following relations:[ h i , h j ] = 0 , [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j , [ e i , f j ] = δ ij h j , (ad e i ) − a ij e j = 0 , (ad f i ) − a ij f j = 0 , ( i = j ) , ) (B.14)where the action (ad x ) N to y is defined as(ad x ) N ( y ) = [ x, [ x, . . . [ x, | {z } N y ] . . . ]] = P Nk =0 ( − k (cid:18) Nk (cid:19) x N − k yx k . (B.15)In sl (2 , C ), we have e = e = (cid:20) (cid:21) , f = f = (cid:20) (cid:21) , h = h = (cid:20) − (cid:21) , (cid:20) h, (cid:18) ef (cid:19)(cid:21) =2 (cid:18) e − f (cid:19) , [ e, f ]= h, h =1 . (B.16)Finally, following de Kerf, B¨auerle and Kroode [89], we give more general definition ofthe 2- Cocycles on Lie algebras. Let M be a Lie algebra and be over the field F (= R or C ). For the Cartesian product M × M := { ( x, y ) | x, y ∈ M } bilinear map φ : M × M → F iscalled on M , if it satisfies the two conditions given below, φ ( x, y ) = − φ ( y, x ) , φ ( x, [ y, z ])+ φ ( y, [ z, x ])+ φ ( z, [ x, y ]) = 0 , for all x, y, z ∈ M. (B.17)The condition (B.17) is called the Jacobi identity for the 2-cocycles . Acknowledgements
One of the authors (S. N.) expresses his sincere thanks to Professor Constan¸ca Providˆenciafor kind and warm hospitality extended to him at CFisUC, Departamento de F´ısica, Uni-versidade de Coimbra, Portugal. This work was supported by FCT (Portugal) under theProject UID/FIS/04564/2016 and UID/FIS/04564/2019.44 eferences [1] M. Yamamura and A. Kuriyama, Time-Dependent Hartree-Fock Method and ItsExtension,
Prog. Theor. Phys. Suppl. (1987).[2] A. Bohr and B. Mottelson, Collective and Individual-Particle Aspects of NuclearStructure, Mat. Fys. Medd. Dan. Vid. Selsk. (1953) No.16[3] R. Arvieu and M. Verononi, Quasi-particles and collective states of spherical nuclei, Compt. Rend. (1960) 992-994; 2155-2157,M. Baranger, Extension of the Shell Model for Heavy Spherical Nuclei,
Phys. Rev. (1960) 957-968.T. Marumori, On the Collective Motion in Even-Even SphericalNuclei,
Prog. Theor. Phys. (1960) 331-356.[4] B.R. Mottelson, Nucear Structure, The Many Body Ploblem, Le Probl`eme `a n Corps,Lectures at Les Houches Summer School, Paris, Dunod (1959) 283-315.[5] N.N. Bogoliubov, The Compensation Principle and the Self-Consistent Field Method, Soviet Phys.-Uspekhi (1959) 236-254.[6] S.T. Belyaev and V.G. Zelevinsky, Anharmonic Effects of Quadrupole Oscillationsof Spherical Nuclei, Nucl. Phys. (1962) 582-604.[7] T. Marumori,M.Yamamura and A.Tokunaga, On the Anharmonic Effects on the Col-lective Oscillations in Spherical Even Nuclei. I, Prog.Theor.Phys. (1964)1009-1025.[8] J. da Providencia and J. Weneser, Nuclear Ground-State Correlations and BosonExpansions, Phys. Rev. C (1970) 825-833,E.R. Marshalek, On the relation between Beliaev-Zelevinsky and Marumori bosonexpansions, Nucl. Phys. A (1971) 401-409.[9] H. Fukutome, M. Yamamura and S. Nishiyama, A New Fermion Many-Body TheoryBased on the SO(2N+1) Lie Algebra of the Fermion Operators,
Prog. Theor. Phys. (1977) 1554-1571.[10] M. Yamamura and S. Nishiyama, An a priori Quantized Time-Dependent Hartree-Bogoliubov Theory.-A Generalization of the Schwinger Representation of Quasi-Spinto the Fermion Pair Algebra-, Prog. Theor. Phys. (1976) 124-134.[11] H. Fukutome,The Group Theoretical Structure of Fermion Many-Body Systems Aris-ing from the Canonical Anticommutation Relation. I -Lie Algebras of Fermion Opera-tors and Exact Generator Coordinate Representations of State Vectors- , Prog. Theor.Phys. (1981) 809-827.[12] M. Nogami, Dynamical Hartree Field and Collective motion, Soryushiron Kenkyu(Kyoto), in Japanese, (1955-1956) 600-608.[13] E.R. Marshalek and G. Holzwarth, Boson expansion and Hartree-Bogoliubov Theory, Nucl. Phys. A (1972) 438-448.[14] D. R. Inglis, Nuclear Moments of Inertia due to Nucleon Motion in a Rotating
Phys.Rev. (1956) 1786-1795.[15] Y. Shono and H. Tanaka, Nuclear Collective Motion and the Effective Two-BodyPotential,
Prog. Theor. Phys. (1959) 17-191.[16] D.J. Thouless and J.G. Valatin, Time-dependent Hartree-Fock equations and rota-tional states of nuclei, Nucl. Phys. (1962) 211-230.[17] M. Baranger and M. V´en´eroni, An adiabatic time-dependent Hartree-Fock theory ofcollective motion in finite systems, Ann. of Phys. (1978) 123-200.4518] D.M. Brink, M.J. Giannoni and M. Veneroni, Derivation of an Adiabatic Time-Dependent Hartree-Fock Formalism from a Variational Principle,
Nucl. Phys. A (1976) 237.[19] K. G¨oke and P.G. Reinhard, A consistent microscopic theory of collective motion inthe framework of an ATDHF approach,
Ann. of Phys. (1978) 328-355.A.K. Mukherjee and M.K. Pal, Evaluation of the optimal path in ATDHF theory,
Nucl. Phys. A (1982) 289-304.[20] F.H. Villars, Adiabatic Time-Dependent Hartree-Fock Theory in Nuclear Physics,
Nucl. Phys. A (1977) 269-296.[21] G. Holzwarth and T. Yukawa, Choice of the constraining operator in the constrainedHartree-Fock method,
Nucl. Phys. A (1974) 125-140.[22] D. J. Rowe and R. Bassermann, Coherent state theory of large amplitude collectivemotion,
Can. J. Phys. (1976) 1941-1968.[23] T. Marumori, T. Maskawa, F. Sakata and A. Kuriyama, Self-consistent CollectiveCoordinate Method for the Large-Amplitude Nuclear Collective Motion, Prog. Theor.Phys. (1980) 1294.[24] S. Nishiyama and J. da Providˆencia, Exact canonically conjugate momenta toquadrupole-type collective coordinates and derivation of nuclear quadrupole-type col-lective Hamiltonian, Nucl. Phys. A (2014) 51-88.[25] T. Marumori, J. Yukawa and R. Tanaka, On the Foundation of the Unified NuclearModel, I,
Prog. Theor. Phys. (1955) 442-454.[26] S. Tomonaga, Elementary Theory of Quantum-Mechanical Collective Motion of Par-ticles I, Prog. Theor. Phys. (1955) 467-481, ibidem II, (1955) 482-496.[27] S. Nishiyama and J. da Providˆencia, Description of collective motion in two-dimensional nuclei; Tomonaga’s method revisited, Nucl. Phys. A (2015) 1-17.[28] D. Candlin, On Sums over Trajectories for Systems wth Fermi Statistics,
Il NuovoCimento (1956) 231-239.[29] F.A. Berezin,The Method of Second Quantization,Pure and Applied Physics,A Seriesof Monographs and Textbooks Academic Press NewYork and London 1966.[30] R. Casalbuoni, The Classical Mechanics for Bose-Fermi Systems,
Nuovo Cim A (1976) 389-431.[31] P.A M. Dirac,The Principles of Quantum Mechanics, 4th Edition, Oxford UniversityPress, 1958.[32] M. Yamamura and A. Kuriyama, A Microscopic Theory of Collective and Indepen-dent-Particle Motions, Prog. Theor. Phys. (1981) 550-564.[33] P.J. Olver, Applications of Lie Groups to Differential Equations , Second Edition,Graduates texts in mathematics; 107, Springer-Verlag, New York, 1993.[34] J. Ablowitz, J. Kaup, C. Newell and H. Segur, The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems.
Studies in Applied Mathematics , Vol.LIII,No. 4 (1974) 249.[35] D.H. Sattinger, Gauge Theories For Soliton Problems in
Nonlinear Problems: Presentand Future , pp.51-64, eds. by A.R. Bishop, D.K. Cambell and B. Nicolaenko. North-Holland Publishing company, 1982.[36] M. O. Stephen,
Geometric Perturbation Theory in Physics , World Scientific Publish-ing Co. Pte. Ltd., 1984. 4637] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, (a) Transformation groups for solitonequations, Nonlinear Integrable Systems-Classical Theory and Quantum Theory-,Ed. by M. Jimbo and T. Miwa World Scientific Publishing Co. Pte. Ltd., 1983),pp.39-119. (b) Mathematical science of soliton, in Japanese, Monographs in
IwanamiLecture on Applied Mathematics (1993) pp.1-112. Iwanami Publishing Company.[38] L.A. Dickey, Soliton Equations and Hamiltonian System, World Scientific PublishingCo. Pte. Ltd., 1991.[39] G.M. D’Ariano and M.G. Rasetti, Soliton equations, τ -functions and coherent states,Integrable Systems in Statistical Mechanics, Ed. by G.M. D’Ariano, A. Montorsiand M.G. Rasetti World Scientific Publishing Co. Pte. Ltd., 1985, pp.143-152. G.M.D’Ariano and M.G. Rasetti, Soliton Equations and Coherent States, Phys. Lett. A (1985) 291-294.[40] E. Bettelheim, A.G. Abanov and P.B. Wiegmann, Orthogonality Catastrophe andShock Waves in a Nonequilibrium Fermi Gas,
Phys. Rev. Lett. (2006) 246402.[41] P. Ring and P. Schuck, The nuclear many-body problem , Springer, Berlin, 1980.[42] H. Fukutome, Unrestricted Hartree-Fock Theory and Its Applications to Moleculesand Chemical Reactions,
Int. J. Quantum Chem. (1981) 955.[43] D. J. Thouless, Stability conditions and nuclear rotations in the Hartree-Fock theory, Nucl. Phys. (1960) 225-232.[44] A.M. Perelomov, Coheret States for Arbitrary Lie Group, Comm. Math. Phys. (1972) 222; Generalized coherent states and some of their applications, Soviet Phys.-Uspekhi. (1977) 703.[45] A. Pressley and G. Segal, Loop Groups , Clarendon Press, Oxford, 1986.[46] R. Hirota, Direct method of finding exact solutions of nonlinear evolution equation,Lecture Notes in Mathematics, ed. by Miura R.M. Springer, New York, 1976 p.40.[47] J. Lepowsky and R.L. Wilson, Construction of the Affine Lie Algebra A (1)1 , Commun.Math. Phys. (1978) 43-53.[48] M. Tajiri and Y. Watanabe, Periodic Wave Solutions as Imbricate Series of RationalGrowing Modes: Solutions to the Boussinesq Equation, J. Phys. Soc. Jpn. (1997)1943-1949; Breather solutions to the focusing nonlinear Schr¨odinger equation, Phys.Rev. E (1998) 3510-3519.[49] S.Nishiyama,Microscopic Theory of Large-Amplitude Collective Motions Based on theSO(2N+1) Lie Algebra of the Fermion Operators, Nuovo Cimento A (1988) 239-256.[50] S. Nishiyama, Time Dependent Hartree-Bogolibov Equation on the Coset SpaceSO(2N+2)/U(N+1) and QuasiI AntiI-Commutation Relation Approximation, Int.J. Mod. Phys. E (1998) 677-707.[51] J.P. Boyd, New directions in solitons and nonlinear periodic waves: Polycnoidalwaves, imbricated solitons, weakly non-local solitary waves and numerical boundaryvalue algorithms, in Advances in Applied Mechanics edited by J. W. Hutchinson andT. Y. Wu, Vol. Academic Press, 1989, 1-82.[52] S. Nishiyama, Path Integral on the Coset Space of the SO(2N) Group and the Time-Dependent Hartree-Bogoliubov Equation,
Prog. Theor. Phys. (1981) 348-350.[53] S. Nishiyama, Note on the New Type of the SO(2N+1) Time-Dependent Hartree-Bogoliubov Equation, Prog. Theor. Phys. (1982) 680-683.An Equation for the Quasi-Particle RPA Based on the SO(2N+1) Lie Algebra of theFermion Operators, ibidem (1983) 1811-1814.4754] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm.Pure Appl. Math. (1968) 467-490.[55] M. Boiti, J.J.-P. L´eon, L. Martina and F. Pempinelli, Scattering of Localized Solu-tions in the Plane, Phys. Lett. A (1988) 432-439.[56] A.S. Fokas and P.M. Santini, Coherent Structures in Multidimensions,
Phys. Rev.Lett. (1989) 1329-1333; Dromions and a Boundary Value Problem for the Davey-Stewartson Equation, Physica D (1990) 99-130.[57] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc.R. Soc. London A (1974) 101-110.[58] J.Hietrinta and R.Hirota,Mutidromion Soltions to the Davey-Stewartson Equa-tion,
Phys.Lett.A (1990) 237-244.M. Jaulent,M.A.Manna and L.Martinez-Alonso,Fermionic analysis of Davey-Stewartson dromions,
Phys.Lett.A (1990) 303-307.[59] R. Hernandes Heredero, L. Martinez-Alonso and E. Medina Reus, Fusion and fissionof dromions in the Davey-Stewartson Equation,
Phys. Lett. A (1991) 37-41.[60] S. Nishiyama and T. Komatsu, Integrability Conditions for a Determination of Col-lective Submanifolds. I. - Group-Theoretical Aspects,
Nuovo Cimento A (1984)429-442; Integrability Conditions for a Determination of Collective Submanifolds.II. - On the Validity of the < Maximally Decoupled > Theory, (1984) 255-267;Integrability Conditions for a Determination of Collective Submanifolds.III. - An Investigation of the Nonlinear Time Evolution Arising from the Zero-Curvature Equation, (1987) 513-522:Integrability Conditions for a Determination of Collective Submanifolds. A SolutionProcedure, J. Phys. G: Nucl. Part. Phys. (1989) 1265-1274.[61] B. Schutz, Geometrical methods of mathematical physics , Cambridge University PressCambridge 1980.[62] Bo-Yu Hou and Bo-Yuan,
Differential Geometry for Physicists , World Scientific Pub-lishing Co. Pte. Ltd., 1997.[63] F. Sakata, T. Marumori, Y. Hashimoto, and T. Une, Geometry of the Self-ConsistentCollective-Coordinate Method for the Large-Amplitude Collective Motion:Stability Condition of “Maximally-Decoupled” Collective Submanifold,
Prog. Theor.Phys. (1983) 424-438.[64] H. Fukutome and S. Nishiyama, Time Dependent SO(2N+1) Theory for UnifiedDescription of Bose and Fermi Type Collective Excitations, Prog. Theor. Phys. (1984) 239-251.[65] S.G. Rajeev, Quantum Hydrodynamics in Two Dimensions, Int. J. Mod. Phys. A (1994) 5583-5624.[66] S.G. Rajeev and O.T. Turgut, Geometric Quantization and Two Dimensional QCD, Commun. Math. Phys. (1998) 493-517.[67] E. Topra and O.T. Turgut, Large N limit of SO(2N) scalar gauge theory, J. Math.Phys. (2002) 1340-1352. Large N limit of SO(2 N ) gauge theory of fermions andbosons, ibidem (2002) 3074-3096. Wave Functions of Fermion Many-Body Sys-tems,[68] M. Caseller, R. Megna and S. Sciuto, Generalizations of the Sine-Gordon Equationwith SU p + q /S ( U p × U q ) Structure, Nuovo Cimento A (1981) 339-352.R. D’Auria, T. Regge and S. Sciuto, A general scheme for bidimensional models withassociated linear set, Phys. Lett. (1980) 363-366.4869] D.J. Rowe, A. Ryman and G. Rosensteel, Many-body quantum mechanics as a sym-plectic dynamical system,
Phys. Rev. A (1980) 2362-2373.[70] E. Miller and B. Sturmfels, Pl¨ucker coordinates, Chapter 14 pp. 273-287 in Combi-natorial Commutative Algebra, Graduate Texts in Mathematics, Graduate Texts inMathematics (227), Springer, 2005.[71] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmannmanifolds, RIMS Kokyuroku (1981) 30-46.[72] H. Fukutome, On the SO(2N+1) Regular Representation of Operators and WaveFunctions of Fermion Many-Body Systems,
Prog. Theor. Phys. (1977) 1692-1708.[73] P. Goddard and D. Olive, Kac-Moody and Virasoro Algebras in Relation to QuantumPhysics, Int. J. Mod. Phys. A (1986) 303-414.[74] V.G. Kac and A.K. Raina, Bombay Lectures on Highest Weight Representation ofInfinite Dimensional Lie Algebras , World Scientific Publishing Co. Pte. Ltd., 1987.[75] A.Yu. Orlov and P. Winternitz, (a) P ∞ Algebra of KP, Free Fermions and 2-Cocyclein the Lie Algebra of Pseudo Differential Operators,
Int. J. Mod. Phys. (1997)3159-3193: (b) Algebra of pseudodifferential operators and symmetries of equationsin the Kadomtsev-Petviashvili hierarchy,
J. Math. Phys. (1997) 4644-4674.[76] V. G. Kac and D. Peterson, Lectures on Infinite Wedge Representation and MKPHierarchy , Seminaire de Math.Superieures, Les Presses de L’Universit´e de Montr´eal,102 (1986) 141-186.[77] V.G. Kac, Infinite Dimensional Lie Algebras, 3rd edition, Cambridge UniversityPress, 1990;
Vertex Algebras for Beginners , University Lecture Series , Ameri-can Mathematical Society, Providence, Rhode Island, 1996.[78] P.G. Grinivich and Y.Au. Orlov, Virasolo Action on Riemann Surfaces, Grassman-nians, det ∂ J and Segal-Wilson τ -Functions, in Problems of Modern Quantum FieldTheory , 86-106, Eds. A.A. Belavin, A.U. Klimyk and A.B. Zamolodchikov, (Springer-Verlag, Berlin, 1989).[79] T. Komatsu and S. Nishiyama, Self Consistent Field Method and τ -FunctionalMethod on Group Manifold in Soliton Theory, Proceedings of the Sixth InternationalWigner Symposium , Bogazici University Press-Istanbul 2002, 381-409.[80] T. Komatsu and S. Nishiyama, Self-consistent field method from a τ -functional view-point, J. Phys. Math. Gen. A (2000) 5879-5899;T. Komatsu, Self Consistent Field Method and τ -Functional Method in FermionMany-Body Systems, Doctoral Thesis at Osaka Prefecture University (2000).[81] S. Nishiyama, J. da Providˆencia and T. Komatsu, Self-consistent field-method and τ -functional method on group manifold in soliton theory. II. Laurent coefficients ofsolutions for b sl n and for c su n , J. Math. Phys. (2007) 053502.[82] H. Fukutome, Theory of Resonating Quantum Fluctuations in a Fermion System, Prog. Theor. Phys. (1988) 417-432. S. Nishiyama and H. Fukutome, Resonat-ing Hartree-Bogoliubov Theory for a Superconducting Fermion System with LargeQuantum Fluctuations, ibidem (1991) 1211-1222.[83] S. Nishiyama, Application of the resonating Hartree-Fock theory to the Lipkin model, Nucl. Phys. A (1994) 317-350;S. Nishiyama, M. Ido and K. Ishida, Parity-Projectef Resonating Hartree-Fock Ap-proximation to the Lipkin Model,
Int. J. Mod. Phys. E (1999) 443-460.4984] S. Nishiyama, First-Order Approximation of the Number-Projected SO(2N) Tamm-Dancoff Equation and its Reduction by the Schur Function, Int. J. Mod. Phys. E (1999) 461-483.[85] T. Komatsu and S. Nishiyama, Toward a unified algebraic understanding of conceptsof particle and collective motions in fermion many-body systems, J. Phys. Math.Gen. A (2001) 6481-6493.[86] S. Nishiyama and T. Komatsu, RPA Equation Embedded into Infinite-DimensionalFock Space F ∞ , Physics of Atomic Nuclei (2002) 1076-1082.S. Nishiyama, J. da Providˆencia and T. Komatsu, The RPA equation embedded intoinfinite-dimensional Fock space F ∞ , J. Phys. Math. Gen. A (2005) 6759-6775.[87] V.G. Kac and J.W. van de Leur, The n-component KP hierarchy and representationtheory, in Important developments in soliton theory , eds. Fokas A.S. and ZakharovV.E., Springer Series in Nonlinear Dynamics (1993), pp.302-343.[88] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Operator Approach to theKadomtsev-Petviashvili Equation - Transformation Groups for Soliton Equations III-,
J. Phys. Soc. Jpn. (1981) 3806-3812.[89] E.A. de Kerf, G.G.A. B¨auerle and A.P.E. Kroode, Lie Algebras, Finite and InfiniteDimensional Lie Algebras and Applications in Physics
Part 2, Elsevier Science V.B.,Amsterdam, 1997.[90] H. Bethe, On the Theory of Metals, I. Eigenvalues and Eignefunctions of a LinearChain of Atoms,
Zeits. Physik (1931) 205-226.[91] H.J. Lipkin, N. Meshkov. and A.J. Glick, Validity of Many-Body ApproximationMethods for a Solvable Model (I). Exact solutions and Perturbation Theory, Nucl.Phys. (1965) 188-198.[92] H.Morita,H.Ohnishi,J.daProvidˆencia and S.Nishiyama, Exact solutions for the LMGmodel Hamiltonian based on the Bethe ansatz, Nucl.Phys.B [FS] (2006) 337-350.[93] D.J. Rowe, T. Song. and H. Chen., Unified pair-coupling theory of fermion systems,
Phys. Rev. C (1991) R598-R601. H. Chen, T. Song and D.J. Rowe, The pair-coupling model, Nucl. Phys. A (1995) 181-204.[94] R.W. Richardson, Exact Eigenstates of the Pairing-Force Hamiltonian. II,
J. Math.Phys. (1965) 1034-1051.[95] D.E. Littlewood, The theory of group characters and matrix representation of groups,Clarendon, Oxford, 1958.[96] I. G. MacDonald, Symmetric Functions and Hall Polynomials, Oxford University,Oxford,1979.[97] F. Pan and J.P. Draayer, (a) New algebraic approach for an exact solution of the nu-clear mean-field plus orbit-dependent pairing hamiltonian, Phys.Lett.B (1998)7-13.(b) Exact solutions for some nuclear many-body problems, Ann. Phys. (NY) (1999) 120-140.[98] F. Pan, D. Zhou, L. Dai and J.P. Draayer, Exact solution of the mean-field plusseparable pairing model reexamined,
Phys. Rev. C (2017) 034308-1-8.[99] P. Mansfield, Solution of the Initial Value Problem for the sine-Gordon EquationUsing a Kac-Moody Algebra, Commun. Math. Phys. (1985) 525-537.[100] M. Gaudin, Diagonalisatiom D’une Classe D’hamiltoniens de Spin, J. Physique (1976) 1087-1098; La Fonction d’Onde de Bethe , Masson, Paris, 1983.50101] E.K. Sklyanin, Generating Function of Correlators in the sl Guadin Model,
Lett.Math. Phys. (1999) 275-292.[102] G. Oritz, R. Somma, J. Dukelsky and S. Rombouts, Exactly-solvable models derivedfrom a generalized Gaudin algebra, Nucl. Phys. B [FS] (2005) 421-457.[103] H.R. Larsson, C.A. Jim´enez-Hoyos and G.K-L. Chan, Minimal matrix product statesand generalizations of mean-field and geminal wavefunctions,
J. Chem. Theory Com-put. (2020). DOI:10.102/acs. jctc. 0c00463.[104] J. Daboul, P. Slodowy and C. Daboul, The hydrogen algebra as centerless twistedKac-Moody algebra,
Phys. Lett. B (1993) 321-328.[105] A. Klein, N.R. Walet and G.D. Dang, Classical Theory of Collective Motion in theLarge Amplitude, Small Velocity Regime,
Ann. of Phys. (1991) 90-148.[106] J.T. Ottesen, Infinite Dimensional Groups and Algebras in Quantum Physics, LectureNote in Physics, New Series m: Monographs; 27,Berlin: Springer, 1995.[107] S. Nishiyama., J. da Providˆencia, C. Providˆencia, and F. Cordeiro, Extended super-symmetric σ -model based on the SO(2N+1) Lie algebra of the fermion operators, Nucl. Phys. B (2008) 121-145.[108] M. Adler, P. van Moerbeke and S. Birkhoff, B¨acklund Transformations and Regular-ization of Isospectral Operators,
Adv. in Math. (1994) 140-204.[109] T. Taniuti, Part 1. General Theory, Reductive Perturbation Method and Far Fieldsof Wave Equations,
Prog. Theor. Phys. Suppl. (1974) 1-35.[110] M. Nogami and C.S. Warke, Exactly solvable time-dependent Hartree-Fock equa-tions, Phys. Rev. C (1978) 1905-1913.[111] S. Nishiyama and J. da Providˆencia, SO(2N)/U(N) Riccati-Hartree-Bogoliubov equa-tion based on the SO(2N) Lie algebra of the fermion operators, Int. J. Geom. MethodsMod. Phys. (2015) 1550035.[112] S. Nishiyama and J. da Providˆencia, Remarks on the mean-field theory based on theSO(2N+1) Lie algebra of the fermion operators, Int. J. Geom. Methods Mod. Phys. (2019) 1950184; Mean-field theory based on the Jacobi hsp := semi-direct sum h N ⋊ sp (2 N, R ) C algebra of boson operators, J. Math. Phys. (2019) 081706.[113] J. Mickelsson, Current Algebras and Groups, Plenum Press, New York, 1989.[114] S. Nishiyama, H. Morita and H. Ohnishi, Group-theoretical deduction of a dyadicTamm-Dancoff equation by using a matrix-valued generator coordinate, J. Phys.Math. Gen. A (2004) 10585-10607.[115] S. Nishiyama and J. da Providˆencia, Modified Non-Euclidian Transformation on theSO(2N+2)/U(N+1) Grassmannian and SO(2N+1) Random Phase Approximationfor Unified Description of Bose and Fermi Type Collective Excitations, Int. J. Geom.Methods Mod. Phys. (2016) 1650043.[116] A.A. Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics Vol-ume 64: American Mathematical Society Providence, Rhode Island, 2004.
Elements of the Theory of Representations , Springer-Verlag, New York, 1976.[117] M. Jimbo,