Effective Sine(h)-Gordon-like equations for pair-condensates composed of bosonic or fermionic constituents
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Effective Sine(h)-Gordon-like equations for pair-condensatescomposed of bosonic or fermionic constituents
Bernhard Mieck Abstract
An effective coherent state path integral for super-symmetric pair condensates is investigatedwith specification on the nontrivial coset integration measure. The non-Euclidean integrationmeasure prevents straightforward classical equations and solutions of the independent, anomalousfield variables which follow from variations of the actions in the exponential phase weight factors.We examine a transformation with a suitable super-Jacobi matrix for the change of coset integrationmeasure to ’flat’ Euclidean path integration fields of pair condensates. The independent parameterfields of the super-symmetric anomalous terms are given by those of the Osp(
S, S | L ) / U( L | S ) cosetsuper-manifold. The described, effective coherent state path integral of pair condensates is obtainedby a gradient expansion after a Hubbard-Stratonovich transformation (HST) of the original pathintegral with super-fields of bosonic and fermionic atoms. A modified HST of bosonic and fermionicsuper-fields converts the original path integral into one with ’Nambu’ doubled, super-symmetricself-energies. Due to the addition of source fields, we consider a spontaneous symmetry breakingof the total Osp( S, S | L ) super-group to the Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) coset decompositionwith the super-unitary U( L | S ) group as the invariant subgroup of the background density field.The nontrivial coset integration measure, determined by the square root ( SDET( ˆ G Osp / U ) ) / ofthe super-determinant of the Osp( S, S | L ) / U( L | S ) coset metric tensor ˆ G Osp / U , is eliminated bythe ’ inverse square root ’ of the coset metric tensor ( ˆ G Osp / U ) − / as the appropriate super-Jacobimatrix; this results into Euclidean path integration variables for the pair condensate fields. Adiagonal construction of the coset metric tensor ˆ G Osp / U allows a straightforward application ofthe super-Jacobi matrix ( ˆ G Osp / U ) − / to the pair condensate fields of the Osp( S, S | L ) / U( L | S )coset super-manifold which also appears with this coset metric tensor in the gradients and kineticenergies of the actions. Therefore, we acquire a considerable simplification of the effective coherentstate path integral in terms of anomalous, ’ Euclidean path integration variables ’. In analogy andin the sense of statistical thermodynamics, the particular property of being a ’ state variable ’ isverified for the modulus of eigenvalues of the coset order-parameter matrix for anomalous fields.On the contrary, the phase values of eigenvalues of the anomalous coset order-parameter matrixdepend on the chosen time contour path or previous history of transformation fields and thereforecorrespond to the path dependent ’ heat ’ or ’ work ’ variables of thermodynamics in a transferredsense. According to the transformation to Euclidean, anomalous path integration variables, firstorder variations of fields can be performed for classical equations with inclusion of second and highereven order variations for universal fluctuations determined by the coset metric tensor ˆ G Osp / U .Furthermore, we mention how to extend finite order gradient expansions to infinite order by usinga suitable integral representation for the logarithm of an operator and similarly for its inverse. Keywords : super-symmetry, spontaneous symmetry breaking, nonlinear sigma model, coherentstate path integral, Keldysh time contour, many-particle physics.
PACS : e-mail: ”[email protected]”;freelance activity during 2007-2009; current location : Zum Kohlwaldfeld 16, D-65817 Eppstein, Germany. CONTENTS
Contents
S, S | L ) / U( L | S ) ⊗ U( L | S ) . . . . . . . . . . . . . 212.3 Effective action for pair condensates with coupling coefficients of the background field . 242.4 Scaling of physical parameters and quantities to dimensionless values and fields . . . . . 29 Generating functionals, as coherent state path integrals, allow various kinds of approximations or evenexact solutions apart from being representations of many-particle quantum mechanics [1]-[4]. Coherentstate path integrals can be examined by Monte Carlo methods according to appropriate importancesampling and stationary phase filtering or even by exact solutions in the case of integrable systems [1, 2].At zero temperature they consist of a weight factor, usually an exponential phase comprising a classicalaction, so that certain classical field configurations can contribute a dominant part in the weightingwith the exponential. These dominant contributions are usually obtained by the stationary phase orfirst order variation of the actions in the exponent of the weight factor. In principle this variation canbe extended to second or even higher order variations around the solutions of classical fields from thefirst order variation.However, this process of variations for approximating by classical solutions becomes nontrivial in thecase of non-Euclidean integration measures of the field variables. In this case one can apply the methodof steepest descent for a polynomial-like integration measure with the exponential of classical actions [5]or one transforms the whole factor of the ’ integration measure ’ to its ’ exponential(logarithm(integrationmeasure)) ’ form so that it has an equivalent weight as the classical action terms in the exponents.This method is straightforward; but, one can also try to determine a more sophisticated transforma-tion of the path field variables so that the nontrivial integration measure is eliminated for Euclideanintegration variables by inclusion of an additional Jacobi-determinant. The functional dependence ofthe classical actions with the original fields is then altered by the corresponding Jacobi-matrix of thenew Euclidean integration field variables. Both methods, the method of steepest descent (with ’ ex-ponential(logarithm(integration measure)) ’ or the removal of nontrivial integration measures by trans-formations with a suitable Jacobi-matrix, are in general far from being equivalent. In spite of the’ exponential(logarithm(integration measure)) ’ form, the fields of the nontrivial integration measure inthe variations of steepest descent method contribute in a different manner than the fields of the actionsweighted by exponentials. In the case of the considered coherent state path integral [6], it is inevitablethat the ’ exponential(logarithm(integration measure)) ’ in the method of steepest descent has its firstcontributions from second and all higher even order variations of the fields whereas the main otheractions have only non-vanishing terms in odd-numbered order of variations with the independent fieldson the time contour. Therefore, the simple method of steepest descent causes inconsistent treatment inthe case of nontrivial path integration measures and their variations on the time contour in comparisonto the variations on the time contour dependent fields in the actions of the exponentials.We illustrate this problem in analogy to a multidimensional integral Z [ ~x ], ( ~x = { x , . . . , x N } ), withan action A [ ~x ] Z [ ~x ] = Z d [ ~x ] q det (cid:0) ˆ g ( ~x ) (cid:1) exp (cid:8) ı A [ ~x ] (cid:9) , (1.1)where the Euclidean integration measure d [ ~x ] is modified by the square root of a metric tensor ˆ g ij ( ~x ) as the nontrivial integration measure (cid:0) ds (cid:1) = dx i ˆ g ij ( ~x ) dx j . (1.2)The transformation to Euclidean variables d [ ~y ] is related to the inverse square root of the metrictensor ˆ g − / ( ~x ) as the appropriate Jacobi matrix where the symmetry of the metric tensor allows adecomposition into orthogonal matrices ˆ O ij ( ~x ) and real eigenvalues ˆ λ k ( ~x ) (cid:0) ds (cid:1) = dx i ˆ g ij ( ~x ) dx j = dx i ˆ O Tik ( ~x ) ˆ λ k ( ~x ) ˆ O kj ( ~x ) dx j = (1.3)= dx i (cid:16) ˆ O T ( ~x ) · ˆ λ / ( ~x ) (cid:17) ki | {z } dy k (cid:16) ˆ λ / ( ~x ) · ˆ O ( ~x ) (cid:17) kj dx j | {z } dy k = dy k dy k = dy k dy k ; dy j = (cid:16) ˆ λ / ( ~x ) · ˆ O ( ~x ) (cid:17) ji dx i ; (1.4)ˆ O ji ( ~x ) dx i = (cid:16) ˆ λ − / ( ~x ) · d~y (cid:17) j ; (1.5)= ⇒ y j = y j ( ~x ) = ⇒ x i = x i ( ~y ) . This yields with the additional Jacobi matrix ˆ J ik = ( ∂x i /∂y k ) = ( ˆ O T ( ~x ) · ˆ λ − / ( ~x ) ) ik Euclidean inte-gration variables ~y for Z [ ~x ( ~y )]ˆ J ik = ∂x i ∂y k = (cid:16) ˆ O T ( ~x ) · ˆ λ − / ( ~x ) (cid:17) ik ; (1.6)det (cid:0) ˆ J ik (cid:1) = det h(cid:0) ˆ O T ( ~x ) · ˆ λ − / ( ~x ) (cid:1) ik i = det (cid:2) ˆ g − / ( ~x ) (cid:3) = (cid:16) det (cid:2) ˆ g ( ~x ) (cid:3)(cid:17) − / ; (1.7) Z [ ~x ( ~y )] = Z d [ ~y ] det h ˆ g − / ( ~x ) i q det (cid:0) ˆ g ( ~x ) (cid:1)| {z } ≡ exp (cid:8) ı A [ ~x ( ~y )] (cid:9) ; (1.8)= Z ′ [ ~y ] = Z d [ ~y ] exp { ı A ′ [ ~y ] } ; A ′ [ ~y ] = A [ ~x ( ~y )] ; Z ′ [ ~y ] = Z [ ~x ( ~y )] . (1.9)The functional Taylor expansion of the action is then achieved straightforwardly where ’classical equa-tions’ are determined in a transferred sense from the vanishing of the first order variation which canbe improved by Gaussian integrals of the second order variation for fluctuations around the ’classicalsolutions’. In order to obtain the transformation to Euclidean fields ~y , it is of particular importance thatthe metric tensor ˆ g ij ( ~x ) can be diagonalized to the eigenvalues ˆ λ k ( ~x ). By taking the (inverse) squareroot of eigenvalues [ˆ λ k ( ~x )] ± / , one acquires the (inverse) square root of the metric tensor [ˆ g ( ~x )] ± / incombination with the orthogonal matrix ˆ O ji ( ~x ) as the eigenvectors.In this paper we investigate an analogous problem, but in the more involved context of a super-symmetric coherent state path integral [6] with a nontrivial integration measure which also containsanti-commuting integration fields. The measure is given as the square root of the super-determinant[SDET( ˆ G Osp / U )] / of the metric tensor ˆ G Osp / U in a coset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) .1 Variation of classical actions in coherent state path integrals with nontrivial integration measures S, S | L ) with the super-unitary U( L | S ) group as subgroup [7]-[17]. The independent field degrees of freedom of the final effective actions are restricted to the anoma-lous molecular- and BCS- pair condensates in a spontaneous symmetry breaking (SSB) [18, 19] with thecoset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ). We briefly describe the steepest descent methodby exponentiating and taking the logarithm of the coset integration measure; furthermore, a detailedaccount is outlined for the transformation to a Euclidean coherent state path integration measure withthe inverse square root of the metric tensor ( ˆ G Osp / U ) − / as the appropriate super-Jacobi-matrix. Thelatter transformation completely removes the coset integration measure and yields nontrivial classicalfield dependence in the actions, but results in simple Euclidean path integration measures of the inde-pendent fields. According to the Euclidean path integration measure, the variations with the classicalfields in the actions of the exponents allow a consistent treatment of the non-equilibrium time contourintegrals in the coherent state path integrals [20]-[25].One may expect that all the transformations of the anomalous, coset fields only involve spatiallyand time-like local expressions as one transforms to the ’flat’ Euclidean path integration fields with thesuper-Jacobi matrix given by the inverse square root ( ˆ G Osp / U ) − / of the coset metric tensor ˆ G Osp / U .However, we verify that one has to take into account previous values or time contour histories in thecase of phase-valued transformations of the eigenvalues of the coset order-parameter matrix; this isin contrast to the absolute values of eigenvalues of the order-parameter matrix which only yield localspace-time expressions in the transformations. Therefore, the absolute values of eigenvalues of thecoset order-parameter are similar to ’ state variables ’ in the sense of thermodynamics; on the contrary,the phase values of the eigenvalues of the coset order-parameter matrix require the detailed previoustime contour history in order to achieve the transformed, Euclidean path integration variables. Inconsequence, one can compare the transformation of the phases of the coset eigenvalues with the path-dependent ’ work ’ or ’ heat ’ variables of thermodynamics in a transferred sense. This observed propertyof our transformations to Euclidean fields is in accordance with other models, as the transition fromincoherent to coherent laser light, where the phase of the laser light is treated separately (as e.g. in aphase diffusion model) or in analogy to a second order phase transition for the laser threshold [30]-[37].Section 1.2 is devoted to the issue of finite versus infinite order gradient expansion of a (super-)determinant. Finite order gradient expansions have the advantage to be related to known, integrable,classical Sine(h)-Gordon-like equations; however, as one only takes into account gradually varyingspatial gradients of coset matrices, it turns out that the ’inverse’ of these slowly altering gradientsis inevitably involved yielding also strongly varying fields in coordinate space. Therefore, we pointout a suitable integral representation for the logarithm and similarly for the inverse of an operator[48] so that infinite order gradients are considered in a reliable manner [62, 45]. In sections 1.3, 1.4general properties of super-matrices are reviewed for the considered case of super-symmetric coherentstate path integrals with ’Nambu’ doubled super-matrices for the self-energy (compare Refs. [18, 19] forthe doubling of fields and see Refs. [7]-[17] for more details concerning super-groups with their super-algebras). We define the underlying Hamiltonian with the combination of Bose- and Fermi-operators andintroduce symmetry breaking source fields for a coherent BEC-wavefunction and for coherent molecular-and BCS- pair condensates (compare Refs. [26]-[29] for similar cases in many-body theory). In Ref.[6] the various steps and the analysis of super-symmetries Osp( S, S | L ) / U( L | S ) are outlined for thetransformation to a coherent state path integral with the ’Nambu’ doubled self-energy δ e Σ abαβ ( ~x, t p ) e K taking values in the ortho-symplectic super-algebra osp( S, S | L ). A gradient expansion, combined witha coset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ), reduces the independent angular momentumfield degrees of freedom to anomalous terms whose effective Sine(h)-Gordon-like actions are determinedby a real, scalar self-energy density as background field for effective coupling constants. Since thepresent paper aims at the removal of the nontrivial coset integration measure (SDET( ˆ G Osp / U )) / bya transformation with the inverse square root of the coset metric tensor ( ˆ G Osp / U ) − / , we also traceout the detailed parametrization of the self-energy as an exact element of the ortho-symplectic algebraosp( S, S | L ) in section 2.1. The osp( S, S | L ) self-energy generator is separated by a coset decompositioninto u( L | S ) density terms as subalgebra and into osp( S, S | L ) / u( L | S ) related anomalous molecular- andBCS- terms whose nontrivial integration measure is briefly outlined in section 2.2. Section 2.3 containsthe effective actions of the coset matrices for pair condensates following from the gradient expansionwith averaging of coupling parameters according to the background self-energy density field. In section2.4 we apply a scaling to dimensionless fields and parameters of the actions in the exponentials ofcoherent state path integrals with non-Euclidean path integration measure. After general symmetryconsiderations in section 3.1, sections 3.2.1 to 3.2.3 finally encompass the suitable transformations withthe inverse square root of the metric tensor ( ˆ G Osp / U ) − / of Osp( S, S | L ) / U( L | S ) in order to replace thenontrivial coset integration measure by Euclidean path integration measures of the independent fields.In section 3.3 diagonal elements of coset matrices as in ˆ T − ( ~x, t p ) ( ∂ ˆ T ( ~x, t p ) ) are related to the diagonalelements of the new transformed field variables for anomalous field degrees of freedom having Euclideanpath integration measures. Furthermore, we describe the problem for the ’ path-dependent ’ phase valuesof the coset order-parameter matrix where one has to include nonlocal time contour dependent historiesfor the transformation to Euclidean fields. Section 4.1 comprises the variations of the effective actions forclassical field equations with the Euclidean path integration variables. In section 4.2 a brief summary isincluded how the transformations to Euclidean coherent state path fields effect the observables followingfrom differentiation with respect to the source fields. We also point out again for the possible extensionsof the few classical integrable systems to chaotic cases which may be classified in terms of r-s matricesand symmetry breaking extensions of quantum groups [53]-[60]. There always appears the problem whether the restriction to a finite order gradient expansion is sufficientfor considering a functional determinant in the ’det( ˆ O ) = exponential { trace logarithm( ˆ O } ’ kind. As onetakes only terms with derivatives for stable, static energy configurations in 3(+1) spatial dimensions,one has to expand from second up to fourth order gradients so that one cannot scale the particularconfiguration to arbitrary small or large sizes in the three dimensional coordinate space integrationsover the static Hamiltonian density (’Derrick’s theorem’ [61]). The spatially two-dimensional case isexpected to contribute to the Goldstone modes in a SSB with second order gradients as a lowest orderapproximation. Since we reduce the expansion up to second order gradients in the present paper, wehave only extracted the Goldstone modes of the SSB Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ). Following Ref.[6], one can straightforwardly continue with an expansion to higher order gradients according to the rulesand principles of chapter 4 in [6]. However, one has to decide which transport coefficients, composed ofthe background field, should remain from the gradient expansion as the unsaturated operators act to the .2 Finite versus infinite order gradient expansion of determinants (cid:0) ln ˆ O (cid:1) = (cid:18) Z + ∞ dv exp {− v ˆ1 } − exp {− v ˆ O } v (cid:19) ; (1.10) (cid:0) ˆ O − (cid:1) = (cid:18) Z + ∞ dv exp (cid:8) − v ˆ O (cid:9)(cid:19) . (1.11)As we use the particular form (1.12) for the gradient operator (ˆ1 + δ ˆ H ( ˆ T − , ˆ T ) h ˆ H i − ) with meanfield approximation h . . . i for the anomalous doubled, one-particle operator ˆ H , we obtain the relationˆ T − h ˆ H i ˆ T h ˆ H i − which results for vanishing source term ˆ J ≡ δ ˆ H ( ˆ T − , ˆ T ) = ˆ T − h ˆ H i ˆ T − h ˆ H i ; (1.12)ˆ O = (cid:18) ˆ1 + (cid:16) δ ˆ H ( ˆ T − , ˆ T ) + e J ( ˆ T − , ˆ T ) (cid:17) h ˆ H i − (cid:19) (1.13)= ˆ T − h ˆ H i ˆ T h ˆ H i − + e J ( ˆ T − , ˆ T ) h ˆ H i − ; A SDET [ ˆ
T , h ˆ H i ; ˆ J ≡
0] = 12 tr STR ln h T − h ˆ H i ˆ T h ˆ H i − i (1.14)= 12 Z + ∞ dv tr STR (cid:20) exp {− v ˆ1 } − exp {− v ˆ T − h ˆ H i ˆ T h ˆ H i − } v (cid:21) . If one assumes slowly varying finite order gradients of ˆ T − h ˆ H i ˆ T , one will also obtain unintented,extraordinary large spatial and time-like variations with ˆ T h ˆ H i − ˆ T − according to the additional traceoperation on the logarithm. In order to circumvent this problem, we use in Refs. [62, 45] the particularintegral representation (1.10,1.11) for the logarithm of an operator (and similarly for the inverse) whichgives a simple representation of the logarithm with the (coset matrix weighted) combination of h ˆ H i andits inverse h ˆ H i − in an exponential. One can emphasize this point by a gauge transformation of thecoset decomposition so that the mean field operator h ˆ H i is simplified to pure spatial gradient operators(compare Ref. [62]).If we suppose finite, positive eigenvalues for the total operator ˆ O = ˆ T − h ˆ H i ˆ T h ˆ H i − , the inversefactorials 1 /n ! of exp {− v ˆ T − h ˆ H i ˆ T h ˆ H i − } cause a meaningful expansion and convergence instead of a pure logarithm ln( ˆ T − h ˆ H i ˆ T h ˆ H i − ) with reciprocal integer numbers in the expansion. Therefore,one can also rely on the integral representations (1.10,1.11) for the logarithm and for the inverse ofan operator ˆ O and can apply these relations for reducing the path integral part with coset matri-ces ˆ T ( ~x, t p ) = exp {− ˆ Y ( ~x, t p ) } and coset generator ˆ Y ( ~x, t p ) for molecular and BCS-pair condensates.We can even choose the eigenbasis of the mean field approximated, one-particle operator h ˆ H i or ofits anomalous doubled version h ˆ H i instead of the d+1 dimensional coordinate representation. Thisparticular matrix representation for ˆ T in terms of the eigenbasis of h ˆ H i allows to calculate observ-ables as correlation functions of anomalous super-field combinations h ψ ~x,α ( t p ) ψ ~x ′ ,β ( t ′ p ) i , density terms h ψ ∗ ~x,α ( t p ) ψ ~x ′ ,β ( t p ) i and eigenvalue correlations of molecular and BCS-terms. In this section we briefly define the super-fields from their corresponding bosonic and fermionic oper-ators. We introduce the complex conjugation, super-transposition, super-trace and super-determinantin analogy to the operations on ordinary matrices [7]-[17]. The basic constituents are determined bysuper-fields ψ ~x,α ( t ) which have L (= 2 l + 1) odd-numbered bosonic and S (= 2 s + 1) even-numberedfermionic angular momentum degrees of freedom. The summations over these bosonic and fermionicangular momentum degrees of freedom are abbreviated by the first greek letters α, β, γ . . . in bilin-ear or quartic relations of the super-fields. We specify these N = L + S component super-fields in(1.15) with internal bosonic vector ~b ~x ( t ) = { b ~x,m ( t ) } and internal Grassmann-valued, fermionic vector ~α ~x ( t ) = { α ~x,r ( t ) } α , β , . . . = − l, . . . , + l | {z } L bosons ; − s, . . . , + s | {z } S fermions ; l = 0 , , , . . . ; s = 12 , , , . . . ; (1.15) N = L + S ; L = 2 l + 1 ; S = 2 s + 1 ; ψ ~x,α ( t ) = (cid:18) ~b ~x ( t ) ~α ~x ( t ) (cid:19) ; ~b ~x ( t ) = (cid:8) b ~x,m ( t ) (cid:9) = n b ~x, − l ( t ) , . . . , b ~x, + l ( t ) o T ; ~α ~x ( t ) = (cid:8) α ~x,r ( t ) (cid:9) = n α ~x, − s ( t ) , . . . , α ~x, + s ( t ) o T ; ψ + ~x,α ( t ) = (cid:16) b ∗ ~x, − l ( t ) , . . . , b ∗ ~x, + l ( t ) ; α ∗ ~x, − s ( t ) , . . . , α ∗ ~x, + s ( t ) (cid:17) . These coherent super-fields are applied on the non-equilibrium time contour to define the coherent statepath integral following from the Hamilton operator (1.17) with combined Bose- and Fermi-operators in(1.16) ˆ ψ ~x,α = n ˆ ~b ~x , ˆ ~α ~x o T ; ˆ ψ + ~x,α = n ˆ ~b + ~x , ˆ ~α + ~x o ; (1.16)ˆ H ( ˆ ψ + , ˆ ψ, t ) = X ~x X α ˆ ψ + ~x,α ˆ h ( ~x ) ˆ ψ ~x,α + X ~x,~x ′ X α,β ˆ ψ + ~x ′ ,β ˆ ψ + ~x,α V | ~x ′ − ~x | ˆ ψ ~x,α ˆ ψ ~x ′ ,β + (1.17) The spatial sum P ~x . . . is dimensionless and is scaled with the system volume so that P ~x is equivalent to R L d ( d d x/L d ) . . . . .3 Bosonic and fermionic operators with their coherent state field representations X ~x,α (cid:16) j ∗ ψ ; α ( ~x, t ) ˆ ψ ~x,α + ˆ ψ + ~x,α ( t ) j ψ ; α ( ~x, t ) (cid:17) ++ 12 X ~x str α,β " e j + ψψ ; N × N ( ~x, t ) (cid:18) ˆ c ~x,L × L ˆ η T~x,L × S ˆ η ~x,S × L ˆ f ~x,S × S (cid:19) + (cid:18) ˆ c + ~x,L × L ˆ η + ~x,L × S ˆ η ∗ ~x,S × L ˆ f + ~x,S × S (cid:19) e j ψψ ; N × N ( ~x, t ) .Aside from the one-particle operator ˆ h ( ~x ) (1.18) with kinetic energy, trap- and chemical potential u ( ~x ), µ , we include a short-ranged quartic interaction potential V | ~x ′ − ~x | with super-symmetry between Bose-and Fermi-particles. These two operator terms obey a global super-unitary invariance U( L | S ) so thata super-symmetry results between bosonic and fermionic angular momentum degrees of freedom. Weassume that this super-symmetry may be achieved by appropriate tuning of Feshbach resonances withsimilar effective masses and similar properties concerning the trap potential [38]-[42]ˆ h ( ~x ) = ˆ ~p m + u ( ~x ) − µ ; (1.18) j ψ ; N ( ~x, t ) = n j ψ ; B,L ( ~x, t ) ; j ψ ; F,S ( ~x, t ) o T . (1.19)Apart from the U( L | S ) symmetry breaking source field j ψ ; α ( ~x, t ) (1.19) for a coherent BEC wavefunction,we specialize on the investigation of super-symmetric pair condensates which are created by the N × N = ( L + S ) × ( L + S ) super-symmetric source matrix e j ψψ ; N × N ( ~x, t ) in (1.17). The boson-boson paircondensate terms are denoted by a L × L symmetric operator matrix ˆ c ~x,L × L (1.20) and the fermion-fermion pair condensates (acting as a boson in its entity) are marked by the anti-symmetric operatormatrix ˆ f ~x,S × S (1.21). The fermion-boson mixed operator (1.22) and its transpose (1.23) are abbreviatedby ˆ η ~x,S × L and ˆ η T~x,L × S and have fermionic properties as an entity, due to the composition of a boson andfermion operatorˆ c ~x,L × L = (cid:8) ˆ c ~x,mn (cid:9) = (cid:8) ˆ b ~x,m ˆ b ~x,n (cid:9) ; ˆ c T~x,mn = ˆ c ~x,mn ; m, n = − l, . . . , + l ; (1.20)ˆ f ~x,S × S = (cid:8) ˆ f ~x,rr ′ (cid:9) = (cid:8) ˆ α ~x,r ˆ α ~x,r ′ (cid:9) ; ˆ f T~x,rr ′ = − ˆ f ~x,rr ′ ; r, r ′ = − s, . . . , + s ; (1.21)ˆ η ~x,S × L = (cid:8) ˆ η ~x,rm (cid:9) = (cid:8) ˆ α ~x,r ˆ b ~x,m (cid:9) ; r = − s, . . . , + s ; m = − l, . . . , + l ; (1.22)ˆ η T~x,L × S = (cid:8) ˆ η T~x,mr (cid:9) = (cid:8) ˆ b ~x,m ˆ α ~x,r (cid:9) ; m = − l, . . . , + l ; r = − s, . . . , + s . (1.23)The N × N source matrix e j ψψ ; N × N ( ~x, t ) (1.24) has to respect the symmetry properties of the super-symmetric, paired terms in (1.20-1.23) and therefore has a symmetric even sub-matrix ˆ j B ; L × L ( ~x, t ) forthe boson-boson pair condensates (1.25) and an anti-symmetric even sub-matrix ˆ j F ; S × S ( ~x, t ) for thecorresponding fermion-fermion paired terms (1.26). Furthermore, Grassmann or anti-commuting fieldsˆ j η ; S × L ( ~x, t ), ˆ j Tη ; L × S ( ~x, t ) (1.27) generate the boson-fermion ˆ η T~x,L × S (1.23) or fermion-boson ˆ η ~x,S × L (1.22)pair condensates in a super-trace relation. Appropriate signs have to be taken into account due to theproperty of a super-trace in the fermion-fermion part of a super-matrix e j ψψ ; N × N ( ~x, t ) = (cid:18) ˆ j B ; L × L ( ~x, t ) − ˆ j Tη ; L × S ( ~x, t )ˆ j η ; S × L ( ~x, t ) − ˆ j F ; S × S ( ~x, t ) (cid:19) ; (1.24)ˆ j B ; L × L ( ~x, t ) ∈ C even ; ˆ j TB ; mn ( ~x, t ) = ˆ j B ; mn ( ~x, t ) ; (1.25)0 ˆ j F ; S × S ( ~x, t ) ∈ C even ; ˆ j TF ; rr ′ ( ~x, t ) = − ˆ j F ; rr ′ ( ~x, t ) ; ˆ j F ; rr ( ~x, t ) = 0 ; (1.26)ˆ j η ; S × L ( ~x, t ) = { j η ; rm ( ~x, t ) } ∈ C odd ; m = − l, . . . , + l ; (1.27)ˆ j + η ; L × S ( ~x, t ) = { j + η ; mr ( ~x, t ) } ∈ C odd ; r = − s, . . . , + s . In the following we define the complex conjugation (1.28) of Grassmann variables and the super-transposition (1.30,1.31), the super-trace (1.32), the super-hermitian conjugation (1.33) and the super-determinant (1.34) of graded- or super-matrices as ˆ N , ˆ N (1.29) [7]-[17]. The complex conjugation of aproduct ( ξ . . . ξ i . . . ξ n ) ∗ of anti-commuting variables ξ i changes these n-factors to its reversed order withcomplex conjugated, odd numbers ξ ∗ i . This definition provides the combination ξ ∗ i ξ i of a Grassmannnumber ξ i and its complex conjugate ξ ∗ i with the property of an even, real (but nilpotent) variable( ξ . . . ξ i . . . ξ n ) ∗ = ξ ∗ n . . . ξ ∗ i . . . ξ ∗ ; ( ξ ∗ i ) ∗ = ξ i ; ( ξ ∗ i ξ i ) ∗ = ξ ∗ i ( ξ ∗ i ) ∗ = ξ ∗ i ξ i . (1.28)Super-matrices as ˆ N , ˆ N (1.29) consist of the even boson-boson blocks ˆ c , ˆ c and the even fermion-fermion blocks ˆ f , ˆ f . Sub-matrices of anti-commuting variables are placed in the non-diagonal fermion-boson blocks ˆ χ , ˆ χ and boson-fermion blocks ˆ η T , ˆ η T . Under a super-transposition ’st’ of the super-matrices ˆ N , ˆ N (1.30), the even parts ˆ c , ˆ c and ˆ f , ˆ f are transposed in the manner of ordinarymatrices whereas the fermion-boson blocks ˆ χ , ˆ χ and boson-fermion blocks ˆ η T , ˆ η T are exchanged withtransposition and with the inclusion of an additional minus sign in the resulting fermion-boson blocks − ˆ η , − ˆ η (1.30). This definition of super-transposition preserves the property of ordinary matrices tobe reversed under transposition in a product of matrices (1.31)ˆ N = (cid:18) ˆ c ˆ η T ˆ χ ˆ f (cid:19) ; ˆ N = (cid:18) ˆ c ˆ η T ˆ χ ˆ f (cid:19) ; (1.29)ˆ N st = (cid:18) ˆ c T ˆ χ T − ˆ η ˆ f T (cid:19) ; ˆ N st = (cid:18) ˆ c T ˆ χ T − ˆ η ˆ f T (cid:19) ; (1.30)( ˆ N · ˆ N ) st = ˆ N st · ˆ N st . (1.31)The super-trace ’str’ of a super-matrix ˆ N comprises the traces of the even boson-boson part ˆ c and theeven fermion-fermion part ˆ f (1.32). However, an additional minus sign has to be included in the traceof the fermion-fermion part so that the cyclic invariance of a product of super-matrices is maintainedin a super-trace relation as in a trace with the product of several ordinary matricesstr[ ˆ N ] = str (cid:18) ˆ c ˆ η T ˆ χ ˆ f (cid:19) = tr[ˆ c ] − tr[ ˆ f ] ; str[ ˆ N ˆ N ] = str[ ˆ N ˆ N ] . (1.32)The super-hermitian conjugation (1.33) of super-matrices ˆ N , ˆ N (1.29) does not involve additionalminus signs as the super-transposition (1.30,1.31). In comparison to (1.31), the property ( ˆ N ˆ N ) + =ˆ N +2 ˆ N +1 (reversal of a product under super-hermitian conjugation) is already contained without addi-tional minus signs because the complex conjugation (1.28) of a product of anti-commuting numbers isdefined with an exchange of the factors to its reversed orderˆ N +1 = (cid:18) ˆ c +1 ˆ χ +1 ˆ η ∗ ˆ f +1 (cid:19) ; ˆ N +2 = (cid:18) ˆ c +2 ˆ χ +2 ˆ η ∗ ˆ f +2 (cid:19) ; ( ˆ N ˆ N ) + = ˆ N +2 ˆ N +1 . (1.33) .4 The super-symmetric coherent state path integral N is generalized from the relationdet( ˆ M ) = exp { tr ln( ˆ M ) } of ordinary matrices ˆ M . The ordinary trace relation ’tr’ in the exponentis generalized with the super-trace ’str’ (1.32) for super-matrices ˆ N , consisting of the even boson-boson, fermion-fermion blocks ˆ c L × L , ˆ f S × S and the odd fermion-boson, boson-fermion blocks ˆ χ S × L ,ˆ η TL × S . Using the properties of a super-trace ’str’ (as cyclic invariance), one can transform the generalizedrelation sdet( ˆ N N × N ) = exp { str ln( ˆ N N × N ) } (1.34) to ordinary L × L and S × S determinants where thedeterminant det( ˆ f S × S ) of the even fermion-fermion section appears in the denominator because of theadditional negative sign in the fermion-fermion section of a super-tracesdet (cid:0) ˆ N N × N (cid:1) ! = exp (cid:26) str ln (cid:18) ˆ c L × L ˆ η TL × S ˆ χ S × L ˆ f S × S (cid:19)(cid:27) (1.34)= exp (cid:26) str ln (cid:18) ˆ c L × L
00 ˆ f S × S (cid:19) (cid:18) ˆ1 L × L ˆ c − L × L ˆ η TL × S ˆ f − S × S ˆ χ S × L ˆ1 S × S (cid:19)(cid:27) = det (cid:0) ˆ c L × L − ˆ η TL × S ˆ f − S × S ˆ χ S × L (cid:1) det (cid:0) ˆ f S × S (cid:1) . In the case of a product of super-matrices, the property sdet( ˆ N ˆ N ) = sdet( ˆ N ) sdet( ˆ N ) for thefactorization of the super-determinant holds in a similar manner as in the case with ordinary matricesbecause of the cyclic invariance in the super-trace (1.32). In the remainder we consider the time contour integral (1.35) with the time variable t p on the twobranches p = ± for the time development of the Hamiltonian (1.17) in forward R ∞−∞ dt + . . . and back-ward R −∞∞ dt − . . . direction [20]-[25]. The negative sign of the backward propagation R −∞∞ dt − . . . = − R ∞−∞ dt − . . . will be frequently taken into account by the time contour metric-symbol η p = ± = p = ± Z C dt p . . . = Z + ∞−∞ dt + . . . + Z −∞ + ∞ dt − . . . = Z + ∞−∞ dt + . . . − Z + ∞−∞ dt − . . . (1.35)= X p = ± Z ∞−∞ dt p η p . . . ; ( η p = ± = ± ) . The corresponding coherent state path integral Z [ˆ J , j ψ , e j ψψ ] [20]-[25],[6] of the Hamiltonian (1.17) withits symmetry breaking source fields j ψ ; N ( ~x, t ) (1.19) and e j ψψ ; N × N ( ~x, t ) (1.24-1.27) is given in relation(1.36) with inclusion of the time contour integrals (1.35) in the exponentials Z [ˆ J , j ψ , e j ψψ ] = Z d [ ψ ~x,α ( t p )] exp (cid:26) − ı ~ Z C dt p X ~x X α ψ ∗ ~x,α ( t p ) ˆ H p ( ~x, t p ) ψ ~x,α ( t p ) (cid:27) (1.36) × exp (cid:26) − ı ~ Z C dt p X ~x,~x ′ X α,β ψ ∗ ~x ′ ,β ( t p ) ψ ∗ ~x,α ( t p ) V | ~x ′ − ~x | ψ ~x,α ( t p ) ψ ~x ′ ,β ( t p ) (cid:27) × exp (cid:26) − ı ~ Z C dt p X ~x X α (cid:16) j ∗ ψ ; α ( ~x, t p ) ψ ~x,α ( t p ) + ψ ∗ ~x,α ( t p ) j ψ ; α ( ~x, t p ) (cid:17)(cid:27) × exp ( − ı ~ Z C dt p X ~x str α,β " (cid:18) ˆ j + B ; L × L ( ~x, t p ) ˆ j + η ; L × S ( ~x, t p ) − ˆ j ∗ η ; S × L ( ~x, t p ) − ˆ j + F ; S × S ( ~x, t p ) (cid:19) (cid:18) ˆ c L × L ( ~x, t p ) ˆ η TL × S ( ~x, t p )ˆ η S × L ( ~x, t p ) ˆ f S × S ( ~x, t p ) (cid:19) ++ (cid:18) ˆ c + L × L ( ~x, t p ) ˆ η + L × S ( ~x, t p )ˆ η ∗ S × L ( ~x, t p ) ˆ f + S × S ( ~x, t p ) (cid:19) (cid:18) ˆ j B ; L × L ( ~x, t p ) − ˆ j Tη ; L × S ( ~x, t p )ˆ j η ; S × L ( ~x, t p ) − ˆ j F ; S × S ( ~x, t p ) (cid:19) × exp (cid:26) − ı ~ Z C dt (1) p dt (2) p X ~x,~x ′ X α,β Ψ + b~x ′ ,β ( t (2) p ) ˆ J ba~x ′ ,β ; ~x,α ( t (2) p ; t (1) p ) Ψ a~x,α ( t (1) p ) (cid:27) .Due to a missing potential for disorder with an ensemble average, the super-symmetric coherent statefields ψ ~x,α ( t p ), ψ ∗ ~x,α ( t p ) only couple on a single specific branch of the time contour without any combina-tions between forward ’+’ and backward ’-’ propagation (compare with the Refs. [43]-[45] in the case ofdisorder). The one-particle operator ˆ h ( ~x ) (1.18) is completed to ˆ H p ( ~x, t p ) with the time contour deriva-tive ˆ E p = ı ~ ∂/∂t p and the imaginary time contour increment ı ε p = ± = ( ± ) ı ε , ( ε > + ) for appropriateconvergence properties of Green functions with propagation according to suitable time directionsˆ H p ( ~x, t p ) = − ˆ E p − ı ε p + ˆ h ( ~x ) = − ı ~ ∂∂t p − ı ε p + ˆ ~p m + u ( ~x ) − µ . (1.37)A further source matrix ˆ J ba~x ′ ,β ; ~x,α ( t (2) p , t (1) p ) is incorporated in the coherent state path integral Z [ˆ J , j ψ , e j ψψ ](1.36) because it combines ’Nambu’ doubled coherent state fields Ψ a (=1 / ~x,α ( t p ) = { ψ ~x,α ( t p ) ; ψ ∗ ~x,α ( t p ) } T [18, 19]. Therefore, it is possible to generate anomalous terms as h ψ ~x,β ( t p ) ψ ~x,α ( t p ) i by a single dif-ferentiation of Z [ˆ J , j ψ , e j ψψ ] (1.36) with respect to ˆ J ~x,β ; ~x,α ( t p , t p ). Furthermore, one has to distinguishbetween source fields j ψ ; α ( ~x, t p ), e j ψψ ; αβ ( ~x, t p ) with a dependence on the time contour branch ’ p = ± ’for generating observables by differentiation of (1.36) and the corresponding ’condensate seed’ fields j ψ ; α ( ~x, t ), e j ψψ ; αβ ( ~x, t ) for SSB [26]-[29]. The latter ’condensate seeds’ follow by setting the correspond-ing time branches to equivalent finite values (1.38,1.39); this has to be performed at the final end ofcalculations with Z [ˆ J , j ψ , e j ψψ ] (1.36) after the prevailing observables have been determined by differ-entiation with respect to j ψ ; α ( ~x, t p ), e j ψψ ; αβ ( ~x, t p ) or with respect to ˆ J ba~x ′ ,β ; ~x,α ( t (2) p , t (1) p ). The last sourcematrix ˆ J ba~x ′ ,β ; ~x,α ( t (2) p , t (1) p ) has then to be set to zero with remaining finite ’condensate seeds’ j ψ ; α ( ~x, t ), e j ψψ ; αβ ( ~x, t ) for the creation of a coherent BEC-wavefunction and for the creation of pair condensateswith super-symmetry on the coset space Osp( S, S | L ) / U( L | S ) j ψ ; α ( ~x, t p ) := j ψ ; α ( ~x, t ) = 0 ; ’condensate seed’ for h ψ ~x,α ( t p ) i ; (1.38) e j ψψ ; αβ ( ~x, t p ) := e j ψψ ; αβ ( ~x, t ) = 0 ; ’condensate seed’ for h ψ ~x,β ( t p ) ψ ~x,α ( t p ) i . (1.39)The even coherent state fields ˆ c L × L ( ~x, t p ), ˆ f S × S ( ~x, t p ) and odd coherent fields ˆ η S × L ( ~x, t p ), ˆ η TL × S ( ~x, t p )(1.36), corresponding to the operators ˆ c ~x,L × L , ˆ f ~x,S × S and ˆ η ~x,S × L , ˆ η T~x,L × S (1.20-1.23), involve super-symmetric combinations of anomalous terms as ’ ψ ~x,β ( t p ) ψ ~x,α ( t p )’ so that a ’Nambu’-doubling (1.40) .4 The super-symmetric coherent state path integral a (=1 / ~x,α ( t p ) = { ψ ~x,α ( t p ) ; ψ ∗ ~x,α ( t p ) } T of coherent state fields has to be taken into account in transforma-tions of Z [ˆ J , j ψ , e j ψψ ] (1.36)Ψ a (=1 / ~x,α ( t p ) = (cid:18) ψ ~x,α ( t p ) ψ ∗ ~x,α ( t p ) (cid:19) = (cid:26) ~b ~x ( t p ) , ~α ~x ( t p ) | {z } a =1 ; ~b ∗ ~x ( t p ) , ~α ∗ ~x ( t p ) | {z } a =2 (cid:27) T . (1.40)According to the presence of anomalous terms, an order-parameter ˆΦ ab~x,α ; ~x ′ ,β ( t p ) has to respect the sym-metries of the dyadic product of ’Nambu’ doubled super-fields (1.41) with ’Nambu’ indices a, b = 1 , ~x,α ; ~x ′ ,β ( t p ), ˆΦ ~x,α ; ~x ′ ,β ( t p ), this guarantees the inclusion of pair condensateterms in the off-diagonal blocks with super-matrices ˆΦ ~x,α ; ~x ′ ,β ( t p ), ˆΦ ~x,α ; ~x ′ ,β ( t p ) so that the appropriatesuper-symmetries allow for a coset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) of the ortho-symplecticsuper-group Osp( S, S | L ) with the super-unitary subgroup U( L | S )ˆΦ ab~x,α ; ~x ′ ,β ( t p ) = Ψ a~x,α ( t p ) ⊗ Ψ + b~x ′ ,β ( t p ) = (cid:18) ψ ~x,α ( t p ) ψ ∗ ~x,α ( t p ) (cid:19) a ⊗ (cid:16) ψ ∗ ~x ′ ,β ( t p ) ; ψ ~x ′ ,β ( t p ) (cid:17) b (1.41)= (cid:18) h ψ ~x,α ( t p ) ψ ∗ ~x ′ ,β ( t p ) i h ψ ~x,α ( t p ) ψ ~x ′ ,β ( t p ) ih ψ ∗ ~x,α ( t p ) ψ ∗ ~x ′ ,β ( t p ) i h ψ ∗ ~x,α ( t p ) ψ ~x ′ ,β ( t p ) i (cid:19) ab = ˆΦ ~x,α ; ~x ′ ,β ( t p ) ˆΦ ~x,α ; ~x ′ ,β ( t p )ˆΦ ~x,α ; ~x ′ ,β ( t p ) ˆΦ ~x,α ; ~x ′ ,β ( t p ) ! ab .In comparison to the N × N = ( L + S ) × ( L + S ) super-matrices ˆ N , ˆ N in Eqs. (1.29-1.34), we havetherefore to consider 2 N × N super-matrices ˆΦ abαβ consisting of four N × N = ( L + S ) × ( L + S ) sub-super-matrices ˆΦ αβ , ˆΦ αβ and ˆΦ αβ , ˆΦ αβ . Each of the four sub-super-matrices is composed of an even L × L ( S × S ) boson-boson (fermion-fermion) block and the two odd parts in the S × L fermion-boson and L × S boson-fermion blocks. Consequently, we have to generalize operations as the super-transposition ’st’ of N × N = ( L + S ) × ( L + S ) super-matrices to those of 2 N × N super-matrices being partitioned intofour N × N sub-super-matrices. The super-transposition ’st’ and super-trace ’str’ are straightforwardlyextended to the super-transposition ’ST’ (1.42) and super-trace ’STR’ (1.43) of 2 N × N super-matrices.The total, even, 2 L × L boson-boson, 2 S × S fermion-fermion sections and the total, odd, 2 S × L fermion-boson, 2 L × S boson-fermion sections are consistently split into four parts, respectively, andare distributed to four N × N sub-super-matrices. The super-hermitian conjugation (1.44) of 2 N × N super-matrices ˆΦ abαβ follows by taking the super-hermitian conjugate of the block diagonal parts ˆΦ αβ ,ˆΦ αβ as in (1.33) and also of ˆΦ αβ , ˆΦ αβ ; in addition the latter super-hermitian conjugated, off-diagonal N × N blocks have to exchange their places (cid:16) ˆΦ abαβ (cid:17) ST = ˆΦ αβ ˆΦ αβ ˆΦ αβ ˆΦ αβ ! ST = (cid:0) ˆΦ αβ (cid:1) st (cid:0) ˆΦ αβ (cid:1) st (cid:0) ˆΦ αβ (cid:1) st (cid:0) ˆΦ αβ (cid:1) st ! ; (1.42)STR a,α ; b,β h ˆΦ abαβ i = str α,β h ˆΦ αβ i + str α,β h ˆΦ αβ i = m =+ l X m = − l ˆΦ mm − r =+ s X r = − s ˆΦ rr + m =+ l X m = − l ˆΦ mm − r =+ s X r = − s ˆΦ rr ; (1.43) (cid:16) ˆΦ abαβ (cid:17) + = ˆΦ αβ ˆΦ αβ ˆΦ αβ ˆΦ αβ ! + = (cid:0) ˆΦ αβ (cid:1) + (cid:0) ˆΦ αβ (cid:1) + (cid:0) ˆΦ αβ (cid:1) + (cid:0) ˆΦ αβ (cid:1) + ! . (1.44)4 In a similar manner the super-determinant ’sdet( ˆ N )’ is extended to a super-determinant ’SDET( ˆΦ abαβ )’(1.45) of 2 N × N super-matrices by substituting the super-trace ’str’ (1.32) in relation (1.34) with thesuper-trace ’STR’ (1.43) of 2 N × N super-matrices, having symmetries as the dyadic product (1.41)of ’Nambu’ doubled coherent state fieldsSDET (cid:16) ˆΦ abαβ (cid:17) = exp n STR a,α ; b,β ln (cid:0) ˆΦ abαβ (cid:1)o . (1.45)In Ref. [6] we describe in detail how to transform the coherent state path integral Z [ˆ J , j ψ , e j ψψ ] (1.36) with’Nambu’ doubled super-fields, doubled one-particle and interaction parts to doubled self-energies usingHubbard-Stratonovich transformations (HST) [46]. The properties of a Osp( S, S | L ) / U( L | S ) ⊗ U( L | S )coset decomposition are analyzed in general and require anti-hermitian anomalous terms in the self-energy matrix δ e Σ abαβ ( ~x, t p ) for an appropriate parametrization . Additionally we have to incorporate areal, scalar background field σ (0) D ( ~x, t p ) as a self-energy density term for P N = L + Sα =1 ψ ∗ ~x,α ( t p ) ψ ~x,α ( t p ) in theHST transformations. The U( L | S ) density terms, as subgroup of Osp( S, S | L ) in δ ˆΣ aaαβ ( ~x, t p ) e K , onlycontribute as ’hinge’-fields in the spontaneous symmetry breaking according to the coset decompositionOsp( S, S | L ) / U( L | S ) ⊗ U( L | S ). After a complete ’Nambu’ doubling and suitable HST’s of Z [ˆ J , j ψ , e j ψψ ](1.36), we obtain in Ref. [6] a coherent state path integral Z [ˆ J , J ψ , ı ˆ J ψψ ] (1.46) which depends on the real,scalar self-energy density σ (0) D ( ~x, t p ) as background field and on the 2 N × N super-symmetric self-energy δ e Σ abαβ ( ~x, t p ) e K (1.51) with anti-hermitian anomalous terms ( δ e Σ a = bαβ ( ~x, t p ) ) + = − δ e Σ b = aαβ ( ~x, t p ) in the off-diagonals ( a = b ). Corresponding to the short-ranged interaction potential V | ~x ′ − ~x | , the spatially nonlocalself-energies, resulting from the HST’s, are approximated to their local form with an effective, constantinteraction parameter V . This approximation is justified by the assumption that the strong oscillationslead to a cancellation of phases for exceeding interaction range. Introducing the 2 N × N ’Nambu’doubled super-matrix e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (1.47), we achieve in Ref. [6] the coherent state path integral Z [ˆ J , J ψ , ı ˆ J ψψ ] (1.46) where the source fields j ψ ; α ( ~x, t p ), e j ψψ ; αβ ( ~x, t p ) are converted to their ’Nambu’doubled form J a (=1 / ψ ; α ( ~x, t p ) = { j ψ ; α ( ~x, t p ) ; j ∗ ψ ; α ( ~x, t p ) } (1.48) and to ı ˆ J a = bψψ ; αβ ( ~x, t p ) (1.49,1.50). Wehave also to include various ’Nambu’ metric tensors ˆ K a = { (ˆ1 L × L , ˆ1 S × S ) a =1 ; (ˆ1 L × L , − ˆ1 S × S ) a =2 } , e K a = { (ˆ1 L × L , ˆ1 S × S ) a =1 ; ( − ˆ1 L × L , ˆ1 S × S ) a =2 } and ˆ I a = { (ˆ1 L × L , ˆ1 S × S ) a =1 ; (ˆ ı L × L , ˆ ı S × S ) a =2 } so thatthe parametrization and propagation of the self-energy fields in the exponentials are confined to theortho-symplectic super-group Osp( S, S | L ) Z [ˆ J , J ψ , ı ˆ J ψψ ] = Z d [ σ (0) D ( ~x, t p )] exp (cid:26) ı ~ V Z C dt p X ~x σ (0) D ( ~x, t p ) σ (0) D ( ~x, t p ) (cid:27) × Z d [ δ e Σ( ~x, t p ) e K ] × exp (cid:26) ı ~ V Z C dt p X ~x STR a,α ; b,β h(cid:16) δ e Σ( ~x, t p ) − ı ˆ J ψψ ( ~x, t p ) (cid:17) e K (cid:16) δ e Σ( ~x, t p ) − ı ˆ J ψψ ( ~x, t p ) (cid:17) e K i(cid:27) In the remainder the tilde ’ e ’ of δ e Σ N × N refers to a self-energy with anti-hermitian anomalous terms ı δ ˆΣ N × N , ı δ ˆΣ N × N in comparison to δ ˆΣ N × N with hermitian pair condensates δ ˆΣ N × N , δ ˆΣ N × N ; δ ˆΣ N × N = (cid:0) δ ˆΣ N × N (cid:1) + . We markthe ’Nambu’ doubled self-energy δ e Σ abαβ ( ~x, t p ) e K (1.51) with a ’ δ ’ in order to distinguish from the total sum e Σ abαβ ( ~x, t p ) e K (2.1) with the background field σ (0) D ( ~x, t p ) as the dominant contribution. The metric e K (1.52) has to be added to theself-energy for taking values within the ortho-symplectic super-algebra osp( S, S | L ). .4 The super-symmetric coherent state path integral × ( SDET (cid:20) e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (cid:21)) − / × (1.46) × exp (cid:26) ı ~ Ω Z C dt p dt ′ q X ~x,~x ′ N x J + bψ ; β ( ~x ′ , t ′ q ) ˆ I e K e M − ; ba~x ′ ,β ; ~x,α ( t ′ q , t p ) ˆ I J aψ ; α ( ~x, t p ) (cid:27) ; e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) = δ ~x,~x ′ η p δ p,q δ t p ,t ′ q " ˆ H p ( ~x, t p ) + σ (0) D ( ~x, t p ) ˆ H Tp ( ~x, t p ) + σ (0) D ( ~x, t p ) ! ++ δ ˆΣ αβ ( ~x, t p ) ı δ ˆΣ αβ ( ~x, t p ) ı δ ˆΣ αβ ( ~x, t p ) δ ˆΣ αβ ( ~x, t p ) ! e K abαβ + ˆ I ˆ K η p ˆ J ab~x,α ; ~x ′ ,β ( t p , t ′ q )Ω N x η q ˆ K ˆ I e K | {z } e J ab~x,α ; ~x ′ ,β ( t p ,t ′ q ) ; (1.47) J a (=1 / ψ ; α ( ~x, t p ) = n j ψ ; α ( ~x, t p ) | {z } a =1 ; j ∗ ψ ; α ( ~x, t p ) | {z } a =2 o T ; (1.48)ˆ J a = bψψ ; αβ ( ~x, t p ) = (cid:18) j ψψ ; αβ ( ~x, t p )ˆ j + ψψ ; αβ ( ~x, t p ) 0 (cid:19) ; (1.49)ˆ j ψψ ; αβ ( ~x, t p ) = (cid:18) ˆ j B ; L × L ( ~x, t p ) ˆ j Tη ; L × S ( ~x, t p )ˆ j η ; S × L ( ~x, t p ) ˆ j F ; S × S ( ~x, t p ) (cid:19) ; (1.50)ˆ j TB ; L × L ( ~x, t p ) = ˆ j B ; L × L ( ~x, t p ) ; ˆ j TF ; S × S ( ~x, t p ) = − ˆ j F ; S × S ( ~x, t p ) . Apart from the self-energy density σ (0) D ( ~x, t p ) as background field in (1.46,1.47), the self-energy super-matrix δ e Σ abαβ ( ~x, t p ) e K (1.51) only enters into the coherent state path integral (1.46,1.47) with indepen-dent field degrees of freedom confined to the parameters of the ortho-symplectic osp( S, S | L ) super-algebra. It has to be noted that the self-energy super-matrix δ e Σ abαβ ( ~x, t p ) has to include the appropriatemetric e K (1.52) in order to become an exact element of osp( S, S | L ) δ e Σ abαβ ( ~x, t p ) e K = δ ˆΣ αβ ( ~x, t p ) ı δ ˆΣ αβ ( ~x, t p ) ı δ ˆΣ αβ ( ~x, t p ) δ ˆΣ αβ ( ~x, t p ) ! abαβ e K ; (1.51) e K = n ˆ1 L × L , ˆ1 S × S | {z } a =1 ; e κ N × N z }| { − ˆ1 L × L , ˆ1 S × S | {z } a =2 o ; e κ N × N = n − ˆ1 L × L | {z } BB , ˆ1 S × S | {z } F F o . (1.52)The density terms δ ˆΣ αβ ( ~x, t p ), δ ˆΣ αβ ( ~x, t p ), referring to the super-unitary U( L | S ) group, are elimi-nated in combination of the coset decomposition with the gradient expansion and have the effect of’hinge’ functions in a SSB. According to the symmetry examination in Ref. [6], the coset decompositionOsp( S, S | L ) / U( L | S ) ⊗ U( L | S ) requires anti-hermitian anomalous terms δ e Σ αβ ( ~x, t p ) = ı δ ˆΣ αβ ( ~x, t p ),6 δ e Σ αβ ( ~x, t p ) = ı δ ˆΣ αβ ( ~x, t p ), δ ˆΣ αβ ( ~x, t p ) = ( δ ˆΣ αβ ( ~x, t p ) ) + (1.51,1.52) in order to obtain the correctnumber of independent field degrees of freedom as the independent parameters of Osp( S, S | L ). Thegradient expansion of the super-matrix e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (1.47) with the coset fields in ˆ T ( ~x, t p ) resultsin effective actions A ′ N − (cid:2) ˆ T ; J ψ (cid:3) , A ′ N (cid:2) ˆ T ; J ψ (cid:3) , A ′ N +1 (cid:2) ˆ T (cid:3) (1.53) which can be classified according to aparameter N = ~ Ω N x (Ω = 1 / ∆ t , N x = ( L/ ∆ x ) d ), denoting the total number of spatial points onan underlying grid and specifying the maximum possible energy ~ Ω corresponding to the discrete timesteps ∆ t . The nontrivial coset integration measure is indicated by d [ ˆ T − ( ~x, t p ) d ˆ T ( ~x, t p )] in Z [ˆ J , J ψ , ı ˆ J ψψ ](1.53) (see section 2.2). The source action A ˆ J ψψ (cid:2) ˆ T (cid:3) , following from ı ˆ J a = bψψ ; αβ ( ~x, t p ) (1.49,1.50), is inde-pendent from gradients and the background field σ (0) D ( ~x, t p ) and can be simplified by using propertiesof Vandermonde matrices [47]. The action, resulting for the additional source matrix ˆ J ba~x ′ ,β ; ~x,α ( t ′ q , t p )within up to second order of the gradient expansion, is denoted by A ′ (cid:2) ˆ T ; ˆ J (cid:3) and is further investigatedin section 4.2 Z [ˆ J , J ψ , ı ˆ J ψψ ] = Z d (cid:2) ˆ T − ( ~x, t p ) d ˆ T ( ~x, t p ) (cid:3) exp n ı A ˆ J ψψ (cid:2) ˆ T (cid:3)o (1.53) × exp n − A ′ N − (cid:2) ˆ T ; J ψ (cid:3) − A ′ N (cid:2) ˆ T ; J ψ (cid:3) − A ′ N +1 (cid:2) ˆ T (cid:3)o × exp n − A ′ (cid:2) ˆ T ; ˆ J (cid:3)o .It remains to identify the various classical actions in (1.53) with the nontrivial coset integration mea-sure which has to be replaced by Euclidean path integration fields. This is accomplished in sections2.2 and 2.3; however, we have in advance to describe the precise parameters of anomalous fieldsˆ T ( ~x, t p ) following from the total self-energy matrix δ e Σ abαβ ( ~x, t p ) e K (1.51,1.52) in the coset decompo-sition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) (section 2.1).Instead of the gradient expansion for effective actions in Z [ˆ J , J ψ , ı ˆ J ψψ ] (1.53), we remark an alterna-tive solution (1.54) of the original coherent state path integral (1.46) with super-matrix e M ab~x,α ; ~x ′ ,β ( t p , t ′ q )(1.47). This solution follows from the functional variation of (1.46) with respect to the self-energysuper-matrix δ e Σ abαβ ( ~x, t p ) e K (1.51) as a osp( S, S | L ) super-generator0 ≡ ∆ (cid:16) Z [ˆ J , J ψ , ı ˆ J ψψ ] (cid:17) ∆ (cid:16) δ e Σ( ~x, t p ) e K (cid:17) abαβ = (cid:28) N ı η p V h(cid:16) δ e Σ( ~x, t p ) − ı ˆ J ψψ ( ~x, t p ) (cid:17) e K i baβα + (1.54) − η p e M − ; ba~x,β ; ~x,α ( t p + δt p , t p + δt ′ p ) − ı ~ Z C dt (1) q dt (2) q X ~y ,~y N x η p e M − ; ba ~x,β ; ~y ,α ( t p + δt p , t (1) q ) ×× (cid:16) ˆ I J a ψ ; α ( ~y , t (1) q ) ⊗ J + b ψ ; β ( ~y , t (2) q ) ˆ I e K (cid:17) e M − ; b a~y ,β ; ~x,α ( t (2) q , t p + δt ′ p ) (cid:29) Z [ˆ J ,J ψ ,ı ˆ J ψψ ] . One can apply continued fractions of δ e Σ abαβ ( ~x, t p ) e K (1.51) for solving the mean field equation (1.54)(compare Ref. [43]). This process considerably simplifies in case of spatial symmetries and underrestriction to stationary solutions. We note that the matrix e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (1.47) is not only of centralimportance for the gradient expansion with the anomalous terms, but also for the saddle point equation(1.54) because it consists of the background field σ (0) D ( ~x, t p ) apart from the self-energy super-matrix7 δ e Σ abαβ ( ~x, t p ) e K (1.51). The exact mean field equation (1.54) can be approximated by averaging of theinverse e M − ; ab~x,α ; ~x ′ ,β ( t p , t ′ q ) of the super-matrix (1.47) as one-point Green function with the background field σ (0) D ( ~x, t p ). This averaging process of e M − ; ab~x,α ; ~x ′ ,β ( t p , t ′ q ) by σ (0) D ( ~x, t p ) becomes itself more accessible bytaking a saddle point solution for the background density field in order to approximate (1.54). According to super-group properties of Osp(
S, S | L ), the sum of self-energies σ (0) D ˆ1 N × N and δ e Σ N × N e K is factorized into density terms δ ˆΣ D ; N × N , δ ˆΣ D ; N × N (2.6,2.7) and super-matrices ˆ T ( ~x, t p ) (2.2-2.5) forpair condensates within the coset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) (2.1). The self-energydensity matrices δ ˆΣ D ; N × N , δ ˆΣ D ; N × N e κ are related to the super-unitary group U( L | S ) and only act as’hinge’ fields for the SSB and the gradient expansion. The background self-energy density σ (0) D ( ~x, t p ) isinvariant under U( L | S ) subgroup transformations and has therefore to be considered as the invariantground or vacuum state in the SSB of Osp( S, S | L ) with U( L | S ) as subgroup e Σ N × N ( ~x, t p ) e K = σ (0) D ( ~x, t p ) ˆ1 N × N + δ e Σ N × N ( ~x, t p ) e K = (2.1)= ˆ T ( ~x, t p ) σ (0) D ( ~x, t p ) ˆ1 N × N + δ ˆΣ D ; N × N ( ~x, t p ) 00 σ (0) D ( ~x, t p ) e κ + δ ˆΣ D ; N × N ( ~x, t p ) ! e K ˆ T − ( ~x, t p )= σ (0) D ( ~x, t p ) ˆ1 N × N + ˆ T N × N ( ~x, t p ) (cid:18) δ ˆΣ D ; N × N ( ~x, t p ) 00 δ ˆΣ D ; N × N ( ~x, t p ) e κ (cid:19)| {z } δ ˆΣ D ;2 N × N ( ~x,t p ) e K ˆ T − N × N ( ~x, t p ) . The independent field degrees of freedom for pair condensates, originally defined by ı δ ˆΣ αβ , ı δ ˆΣ αβ in (1.51), are described by the osp( S, S | L ) / u( L | S ) super-generator ˆ Y N × N ( ~x, t p ) (2.3) or its expo-nential form ˆ T N × N ( ~x, t p ) = exp {− ˆ Y N × N ( ~x, t p ) } (2.2) for the coset manifold Osp( S, S | L ) / U( L | S ).The super-generator ˆ Y N × N ( ~x, t p ) (2.3) consists of the sub-generators ˆ X N × N ( ~x, t p ) (2.4) and its super-hermitian conjugate e κ ˆ X + N × N in the off-diagonal blocks ( a = b ). They are itself composed of the evencomplex symmetric matrix ˆ c D ; L × L ( ~x, t p ) for molecular condensates and the even complex anti-symmetricmatrix ˆ f D ; S × S ( ~x, t p ) for BCS terms (2.5). The Grassmann valued field degrees of freedom for anomalousterms are given by the matrix ˆ η D ; S × L and its transpose ˆ η TD ; L × S in the fermion-boson and boson-fermionblocks ˆ T abαβ ( ~x, t p ) = exp ( ı (cid:18) ı ˆ X N × N ( ~x, t p ) ı e κ ˆ X + N × N ( ~x, t p ) 0 (cid:19) ) (2.2)= exp n − ˆ Y N × N ( ~x, t p ) o ;8 ˆ Y abαβ ( ~x, t p ) = X αβ ( ~x, t p ) e κ N × N ˆ X + αβ ( ~x, t p ) 0 ! ab ; (2.3)ˆ X αβ ( ~x, t p ) = (cid:18) − ˆ c D ; L × L ( ~x, t p ) ˆ η TD ; L × S ( ~x, t p ) − ˆ η D ; S × L ( ~x, t p ) ˆ f D ; S × S ( ~x, t p ) (cid:19) ; (2.4)ˆ c TD ; L × L ( ~x, t p ) = ˆ c D ; L × L ( ~x, t p ) ; ˆ f TD ; S × S ( ~x, t p ) = − ˆ f D ; S × S ( ~x, t p ) . (2.5)This parametrization with ˆ Y N × N ( ~x, t p ) (2.3) takes into account the exact structure of symmetry break-ing source terms for super-symmetric pair condensates which have been introduced into the originalHamiltonian (1.17) for the coherent state path integrals. The U( L | S ) self-energy density matrices δ ˆΣ D ; N × N ( ~x, t p ) (2.6) and its super-transposed copy δ ˆΣ D ; N × N ( ~x, t p ) e κ (2.7,2.9) contain the independentfield degrees following from the dyadic product density parts ψ ~x,α ( t p ) ⊗ ψ ∗ ~x,β ( t p ) and ψ ∗ ~x,α ( t p ) ⊗ ψ ~x,β ( t p )within the HST transformations δ ˆΣ D ; αβ ( ~x, t p ) = (cid:18) δ ˆ B D ; L × L ( ~x, t p ) δ ˆ χ + D ; L × S ( ~x, t p ) δ ˆ χ D ; S × L ( ~x, t p ) δ ˆ F D ; S × S ( ~x, t p ) (cid:19) ; (2.6) δ ˆΣ D ; αβ ( ~x, t p ) = (cid:18) δ ˆ B TD ; L × L ( ~x, t p ) δ ˆ χ TD ; L × S ( ~x, t p ) δ ˆ χ ∗ D ; S × L ( ~x, t p ) − δ ˆ F TD ; S × S ( ~x, t p ) (cid:19) ; (2.7) δ ˆ B + D ; L × L ( ~x, t p ) = δ ˆ B D ; L × L ( ~x, t p ) ; δ ˆ F + D ; S × S ( ~x, t p ) = δ ˆ F D ; S × S ( ~x, t p ) ; (2.8) δ ˆΣ D ; N × N ( ~x, t p ) = − (cid:16) δ ˆΣ D ; N × N ( ~x, t p ) e κ (cid:17) st . (2.9)The even density terms of the boson-boson, fermion-fermion blocks (2.8) are given by hermitian matrices δ ˆ B D ; L × L ( ~x, t p ), δ ˆ F D ; S × S ( ~x, t p ). The odd density terms in the fermion-boson, boson-fermion sections aredetermined by δ ˆ χ D ; S × L ( ~x, t p ) and its super-hermitian conjugate δ ˆ χ + D ; L × S ( ~x, t p ) (2.6,2.7). These self-energy densities δ ˆΣ D ; N × N ( ~x, t p ), δ ˆΣ D ; N × N ( ~x, t p ) e κ or δ ˆΣ aaD ;2 N × N ( ~x, t p ) e K act as ’hinge’ fields in the SSBand can be factorized to real N = L + S eigenvalues δ ˆ λ N × N or its ’Nambu’ doubled form δ ˆΛ N × N whichcomprise the maximal abelian Cartan subalgebra of rank N for the U( L | S ) or Osp( S, S | L ) super-group(2.10-2.12). The remaining field degrees of freedom ˆ Q N × N ( ~x, t p ), ˆ Q N × N ( ~x, t p ) (2.13-2.15) for the self-energy density matrices have their parameters within the ladder operators of the super-unitary algebrau( L | S ). Since N = L + S real parameter fields are already contained in the eigenvalues δ ˆ λ α ( ~x, t p ) (2.12),the diagonal values of ˆ B D ; mm = 0 ( m = 1 , . . . , L ) and ˆ F D ; ii = 0 ( i = 1 , . . . , S ) have to vanish in thegenerators of ˆ Q N × N ( ~x, t p ) (2.14), ˆ Q N × N ( ~x, t p ) (2.15). In consequence the ladder operators with theirindependent fields only remain from the super-unitary U( L | S ) algebra within the eigenvector matricesˆ Q N × N , ˆ Q N × N of the block diagonal self-energy densities δ ˆΣ aaD ;2 N × N e Kδ ˆΣ D ;2 N × N ( ~x, t p ) e K = ˆ Q − N × N ( ~x, t p ) δ ˆΛ N × N ( ~x, t p ) ˆ Q N × N ( ~x, t p ) ; (2.10) δ ˆΛ N × N ( ~x, t p ) = δ ˆΛ aα ( ~x, t p ) = diag n δ ˆ λ N × N ( ~x, t p ) ; − δ ˆ λ N × N ( ~x, t p ) o ; (2.11) δ ˆ λ N × N ( ~x, t p ) = δ ˆ λ α ( ~x, t p ) = n δ ˆ λ B ;1 , . . . , δ ˆ λ B ; m , . . . , δ ˆ λ B ; L ; δ ˆ λ F ;1 , . . . , δ ˆ λ F ; i , . . . , δ ˆ λ F ; S o ;(2.12)ˆ Q N × N ( ~x, t p ) = (cid:18) ˆ Q N × N
00 ˆ Q N × N (cid:19) ; (cid:0) ˆ Q N × N (cid:1) st = ˆ Q , + N × N = ˆ Q , − N × N ; (2.13) .1 The independent field variables for the super-symmetric pair condensates Q N × N ( ~x, t p ) = exp ( ı ˆ B D ; L × L ˆ ω + D ; L × S ˆ ω D ; S × L ˆ F D ; S × S !) ; ˆ B + D ; L × L = ˆ B D ; L × L ; (2.14)ˆ Q N × N ( ~x, t p ) = exp ( ı − ˆ B TD ; L × L ˆ ω TD ; L × S − ˆ ω ∗ D ; S × L − ˆ F TD ; S × S !) ; ˆ F + D ; S × S = ˆ F D ; S × S ; (2.15)ˆ B D ; mm = 0 ( m = 1 , . . . , L ) ; ˆ F D ; ii = 0 ( i = 1 , . . . , S ) . In analogy one can diagonalize the off-diagonal blocks ˆ X N × N ( ~x, t p ), e κ ˆ X + N × N ( ~x, t p ) (2.2-2.5) of thesuper-generator ˆ Y N × N ( ~x, t p ) for the pair condensates (2.16-2.20). The diagonal blocks of ˆ Y DD ;2 N × N in the anomalous sectors (2.16,2.17) consist of the diagonal matrices ˆ X DD ; N × N , e κ ˆ X + DD ; N × N (2.18-2.20)which are rotated with the parameters ˆ C D ; L × L , ˆ G D ; S × S , ˆ ξ D ; S × L , ˆ ξ ∗ D ; S × L by the matrices ˆ P N × N , ˆ P N × N (orˆ P aa N × N ) (2.21-2.23) to the generators ˆ X N × N ( ~x, t p ), e κ ˆ X + N × N ( ~x, t p ) (2.3,2.4). The complex L parameters c m within the diagonal matrix ˆ c L × L describe the anomalous molecular terms and are factorized intoits modulus | c m | and phase ϕ m (2.19). The parameters ˆ f S × S ( ~x, t p ) (2.20) of the fermionic degreeshave to consider the anti-symmetric form ˆ f D ; S × S of BCS terms (2.5) so that one has to introduce thequaternion algebra (2.25) with anti-symmetric Pauli matrix ( τ ) µν and complex field variables f r ascorresponding anti-symmetric eigenvalues for the BCS terms (2.20). The rotation matrices ˆ P aa N × N (2.21-2.23) include the remaining field degrees of freedom of ˆ X N × N ( ~x, t p ), e κ ˆ X + N × N ( ~x, t p ) (2.4) witheven hermitian matrices ˆ C D ; L × L , ˆ G D ; S × S (2.24,2.25) for the boson-boson, fermion-fermion parts where L diagonal real parameters (or S/ τ ) µν ) have to be excluded fromthe ladder operators. They are already contained within the L complex eigenvalues c m or S/ τ ) µν f r ˆ Y N × N ( ~x, t p ) = ˆ P − N × N ( ~x, t p ) ˆ Y DD ;2 N × N ( ~x, t p ) ˆ P N × N ( ~x, t p ) (2.16)= (cid:18) P , − ˆ X DD ˆ P ˆ P , − e κ ˆ X + DD ˆ P (cid:19) ab ;ˆ Y DD ;2 N × N ( ~x, t p ) = (cid:18) X DD ; N × N ( ~x, t p ) e κ ˆ X + DD ; N × N ( ~x, t p ) 0 (cid:19) ; (2.17)ˆ X DD ; N × N ( ~x, t p ) = − ˆ c L × L ( ~x, t p ) 00 ˆ f S × S ( ~x, t p ) ! ; (2.18)ˆ c L × L ( ~x, t p ) = diag n c ( ~x, t p ) , . . . , c m ( ~x, t p ) , . . . , c L ( ~x, t p ) o ; (2.19) c m ( ~x, t p ) = | c m ( ~x, t p ) | exp { ı ϕ m ( ~x, t p ) } ; ( c m ( ~x, t p ) ∈ C even ) ; The range of indices for the angular momentum degrees of freedom for the fermions and bosons is adapted from − s, . . . , + s and − l, . . . , + l to the range 1 , . . . , S = 2 s + 1 and 1 , . . . , L = 2 l + 1. Furthermore, two notations for the indexrange of the fermions are used in parallel in the remainder : (i) The first notation labels the angular momentum degreesof freedom from i, j = 1 , . . . , S = 2 s + 1, especially in the density parts. (ii) The second one regards the quaternionicstructure of the fermion-fermion parts concerning the anomalous sectors and has a 2 × µ, ν = 1 , r, r ′ = 1 , . . . , S/ δλ F ; rµ corresponds to δλ F ; i =2( r − µ . ˆ f S × S ( ~x, t p ) = diag n(cid:0) τ (cid:1) µν f ( ~x, t p ) , . . . , (cid:0) τ (cid:1) µν f r ( ~x, t p ) , . . . , (cid:0) τ (cid:1) µν f S/ ( ~x, t p ) o ; (2.20) f r ( ~x, t p ) = | f r ( ~x, t p ) | exp { ı φ r ( ~x, t p ) } ;( f r ( ~x, t p ) ∈ C even ) ; ( r = 1 , . . . , S/ , ( µ, ν = 1 ,
2) ;ˆ P N × N ( ~x, t p ) = (cid:18) ˆ P N × N ( ~x, t p ) 00 ˆ P N × N ( ~x, t p ) (cid:19) ; (cid:0) ˆ P N × N (cid:1) st = ˆ P , + N × N = ˆ P , − N × N ; (2.21)ˆ P N × N ( ~x, t p ) = exp ( ı ˆ C D ; L × L ( ~x, t p ) ˆ ξ + D ; L × S ( ~x, t p )ˆ ξ D ; S × L ( ~x, t p ) ˆ G D ; S × S ( ~x, t p ) !) ; (2.22)ˆ P N × N ( ~x, t p ) = exp ( ı − ˆ C TD ; L × L ( ~x, t p ) ˆ ξ TD ; L × S ( ~x, t p ) − ˆ ξ ∗ D ; S × L ( ~x, t p ) − ˆ G TD ; S × S ( ~x, t p ) !) ; (2.23)ˆ C + D ; L × L ( ~x, t p ) = ˆ C D ; L × L ( ~x, t p ) ; ˆ C D ; mm ( ~x, t p ) = 0 ; ( m = 1 , . . . , L ) ; (2.24)ˆ G + D ; S × S ( ~x, t p ) = ˆ G D ; S × S ( ~x, t p ) ; ˆ G D ; rµ,rν ( ~x, t p ) = 0 ; ( r = 1 , . . . , S/ , ( µ, ν = 1 ,
2) ; (2.25)ˆ G D ; S × S ( ~x, t p ) = ˆ G D ; rµ,r ′ ν ( ~x, t p ) = X k =0 ( τ k ) µν ˆ G ( k ) D ; rr ′ ( ~x, t p ) ; (cid:0) ˆ G ( k ) D ; rr ′ ( ~x, t p ) (cid:1) + = ˆ G ( k ) D ; rr ′ ( ~x, t p ) . In section 4.1 we have to require that the diagonal matrix elements (the quaternionic diagonal, anti-symmetric matrix elements) of the boson-boson part (fermion-fermion part) have to vanish in thegauge combination ( ∂ ˆ P aaN × N ( ~x, t p ) ) ˆ P − aaN × N ( ~x, t p ) of the block diagonal matrices ˆ P aaN × N ( ~x, t p ) with theirderivatives (2.26,2.27) 0 ! = h(cid:0) ∂ ˆ P aaN × N ( ~x, t p ) (cid:1) ˆ P − aaN × N ( ~x, t p ) i BB ; mm ; (2.26)0 ! = h(cid:0) ∂ ˆ P aaN × N ( ~x, t p ) (cid:1) ˆ P − aaN × N ( ~x, t p ) i F F ; rµ,rν . (2.27)This can be accomplished by a gauge transformation (2.28) of ˆ P aaN × N ( ~x, t p ) with a diagonal (quaterniondiagonal) matrix ˆ P aaDD ; N × N ( ~x, t p ) which has only non-vanishing matrix elements ˆ C D ; mm = 0, ˆ G D ; rµ,rν =0 along the diagonals, just in opposite to ˆ P aaN × N ( ~x, t p ) (2.24,2.25). These diagonal, real ˆ C D ; mm andhermitian 2 × G D ; rµ,rν (2.31,2.32) have to depend on the off-diagonal parameters of theladder operators in ˆ P N × N , ˆ P N × N and have to be chosen with suitable dependence in such a mannerthat the block diagonal, gauge transformed super-matrices ˆ P aaN × N = ˆ P aaDD ; N × N ˆ P aaN × N (2.28) fulfill theproperty of P N = L + Sβ =1 ( ∂ ˆ P aaαβ ) ˆ P − aaβα ≡ P N = L + Sβ =1 ( ∂ ˆ P aaαβ ) ˆ P − aaβα ≡ α (with ’ α ’ denoting a quaternion matrix element in the fermion-fermion section !)ˆ P aaN × N ( ~x, t p ) → ˆ P aaN × N ( ~x, t p ) = ˆ P aaDD ; N × N ( ~x, t p ) ˆ P aaN × N ( ~x, t p ) ; (2.28)ˆ P DD ; N × N ( ~x, t p ) = exp (cid:26) ı (cid:18) ˆ C D ; mm ( ~x, t p ) 00 ˆ G D ; rµ,rν ( ~x, t p ) (cid:19)(cid:27) ; (2.29)ˆ P DD ; N × N ( ~x, t p ) = exp ( − ı ˆ C D ; mm ( ~x, t p ) 00 ˆ G TD ; rµ,rν ( ~x, t p ) !) ; (2.30) .2 The coset integration measure of Osp ( S, S | L ) / U ( L | S ) ⊗ U ( L | S ) 21ˆ C D ; mm ( ~x, t p ) = ˆ C D ; mm (cid:16) ˆ C D ; m = n ; ˆ G D ; rµ,r ′ ν , ( r = r ′ ); ˆ ξ D ; S × L ; ˆ ξ ∗ D ; S × L (cid:17) ; (2.31)ˆ G D ; rµ,rν ( ~x, t p ) = ˆ G D ; rµ,rν (cid:16) ˆ C D ; m = n ; ˆ G D ; rµ,r ′ ν , ( r = r ′ ); ˆ ξ D ; S × L ; ˆ ξ ∗ D ; S × L (cid:17) ; (2.32) (cid:0) ∂ ˆ P aaN × N ( ~x, t p ) (cid:1) ˆ P − aaN × N ( ~x, t p ) = ˆ P aaDD ; N × N ( ~x, t p ) (cid:0) ∂ ˆ P aaN × N ( ~x, t p ) (cid:1) ˆ P − aaN × N ( ~x, t p ) ˆ P − aaDD ; N × N ( ~x, t p )+ (cid:0) ∂ ˆ P aaDD ; N × N ( ~x, t p ) (cid:1) ˆ P − aaDD ; N × N ( ~x, t p ) ; (2.33)ˆ P − aaDD ; mm ( ~x, t p ) (cid:0) ∂ ˆ P aaDD ; mm ( ~x, t p ) (cid:1) = − N = L + S X α =1 (cid:0) ∂ ˆ P aam,α ( ~x, t p ) (cid:1) ˆ P − aaα,m ( ~x, t p ) ; (2.34) X λ =1 , ˆ P − aaDD ; rµ,rλ ( ~x, t p ) (cid:0) ∂ ˆ P aaDD ; rλ,rν ( ~x, t p ) (cid:1) = − N = L + S X α =1 (cid:0) ∂ ˆ P aarµ,α ( ~x, t p ) (cid:1) ˆ P − aaα,rν ( ~x, t p ) . (2.35) ( S, S | L ) / U ( L | S ) ⊗ U ( L | S ) The coset decomposition Osp(
S, S | L ) / U( L | S ) ⊗ U( L | S ), as described in section 2.1 for the self-energysuper-matrix δ e Σ abαβ ( ~x, t p ) e K , involves a nontrivial integration measure. The corresponding super-Jacobi matrix of this transformation follows from the square root ˆ G / / U of the Osp( S, S | L ) / U( L | S )metric tensor ˆ G Osp / U for the invariant length ( ds Osp / U ( δ ˆ λ )) (2.36). We neglect anomalies, whichare caused by the anticommuting variables, and introduce the super-determinant SDET( ˆ G / / U ) =(SDET( ˆ G Osp / U )) / of this super-Jacobi-matrix ˆ G / / U for the change of integration measure from d [ σ (0) D ( t p )] d [ δ e Σ N × N ( t p ) e K ] to d [ σ (0) D ( t p )] d [ d ˆ Q ˆ Q − ; δ ˆ λ ] d [ ˆ T − d ˆ T ; δ ˆ λ ]. This change of integrationmeasure can be particularly obtained by diagonalizing the metric tensor ˆ G Osp / U with the eigenvaluesˆ c L × L ( ~x, t p ), ˆ f S × S ( ~x, t p ) ( c m ( ~x, t p ), f r ( ~x, t p )) (2.16-2.20) and eigenvector matrices ˆ P aaN × N ( ~x, t p ) (2.21-2.25,2.28-2.35) of the coset matrices ˆ X N × N ( ~x, t p ), e κ ˆ X + N × N ( ~x, t p ) (2.2-2.5) for the independent anomalousterms (cid:16) ds Osp / U ( δ ˆ λ ) (cid:17) = − α,β h(cid:0) e T − d e T (cid:1) αβ (cid:0) e T − d e T (cid:1) βα (cid:0) δ e λ β + δ e λ α (cid:1) i ; (2.36) (cid:0) e T − d e T (cid:1) abαβ = (cid:0) ˆ P ˆ T − d ˆ T ˆ P − (cid:1) abαβ − (cid:0) ˆ P ˆ Q − d ˆ Q ˆ P − (cid:1) abαβ ; (2.37) d (cid:2) σ (0) D ( t p ) (cid:3) d (cid:2) δ e Σ N × N ( t p ) e K (cid:3) = (2.38)= d (cid:2) σ (0) D ( t p ) (cid:3) d (cid:2) d ˆ Q ( t p ) ˆ Q − ( t p ); δ ˆ λ ( t p ) (cid:3) d (cid:2) ˆ T − ( t p ) d ˆ T ( t p ); δ ˆ λ ( t p ) (cid:3) . Repeated application of Eq. (2.39) (for the variation δ ˆ B of general generators ˆ B in the exponent)allows to simplify the combination ( ˆ P ˆ T − d ˆ T ˆ P − ) abαβ of coset matrices to a relation (2.40) with theireigenvalues ˆ X DD , e κ ˆ X + DD , ˆ Y DD for the pair condensates by an additional integration over a parameter v ∈ [0 ,
1] with ˆ Y DD [48, 6]exp (cid:8) ˆ B (cid:9) δ (cid:16) exp (cid:8) − ˆ B (cid:9)(cid:17) = − Z dv exp (cid:8) v ˆ B (cid:9) δ ˆ B exp (cid:8) − v ˆ B (cid:9) ; (2.39)2 (cid:0) ˆ P ˆ T − d ˆ T ˆ P − (cid:1) abαβ = (cid:16) ˆ P exp (cid:8) ˆ Y (cid:9) d (cid:16) exp (cid:8) − ˆ Y (cid:9)(cid:17) ˆ P − (cid:17) abαβ (2.40)= − Z dv (cid:16) exp (cid:8) v ˆ Y DD (cid:9) d ˆ Y ′ exp (cid:8) − v ˆ Y DD (cid:9)(cid:17) abαβ ; d ˆ Y ′ = ˆ P d ˆ Y ˆ P − ; str (cid:2) d ˆ Y ′ d ˆ Y ′ (cid:3) = str (cid:2) d ˆ Y d ˆ Y (cid:3) . (2.41)The integration measure of the ˆ P , ˆ P − rotated, independent coset elements d ˆ Y ′ (2.41) is equivalent tothe original anomalous terms within matrix d ˆ Y . Therefore, one can perform the integrations overthe parameter v ∈ [0 ,
1] with the eigenvalues ˆ Y DD of ˆ Y straightforwardly to obtain the metric tensorˆ G Osp / U for ( ds Osp / U ( δ ˆ λ )) (2.36-2.38). In the following we list the results for the integration measurein terms of the diagonal coset metric tensor, determined by c m , f r , and specify in relations (2.42-2.45)the integration measure for the block diagonal self-energy densities (2.6-2.15) δ ˆΣ D ;2 N × N e K in the cosetdecomposition δ ˆΣ D ;2 N × N e K = ˆ Q − N × N δ ˆΛ N × N ˆ Q N × N ; (2.42) d (cid:2) d ˆ Q ˆ Q − ; δ ˆ λ (cid:3) = d (cid:2) δ ˆΣ D e K (cid:3) ; (2.43) d (cid:2) d ˆ Q ˆ Q − , δ ˆΛ (cid:3) = d (cid:2) δ ˆΣ D e K (cid:3) = Y { ~x,t p } "(cid:26) ( L + S ) / (cid:18) L Y m =1 d (cid:0) δλ B ; m (cid:1)(cid:19)(cid:18) S Y i =1 d (cid:0) δλ F ; i (cid:1)(cid:19)(cid:27) × (2.44) × (cid:26) L Y m =1 L Y n = m +1 (cid:18) (cid:0) d ˆ Q ˆ Q , − (cid:1) BB ; mn ∧ (cid:0) d ˆ Q ˆ Q , − (cid:1) BB ; nm ı (cid:0) δ ˆ λ B ; n − δ ˆ λ B ; m (cid:1) (cid:19)(cid:27) × (cid:26) S Y i =1 S Y i ′ = i +1 (cid:18) (cid:0) d ˆ Q ˆ Q , − (cid:1) F F ; ii ′ ∧ (cid:0) d ˆ Q ˆ Q , − (cid:1) F F ; i ′ i ı (cid:0) δ ˆ λ F ; i ′ − δ ˆ λ F ; i (cid:1) (cid:19)(cid:27) × (cid:26) L Y m =1 S Y i ′ =1 (cid:18) (cid:0) d ˆ Q ˆ Q , − (cid:1) BF ; mi ′ (cid:0) d ˆ Q ˆ Q , − (cid:1) F B ; i ′ m (cid:0) δ ˆ λ F ; i ′ − δ ˆ λ B ; m (cid:1) − (cid:19)(cid:27) ; d (cid:2) δ ˆΣ D e K (cid:3) = d (cid:2) d ˆ Q ˆ Q − , δ ˆΛ (cid:3) = Y { ~x,t p } "(cid:26) ( L + S ) / (cid:18) L Y m =1 d (cid:0) δ ˆ B D ; mm (cid:1)(cid:19)(cid:18) S Y i =1 d (cid:0) δ ˆ F D ; ii (cid:1)(cid:19)(cid:27) × (2.45) × (cid:26) L Y m =1 L Y n = m +1 (cid:18) d (cid:0) δ ˆ B ∗ D ; mn (cid:1) ∧ d (cid:0) δ ˆ B D ; mn (cid:1) ı (cid:19)(cid:27) × (cid:26) S Y i =1 S Y i ′ = i +1 (cid:18) d (cid:0) δ ˆ F ∗ D ; ii ′ (cid:1) ∧ d (cid:0) δ ˆ F D ; ii ′ (cid:1) ı (cid:19)(cid:27) ×× (cid:26) L Y m =1 S Y i ′ =1 (cid:18) d (cid:0) δ ˆ χ ∗ D ; i ′ m (cid:1) d (cid:0) δ ˆ χ D ; i ′ m (cid:1)(cid:19)(cid:27) . In the remainder ˆ P , ˆ P − transformed coset elements are marked by an additional prime ” ’ ” , as e. g. for d ˆ Y ′ =ˆ P d ˆ Y ˆ P − . .2 The coset integration measure of Osp ( S, S | L ) / U ( L | S ) ⊗ U ( L | S ) 23One has to consider that the original invariant length ( ds Osp / U ( δ ˆ λ )) (2.36) of the coset integrationmeasure d [ ˆ T − d ˆ T ; δ ˆ λ ] (2.38,2.46) also incorporates the eigenvalues δλ α of the density terms. However,the eigenvalues δλ α of the densities factorize into a polynomial P ( δ ˆ λ ) (2.47) and separate from thecoset integration d [ ˆ T − d ˆ T ] which solely depends on the field variables of ˆ X N × N , e κ ˆ X + N × N weighted byfunctions of their eigenvalues c m , f r d (cid:2) ˆ T − d ˆ T ; δ ˆ λ (cid:3) = P (cid:0) δ ˆ λ (cid:1) d (cid:2) ˆ T − d ˆ T (cid:3) ; (2.46) P (cid:0) δ ˆ λ (cid:1) = Y { ~x,t p } "(cid:26)(cid:18) L Y m =1 (cid:0) δ ˆ λ B ; m (cid:1) (cid:19)(cid:18) S/ Y r =1 (cid:0) δ ˆ λ F ; r + δ ˆ λ F ; r (cid:1) (cid:19)(cid:27) × (cid:26) L Y m =1 L Y n = m +1 (cid:18)(cid:0) δ ˆ λ B ; n + δ ˆ λ B ; m (cid:1) (cid:19)(cid:27) (2.47) × (cid:26) S/ Y r =1 S/ Y r ′ = r +1 (cid:18)(cid:0) δ ˆ λ F ; r + δ ˆ λ F ; r ′ (cid:1) (cid:0) δ ˆ λ F ; r + δ ˆ λ F ; r ′ (cid:1) × (cid:0) δ ˆ λ F ; r + δ ˆ λ F ; r ′ (cid:1) (cid:0) δ ˆ λ F ; r + δ ˆ λ F ; r ′ (cid:1) (cid:19)(cid:27) × (cid:26) L Y m =1 S/ Y r ′ =1 (cid:18)(cid:0) δ ˆ λ F ; r ′ + δ ˆ λ B ; m (cid:1) (cid:0) δ ˆ λ F ; r ′ + δ ˆ λ B ; m (cid:1) (cid:19) − (cid:27) . The actual coset integration measure d [ ˆ T − d ˆ T ] is listed in relation (2.48) where the polynomial P ( δ ˆ λ )(2.47) has been isolated and been shifted to the action terms, which are determined by integrations overthe self-energy densities δ ˆΣ D ;2 N × N e K (2.42-2.45). After their removal by integration, these self-energydensities or ’hinge’ fields of the SSB yield the action term A ˆ J ψψ [ ˆ T ] of the ’condensate seeds’ with thesource matrix ı ˆ J a = bψψ ; αβ ( ~x, t ) for the pair condensates d (cid:2) ˆ T − d ˆ T (cid:3) = Y { ~x,t p } "( L Y m =1 (cid:18) d ˆ c ∗ D ; mm ∧ d ˆ c D ; mm ı (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) | c m | (cid:1) | c m | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:19)) (2.48) × ( L Y m =1 L Y n = m +1 (cid:18) d ˆ c ∗ D ; mn ∧ d ˆ c D ; mn ı (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) | c m | + | c n | (cid:1) | c m | + | c n | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) | c m | − | c n | (cid:1) | c m | − | c n | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:19)) × ( S/ Y r =1 (cid:18) d ˆ f (2) ∗ D ; rr ∧ d ˆ f (2) D ; rr ı sinh (cid:0) | f r | (cid:1) | f r | (cid:19)) × ( S/ Y r =1 S/ Y r ′ = r +1 3 Y k =0 (cid:18) d ˆ f ( k ) ∗ D ; rr ′ ∧ d ˆ f ( k ) D ; rr ′ ı (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:0) | f r | + | f r ′ | (cid:1) | f r | + | f r ′ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:0) | f r | − | f r ′ | (cid:1) | f r | − | f r ′ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:19)) × ( L Y m =1 S/ Y r ′ =1 d ˆ η ∗ D ; r ′ ,m d ˆ η D ; r ′ ,m d ˆ η ∗ D ; r ′ ,m d ˆ η D ; r ′ ,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:0) | f r ′ | + ı | c m | (cid:1) | f r ′ | + ı | c m | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:0) | f r ′ | − ı | c m | (cid:1) | f r ′ | − ı | c m | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! ) .4 The effective actions (2.49) of coset matrices ˆ T ( ~x, t p ) (2.2-2.5) with their independent fields for anoma-lous terms in the super-generator ˆ Y ( ~x, t p ) follow from a gradient expansion of the super-matrix (1.47) e M ab~x,α ; ~x ′ ,β ( t p , t ′ q ) in the coherent state path integral Z [ˆ J , J ψ , ı ˆ J ψψ ] (1.46). It is of central importancethat the coset decomposition allows a factorization of the integration measure into density terms andfields for the pair condensates. In the following we give the result of the gradient expansion combinedwith the coset decomposition and classify the effective, remaining actions for anomalous fields ˆ Y ( ~x, t p )(2.2-2.5,2.16-2.25) according to the parameter N = ~ Ω N x (Ω = 1 / ∆ t , N x = ( L/ ∆ x ) d ). This parameter N arises e. g. in the course of the gradient expansion of the super-determinant when one has to intro-duce an appropriate integration for the discrete spatial and time-like points on an underlying grid withintervals ∆ x and ∆ tZ [ˆ J , J ψ , ı ˆ J ψψ ] = Z d (cid:2) ˆ T − ( ~x, t p ) d ˆ T ( ~x, t p ) (cid:3) exp n ı A ˆ J ψψ (cid:2) ˆ T (cid:3)o (2.49) × exp n − A ′ N − (cid:2) ˆ T ; J ψ (cid:3) − A ′ N (cid:2) ˆ T ; J ψ (cid:3) − A ′ N +1 (cid:2) ˆ T (cid:3)o × exp n − A ′ (cid:2) ˆ T ; ˆ J (cid:3)o .The effective action A ′ N − (cid:2) ˆ T ; J ψ (cid:3) (2.49,2.54) of order N − is composed of the gradients (2.50) withsuper-matrices ˆ Z = ˆ T ˆ S ˆ T − (2.51,2.52), following from the expansion of the super-determinant, andthe gradients ( e ∂ i ˆ T ) ˆ T − , resulting from the expansion of the inverse super-matrix e M − ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (1.47)with the ’Nambu’ doubled source fields J + aψ ; α ( ~x, t p ) . . . J aψ ; α ( ~x, t p ). The combination of coset matricesˆ T , ˆ T − to the super-matrix ˆ Z = ˆ T ˆ S ˆ T − with metric ˆ S = { ˆ S aa δ αβ } ( ˆ S = +1 ; ˆ S = −
1) (2.52)completely restricts the ’Nambu’ doubled super-trace ’STR’ (1.43) to terms of the pair condensates inthe off-diagonal blocks of ( ˆ T − ( e ∂ i ˆ T )) a = bαβ (2.53) with super-trace ’str’ (1.32) e ∂ i := ~ √ m ∂∂x i ; (2.50)ˆ Z ( ~x, t p ) = ˆ T ( ~x, t p ) ˆ S ˆ T ( ~x, t p ) ; (2.51)ˆ S = (cid:8) ˆ S a δ ab δ αβ (cid:9) = n +ˆ1 N × N | {z } a =1 ; − ˆ1 N × N | {z } a =2 o ; (2.52)STR a,α ; b,β h(cid:16) e ∂ i ˆ Z ( ~x, t p ) (cid:17) (cid:16) e ∂ j ˆ Z ( ~x, t p ) (cid:17)i = (2.53)= − a = b X a,b =1 , str α,β nh ˆ T − ( ~x, t p ) (cid:0) e ∂ i ˆ T ( ~x, t p ) (cid:1)i a = bαβ h ˆ T − ( ~x, t p ) (cid:0) e ∂ j ˆ T ( ~x, t p ) (cid:1)i b = aβα o . The effective coupling functions c ij ( ~x, t p ) (2.55-2.58) for gradients of ˆ Z and d ij ( ~x, t p ) (2.59) for ( e ∂ i ˆ T ) ˆ T − are achieved from the action of ’ unsaturated ’ gradient operators ’ e ∂ i ’ onto Green functions of the .3 Effective action for pair condensates with coupling coefficients of the background field σ (0) D ( ~x, t p ) and by the average h . . . i ˆ σ (0) D with the corresponding background functional Z [ j ψ ; ˆ σ (0) D ] (see Eqs. (2.74-2.76) and Ref. [6] with chapter 4 and appendix B) A ′ N − (cid:2) ˆ T ; J ψ (cid:3) = 14 1 N Z C dt p ~ X ~x c ij ( ~x, t p ) STR h(cid:16) e ∂ i ˆ Z ( ~x, t p ) (cid:17) (cid:16) e ∂ j ˆ Z ( ~x, t p ) (cid:17)i + (2.54) − ı N Z C dt p ~ X ~x X a,b =1 , N = L + S X α,β =1 d ij ( ~x, t p ) ×× J + ,bψ ; β ( ~x, t p ) N (cid:18) ˆ I e K (cid:16) e ∂ i ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) (cid:16) e ∂ j ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) ˆ I (cid:19) baβα J aψ ; α ( ~x, t p ) N ; c ij ( ~x, t p ) = c (1) ,ij ( ~x, t p ) + c (2) ,ij ( ~x, t p ) ; (2.55)˘ v ( ~x, t p ) = ˘ u ( ~x ) + ˘ σ (0) D ( ~x, t p ) = u ( ~x ) + σ (0) D ( ~x, t p ) N ; (2.56) c (1) ,ij ( ~x, t p ) = − (cid:28)(cid:16) e ∂ i e ∂ j ˘ v ( ~x, t p ) (cid:17)(cid:29) ˆ σ (0) D − δ ij d X k =1 (cid:28)(cid:16) e ∂ k e ∂ k ˘ v ( ~x, t p ) (cid:17)(cid:29) ˆ σ (0) D ; (2.57) c (2) ,ij ( ~x, t p ) = 2 (cid:28)(cid:16) e ∂ i ˘ v ( ~x, t p ) (cid:17) (cid:16) e ∂ j ˘ v ( ~x, t p ) (cid:17)(cid:29) ˆ σ (0) D − δ ij d X k =1 (cid:28)(cid:16) e ∂ k ˘ v ( ~x, t p ) (cid:17) (cid:29) ˆ σ (0) D ; (2.58) d ij ( ~x, t p ) = 2 (cid:28) (cid:16) e ∂ i ˘ v ( ~x, t p ) (cid:17) (cid:16) e ∂ j ˘ v ( ~x, t p ) (cid:17) − (cid:16) e ∂ i e ∂ j ˘ v ( ~x, t p ) (cid:17)(cid:29) ˆ σ (0) D . (2.59)The action A ′ N (cid:2) ˆ T ; J ψ (cid:3) (2.60) of order N does not contain coupling parameters as c ij ( ~x, t p ), d ij ( ~x, t p ) ofthe background field apart from the average of h σ (0) D ( ~x, t p ) i ˆ σ (0) D for an effective potential which modifiesthe trap potential u ( ~x ). It is also composed of a part following from the gradient expansion of thesuper-determinant and a part for the coherent BEC wavefunction with the ’Nambu’ doubled bilinearsource fields J + aψ ; α ( ~x, t p ) . . . J aψ ; α ( ~x, t p ) A ′ N (cid:2) ˆ T ; J ψ (cid:3) = − Z C dt p ~ X ~x ( STR (cid:20) ˆ T − ( ~x, t p ) ˆ S (cid:16) ˆ E p ˆ T ( ~x, t p ) (cid:17) + ˆ T − ( ~x, t p ) (cid:16) e ∂ i e ∂ i ˆ T ( ~x, t p ) (cid:17)(cid:21) ++ (cid:16) u ( ~x ) − µ − ı ε p + (cid:10) σ (0) D ( ~x, t p ) (cid:11) ˆ σ (0) D (cid:17) STR (cid:20)(cid:16) ˆ T − ( ~x, t p ) (cid:17) − ˆ1 N × N (cid:21)) + (2.60) − ı Z C dt p ~ X ~x X a,b =1 , N = L + S X α,β =1 J + ,bψ ; β ( ~x, t p ) N (cid:20) ˆ I e K (cid:18)(cid:16) e ∂ i e ∂ i ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) ++ ˆ T ( ~x, t p ) ˆ S ˆ T − ( ~x, t p ) (cid:16) ˆ E p ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) + − (cid:16) e ∂ i ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) (cid:16) e ∂ i ˆ T ( ~x, t p ) (cid:17) ˆ T − ( ~x, t p ) (cid:19) ˆ I (cid:21) baβα J aψ ; α ( ~x, t p ) N . The action A ′ N +1 (cid:2) ˆ T (cid:3) (2.61) of order N +1 does not involve any gradients and has completely differentproperties for the variation of classical field solutions, due to the additional metric η p in the time contourintegral A ′ N +1 (cid:2) ˆ T (cid:3) = N Z C dt p ~ η p X ~x STR (cid:20)(cid:16) ˆ T − ( ~x, t p ) (cid:17) − ˆ1 N × N (cid:21) . (2.61)According to the additional contour metric η p in (2.61), the two branches of the time contour integral in A ′ N +1 (cid:2) ˆ T (cid:3) are added whereas the two branches of time contour integrals in A ′ N − (cid:2) ˆ T ; J ψ (cid:3) (2.54), A ′ N (cid:2) ˆ T ; J ψ (cid:3) (2.60) are subtracted. Therefore, the variation δ ˆ Y ( ~x, t p ) (2.62-2.64) of classical fields in A ′ N +1 (cid:2) ˆ T (cid:3) (2.61)has its first contribution in the second order variation with δ ˆ Y ( ~x, t p ) for the independent, anomalousfields y κ ( ~x, t p ) in ˆ Y ( ~x, t p ) = y κ ( ~x, t p ) ˆ Y ( κ ) with coset super-generators ˆ Y ( κ ) (concerning variation ofactions with contour time integrals in coherent state path integrals see Refs. [49, 50])ˆ Y ( ~x, t p = ± ) = ˆ Y ( ~x, t ) + δ ˆ Y ( ~x, t p = ± ) = ˆ Y ( ~x, t ) ± δ ˆ Y ( ~x, t ) ; (2.62)ˆ Y ( ~x, t p = ± ) = y κ ( ~x, t p = ± ) ˆ Y ( κ ) ; (2.63) y κ ( ~x, t p = ± ) = y κ ( ~x, t ) + δy κ ( ~x, t p = ± ) = y κ ( ~x, t ) ± δy κ ( ~x, t ) . (2.64)The variations of A ′ N − (cid:2) ˆ T ; J ψ (cid:3) (2.54), A ′ N (cid:2) ˆ T ; J ψ (cid:3) (2.60) already contribute in first order of δ ˆ Y ( ~x, t p = ± )(2.62-2.64) and allow for classical field solutions following from first order variations to a stationary phasein the coherent state path integral (2.49). The second and all higher even order variations of A ′ N +1 (cid:2) ˆ T (cid:3) (2.61) modify these classical, first order variational solutions of A ′ N − (cid:2) ˆ T ; J ψ (cid:3) , A ′ N (cid:2) ˆ T ; J ψ (cid:3) and can beregarded as general fluctuation terms with universal properties , entirely determined by the symmetriesof the coset decomposition for the anomalous fields. This property of contributing only from secondand higher even order variations with δ ˆ Y ( ~x, t p = ± ) also holds for the coset integration measure (2.48)and causes the inconsistent treatment in comparison to the other main actions A ′ N − (cid:2) ˆ T ; J ψ (cid:3) (2.54), A ′ N (cid:2) ˆ T ; J ψ (cid:3) (2.60). The transformation with the inverse square root ˆ G − / / U of the coset metric tensorremoves this artificial problem and yields Euclidean path integration measures for the independent,anomalous fields. However, one obtains a different dependence of the pair condensate fields in theactions A ′ N − (cid:2) ˆ T ; J ψ (cid:3) (2.54), A ′ N (cid:2) ˆ T ; J ψ (cid:3) (2.60), according to the transformation with the super-Jacobimatrix ˆ G − / / U . The functional dependence of anomalous fields is also changed by this transformation inthe action term A ′ N +1 (cid:2) ˆ T (cid:3) (2.61) which, however, cannot be eliminated as the coset integration measure(2.48). In spite of the transformation with the inverse square root of the coset metric tensor, the actionterm A ′ N +1 (cid:2) ˆ T (cid:3) (2.61) only allows non-vanishing variations with δ ˆ Y ( ~x, t p = ± ) in second and higher evenorders and modifies the classical solutions of the first order variations of A ′ N − (cid:2) ˆ T ; J ψ (cid:3) , A ′ N (cid:2) ˆ T ; J ψ (cid:3) by universal fluctuations . These universal fluctuations are solely determined by the coset decompositionOsp( S, S | L ) / U( L | S ) ⊗ U( L | S ), due to the absence of any background field dependencies.The pair condensate action term A ˆ J ψψ [ ˆ T ] (2.65) in Z [ˆ J , J ψ , ı ˆ J ψψ ] (2.49) arises from the integrationof the quadratic self-energy density action A (cid:2) ˆ T , δ ˆΣ D ; ı ˆ J ψψ (cid:3) (2.66) over the independent density fieldsof δ ˆΣ D e K (2.45) with inclusion of the polynomial P ( δ ˆ λ ) (2.47) of the eigenvalues δ ˆ λ (2.11,2.12) for the .3 Effective action for pair condensates with coupling coefficients of the background field δ ˆΣ D or δ ˆΣ D e κ (2.6-2.9). The eigenvalues δλ α can be discerned as the parame-ters or variables of the U( L | S ) related density terms δ ˆΣ D , δ ˆΣ D e κ (2.6-2.15) within the characteristiceigenvalue equations of the super-determinants (2.67,2.68) so that the integration measure d [ δ ˆΣ D e K ](2.45) with additional polynomial P ( δ ˆ λ ( δ ˆΣ D e K )) (2.47) can be used to specify the effective pair ’con-densate seed’ action A ˆ J ψψ (cid:2) ˆ T (cid:3) (2.65). Alternatively, the factorization of δ ˆΣ D e K into eigenvalues δ ˆ λ α and eigenvectors ˆ Q aaαβ , ˆ Q − aaαβ (2.6-2.15) can be applied to determine the action A ˆ J ψψ (cid:2) ˆ T (cid:3) (2.65) afterintegration of A (cid:2) ˆ T , ˆ Q − δ ˆΛ ˆ Q ; ı ˆ J ψψ (cid:3) (2.69) over the eigenvalues and eigenvectors with inclusion of P ( δ ˆ λ ) (2.47). The latter method of factorization with eigenvalues allows to disentangle the integrationswith properties of Vandermonde matrices and Gaussian weights for orthogonal Hermite- (or relatedLaguerre) polynomials [47]exp n ı A ˆ J ψψ (cid:2) ˆ T (cid:3)o = Z d (cid:2) δ ˆΣ D ( ~x, t p ) e K (cid:3) P (cid:0) δ ˆ λ ( ~x, t p ) (cid:1) exp n ı A (cid:2) ˆ T , δ ˆΣ D ; ı ˆ J ψψ (cid:3)o = (2.65)= Z d (cid:2) d ˆ Q ( ~x, t p ) ˆ Q − ( ~x, t p ); δ ˆ λ ( ~x, t p ) (cid:3) P (cid:0) δ ˆ λ ( ~x, t p ) (cid:1) exp n ı A (cid:2) ˆ T , ˆ Q − δ ˆΛ ˆ Q ; ı ˆ J ψψ (cid:3)o ; A (cid:2) ˆ T , δ ˆΣ D ; ı ˆ J ψψ (cid:3) = 14 ~ V Z C dt p X ~x (cid:26) STR h δ ˆΣ D ;2 N × N ( ~x, t p ) e K δ ˆΣ D ;2 N × N ( ~x, t p ) e K i + (2.66) − h ı ˆ J ψψ ( ~x, t p ) e K ˆ T ( ~x, t p ) δ ˆΣ D ;2 N × N ( ~x, t p ) e K ˆ T − ( ~x, t p ) i ++ STR h ı ˆ J ψψ ( ~x, t p ) e K ı ˆ J ψψ ( ~x, t p ) e K i(cid:27) ;sdet n δ ˆΣ D ; αβ − δλ δ αβ o = 0 ; sdet n δ ˆΣ D ; αβ e κ − (cid:0) − δλ (cid:1) δ αβ o = 0 ; (2.67) δ ˆΣ D ; N × N ( ~x, t p ) = − (cid:16) δ ˆΣ D ; N × N ( ~x, t p ) e κ (cid:17) st ; (2.68) A (cid:2) ˆ T , ˆ Q − δ ˆΛ ˆ Q ; ı ˆ J ψψ (cid:3) = (2.69)= 14 ~ V Z C dt p X ~x STR h(cid:0) δ e Σ( ~x, t p ) − ı ˆ J ψψ ( ~x, t p ) (cid:1) e K (cid:0) δ e Σ( ~x, t p ) − ı ˆ J ψψ ( ~x, t p ) (cid:1) e K i = 14 ~ V Z C dt p X ~x (cid:26) h(cid:0) δ ˆ λ N × N ( ~x, t p ) (cid:1) i + − h ı ˆ J ψψ ( ~x, t p ) e K ˆ T ( ~x, t p ) ˆ Q − ( ~x, t p ) δ ˆΛ( ~x, t p ) ˆ Q ( ~x, t p ) ˆ T − ( ~x, t p ) i ++ STR h ı ˆ J ψψ ( ~x, t p ) e K ı ˆ J ψψ ( ~x, t p ) e K i(cid:27) . It remains to outline the averaging procedure h . . . i ˆ σ (0) D of the coupling coefficients c ij ( ~x, t p ), d ij ( ~x, t p )with the generating functional Z [ j ψ ; ˆ σ (0) D ] (2.74) of the background field σ (0) D ( ~x, t p ). Aside from the8 quadratic term of σ (0) D ( ~x, t p ) following from the HST’s, the Hamilton operator ˆ H [ˆ σ (0) D ] (2.70) specifiesthe determinant and the coherent BEC-wavefunction parts with the source fields j ψ ; α ( ~x, t p ) in Z [ j ψ ; ˆ σ (0) D ](2.74). If the number L = 2 l + 1, ( l = 0 , , , . . . ) of bosonic angular momentum degrees of freedomexceeds those of fermionic angular momentum degrees of freedom ( L > S = 2 s + 1), ( s = , , , . . . ),the determinant of the operator ˆ H [ˆ σ (0) D ] (2.70) appears in the denominator ( det( ˆ H [ˆ σ (0) D ]) ) − ( L − S ) witha power ( L − S ) >
0. In this extraordinary case ( L − S ) > V <
0, the background generating functional Z [ j ψ ; ˆ σ (0) D ] (2.74) may describe the experimentally observ-able, considerable increase of the coherent bosonic BEC-wavefunctions towards the collapse, due to theappearance of effective zero eigenvalues of ˆ H [ˆ σ (0) D ] (2.70) in the propagation with ( det( ˆ H [ˆ σ (0) D ]) ) − ( L − S ) in Z [ j ψ ; ˆ σ (0) D ] [51, 52]. We list in relations (2.70-2.73) the Hamilton operator ˆ H [ˆ σ (0) D ] (2.70), its corre-sponding Green function ˆ g (0) [ˆ σ (0) D ] (2.71) and the definitions of the trace ’tr’ (2.72) and unit operator ’ˆ1(2.73) in the considered Hilbert space with the complete set of states concerning spatial points and thecontour time (compare with Ref. [6], chapter 4.2)ˆ H [ˆ σ (0) D ] = (cid:18) ˆ η (cid:0) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ | {z } ˆ h p (cid:1) + ˆ σ (0) D (cid:19) ; (2.70)ˆ g (0) [ˆ σ (0) D ] = (cid:16) ˆ H [ˆ σ (0) D ] (cid:17) − = (cid:18) ˆ η (cid:0) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ | {z } ˆ h p (cid:1) + ˆ σ (0) D (cid:19) − ; (2.71)tr h . . . i = Z C dt p ~ η p X ~x N h ~x, t p | . . . | ~x, t p i (2.72)= Z ∞−∞ dt + ~ X ~x N h ~x, t + | . . . | ~x, t + i + Z ∞−∞ dt − ~ X ~x N h ~x, t − | . . . | ~x, t − i ;ˆ1 = Z C dt p ~ η p X ~x N | ~x, t p ih ~x, t p | (2.73)= Z ∞−∞ dt + ~ X ~x N | ~x, t + ih ~x, t + | + Z ∞−∞ dt − ~ X ~x N | ~x, t − ih ~x, t − | ; Z [ j ψ ; ˆ σ (0) D ] = Z d [ˆ σ (0) D ( ~x, t p )] exp (cid:26) ı ~ V Z C dt p X ~x σ (0) D ( ~x, t p ) σ (0) D ( ~x, t p ) (cid:27) (2.74) × exp (cid:26) − ( L − S ) tr (cid:20) ln (cid:18) ˆ η (cid:0) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ | {z } ˆ h p (cid:1) + ˆ σ (0) D (cid:19)(cid:21)(cid:27) × exp (cid:26) ı N = L + S X α =1 N h j ψ ; α | ˆ η ˆ g (0) [ˆ σ (0) D ] ˆ η | j ψ ; α i (cid:27) .The averaging procedure h . . . i ˆ σ (0) D (2.75) for the coupling coefficients c ij ( ~x, t p ), d ij ( ~x, t p ) has therefore tobe performed with the generating function Z [ j ψ ; ˆ σ (0) D ] (2.74) of the background field σ (0) D ( ~x, t p ) according .4 Scaling of physical parameters and quantities to dimensionless values and fields (cid:28)(cid:18) functional of σ (0) D ( ~x, t p ) with gradient terms (cid:19)(cid:29) ˆ σ (0) D = (2.75)= Z d [ˆ σ (0) D ( ~x, t p )] exp (cid:26) ı ~ V Z C dt p X ~x σ (0) D ( ~x, t p ) σ (0) D ( ~x, t p ) (cid:27) × exp (cid:26) − ( L − S ) tr ln (cid:18) ˆ η (cid:0) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ | {z } ˆ h p (cid:1) + σ (0) D (cid:19)(cid:27) × exp (cid:26) ı N = L + S X α =1 N h j ψ ; α | ˆ η ˆ g (0) [ˆ σ (0) D ] ˆ η | j ψ ; α i (cid:27) × (cid:18) functional of σ (0) D ( ~x, t p ) with gradient terms (cid:19) .Instead of functional averaging by Z [ j ψ ; ˆ σ (0) D ] according to Eq. (2.75), one can also apply a saddle pointequation or first order variation with the background field in order to obtain a mean field solution for σ (0) D ( ~x, t p ) (2.76). This mean field solution can then be substituted for the functional dependence ofthe coupling coefficients c ij ( ~x, t p ), d ij ( ~x, t p ) on the background field according to the defining relations(2.55-2.59). One can expect a good approximation by this mean field solution because the generatingfunction Z [ j ψ ; ˆ σ (0) D ] is only determined by the background field σ (0) D ( ~x, t p ) which is itself related to thedifference of boson-boson and fermion-fermion densities due to the U(1) symmetries in Z [ j ψ ; ˆ σ (0) D ]0 ≡ ıV N σ (0) D ( ~x, t p ) − (cid:0) L − S (cid:1) (cid:20) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ + σ (0) D ( ~x, t p ) (cid:21) − ~x,~x ( t p + δt p , t p + δt ′ p )+ − ı Z C dt (1) q ~ dt (2) q ~ X ~y ,~y N N = L + S X α =1 × (2.76) × j + ψ ; α ( ~y , t (2) q ) (cid:20) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ + σ (0) D ( ~x, t p ) (cid:21) − ~y ,~x ( t (2) q , t p + δt ′ p ) ×× (cid:20) − ˆ E p − ı ε p + ˆ ~p m + u (ˆ ~x ) − µ + σ (0) D ( ~x, t p ) (cid:21) − ~x,~y ( t p + δt p , t (1) q ) j ψ ; α ( ~y , t (1) q ) . A typical property of classical equations concerns the re-scaling of physical parameters and quantitiesto dimensionless values and fields, as e.g. in the Gross-Pitaevskii or nonlinear Schr¨odinger equation. Inthe considered, effective coherent state path integral Z [ˆ J , J ψ , ı ˆ J ψψ ] (2.49), we therefore scale the actions A ′ N − [ ˆ T ; J ψ ] (2.54), A ′ N [ ˆ T ; J ψ ] (2.60) and A ′ N +1 [ ˆ T ] (2.61) with the pair condensate fields in dependenceon discrete spatial and time-like coordinates to dimensionless values. In consequence, one can performthe first and higher order variations of the actions (under the presumed Euclidean path integration fields)0 in classical correspondence to the coherent state path integral which represents many-body quantummechanics. We list in Eq. (2.77) the parameters Ω and N x , which indicate the maximum energy ~ Ωand number of discrete spatial points, and combine them to the parameter N which inevitably appearswith the space and time contour integrals. Furthermore, we have to scale all energy parameters andpotentials to dimensionless quantities with the parameter N Ω = 1 / ∆ t , N x = ( L/ ∆ x ) d → N = ~ Ω N x ; (2.77) ε p , µ , V , u ( ~x ) → ˘ ε p = ε p / N , ˘ µ = µ / N , ˘ V = V / N , ˘ u ( ~x ) = u ( ~x ) / N ; (2.78) σ (0) D ( ~x, t p ) → ˘ σ (0) D ( ~x, t p ) = σ (0) D ( ~x, t p ) / N . (2.79)The dimensionless, scaled quantities are denoted by the additional symbol ’ ˘ ’ above the corresponding,original physical parameter or physical quantity symbol. Similarly, the contour time t p = t ˘ t p , thecontour time derivative ˆ E p = N ˘ E p and the spatial coordinates ~x = x ˘ ~x with their gradients ∂ i = ˘ ∂ i /x are scaled by the parameters t = ~ / N , x = t · / (2 m N ) / to dimensionless quantitiesˆ E p = ı ~ ∂∂t p ; ~ ω p → ı ∂∂ ˘ t p = ˆ E p / N = ˘ E p = ı ˘ ∂ ˘ t p ; ˘ ω p = ~ ω p / N ; (2.80) t = ~ / N ; d ˘ t p = N dt p / ~ = dt p /t ; (2.81) ~ m ∂∂~x · ∂∂~x → (cid:18) ~ m − N (cid:19) ∂∂~x · ∂∂~x = x ∂∂~x · ∂∂~x = ∂∂ ˘ ~x · ∂∂ ˘ ~x ; ˘ ~x = ~x/x ; (2.82) d d x/L d → ( x /L ) d d d ˘ x ; x = (cid:16) ~ m − .(cid:0) N (cid:1)(cid:17) / = t (cid:16) .(cid:0) m N (cid:1)(cid:17) / . (2.83)Application of (2.77-2.83) for the re-scaling of A ′ N − [ ˆ T ; J ψ ] (2.54), A ′ N [ ˆ T ; J ψ ] (2.60), A ′ N +1 [ ˆ T ] (2.61)yields the action A ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ] (2.84) with Lagrangian L ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ] (2.85) for the anomalous fields inthe coset super-generator ˆ Y (˘ ~x, ˘ t p ). The action term A ˆ J ψψ [ ˆ T ] in (2.84) creates these anomalous fieldsfrom the vacuum state through the ’condensate seed matrix’ ˆ J a = bψψ ; αβ ( ~x, t p ). However, instead of a detailedcreation process by ˆ J a = bψψ ; αβ ( ~x, t p ), we simply assume suitable, initial conditions of the pair condensatefields in ˆ Y (˘ ~x, ˘ t p ) whose dynamics are determined by the action A ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ] (2.84) or L ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ](2.85) Z [ˆ J , J ψ , ı ˆ J ψψ ] = Z d (cid:2) ˆ T − (˘ ~x, ˘ t p ) d ˆ T (˘ ~x, ˘ t p ) (cid:3) exp n ı A ˆ J ψψ (cid:2) ˆ T (cid:3)o × exp n − A ′ (cid:2) ˆ T ; ˆ J (cid:3)o (2.84) × exp n − A ( d ) (cid:2) ˆ Z, ˆ T ; ˘ J ψ (cid:3)o ; A ( d ) (cid:2) ˆ Z, ˆ T ; ˘ J ψ (cid:3) = Z C d ˘ t p Z d d ˘ x (cid:18) x L (cid:19) d L ( d ) (cid:2) ˆ Z, ˆ T ; ˘ J ψ (cid:3) . (2.85)The action A ′ [ ˆ T ; ˆ J ] in Z [ˆ J , J ψ , ı ˆ J ψψ ] (2.84) specifies the observable quantities by differentiation withrespect to ˆ J ab~x,α ; ~x ′ ,β ( t p , t ′ q ) (compare section 4.2) which has afterwards to be set to zero. Therefore, A ′ [ ˆ T ; ˆ J ] cannot effect the dynamics of the pair condensate fields as the action A ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ] (2.85).1Relation (2.86) finally contains the complete, re-scaled Lagrangian L ( d ) [ ˆ Z, ˆ T ; ˘ J ψ ] whose dependencieson pair condensates have to be modified by the transformation of the super-Jacobi matrix ˆ G − / / U fromthe coset metric tensor; in consequence, the nontrivial coset integration measure d [ ˆ T − ( ~x, t p ) d ˆ T ( ~x, t p )](2.48) in (2.84) is eliminated for Euclidean path integration fields as the new, independent anomalousfield variables in Z [ˆ J , J ψ , ı ˆ J ψψ ] (2.84,2.85) (see following section 3) L ( d ) (cid:2) ˆ Z, ˆ T ; ˘ J ψ (cid:3) = ˘ c ij (˘ ~x, ˘ t p )4 STR h(cid:16) ˘ ∂ i ˆ Z (˘ ~x, ˘ t p ) (cid:17) (cid:16) ˘ ∂ j ˆ Z (˘ ~x, ˘ t p ) (cid:17)i + (2.86) −
12 STR h ˆ T − (˘ ~x, ˘ t p ) ˆ S (cid:16) ˘ E p ˆ T (˘ ~x, ˘ t p ) (cid:17) + ˆ T − (˘ ~x, ˘ t p ) (cid:16) ˘ ∂ i ˆ T (˘ ~x, ˘ t p ) (cid:17) ˆ T − (˘ ~x, ˘ t p ) (cid:16) ˘ ∂ i ˆ T (˘ ~x, ˘ t p ) (cid:17)i + − (cid:16) ˘ u (˘ ~x ) − ˘ µ − ı ˘ ε p + (cid:10) ˘ σ (0) D (˘ ~x, ˘ t p ) (cid:11) ˆ σ (0) D (cid:17) STR h(cid:16) ˆ T − (˘ ~x, ˘ t p ) (cid:17) − ˆ1 N × N i + − ı (cid:16) ˘ d ij (˘ ~x, ˘ t p ) − δ ij (cid:17) ×× ˘ J + ,bψ ; β (˘ ~x, ˘ t p ) (cid:20) ˆ I e K (cid:16) ˘ ∂ i ˆ T (˘ ~x, ˘ t p ) (cid:17) ˆ T − (˘ ~x, ˘ t p ) (cid:16) ˘ ∂ j ˆ T (˘ ~x, ˘ t p ) (cid:17) ˆ T − (˘ ~x, ˘ t p ) ˆ I (cid:21) baβα ˘ J aψ ; α (˘ ~x, ˘ t p ) + − ı J + ,bψ ; β (˘ ~x, ˘ t p ) (cid:20) ˆ I e K ˆ T (˘ ~x, ˘ t p ) ˆ S ˆ T − (˘ ~x, ˘ t p ) (cid:16) ˘ E p ˆ T (˘ ~x, ˘ t p ) (cid:17) ˆ T − (˘ ~x, ˘ t p ) ˆ I (cid:21) baβα ˘ J aψ ; α (˘ ~x, ˘ t p ) ++ η p (cid:20)(cid:16) ˆ T − (˘ ~x, ˘ t p ) (cid:17) − ˆ1 N × N (cid:21) . It is the aim of this section to transform the ’ ˆ P , ˆ P − ’ rotated derivative ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − involving sine-(sinh-) functions of eigenvalues to a Euclidean form ( ∂ ˆ Z abαβ ) (3.1) (compare appendix A and C in Ref.[6]). The general symbolic derivative ’ ∂ ’, appearing in this section, representatively replaces a partial,spatial gradient ’ ˘ ∂ i ’ or time contour derivative ’ ˘ ∂ ˘ t p ’, a variation symbol ’ δ ’ for stationary phases or atotal derivative ’ d ’. The transformed ’Nambu’ doubled super-matrix ( ∂ ˆ Z abαβ ) (3.1) of Euclidean formis composed of four sub-super-matrices with densities ( ∂ ˆ Y αβ ), ( ∂ ˆ Y αβ ) and anomalous terms ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ) (cid:0) ∂ ˆ Z abαβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) abαβ = (cid:0) ∂ ˆ Y αβ (cid:1) (cid:0) ∂ ˆ X αβ (cid:1)e κ (cid:0) ∂ ˆ X + αβ (cid:1) (cid:0) ∂ ˆ Y αβ (cid:1) ! ab . (3.1)Apart from the dependence of the densities ( ∂ ˆ Y αβ ), ( ∂ ˆ Y αβ ) on the pair condensate fields in ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ), the ’Nambu’ doubled super-matrix ( ∂ ˆ Z abαβ ) (3.1) has similar symmetries between matrix2 entries as the self-energy δ e Σ abαβ ( ~x, t p ) e K (1.51,1.52) and is also confined to taking values within theortho-symplectic super-algebra osp( S, S | L ). The super-matrix ( ∂ ˆ X αβ ) (3.2) and its super-hermitianconjugate e κ ( ∂ ˆ X + αβ ) (3.3) consist of the symmetric, complex, even matrices ( ∂ ˆ b mn ), ( ∂ ˆ b + mn ) for themolecular pair condensates and the anti-symmetric, complex, even matrices ( ∂ ˆ a rµ,r ′ ν ), ( ∂ ˆ a + rµ,r ′ ν ) forthe BCS pair condensate terms (3.4). This is in accordance with the N × N super-matrices ˆ X N × N , e κ ˆ X + N × N (2.3,2.4) and their symmetric, complex, even boson-boson parts ˆ c D ; L × L , ˆ c + D ; L × L and anti-symmetric, complex, even fermion-fermion parts ˆ f D ; S × S , ˆ f + D ; S × S (2.5). The odd parts ˆ η D ; S × L , ˆ η TD ; L × S (2.4) (respectively their complex conjugates ˆ η ∗ D ; S × L , ˆ η + D ; L × S ) are substituted by ˆ ζ rµ,n , ˆ ζ Tm,r ′ ν (and ˆ ζ ∗ rµ,n ,ˆ ζ + m,r ′ ν ) in ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ) (compare footnote 4 for the indexing and numbering of the anomalousfermion-fermion parts by 2 × (cid:0) ∂ ˆ X αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = − (cid:0) ∂ ˆ b mn (cid:1) (cid:0) ∂ ˆ ζ Tm,r ′ ν (cid:1) − (cid:0) ∂ ˆ ζ rµ,n (cid:1) (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) ! αβ ; (3.2) e κ (cid:0) ∂ ˆ X + αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = (cid:0) ∂ ˆ b ∗ mn (cid:1) (cid:0) ∂ ˆ ζ + m,r ′ ν (cid:1)(cid:0) ∂ ˆ ζ ∗ rµ,n (cid:1) (cid:0) ∂ ˆ a + rµ,r ′ ν (cid:1) ! αβ ; (3.3) (cid:0) ∂ ˆ b mn (cid:1) = (cid:0) ∂ ˆ b Tmn (cid:1) ; (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) = − (cid:0) ∂ ˆ a Trµ,r ′ ν (cid:1) = X k =0 (cid:0) τ k (cid:1) µν (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) . (3.4)The density part ( ∂ ˆ Y αβ ) (3.5,3.1) has in its entity as a N × N super-matrix anti-hermitian properties,with the even, hermitian boson-boson and fermion-fermion matrices ( ∂ ˆ d mn ), ( ∂ ˆ g rµ,r ′ ν ) (3.6). The oddparts of ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) (similar to δ ˆ χ D ; S × L , δ ˆ χ + D ; L × S of δ ˆΣ D ; N × N ) are represented by ( ∂ ˆ ξ rµ,n ),( ∂ ˆ ξ + m,r ′ ν ) so that one obtains an anti-hermitian property of ( ∂ ˆ Y αβ ) because of the total imaginary factor (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = ı (cid:0) ∂ ˆ d mn (cid:1) (cid:0) ∂ ˆ ξ + m,r ′ ν (cid:1)(cid:0) ∂ ˆ ξ rµ,n (cid:1) (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) ! αβ ; (3.5) (cid:0) ∂ ˆ Y αβ (cid:1) + = − (cid:0) ∂ ˆ Y αβ (cid:1) ; (cid:0) ∂ ˆ d + mn (cid:1) = (cid:0) ∂ ˆ d mn (cid:1) ; (cid:0) ∂ ˆ g + rµ,r ′ ν (cid:1) = (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) . (3.6)The ’22’ N × N super-matrix ( ∂ ˆ Y αβ ) (3.7,3.1) is related by super-transposition ’st’ (1.29-1.31) to the ’11’ N × N density super-matrix ( ∂ ˆ Y αβ ) (3.5,3.6) with inclusion of an additional minus sign. It is composedof the same variables with ( ∂ ˆ d TL × L ), ( ∂ ˆ g TS × S ) and also contains the same odd parts ( ∂ ˆ ξ ∗ S × L ), ( ∂ ˆ ξ TL × S )apart from the additional super-transposition (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = − ı (cid:0) ∂ ˆ d Tmn (cid:1) − (cid:0) ∂ ˆ ξ Tm,r ′ ν (cid:1)(cid:0) ∂ ˆ ξ ∗ rµ,n (cid:1) (cid:0) ∂ ˆ g Trµ,r ′ ν (cid:1) ! αβ ; (3.7) (cid:0) ∂ ˆ Y αβ (cid:1) st = − (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) ,stαβ = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ . (3.8)According to the coset decomposition, both parts, density (3.5-3.8) and anomalous terms (3.2-3.4) of( ∂ ˆ Z abαβ ) (3.1), depend on the original, independent anomalous fields in ˆ X N × N , e κ ˆ X + N × N (2.2-2.5) and on .2 Removal of the coset integration measure and transformation to Euclidean integration variables c m , f r (2.16-2.25). In consequence there exists a cross-dependence of the density fields( ∂ ˆ Z αβ ) = ( ∂ ˆ Y αβ ), ( ∂ ˆ Z αβ ) = ( ∂ ˆ Y αβ ) over the original matrices ˆ X N × N , e κ ˆ X + N × N of the coset decompositionto the Euclidean pair condensate integration variables in ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ). In appendix C of Ref. [6] we have explicitly computed the general derivative (cid:0) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:1) abαβ in terms of the eigenvalues ˆ Y DD , ˆ X DD , e κ ˆ X + DD (2.16-2.25) and the rotated derivative ˆ P ( ∂ ˆ Y ) ˆ P − (3.11) of the independent anomalous fields ( ∂ ˆ c D ; L × L ), ( ∂ ˆ f D ; S × S ), ( ∂ ˆ η D ; S × L ), ( ∂ ˆ η + D ; L × S ) (2.2-2.5). Usingrelation (3.9) for the derivative of an exponential of a matrix [48, 6], it remains to multiply the rotatedderivative ( ∂ ˆ Y ′ ) = ˆ P ( ∂ ˆ Y ) ˆ P − (3.11) by the diagonal anomalous matrices in the various block parts ofexp {± v ˆ Y DD } with eigenvalues c m and quaternion eigenvalues ( τ ) µν f r (2.16-2.25)exp (cid:8) ˆ B (cid:9) δ (cid:16) exp (cid:8) − ˆ B (cid:9)(cid:17) = − Z dv exp (cid:8) v ˆ B (cid:9) δ ˆ B exp (cid:8) − v ˆ B (cid:9) ; (3.9) − (cid:0) ∂ ˆ Z abαβ (cid:1) = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) abαβ = (cid:16) ˆ P exp (cid:8) ˆ Y (cid:9) (cid:16) ∂ exp (cid:8) − ˆ Y (cid:9)(cid:17) ˆ P − (cid:17) abαβ (3.10)= (cid:16) ˆ P exp (cid:8) ˆ P − ˆ Y DD ˆ P (cid:9) (cid:16) ∂ exp (cid:8) − ˆ P − ˆ Y DD ˆ P (cid:9)(cid:17) ˆ P − (cid:17) abαβ = − Z dv (cid:16) exp (cid:8) v ˆ Y DD (cid:9) (cid:0) ∂ ˆ Y ′ (cid:1) exp (cid:8) − v ˆ Y DD (cid:9)(cid:17) abαβ ; (cid:0) ∂ ˆ Y ′ (cid:1) = ˆ P (cid:0) ∂ ˆ Y (cid:1) ˆ P − ; STR h(cid:0) ∂ ˆ Y ′ (cid:1) (cid:0) ∂ ˆ Y ′ (cid:1)i = STR h(cid:0) ∂ ˆ Y (cid:1) (cid:0) ∂ ˆ Y (cid:1)i . (3.11)After straightforward integration (3.10) of (hyperbolic) trigonometric functions with parameter v ∈ [0 , ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ) (3.2-3.4) on the independent rotated variableswithin ( ∂ ˆ c ′ D ; L × L ), ( ∂ ˆ f ′ D ; S × S ), ( ∂ ˆ η ′ D ; S × L ), ( ∂ ˆ η ′ , + D ; L × S ) (3.11,2.2-2.5) of ( ∂ ˆ X N × N ), e κ ( ∂ ˆ X + N × N ) and withcorresponding eigenvalues c m , f r (2.16-2.25). Similarly one achieves the relation between the densities( ∂ ˆ Y αβ ), ( ∂ ˆ Y αβ ) (3.5-3.8) and the original anomalous fields ( ∂ ˆ X N × N ), e κ ( ∂ ˆ X + N × N ) (2.2-2.5) of the cosetdecomposition. Therefore, the densities ( ∂ ˆ Y αβ ), ( ∂ ˆ Y αβ ) can also be related to the Euclidean, independentanomalous integration variables of ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ). In the following subsections 3.2.1, 3.2.2 and 3.2.3,we apply the results for (cid:0) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:1) abαβ of Ref. [6] with appendix C in order to transformto Euclidean fields. These are separated into the split even boson-boson , fermion-fermion and oddfermion-boson, boson-fermion parts. In this subsection we record the transformation of ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) abBB ; mn to ( ∂ ˆ b mn ), ( ∂ ˆ b ∗ mn ) ( a = b ) andcorresponding density parts ( ∂ ˆ d mn ) ( a = b ) and have furthermore to distinguish between the diagonal4 ( m = n ) and off-diagonal ( m = n ) matrix elements of transformations which are restricted to the total,even boson-boson part − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = (cid:0) ∂ ˆ X αβ (cid:1) = − (cid:0) ∂ ˆ b mn (cid:1) (cid:0) ∂ ˆ ζ Tm,r ′ ν (cid:1) − (cid:0) ∂ ˆ ζ rµ,n (cid:1) (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) ! αβ ; (3.12) (cid:0) ∂ ˆ b mn (cid:1) = (cid:0) ∂ ˆ b Tmn (cid:1) ; (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) = − (cid:0) ∂ ˆ a Trµ,r ′ ν (cid:1) ; (3.13) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = (cid:0) ∂ ˆ Y αβ (cid:1) = ı (cid:0) ∂ ˆ d mn (cid:1) (cid:0) ∂ ˆ ξ + m,r ′ ν (cid:1)(cid:0) ∂ ˆ ξ rµ,n (cid:1) (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) ! αβ ; (3.14) (cid:0) ∂ ˆ d + mn (cid:1) = (cid:0) ∂ ˆ d mn (cid:1) ; (cid:0) ∂ ˆ g + rµ,r ′ ν (cid:1) = (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) . (3.15)In relations (3.16-3.19) we give the detailed transformation (3.16,3.17) from the diagonal, rotated,anomalous molecular condensates ( ∂ ˆ c ′ D ; mm ), ( ∂ ˆ c ′ ∗ D ; mm ) to Euclidean fields ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) and also de-termine the reverse transformations (3.18) from ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) to the original fields ( ∂ ˆ c ′ D ; mm ), ( ∂ ˆ c ′ ∗ D ; mm )in the coset decomposition. The back transformation (3.18) allows to calculate the change of integra-tion measure from d ˆ c ′ D ; mm ∧ d ˆ c ′ ∗ D ; mm or in equivalence from the un-rotated fields d ˆ c D ; mm ∧ d ˆ c ∗ D ; mm to thediagonal, Euclidean elements d ˆ b mm ∧ d ˆ b ∗ mm − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) BB ; mm = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) , ∗ BB ; mm = − (cid:0) ∂ ˆ b mm (cid:1) = (3.16)= − (cid:18)
12 + sin (cid:0) | c m | (cid:1) | c m | (cid:19) (cid:0) ∂ ˆ c ′ D ; mm (cid:1) − (cid:18) − sin (cid:0) | c m | (cid:1) | c m | (cid:19) e ı ϕ m (cid:0) ∂ ˆ c ′ ∗ D ; mm (cid:1) ; (cid:18) (cid:0) ∂ ˆ b mm (cid:1)(cid:0) ∂ ˆ b ∗ mm (cid:1) (cid:19) = (cid:18) + sin (cid:0) | c m | (cid:1) | c m | (cid:19) e ı ϕ m (cid:18) − sin (cid:0) | c m | (cid:1) | c m | (cid:19) e − ı ϕ m (cid:18) − sin (cid:0) | c m | (cid:1) | c m | (cid:19) (cid:18) + sin (cid:0) | c m | (cid:1) | c m | (cid:19) (cid:18) (cid:0) ∂ ˆ c ′ D ; mm (cid:1)(cid:0) ∂ ˆ c ′ ∗ D ; mm (cid:1) (cid:19) ;(3.17) (cid:18) (cid:0) ∂ ˆ c ′ D ; mm (cid:1)(cid:0) ∂ ˆ c ′ ∗ D ; mm (cid:1) (cid:19) = (cid:18) + | c m | sin (cid:0) | c m | (cid:1) (cid:19) e ı ϕ m (cid:18) − | c m | sin (cid:0) | c m | (cid:1) (cid:19) e − ı ϕ m (cid:18) − | c m | sin (cid:0) | c m | (cid:1) (cid:19) (cid:18) + | c m | sin (cid:0) | c m | (cid:1) (cid:19) (cid:18) (cid:0) ∂ ˆ b mm (cid:1)(cid:0) ∂ ˆ b ∗ mm (cid:1) (cid:19) ; (3.18) d ˆ c ′ D ; mm ∧ d ˆ c ′ ∗ D ; mm = d ˆ c D ; mm ∧ d ˆ c ∗ D ; mm = d ˆ b mm ∧ d ˆ b ∗ mm | c m | sin (cid:0) | c m | (cid:1) . (3.19)The matrix in (3.17) with eigenvalues c m (2.19) represents the square root of the coset metric ten-sor ˆ G / / U concerning the boson-boson part, however, in its diagonalized form with eigenvalues c m .The inverse transformation (3.18) of Euclidean fields ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) to ( ∂ ˆ c ′ D ; mm ), ( ∂ ˆ c ′ ∗ D ; mm ) there-fore contains the inverse square root of the coset metric tensor ˆ G − / / U as already mentioned in theintroduction. Both kinds of metric tensors, ˆ G / / U and ˆ G − / / U , are considerably simplified because .2 Removal of the coset integration measure and transformation to Euclidean integration variables X N × N , e κ ˆ X + N × N to their eigenvalues in ˆ X DD , e κ ˆ X + N × N with eigenvectors ’ ˆ P aaαβ , ˆ P − aaαβ ’ (2.16-2.35). Since the transformation of ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) (3.18) is com-posed of the inverse square root of the metric tensor ˆ G − / / U , the corresponding integration measureSDET( ˆ G − / / U ) = ( SDET( ˆ G Osp / U ) ) − / cancels the nontrivial integration measure originally introducedfor the coset decomposition of δ e Σ abαβ ( ~x, t p ) e K to ˆ T δ ˆΣ D ;2 N × N e K ˆ T − (compare section 2.2). Accord-ing to appendix C of Ref. [6], we can also give the transformation of the diagonal elements of theboson-boson density part ( ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − ) BB ; mm . It yields with the sub-metric tensor ( ˆ G − / / U ) BB ; mm (3.18) for the relations between ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) and ( ∂ ˆ c ′ D ; mm ), ( ∂ ˆ c ′ ∗ D ; mm ), the diagonal density elements ı ( ∂ ˆ d mm ) (3.20) in terms of tan( | c m | ) of the eigenvalues c m (2.19) and in terms of the diagonal, Euclideananomalous fields ( ∂ ˆ b mm ), ( ∂ ˆ b ∗ mm ) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) BB ; mm = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) BB ; mm = ı (cid:0) ∂ ˆ d mm (cid:1) = (3.20)= − (cid:16) sin (cid:0) | c m | (cid:1)(cid:17) | c m | (cid:16)(cid:0) ∂ ˆ c ′ ∗ D ; mm (cid:1) e ı ϕ m − (cid:0) ∂ ˆ c ′ D ; mm (cid:1) e − ı ϕ m (cid:17) = 12 tan (cid:0) | c m | (cid:1) (cid:16)(cid:0) ∂ ˆ b mm (cid:1) e − ı ϕ m − (cid:0) ∂ ˆ b ∗ mm (cid:1) e ı ϕ m (cid:17) . We transfer the results of the transformation to Euclidean fields from the diagonal boson-boson partsto the case of off-diagonal matrix elements ( m = n ) (3.21) and introduce the coefficients A BB , B BB (3.22,3.23), depending on the modulus of eigenvalues | c m | , | c n | for specification of ( ∂ ˆ b m = n ), ( ∂ ˆ b ∗ m = n ).The transformation (3.21,3.24) determines the off-diagonal matrix elements of the boson-boson part of( ˆ G / / U ) BB ; m = n whereas relation (3.25) describes the back transformation from ( ∂ ˆ b m = n ), ( ∂ ˆ b ∗ m = n ) to theoriginal, ’ ˆ P aaαβ , ˆ P − aaαβ ’ rotated fields ( ∂ ˆ c ′ D ; m = n ), ( ∂ ˆ c ′ ∗ D ; m = n ) of the coset decomposition. The diagonalizedforms of ˆ G / / U , ˆ G − / / U (3.24,3.25) are simplified by the coefficients A BB , B BB which are defined bythe relations (3.22,3.23) of the eigenvalues | c m | , | c n | . Note that the limit process | c n | → | c m | reproducesthe results (3.16-3.19) of diagonal elements ( m = n ) for the boson-boson part of the metric tensor. Thisholds in particular for the case of the integration measure (3.28) which attains the identical form of(3.19) in case of the limit process | c n | → | c m | . The integration measure (3.28) with eigenvalues | c m | , | c n | for d ˆ b m = n , d ˆ b ∗ m = n results into the Euclidean form after substitution into the original coset integration(2.48) for d ˆ c ′ D ; m = n , d ˆ c ′ ∗ D ; m = n of the coset decomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) BB ; mn m = n = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) , ∗ BB ; mn m = n = (3.21)= − (cid:0) ∂ ˆ b mn (cid:1) = − A BB (cid:0) ∂ ˆ c ′ D ; mn (cid:1) − e ı ( ϕ m + ϕ n ) B BB (cid:0) ∂ ˆ c ′ ∗ D ; mn (cid:1) ; A BB = | c m | cos (cid:0) | c n | (cid:1) sin (cid:0) | c m | (cid:1) − | c n | cos (cid:0) | c m | (cid:1) sin (cid:0) | c n | (cid:1) | c m | − | c n | ; (3.22) B BB = | c n | cos (cid:0) | c n | (cid:1) sin (cid:0) | c m | (cid:1) − | c m | cos (cid:0) | c m | (cid:1) sin (cid:0) | c n | (cid:1) | c m | − | c n | ; (3.23)6 (cid:18) (cid:0) ∂ ˆ b mn (cid:1)(cid:0) ∂ ˆ b ∗ mn (cid:1) (cid:19) = (cid:18) A BB e ı ( ϕ m + ϕ n ) B BB e − ı ( ϕ m + ϕ n ) B BB A BB (cid:19) (cid:18) (cid:0) ∂ ˆ c ′ D ; mn (cid:1)(cid:0) ∂ ˆ c ′ ∗ D ; mn (cid:1) (cid:19) ; (3.24) (cid:18) (cid:0) ∂ ˆ c ′ D ; mn (cid:1)(cid:0) ∂ ˆ c ′ ∗ D ; mn (cid:1) (cid:19) = 1 A BB − B BB (cid:18) A BB − e ı ( ϕ m + ϕ n ) B BB − e − ı ( ϕ m + ϕ n ) B BB A BB (cid:19) (cid:18) (cid:0) ∂ ˆ b mn (cid:1)(cid:0) ∂ ˆ b ∗ mn (cid:1) (cid:19) ; (3.25) A BB A BB − B BB = 12 (cid:18) | c m | − | c n | sin (cid:0) | c m | − | c n | (cid:1) + | c m | + | c n | sin (cid:0) | c m | + | c n | (cid:1) (cid:19) ; (3.26) B BB A BB − B BB = 12 (cid:18) | c m | + | c n | sin (cid:0) | c m | + | c n | (cid:1) − | c m | − | c n | sin (cid:0) | c m | − | c n | (cid:1) (cid:19) ; (3.27) d ˆ c ′ D ; mn ∧ d ˆ c ′ ∗ D ; mn = d ˆ c D ; mn ∧ d ˆ c ∗ D ; mn = d ˆ b mn ∧ d ˆ b ∗ mn | c m | + | c n | sin (cid:0) | c m | + | c n | (cid:1) | c m | − | c n | sin (cid:0) | c m | − | c n | (cid:1) . (3.28)According to appendix C of Ref. [6], we sate the corresponding boson-boson density part for off-diagonalmatrix elements ı ( ∂ ˆ d mn ) (3.29) by using coefficient functions C BB , D BB (3.30-3.32) of the eigenvalues | c m | , | c n | (2.19). Combining the transformation (3.25) with (3.29), the coefficients A BB , B BB and C BB , D BB allow to reduce the dependence of ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) BB ; m = n onto the Euclidean variables ( ∂ ˆ b m = n ),( ∂ ˆ b ∗ m = n ) of anomalous terms for final relation (3.33) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) BB ; mn m = n = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) ,TBB ; mn m = n = ı (cid:0) ∂ ˆ d mn (cid:1) = (3.29)= − e ı ϕ m D BB (cid:0) ∂ ˆ c ′ ∗ D ; mn (cid:1) + e − ı ϕ n C BB (cid:0) ∂ ˆ c ′ D ; mn (cid:1) ; C BB = | c n | − | c n | cos (cid:0) | c n | (cid:1) cos (cid:0) | c m | (cid:1) − | c m | sin (cid:0) | c n | (cid:1) sin (cid:0) | c m | (cid:1) | c n | − | c m | ; (3.30) D BB = | c m | − | c m | cos (cid:0) | c m | (cid:1) cos (cid:0) | c n | (cid:1) − | c n | sin (cid:0) | c m | (cid:1) sin (cid:0) | c n | (cid:1) | c m | − | c n | ; (3.31) D BB (cid:0) | c m | , | c n | (cid:1) = C BB (cid:0) | c n | , | c m | (cid:1) ; (3.32) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) BB ; mn m = n = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) ,TBB ; mn m = n = ı (cid:0) ∂ ˆ d mn (cid:1) = (3.33)= e − ı ϕ n sin (cid:0) | c n | (cid:1) (cid:0) ∂ ˆ b mn (cid:1) − e ı ϕ m sin (cid:0) | c m | (cid:1) (cid:0) ∂ ˆ b ∗ mn (cid:1) cos (cid:0) | c m | (cid:1) + cos (cid:0) | c n | (cid:1) == 12 tan (cid:18) | c m | + | c n | (cid:19) h e − ı ϕ n (cid:0) ∂ ˆ b mn (cid:1) − e ı ϕ m (cid:0) ∂ ˆ b ∗ mn (cid:1)i + −
12 tan (cid:18) | c m | − | c n | (cid:19) h e − ı ϕ n (cid:0) ∂ ˆ b mn (cid:1) + e ı ϕ m (cid:0) ∂ ˆ b ∗ mn (cid:1)i .In case of a limit process | c n | → | c m | , we obtain from Eq. (3.33) the result (3.20) for the diagonal densityelements ( m = n ). .2 Removal of the coset integration measure and transformation to Euclidean integration variables In analogy to the boson-boson part, we list corresponding results of the transformation to Euclideanfield variables for the fermion-fermion parts (appendix C in Ref. [6]). However, one has to exchangethe ordinary matrix elements of the boson-boson parts by matrix elements of the quaternion algebrawith standard 2 × × τ ) µν ( ∂ ˆ a (2) rr ) (3.34), due to the anti-symmetry of ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) F F ; rµ,rν and of ( ∂ ˆ a rµ,rν ) =( τ ) µν ( ∂ ˆ a (2) rr ) for the BCS pair condensate terms − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F F ; rµ,rν = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) , + F F ; rµ,rν = (cid:0) ∂ ˆ a rµ,rν (cid:1) = (cid:0) τ (cid:1) µν (cid:0) ∂ ˆ a (2) rr (cid:1) = (3.34)= (cid:0) τ (cid:1) µν (cid:20)(cid:18)
12 + sinh (cid:0) | f r | (cid:1) | f r | (cid:19) (cid:0) ∂ ˆ f ′ (2) D ; rr (cid:1) + (cid:18) − sinh (cid:0) | f r | (cid:1) | f r | (cid:19) e ı φ r (cid:0) ∂ ˆ f ′ (2) ∗ D ; rr (cid:1)(cid:21) .The complementary diagonal forms of ˆ G / / U , ˆ G − / / U with diagonal elements of the fermion-fermionsection follow from the transformations of ( ∂ ˆ f ′ (2) D ; rr ), ( ∂ ˆ f ′ (2) ∗ D ; rr ) (2.2-2.5) to ( ∂ ˆ a (2) rr ), ( ∂ ˆ a (2) ∗ rr ) and vice versa(3.35,3.36). The change of integration measure is given in (3.37) and yields with the original cosetintegration measure of Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) (2.48) Euclidean integration variables d ˆ a (2) rr ∧ d ˆ a (2) ∗ rr (cid:0) ∂ ˆ a (2) rr (cid:1)(cid:0) ∂ ˆ a (2) ∗ rr (cid:1) ! = (cid:18) + sinh (cid:0) | f r | (cid:1) | f r | (cid:19) e ı φ r (cid:18) − sinh (cid:0) | f r | (cid:1) | f r | (cid:19) e − ı φ r (cid:18) − sinh (cid:0) | f r | (cid:1) | f r | (cid:19) (cid:18) + sinh (cid:0) | f r | (cid:1) | f r | (cid:19) (cid:0) ∂ ˆ f ′ (2) D ; rr (cid:1)(cid:0) ∂ ˆ f ′ (2) ∗ D ; rr (cid:1) ! ; (3.35) (cid:0) ∂ ˆ f ′ (2) D ; rr (cid:1)(cid:0) ∂ ˆ f ′ (2) ∗ D ; rr (cid:1) ! = (cid:18) + | f r | sinh (cid:0) | f r | (cid:1) (cid:19) e ı φ r (cid:18) − | f r | sinh (cid:0) | f r | (cid:1) (cid:19) e − ı φ r (cid:18) − | f r | sinh (cid:0) | f r | (cid:1) (cid:19) (cid:18) + | f r | sinh (cid:0) | f r | (cid:1) (cid:19) (cid:0) ∂ ˆ a (2) rr (cid:1)(cid:0) ∂ ˆ a (2) ∗ rr (cid:1) ! ; (3.36) d ˆ f ′ (2) D ; rr ∧ d ˆ f ′ (2) ∗ D ; rr = d ˆ f (2) D ; rr ∧ d ˆ f (2) ∗ D ; rr = d ˆ a (2) rr ∧ d ˆ a (2) ∗ rr | f r | sinh (cid:0) | f r | (cid:1) . (3.37)We quote the result (3.38) for the quaternionic, diagonal densities ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) F F ; rµ,rν in termsof the coset fields ( ∂ ˆ f ′ (2) D ; rr ), ( ∂ ˆ f ′ (2) ∗ D ; rr ) (2.2-2.5) according to appendix C in Ref. [6]. Incorporating thetransformation (3.36) with diagonal sub-metric tensor ˆ G − / / U , we obtain the diagonal density elements ı ( ∂ ˆ g rµ,rν ) of the fermion-fermion part in dependence on tanh( | f r | ) and the Euclidean fermion-fermionpair condensate fields ( ∂ ˆ a (2) rr ), ( ∂ ˆ a (2) ∗ rr ) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F F ; rµ,rν = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) ,TF F ; rµ,rν = ı (cid:0) ∂ ˆ g rµ,rν (cid:1) = (3.38)= δ µν (cid:16) sinh (cid:0) | f r | (cid:1)(cid:17) | f r | (cid:16)(cid:0) ∂ ˆ f ′ (2) ∗ D ; rr (cid:1) e ı φ r − (cid:0) ∂ ˆ f ′ (2) D ; rr (cid:1) e − ı φ r (cid:17) = − δ µν
12 tanh (cid:0) | f r | (cid:1) (cid:16)(cid:0) ∂ ˆ a (2) rr (cid:1) e − ı φ r − (cid:0) ∂ ˆ a (2) ∗ rr (cid:1) e ı φ r (cid:17) . Concerning the off-diagonal, anomalous fields of the fermion-fermion sections, one has to apply allfour quaternion elements ( ∂ ˆ a ( k ) rr ′ ) with ( τ ) µν , ( τ ) µν , ( τ ) µν , ( τ ) µν and ( ∂ ˆ a ( k ) rr ′ ) = − ( ∂ ˆ a ( k ) r ′ r ) being anti-symmetric for k = 0 , , ∂ ˆ a (2) rr ′ ) = ( ∂ ˆ a (2) r ′ r ) for k = 2 − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F F ; rµ,r ′ ν r = r ′ = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) , + F F ; rµ,r ′ ν r = r ′ = X k =0 (cid:0) τ k (cid:1) µν (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) . (3.39)We abbreviate various terms of eigenvalues | f r | , | f r ′ | (2.20) by the coefficients A F F (3.40), B F F (3.41)and have to distinguish between quaternion elements of ( τ ) µν and ( τ k ) µν (k=1,2,3) for the transforma-tion of the original, anomalous coset fields ( ∂ ˆ f ′ ( k ) D ; rr ′ ), ( ∂ ˆ f ′ ( k ) ∗ D ; rr ′ ) (2.2-2.5) to their Euclidean correspondents A F F = | f r | cosh (cid:0) | f r ′ | (cid:1) sinh (cid:0) | f r | (cid:1) − | f r ′ | cosh (cid:0) | f r | (cid:1) sinh (cid:0) | f r ′ | (cid:1) | f r | − | f r ′ | ; (3.40) B F F = | f r | cosh (cid:0) | f r | (cid:1) sinh (cid:0) | f r ′ | (cid:1) − | f r ′ | cosh (cid:0) | f r ′ | (cid:1) sinh (cid:0) | f r | (cid:1) | f r | − | f r ′ | ; (3.41) (cid:0) ∂ ˆ a (0) rr ′ (cid:1)(cid:0) ∂ ˆ a (0) ∗ rr ′ (cid:1) ! = (cid:18) A F F e ı ( φ r + φ r ′ ) B F F e − ı ( φ r + φ r ′ ) B F F A F F (cid:19) (cid:0) ∂ ˆ f ′ (0) D ; rr ′ (cid:1)(cid:0) ∂ ˆ f ′ (0) ∗ D ; rr ′ (cid:1) ! ; (3.42) (cid:0) ∂ ˆ f ′ (0) D ; rr ′ (cid:1)(cid:0) ∂ ˆ f ′ (0) ∗ D ; rr ′ (cid:1) ! = 1 A F F − B F F (cid:18) A F F − e ı ( φ r + φ r ′ ) B F F − e − ı ( φ r + φ r ′ ) B F F A F F (cid:19) (cid:0) ∂ ˆ a (0) rr ′ (cid:1)(cid:0) ∂ ˆ a (0) ∗ rr ′ (cid:1) ! ; (3.43) (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1)(cid:0) ∂ ˆ a ( k ) ∗ rr ′ (cid:1) ! k =1 , , = (cid:18) A F F − e ı ( φ r + φ r ′ ) B F F − e − ı ( φ r + φ r ′ ) B F F A F F (cid:19) (cid:0) ∂ ˆ f ′ ( k ) D ; rr ′ (cid:1)(cid:0) ∂ ˆ f ′ ( k ) ∗ D ; rr ′ (cid:1) ! ; (3.44) (cid:0) ∂ ˆ f ′ ( k ) D ; rr ′ (cid:1)(cid:0) ∂ ˆ f ′ ( k ) ∗ D ; rr ′ (cid:1) ! k =1 , , = 1 A F F − B F F (cid:18) A F F e ı ( φ r + φ r ′ ) B F F e − ı ( φ r + φ r ′ ) B F F A F F (cid:19) (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1)(cid:0) ∂ ˆ a ( k ) ∗ rr ′ (cid:1) ! ; (3.45) A F F A F F − B F F = 12 (cid:18) | f r | − | f r ′ | sinh (cid:0) | f r | − | f r ′ | (cid:1) + | f r | + | f r ′ | sinh (cid:0) | f r | + | f r ′ | (cid:1) (cid:19) ; (3.46) B F F A F F − B F F = − (cid:18) | f r | + | f r ′ | sinh (cid:0) | f r | + | f r ′ | (cid:1) − | f r | − | f r ′ | sinh (cid:0) | f r | − | f r ′ | (cid:1) (cid:19) . (3.47)Taking the determinants of the ˆ G − / / U coset sub-metric transformations (3.43,3.45), we acquire theintegration measure SDET( ˆ G − / / U ) of the particular, diagonalized fermion-fermion parts (3.48) whichare eliminated with the sub-metric tensor parts ( SDET( ˆ G Osp / U ) ) / (2.48) of the original coset decom-position to Euclidean integration variables d ˆ a ( k ) rr ′ ∧ d ˆ a ( k ) ∗ rr ′ , ( k = 0 , , , d ˆ f ′ ( k ) D ; rr ′ ∧ d ˆ f ′ ( k ) ∗ D ; rr ′ = d ˆ f ( k ) D ; rr ′ ∧ d ˆ f ( k ) ∗ D ; rr ′ = d ˆ a ( k ) rr ′ ∧ d ˆ a ( k ) ∗ rr ′ | f r | + | f r ′ | sinh (cid:0) | f r | + | f r ′ | (cid:1) | f r | − | f r ′ | sinh (cid:0) | f r | − | f r ′ | (cid:1) . (3.48) .2 Removal of the coset integration measure and transformation to Euclidean integration variables − ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) F F ; rµ,r ′ ν = ı ( ∂ ˆ g rµ,r ′ ν ) (3.49) are given as quaternionmatrix elements and by coefficients C F F (3.51), D F F (3.52) according to appendix C in Ref. [6] − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F F ; rµ,r ′ ν r = r ′ = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) ,TF F ; rµ,r ′ ν r = r ′ = ı (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) = (3.49)= − (cid:0) τ (cid:1) µν h e − ı φ r ′ C F F (cid:0) ∂ ˆ f ′ (0) D ; rr ′ (cid:1) + e ı φ r D F F (cid:0) ∂ ˆ f ′ (0) ∗ D ; rr ′ (cid:1)i + − X k =1 , , (cid:0) ˆ m k (cid:1) µν h e − ı φ r ′ C F F (cid:0) ∂ ˆ f ′ ( k ) D ; rr ′ (cid:1) − e ı φ r D F F (cid:0) ∂ ˆ f ′ ( k ) ∗ D ; rr ′ (cid:1)i ; (cid:0) ˆ m (cid:1) µν = ı (cid:0) ˆ τ (cid:1) µν ; (cid:0) ˆ m (cid:1) µν = (cid:0) ˆ τ (cid:1) µν ; (cid:0) ˆ m (cid:1) µν = − ı (cid:0) ˆ τ (cid:1) µν ; (3.50) C F F = −| f r ′ | + | f r ′ | cosh (cid:0) | f r ′ | (cid:1) cosh (cid:0) | f r | (cid:1) − | f r | sinh (cid:0) | f r ′ | (cid:1) sinh (cid:0) | f r | (cid:1) | f r ′ | − | f r | ; (3.51) D F F = −| f r | + | f r | cosh (cid:0) | f r | (cid:1) cosh (cid:0) | f r ′ | (cid:1) − | f r ′ | sinh (cid:0) | f r | (cid:1) sinh (cid:0) | f r ′ | (cid:1) | f r | − | f r ′ | ; (3.52) C F F (cid:0) | f r | , | f r ′ | (cid:1) = D F F (cid:0) | f r ′ | , | f r | (cid:1) . (3.53)Insertion of relations (3.43,3.45) with coefficients A F F (3.40), B F F (3.41) into (3.49) yields the fermion-fermion density part in terms of tanh( ( | f r | ± | f r ′ | ) / ∂ ˆ a ( k ) rr ′ ),( ∂ ˆ a ( k ) ∗ rr ′ ) combined with the quaternion, 2 × τ k ) µν , ( τ ) µν , ( ˆ m k ) µν (3.50) − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) F F ; rµ,r ′ ν r = r ′ = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ T − (cid:17) ,TF F ; rµ,r ′ ν r = r ′ = ı (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) = (3.54)= − (cid:0) τ (cid:1) µν (cid:20) e − ı φ r ′ sinh (cid:0) | f r ′ | (cid:1) (cid:0) ∂ ˆ a (0) rr ′ (cid:1) + e ı φ r sinh (cid:0) | f r | (cid:1) (cid:0) ∂ ˆ a (0) ∗ rr ′ (cid:1) cosh (cid:0) | f r | (cid:1) + cosh (cid:0) | f r ′ | (cid:1) (cid:21) + − X k =1 , , (cid:0) ˆ m k (cid:1) µν (cid:20) e − ı φ r ′ sinh (cid:0) | f r ′ | (cid:1) (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) − e ı φ r sinh (cid:0) | f r | (cid:1) (cid:0) ∂ ˆ a ( k ) ∗ rr ′ (cid:1) cosh (cid:0) | f r | (cid:1) + cosh (cid:0) | f r ′ | (cid:1) (cid:21) = − (cid:0) τ (cid:1) µν (cid:26) tanh (cid:18) | f r | + | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a (0) rr ′ (cid:1) + e ı φ r (cid:0) ∂ ˆ a (0) ∗ rr ′ (cid:1)i + − tanh (cid:18) | f r | − | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a (0) rr ′ (cid:1) − e ı φ r (cid:0) ∂ ˆ a (0) ∗ rr ′ (cid:1)i(cid:27) + − X k =1 , , (cid:0) ˆ m k (cid:1) µν ( tanh (cid:18) | f r | + | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) − e ı φ r (cid:0) ∂ ˆ a ( k ) ∗ rr ′ (cid:1)i + − tanh (cid:18) | f r | − | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) + e ı φ r (cid:0) ∂ ˆ a ( k ) ∗ rr ′ (cid:1)i) = − X k =0 (cid:0) τ k τ (cid:1) µν (cid:26) tanh (cid:18) | f r | + | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) − (cid:0) − (cid:1) k e ı φ r (cid:0) ∂ ˆ a ( k )+ rr ′ (cid:1)i +0 − tanh (cid:18) | f r | − | f r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) + (cid:0) − (cid:1) k e ı φ r (cid:0) ∂ ˆ a ( k )+ rr ′ (cid:1)i(cid:27) . In the case of transformations in the odd fermion-boson and boson-fermion sections, we have to considerthe two quaternion elements ( τ ) µν and ( τ ) µν and cite the result (3.55,3.56) for ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) F B ; rµ,n from appendix C of Ref. [6]. The coefficients A F B , B F B abbreviate the relations (3.57),(3.58) for theeigenvalues | c n | (2.19), | f r | (2.20); ( κ, µ, ν = 1 , r, r ′ = 1 , . . . , S/ m, n = 1 , . . . , L ) − (cid:0) ∂ ˆ ζ rµ,n (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F B ; rµ,n = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) , + BF ; n,rµ (3.55)= − A F B (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ D ; rκ,n (cid:1) − B F B e ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ ∗ D ; rκ,n (cid:1) ; − (cid:0) ∂ ˆ ζ ∗ rµ,n (cid:1) = − A F B (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ ∗ D ; rκ,n (cid:1) + B F B e − ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ D ; rκ,n (cid:1) ; (3.56) A F B = | c n | cosh (cid:0) | f r | (cid:1) sin (cid:0) | c n | (cid:1) + | f r | cos (cid:0) | c n | (cid:1) sinh (cid:0) | f r | (cid:1) | c n | + | f r | ; (3.57) B F B = | c n | cos (cid:0) | c n | (cid:1) sinh (cid:0) | f r | (cid:1) − | f r | cosh (cid:0) | f r | (cid:1) sin (cid:0) | c n | (cid:1) | c n | + | f r | . (3.58)The diagonalized, 4 × G / / U , ˆ G − / / U of the fermion-boson, boson-fermion sectionsare described in relations (3.59,3.60) by using the coefficients A F B , B F B for abbreviating relations(3.61,3.62) (cid:18) (cid:0) ∂ ˆ ζ rµ,n (cid:1)(cid:0) ∂ ˆ ζ ∗ rµ,n (cid:1) (cid:19) = A F B (cid:0) τ (cid:1) µκ B F B e ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ − B F B e − ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ A F B (cid:0) τ (cid:1) µκ ! (cid:18) (cid:0) ∂ ˆ η ′ D ; rκ,n (cid:1)(cid:0) ∂ ˆ η ′ ∗ D ; rκ,n (cid:1) (cid:19) ; (3.59) (cid:18) (cid:0) ∂ ˆ η ′ D ; rκ,n (cid:1)(cid:0) ∂ ˆ η ′ ∗ D ; rκ,n (cid:1) (cid:19) = e A F B (cid:0) τ (cid:1) µκ − e B F B e ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ e B F B e − ı ( φ r + ϕ n ) (cid:0) τ (cid:1) µκ e A F B (cid:0) τ (cid:1) µκ ! (cid:18) (cid:0) ∂ ˆ ζ rκ,n (cid:1)(cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) (cid:19) ; (3.60) e A F B = A F B A F B + B F B = 12 (cid:18) | f r | − ı | c n | sinh (cid:0) | f r | − ı | c n | (cid:1) + | f r | + ı | c n | sinh (cid:0) | f r | + ı | c n | (cid:1) (cid:19) ; (3.61) e B F B = B F B A F B + B F B = ı (cid:18) | f r | − ı | c n | sinh (cid:0) | f r | − ı | c n | (cid:1) − | f r | + ı | c n | sinh (cid:0) | f r | + ı | c n | (cid:1) (cid:19) . (3.62)The integration measure (3.63) follows from the ’inverse’ of the determinant of transformation (3.60)where the eigenvalue | c n | of the boson-boson part fits into the hyperbolic sinh-function with the eigen-value | f r | of the fermion-fermion section by using an imaginary factor. In consequence, the original, odd,anomalous coset fields d ˆ η ′ ∗ D ; r ,n , d ˆ η ′ D ; r ,n , d ˆ η ′ ∗ D ; r ,n , d ˆ η ′ D ; r ,n (2.2-2.5) are substituted by the odd Euclideanfields d ˆ ζ ∗ r ,n , d ˆ ζ r ,n , d ˆ ζ ∗ r ,n , d ˆ ζ r ,n in combination of the coset integration measure (2.48) d ˆ η ′ ∗ D ; r ,n d ˆ η ′ D ; r ,n d ˆ η ′ ∗ D ; r ,n d ˆ η ′ D ; r ,n = (3.63) .3 Eigenvalues of cosets for anomalous terms and their transformed, Euclidean correspondents ( d ˆ ζ ∗ r ,n d ˆ ζ r ,n (cid:18) sinh (cid:0) | f r | + ı | c n | (cid:1) | f r | + ı | c n | (cid:19) (cid:18) sinh (cid:0) | f r | − ı | c n | (cid:1) | f r | − ı | c n | (cid:19)) ×× ( d ˆ ζ ∗ r ,n d ˆ ζ r ,n (cid:18) sinh (cid:0) | f r | + ı | c n | (cid:1) | f r | + ı | c n | (cid:19) (cid:18) sinh (cid:0) | f r | − ı | c n | (cid:1) | f r | − ı | c n | (cid:19)) .According to Ref. [6] with appendix C, we list the odd density part (3.64) for the fermion-boson,boson-fermion sections by introducing the coefficients C F B (3.65), D F B (3.66) as abbreviation − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F B ; rµ,n = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) ,TBF ; n,rµ = ı (cid:0) ∂ ˆ ξ rµ,n (cid:1) = (3.64)= − e ı φ r C F B (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ ∗ D ; rκ,n (cid:1) + e − ı ϕ n D F B (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ η ′ D ; rκ,n (cid:1) ; C F B = | f r | − | f r | cos (cid:0) | c n | (cid:1) cosh (cid:0) | f r | (cid:1) − | c n | sin (cid:0) | c n | (cid:1) sinh (cid:0) | f r | (cid:1) | c n | + | f r | ; (3.65) D F B = | c n | − | c n | cos (cid:0) | c n | (cid:1) cosh (cid:0) | f r | (cid:1) + | f r | sin (cid:0) | c n | (cid:1) sinh (cid:0) | f r | (cid:1) | c n | + | f r | . (3.66)We apply (3.60) with the A F B , B F B coefficients (3.57,3.58), together with (3.64) and coefficients C F B , D F B (3.65,3.66), and finally obtain relation (3.67) for the odd density parts. One achieves a dependenceon the odd, anomalous Euclidean fields ( ∂ ˆ ζ ∗ r ,n ), ( ∂ ˆ ζ r ,n ), ( ∂ ˆ ζ ∗ r ,n ), ( ∂ ˆ ζ r ,n ) for the odd, fermion-boson,boson-fermion density parts (3.64) in combination of the eigenvalues | f r | , | c n | (2.19,2.20) appearingwith tanh( ( | f r | ± ı | c n | ) / − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) F B ; rµ,n = ı (cid:0) ∂ ˆ ξ rµ,n (cid:1) = (3.67)= e − ı ϕ n sin (cid:0) | c n | (cid:1) (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ rκ,n (cid:1) + e ı φ r sinh (cid:0) | f r | (cid:1) (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) cosh (cid:0) | f r | (cid:1) + cos (cid:0) | c n | (cid:1) = 12 tanh (cid:18) | f r | + ı | c n | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) − ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ rκ,n (cid:1)i ++ 12 tanh (cid:18) | f r | − ı | c n | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) + ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ rκ,n (cid:1)i . In section 3.2 with subsections 3.2.1, 3.2.2, 3.2.3, we have used the results of appendix C in Ref. [6]for relations (3.9-3.11) in order to transform the original, coset fields in ( ∂ ˆ X N × N ), e κ ( ∂ ˆ X + N × N ) (2.2-2.5) and in ( ˆ P ˆ T − ( ∂ ˆ T ) ˆ P − ) a = bαβ to Euclidean integration variables ( ∂ ˆ Z a = bαβ ) (3.1) depending on ( ∂ ˆ X αβ ), e κ ( ∂ ˆ X + αβ ) (3.2-3.4). (In this section we have to specialize on the total derivative ’ d ’ for the pair condensatepath field variables in place of the general symbolic derivative ’ ∂ ’ of section 3.2. The general symbolicderivative ’ ∂ ’ has been applied as abbreviation for partial, spatial or time-contour-like gradients ’ ˘ ∂ i ’, ’ ˘ ∂ ˘ t p ’2 and ’ δ ’-variations of fields for classical equations or total derivatives ’ d ’ for the independent path fieldsof the integration measure.) However, apart from the dependence on Euclidean integration variables( d ˆ b mn ), ( d ˆ a rµ,r ′ ν ), ( d ˆ ζ rµ,n ), (+c.c.) (3.2-3.4), there also appear the eigenvalues c m (2.19), f r (2.20) ofthe original coset decomposition for anomalous fields. Their dependence has to be determined in termsof the new, independent Euclidean pair condensate fields ( d ˆ b mn ), ( d ˆ a rµ,r ′ ν ), ( d ˆ ζ rµ,n ), (+c.c.). Accordingto section 3.2, we therefore list again relations (3.68-3.75) which have all been calculated in terms ofthe anomalous Euclidean fields ( d ˆ X αβ ) and their super-hermitian conjugate e κ ( d ˆ X + αβ ) (cid:0) d ˆ Z abαβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) abαβ = (cid:0) d ˆ Y αβ (cid:1) (cid:0) d ˆ X αβ (cid:1)e κ (cid:0) d ˆ X + αβ (cid:1) (cid:0) d ˆ Y αβ (cid:1) ! ; (3.68) (cid:0) d ˆ X αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) αβ = − (cid:0) d ˆ b mn (cid:1) (cid:0) d ˆ ζ Tm,r ′ ν (cid:1) − (cid:0) d ˆ ζ rµ,n (cid:1) (cid:0) d ˆ a rµ,r ′ ν (cid:1) ! αβ ; (3.69) e κ (cid:0) d ˆ X + αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) αβ = (cid:0) d ˆ b ∗ mn (cid:1) (cid:0) d ˆ ζ + m,r ′ ν (cid:1)(cid:0) d ˆ ζ ∗ rµ,n (cid:1) (cid:0) d ˆ a + rµ,r ′ ν (cid:1) ! αβ ; (3.70) (cid:0) d ˆ b mn (cid:1) = (cid:0) d ˆ b Tmn (cid:1) ; (cid:0) d ˆ a rµ,r ′ ν (cid:1) = − (cid:0) d ˆ a Trµ,r ′ ν (cid:1) ; (3.71) (cid:0) d ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) αβ = ı (cid:0) d ˆ d mn (cid:1) (cid:0) d ˆ ξ + m,r ′ ν (cid:1)(cid:0) d ˆ ξ rµ,n (cid:1) (cid:0) d ˆ g rµ,r ′ ν (cid:1) ! αβ ; (3.72) (cid:0) d ˆ Y αβ (cid:1) + = − (cid:0) d ˆ Y αβ (cid:1) ; (cid:0) d ˆ d + mn (cid:1) = (cid:0) d ˆ d mn (cid:1) ; (cid:0) d ˆ g + rµ,r ′ ν (cid:1) = (cid:0) d ˆ g rµ,r ′ ν (cid:1) ; (3.73) (cid:0) d ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) αβ = − ı (cid:0) d ˆ d Tmn (cid:1) − (cid:0) d ˆ ξ Tm,r ′ ν (cid:1)(cid:0) d ˆ ξ ∗ rµ,n (cid:1) (cid:0) d ˆ g Trµ,r ′ ν (cid:1) ! αβ ; (3.74) (cid:0) d ˆ Y αβ (cid:1) st = − (cid:0) d ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) ,stαβ = (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) αβ . (3.75)Since the coset eigenvalues c m (2.19), f r (2.20) only take part in the transformed actions of Euclideanpath integration variables with their total values of the moduli | c m | , | f r | and phases ϕ m , φ r , but without any derivatives ’ ∂ ’ , we have to relate the coset eigenvalues c m , f r with the total derivative’ d ’ to the Euclidean pair condensate variables. The causal structure of the original time developmentoperators, which result with over-complete sets of states at every time step into coherent state pathintegrals as an underlying lattice theory, also leads to a natural time ordering in our case (2.84-2.86).The causal structure of (2.84-2.86) is determined by the time contour, due to the two branches offorward and backward propagation. At each slice of the time contour propagation along the coherentstate path generating function, we choose independent, Euclidean pair condensate path fields where theindependence of the Euclidean, anomalous path fields at every time slice refers to the spatial distributionand internal indices of the super-matrices. After having chosen the set of independent spatial fields atevery time slice, the exponential phases with the actions assign a weight to the particular chosen setsof Euclidean fields according to the quadratic couplings of the kinetic energies and composed densitiesfrom anomalous variables. Therefore, the total derivative ’ d ’ in (3.68-3.75), which relates the coseteigenvalues c m , f r to the total values of anomalous, Euclidean fields, only contains the partial time .3 Eigenvalues of cosets for anomalous terms and their transformed, Euclidean correspondents (cid:0) d | c m ( t p ) | (cid:1) = (cid:0) ∂ | c m ( t p ) | (cid:1) ∂t p dt p ; (cid:0) dϕ m ( t p ) (cid:1) = (cid:0) ∂ϕ m ( t p ) (cid:1) ∂t p dt p ; (cid:0) d | f r ( t p ) | (cid:1) = (cid:0) ∂ | f r ( t p ) | (cid:1) ∂t p dt p ; (cid:0) dφ r ( t p ) (cid:1) = (cid:0) ∂φ r ( t p ) (cid:1) ∂t p dt p . (3.76)The spatial vector ~x is omitted in relations (3.76) because the Euclidean integration variables of thegenerating function (2.84-2.86) are determined by time contour paths with spatially independent pointswhich could be abbreviated by an additional index ’ ~x ’ apart from the indices m = 1 , . . . , L or r =1 , . . . , S/ X αβ and e κ ˆ X + αβ . In fact, it turns out that the differential, absolute values( d | c m ( t p ) | ), ( d | f r ( t p ) | ) can be integrated to their total values | c m ( t p ) | , | f r ( t p ) | , according to the propertyof a total derivative for a state variable, whereas the phase values ϕ m ( t p ), φ r ( t p ) involve the detailedtime contour path or the past time contour history of the Euclidean, pair condensate path variables inorder to perform the time contour integrals of the phases in (3.76).We consider again the relations (3.9-3.11) for the variation of exponentials of matrices and obtain Eq.(3.77). However, the suitable choice of gauge (3.78) for diagonal elements of ( d ˆ P ) ˆ P − , described in Eqs.(2.26-2.35), gives rise to a vanishing of the diagonal commutator matrix elements [ ˆ Y DD , ( d ˆ P ) ˆ P − ] ab − ,αα between ˆ Y DD (3.79) (the diagonal, original coset generator with eigenvalues c m , f r ) and the diagonal(quaternion diagonal), vanishing elements of ( d ˆ P ) ˆ P − (3.78). In consequence, eigenvalues c m (quater-nion eigenvalues ( τ ) µν f r ) of ˆ Y DD (3.79) are mapped onto the diagonal (quaternion diagonal) anoma-lous matrix elements in ( d ˆ X BB ; mm ) = − ( d ˆ b mm ), e κ ( d ˆ X + BB ; mm ) = ( d ˆ b ∗ mm ) and ( d ˆ X F F ; rµ,rν ) = ( d ˆ a rµ,rν ), e κ ( d ˆ X + F F ; rµ,rν ) = ( d ˆ a + rµ,rν ) (3.80-3.85). The latter fields are also taken as the independent variables forthe various diagonal elements of the densities (3.86-3.89) and block parts in ( d ˆ Y BB ; mm ) = ı ( d ˆ d mm ),( d ˆ Y F F ; rµ,rν ) = ı ( d ˆ g (0) rr ) δ µν and correspondingly in ( d ˆ Y N × N ); (( d ˆ g rµ,r ′ ν ) = P k =0 (cid:0) τ k (cid:1) µν ( d ˆ g ( k ) rr ′ )) − (cid:0) d ˆ Z abαβ (cid:1) = (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) abαβ = − Z dv e v ˆ Y DD ˆ P (cid:16) d ˆ P − ˆ Y DD ˆ P (cid:17) ˆ P − e − v ˆ Y DD (3.77)= − Z dv e v ˆ Y DD (cid:18)(cid:0) d ˆ Y DD (cid:1) + h ˆ Y DD , (cid:0) d ˆ P (cid:1) ˆ P − i − (cid:19) e − v ˆ Y DD (cid:16)(cid:0) d ˆ P (cid:1) ˆ P − (cid:17) BB ; mm ≡ (cid:16)(cid:0) d ˆ P (cid:1) ˆ P − (cid:17) F F ; rµ,rν ≡ (cid:18)h ˆ Y DD , (cid:0) d ˆ P ) ˆ P − i − (cid:19) abαα ≡ (cid:0) d ˆ Y DD (cid:1) = (cid:18) (cid:0) d ˆ X DD (cid:1)e κ (cid:0) d ˆ X + DD (cid:1) (cid:19) ; ( d ˆ X DD ) = (cid:18) − (cid:0) dc m (cid:1) δ m,n
00 ( τ ) µν (cid:0) df r (cid:1) δ r,r ′ (cid:19) ; (3.79) (cid:0) d ˆ Z abBB ; mm (cid:1) = Z dv (cid:16) e v ˆ Y DD (cid:0) d ˆ Y DD (cid:1) e − v ˆ Y DD (cid:17) abBB ; mm (3.80)4 (cid:0) d ˆ Z abF F ; rµ,rν (cid:1) = Z dv (cid:16) e v ˆ Y DD (cid:0) d ˆ Y DD (cid:1) e − v ˆ Y DD (cid:17) abF F ; rµ,rν (3.81) (cid:0) d ˆ Z αβ (cid:1) = − (cid:0) d ˆ b mm (cid:1) δ mn
00 ( τ ) µν (cid:0) d ˆ a (2) rr (cid:1) δ rr ′ ! αβ (3.82) (cid:0) d ˆ Z αβ (cid:1) = (cid:0) d ˆ b ∗ mm (cid:1) δ mn
00 ( τ ) µν (cid:0) d ˆ a (2) ∗ rr (cid:1) δ rr ′ ! αβ (3.83) | c m ( t p ) | − | c m ( −∞ + ) | = Z t p −∞ + dt ′ q (cid:0) ∂ | c m ( t ′ q ) | (cid:1) ∂t ′ q ; ϕ m ( t p ) − ϕ m ( −∞ + ) = Z t p −∞ + dt ′ q (cid:0) ∂ϕ m ( t ′ q ) (cid:1) ∂t ′ q ; (3.84) | f r ( t p ) | − | f r ( −∞ + ) | = Z t p −∞ + dt ′ q (cid:0) ∂ | f r ( t ′ q ) | (cid:1) ∂t ′ q ; φ r ( t p ) − φ r ( −∞ + ) = Z t p −∞ + dt ′ q (cid:0) ∂φ r ( t ′ q ) (cid:1) ∂t ′ q ; (3.85) (cid:0) d ˆ Z αβ (cid:1) = ı (cid:0) d ˆ d mm (cid:1) δ mn δ µν δ rr ′ (cid:0) d ˆ g (0) rr (cid:1) ! αβ ; (3.86) (cid:0) d ˆ Z αβ (cid:1) = − ı (cid:0) d ˆ d mm (cid:1) δ mn δ µν δ rr ′ (cid:0) d ˆ g (0) rr (cid:1) ! αβ ; (3.87)ˆ d mm ( t p ) − ˆ d mm ( −∞ + ) = Z t p −∞ + dt ′ q (cid:0) ∂ | ˆ d mm ( t ′ q ) | (cid:1) ∂t ′ q ; ( m = 1 , . . . , L = 2 l + 1) ; (3.88)ˆ g rµ,rν ( t p ) − ˆ g rµ,rν ( −∞ + ) = δ µν Z t p −∞ + dt ′ q (cid:0) ∂ | ˆ g (0) rr ( t ′ q ) | (cid:1) ∂t ′ q ; ( r = 1 , . . . , S/ s + 1 / . (3.89)According to the relations in section 3.2 and appendix C of Ref. [6], we can apply the known transfor-mations (3.80-3.83) between eigenvalues c m , f r and the new, independent Euclidean elements ( d ˆ b mm ),( d ˆ a (2) rr ) (3.82,3.83) and as well the dependent, diagonal density terms ( d ˆ d mm ), ( d ˆ g (0) rr ′ ) in order to deter-mine the functions (3.84,3.85,3.88,3.89) with following relations (3.90-3.93) (cid:0) d ˆ b mm (cid:1) = d (cid:0) | ˆ b mm | e ı β m (cid:1) = (cid:0) dc ∗ m (cid:1) e ı ϕ m (cid:18) − sin (cid:0) | c m | (cid:1) | c m | (cid:19) + (cid:0) dc m (cid:1) (cid:18)
12 + sin (cid:0) | c m | (cid:1) | c m | (cid:19) ; (3.90) (cid:0) d ˆ a (2) rr (cid:1) = d (cid:0) | ˆ a (2) rr | e ı α r (cid:1) = (cid:0) df ∗ r (cid:1) e ı φ r (cid:18) − sinh (cid:0) | f r | (cid:1) | f r | (cid:19) + (cid:0) df r (cid:1) (cid:18)
12 + sinh (cid:0) | f r | (cid:1) | f r | (cid:19) ; (3.91) ı (cid:0) d ˆ d mm (cid:1) = h(cid:0) dc m (cid:1) e − ı ϕ m − (cid:0) dc ∗ m (cid:1) e ı ϕ m i (cid:16) sin (cid:0) | c m | (cid:1)(cid:17) | c m | ; ˆ d mm ∈ R ; (3.92) ı (cid:0) d ˆ g (0) rr (cid:1) = − h(cid:0) df r (cid:1) e − ı φ r − (cid:0) df ∗ r (cid:1) e ı φ r i (cid:16) sinh (cid:0) | f r | (cid:1)(cid:17) | f r | ; ˆ g (0) rr ∈ R . (3.93)The separation into real and imaginary parts of Eqs. (3.90-3.93) guides us to the relations (3.94-3.97)where one can also observe the additional negative sign of the fermion-fermion density element ( d ˆ g (0) rr ) .3 Eigenvalues of cosets for anomalous terms and their transformed, Euclidean correspondents d ˆ d mm ). This additional negative sign is caused by theU( L | S ) super-symmetry of the original super-symmetric density ψ ∗ ~x,m ( t p ) ψ ~x,m ( t p ) + ψ ∗ ~x,rµ ( t p ) ψ ~x,rµ ( t p )which corresponds to the difference of boson-boson and fermion-fermion densities (cid:0) d ˆ b mm (cid:1) = d (cid:0) | ˆ b mm | e ı β m (cid:1) = e ı ϕ m h(cid:0) d | c m | (cid:1) + ı sin (cid:0) | c m | (cid:1) (cid:0) dϕ m (cid:1)i ; (3.94) (cid:0) d ˆ a (2) rr (cid:1) = d (cid:0) | ˆ a (2) rr | e ı α r (cid:1) = e ı φ r h(cid:0) d | f r | (cid:1) + ı sinh (cid:0) | f r | (cid:1) (cid:0) dφ r (cid:1)i ; (3.95) (cid:0) d ˆ d mm (cid:1) = (cid:16) sin (cid:0) | c m | (cid:1)(cid:17) (cid:0) dϕ m (cid:1) ; (3.96) (cid:0) d ˆ g (0) rr (cid:1) = − (cid:16) sinh (cid:0) | f r || (cid:1)(cid:17) (cid:0) dφ r (cid:1) . (3.97)We introduce new pair condensate integration variables e b mm = | e b mm | e ı e β m (3.98), e a (2) rr = | e a (2) rr | e ı e α r (3.99) and perform integration measure preserving phase rotations with e − ı ϕ m , e − ı e β m and e − ı φ r , e − ı e α r ,respectively e b mm = | e b mm | e ı e β m ; (3.98) e − ı e β m (cid:0) d e b mm (cid:1) = e − ı ϕ m (cid:0) d ˆ b mm (cid:1) = ⇒ (cid:0) d e b ∗ mm (cid:1) ∧ (cid:0) d e b mm (cid:1) = (cid:0) d ˆ b ∗ mm (cid:1) ∧ (cid:0) d ˆ b mm (cid:1) ; e a (2) rr = | e a (2) rr | e ı e α r ; (3.99) e − ı e α r (cid:0) d e a (2) rr (cid:1) = e − ı φ r (cid:0) d ˆ a (2) rr (cid:1) = ⇒ (cid:0) d e a (2) ∗ rr (cid:1) ∧ (cid:0) d e a (2) rr (cid:1) = (cid:0) d ˆ a (2) ∗ rr (cid:1) ∧ (cid:0) d ˆ a (2) rr (cid:1) . Accordingly, we can replace the diagonal, Euclidean, pair condensate integration variables of Eqs.(3.90-3.93) by ( d e b mm ), ( d e a (2) rr ), (+c.c.) and obtain new relations between the coset eigenvalues | c m | , ϕ m , | f r | , φ r and the rotated Euclidean, anomalous elements e b mm = | e b mm | e ı e β m , e a (2) rr = | e a (2) rr | e ı e α r of Eqs.(3.98,3.99) (cid:0) d | e b mm | (cid:1) + ı | e b mm | (cid:0) d e β m (cid:1) = (cid:0) d | c m | (cid:1) + ı sin (cid:0) | c m | (cid:1) (cid:0) dϕ m (cid:1) ; (3.100) (cid:0) d | e a (2) rr | (cid:1) + ı | e a (2) rr | (cid:0) d e α r (cid:1) = (cid:0) d | f r | (cid:1) + ı sinh (cid:0) | f r | (cid:1) (cid:0) dφ r (cid:1) . (3.101)In consequence of measure preserving phase rotations, the total derivatives ( d | e b mm | ), ( d | c m | ) and( d | e a (2) rr | ), ( d | f r | ) result between the absolute values of Euclidean, diagonal variables and the coset eigen-values so that the absolute values of these transformations are related to path-independent ’state vari-ables’ of thermodynamics in a ’transferred sense’. (One can even substitute the contour time ’ t p ’ by theinverse temperature ’ τ ’ and the contour integrals by the inverse temperature path ’0 . . . β = 1 / ( KT )’of grand canonical statistical operators. In this case, thermodynamical state variables of the absolutevalues | e b mm ( τ ) | , | c m ( τ ) | and | e a (2) rr ( τ ) | , | e f r ( τ ) | can be identified after similar HST’s and coset decomposi-tions of coherent state representations of grand canonical ’inverse temperature’ development operatorsso that the analogy becomes exact.) | c m (˘ ~x, ˘ t p ) | − | c m (˘ ~x, − ˘ ∞ + ) | | {z } =0 = | e b mm (˘ ~x, ˘ t p ) | − | e b mm (˘ ~x, − ˘ ∞ + ) | | {z } =0 ; (3.102)6 | f r (˘ ~x, ˘ t p ) | − | f r (˘ ~x, − ˘ ∞ + ) | | {z } =0 = | e a (2) rr (˘ ~x, ˘ t p ) | − | e a (2) rr (˘ ~x, − ˘ ∞ + ) | | {z } =0 . (3.103)The phases ϕ m , φ r (3.100,3.101) of the complex coset eigenvalues c m , f r are path-dependent with respectto the contour time t p because they are not determined by total derivatives and therefore correspondto a kind of ’heat’- or ’work’-variables of thermodynamics, also in a transferred sense ϕ m (˘ ~x, ˘ t p ) − ϕ m (˘ ~x, − ˘ ∞ + ) | {z } =0 = Z ˘ t p − ˘ ∞ + d ˘ t ′ q ∂ϕ m (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | e b mm (˘ ~x, ˘ t ′ q ) | sin (cid:0) | e b mm (˘ ~x, ˘ t ′ q ) | (cid:1) ∂ e β m (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q ; (3.104) φ r (˘ ~x, ˘ t p ) − φ r (˘ ~x, − ˘ ∞ + ) | {z } =0 = Z ˘ t p − ˘ ∞ + d ˘ t ′ q ∂φ r (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | e a (2) rr (˘ ~x, ˘ t ′ q ) | sinh (cid:0) | e a (2) rr (˘ ~x, ˘ t ′ q ) | (cid:1) ∂ e α r (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q . (3.105)The diagonal boson-boson and fermion-fermion densities are also path-dependent, due to the phases( dϕ m ) and ( dφ r ), and are specified by following Eqs. (3.106,3.107), after substitution of (3.100,3.101)into (3.96,3.97)ˆ d mm (˘ ~x, ˘ t p ) − ˆ d mm (˘ ~x, − ˘ ∞ + ) | {z } =0 = Z ˘ t p − ˘ ∞ + d ˘ t ′ q tan (cid:0) | e b mm (˘ ~x, ˘ t ′ q ) | (cid:1) | e b mm (˘ ~x, ˘ t ′ q ) | ∂ e β m (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q ; (3.106)ˆ g (0) rr (˘ ~x, ˘ t p ) − ˆ g (0) rr (˘ ~x, − ˘ ∞ + ) | {z } =0 = − Z ˘ t p − ˘ ∞ + d ˘ t ′ q tanh (cid:0) | e a (2) rr (˘ ~x, ˘ t ′ q ) | (cid:1) | e a (2) rr (˘ ~x, ˘ t ′ q ) | ∂ e α r (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q . (3.107)In summary, we have defined new integration variables e b mm (˘ ~x, ˘ t p ), e a (2) rr (˘ ~x, ˘ t p ) for the diagonal matrixelements of Eqs. (3.82,3.83) and (3.86,3.87) so that, according to the contour time ordering, we choosepartial derivatives of phases ∂ e β m /∂t ′ q , ∂ e α/∂t ′ q which determine the path-dependent phases ϕ m ( ~x, t p ), φ r ( ~x, t p ) (3.104,3.105) and also the path-dependent diagonal density elements ˆ d mm ( ~x, t p ), ˆ g (0) rr ( ~x, t p )(3.106,3.107). The absolute values of the coset eigenvalues | c m | , | f r | transform in a path-independentmanner as corresponding ’state variables’ and are equivalent to the absolute values | e b mm | , | e a (2) rr | of thenew, phase-rotated, Euclidean integration variables (3.98-3.101).It remains to determine the block diagonal ˆ P , ˆ P − super-matrix of Eqs. (3.77,3.108) in terms ofthe anomalous parts ( d ˆ Z a = bαβ ). Since we have accomplished definite relations between coset eigenvaluesof ˆ Y DD and the diagonal elements of pair condensates (ˆ Z a = bαβ ) (also comprising the diagonal parts ofdensities), we can apply relations (3.77,3.78) or (3.108,3.109) in order to calculate ( d ˆ P ) ˆ P − in terms of( d ˆ Z abαβ ) − (cid:0) d ˆ Z abαβ (cid:1) = (cid:16) ˆ P ˆ T − (cid:0) d ˆ T (cid:1) ˆ P − (cid:17) abαβ = − Z dv e v ˆ Y DD ˆ P (cid:16) d ˆ P − ˆ Y DD ˆ P (cid:17) ˆ P − e − v ˆ Y DD (3.108)= − Z dv e v ˆ Y DD (cid:18)(cid:0) d ˆ Y DD (cid:1) + h ˆ Y DD , (cid:0) d ˆ P (cid:1) ˆ P − i − (cid:19) e − v ˆ Y DD (cid:16)(cid:0) d ˆ P (cid:1) ˆ P − (cid:17) BB ; mm ≡ (cid:16)(cid:0) d ˆ P (cid:1) ˆ P − (cid:17) F F ; rµ,rν ≡ (cid:18)h ˆ Y DD , (cid:0) d ˆ P ) ˆ P − i − (cid:19) abαα ≡ . (3.109)This can be achieved by a separation of the block diagonal N × N matrices ˆ P , ˆ P − into subsequentmultiplications of matrices where each matrix factor only contains a generator for a single parameter or(single quaternion parameter for the fermion-fermion parts). As consequence, one has to treat only 2 × × N × N laddergenerators within the block diagonal N × N super-matrices ˆ P , ˆ P − . After the factorization of ( d ˆ P ) ˆ P − into single group parts with generators comprising only single parameters, we use again (3.108) in orderto integrate over v ∈ [0 ,
1) within exp {± v ˆ Y DD } and the commutator [ ˆ Y DD , ( d ˆ P ) ˆ P − ] with the coseteigenvalues. This is a straightforward procedure, but tedious task for general N × N super-matrices;we have also to point out that the resulting relation between ( d ˆ P ) ˆ P − and ( d ˆ Z abαβ ) strongly depends onthe details of the parametrization, as e. g. the chosen sequence of factors with generators having onlya single parameter. In consequence to the previous section 3.3, we rename the diagonal, Euclidean integration variables e b mm , e a rµ,rν = ( τ ) µν e a (2) rr (3.98,3.99), which preserve the Euclidean integration measure of anomalousfields, to their original symbols ˆ b mm = e ı β m | ˆ b mm | and ˆ a rµ,rν = ( τ ) µν ˆ a (2) rr = ( τ ) µν e ı α r | ˆ a (2) rr | . The total,Euclidean integration measure therefore consists of the time contour path fields d ˆ b mn , d ˆ a rµ,r ′ ν , dζ rµ,n ,(+c.c.) or of the terms of the pair condensate matrices ( d ˆ X αβ ) = ( d ˆ Z αβ ), e κ ( d ˆ X + αβ ) = ( d ˆ Z αβ ) (compareEqs. (3.1-3.4)) d (cid:2) (ˆ Z αβ ) , ( d ˆ Z αβ ) (cid:3) = d (cid:2) ( d ˆ X αβ ) , e κ ( d ˆ X + αβ ) (cid:3) = Y { ˘ ~x, ˘ t p } "(cid:26) L Y m =1 L Y n = m (cid:0) d ˆ b ∗ mn (cid:1) ∧ (cid:0) d ˆ b mn (cid:1) ı (cid:27) × (4.1) × (cid:26) S/ Y r =1 (cid:0) d ˆ a (2) ∗ rr (cid:1) ∧ (cid:0) d ˆ a (2) rr (cid:1) ı (cid:27) × (cid:26) S/ Y r =1 S/ Y r ′ = r +1 3 Y k =0 (cid:0) d ˆ a ( k ) ∗ rr ′ (cid:1) ∧ (cid:0) d ˆ a ( k ) rr ′ (cid:1) ı (cid:27) ×× (cid:26) S/ Y r =1 Y µ =1 , L Y n =1 (cid:0) d ˆ ζ ∗ rµ,n (cid:1) (cid:0) d ˆ ζ rµ,n (cid:1)(cid:27) . The coherent state path integral of Eqs. (2.84,2.85) thus takes the form (4.2) where we have transformedthe coset integration measure d [ ˆ T − (˘ ~x, ˘ t p ) ( d ˆ T (˘ ~x, ˘ t p ))] (2.48) in sections 3.2.1-3.2.3 to the Euclideancorrespondents of integration variables (4.1) for the anomalous pair condensate fields Z (cid:2) ˆ J , ˘ J ψ , ı ˘ J ψψ (cid:3) = Z d (cid:2) (ˆ Z αβ ) , ( d ˆ Z αβ ) (cid:3) exp n ı A ˘ J ψψ (cid:2) ˆ T (cid:3)o × exp n − A ′ (cid:2) ˆ T ; ˆ J (cid:3)o × (4.2)8 × exp n − A ( d ) (cid:2) ˆ Z ; ˘ J ψ (cid:3)o . The action A ˘ J ψψ [ ˆ T ] in (4.2) generates the anomalous fields; however, we neglect the detailed processof generation of pair condensates, which depends on temperature, the trap potential and further spe-cial properties in the experiments, and assume initial conditions for the Euclidean, anomalous fieldsˆ Z αβ , ˆ Z αβ . The second action A ′ [ ˆ T ; ˆ J ] in (4.2) determines the observables with the original source fieldˆ J ba~x ′ ,β ; ~x,α ( t ′ q , t p ) which relates observables, obtained by differentiation, to the original coherent state pathintegral (1.36) of super-fields ψ ~x,α ( t p ), ψ ∗ ~x,α ( t p ). Hence, one can track the original observables (1.36)composed of the super-fields ψ ~x,α ( t p ), ψ ∗ ~x,α ( t p ) to the transformed generating function (4.2) with theEuclidean, pair condensate integration variables whose action A ( d ) [ˆ Z ; ˘ J ψ ] contains the coupling coeffi-cients ˘ c ij (˘ ~x, ˘ t p ) (2.55-2.58) and ˘ d ij (˘ ~x, ˘ t p ) (2.59) of the background density field. In terms of the new,Euclidean, pair condensate fields, the action A ( d ) [ˆ Z ; ˘ J ψ ] (2.85,2.86) is altered to relations (4.3,4.4) whichallow variations for classical field equations, avoiding inconsistencies of nontrivial integration measures A ( d ) [ˆ Z ; ˘ J ψ ] = Z C d ˘ t p Z d d ˘ x (cid:18) x L (cid:19) d L ( d ) [ˆ Z ; ˘ J ψ ] ; (4.3) L ( d ) [ˆ Z ; ˘ J ψ ] = − (cid:16) ˘ c ij + 12 δ ij (cid:17) ( a = b ) X a,b =1 , str α,β h(cid:0) ˘ ∂ i ˆ Z a = bαβ (cid:1)(cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1)i − X a =1 , str α,β h(cid:0) ˘ ∂ i ˆ Z aaαβ (cid:1)(cid:0) ˘ ∂ i ˆ Z aaβα (cid:1)i + (4.4) − (cid:16) ˘ u (˘ ~x ) − ˘ µ − ı ˘ ε p + (cid:10) ˘ σ (0) D (˘ ~x, ˘ t p ) (cid:11) ˆ σ (0) D (cid:17)(cid:18) L X m =1 h cos (cid:0) | ˆ b mm | (cid:1) − i − S/ X r =1 h cosh (cid:0) | ˆ a (2) rr | (cid:1) − i(cid:19) + − ı a,α ; b,β h exp (cid:8) Y DD (cid:9) ˆ S (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1)i − ı (cid:16) ˘ d ij − δ ij (cid:17) ˘ J + ψ ˆ I e K ˆ P − (cid:0) ˘ ∂ i ˆ Z (cid:1) (cid:0) ˘ ∂ j ˆ Z (cid:1) ˆ P ˆ I ˘ J ψ + −
12 ˘ J + ψ ˆ I e K ˆ P − exp (cid:8) − Y DD (cid:9) (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S ˆ P ˆ I ˘ J ψ ++ η p (cid:26) L X m =1 h cos (cid:0) | ˆ b mm (˘ ~x, ˘ t p ) | (cid:1) − i − S/ X r =1 h cosh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t p ) | (cid:1) − i(cid:27) . After classification of the independent, Euclidean pair condensate integration variables ( d ˆ Z αβ ), ( d ˆ Z αβ )in (4.1), we list again the matrices ( ∂ ˆ Z abαβ ), ˆ Y DD of (4.4) with the block diagonal U( L | S ) rotation matricesˆ P aaαβ , ˆ P aa, − αβ , defined in (2.21-2.35). Furthermore, we apply the transformations of sections 3.2 and 3.3,especially for the time contour path dependent density terms and phases ϕ m , φ r of the coset eigenvalues c m , f r (cid:0) ∂ ˆ Z abαβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) abαβ = (cid:0) ∂ ˆ Y αβ (cid:1) (cid:0) ∂ ˆ X αβ (cid:1)e κ (cid:0) ∂ ˆ X + αβ (cid:1) (cid:0) ∂ ˆ Y αβ (cid:1) ! ab ; (4.5) Euclidean, pair condensate integration variables .1 Variation for classical field equations with Euclidean integration variables (cid:0) ∂ ˆ X αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = − (cid:0) ∂ ˆ b mn (cid:1) (cid:0) ∂ ˆ ζ Tm,r ′ ν (cid:1) − (cid:0) ∂ ˆ ζ rµ,n (cid:1) (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) ! αβ ; (4.6) e κ (cid:0) ∂ ˆ X + αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = (cid:0) ∂ ˆ b ∗ mn (cid:1) (cid:0) ∂ ˆ ζ + m,r ′ ν (cid:1)(cid:0) ∂ ˆ ζ ∗ rµ,n (cid:1) (cid:0) ∂ ˆ a + rµ,r ′ ν (cid:1) ! αβ ; (4.7) (cid:0) ∂ ˆ b mn (cid:1) = (cid:0) ∂ ˆ b Tmn (cid:1) ; (cid:0) ∂ ˆ a rµ,r ′ ν (cid:1) = − (cid:0) ∂ ˆ a Trµ,r ′ ν (cid:1) = X k =0 (cid:0) τ k (cid:1) µν (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) ; (4.8) density terms in dependence on the Euclidean pair condensate fields (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = ı (cid:0) ∂ ˆ d mn (cid:1) (cid:0) ∂ ˆ ξ + m,r ′ ν (cid:1)(cid:0) ∂ ˆ ξ rµ,n (cid:1) (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) ! αβ ; (4.9) (cid:0) ∂ ˆ Y αβ (cid:1) + = − (cid:0) ∂ ˆ Y αβ (cid:1) ; (cid:0) ∂ ˆ d + mn (cid:1) = (cid:0) ∂ ˆ d mn (cid:1) ; (cid:0) ∂ ˆ g + rµ,r ′ ν (cid:1) = (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) ; (4.10) (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ = − ı (cid:0) ∂ ˆ d Tmn (cid:1) − (cid:0) ∂ ˆ ξ Tm,r ′ ν (cid:1)(cid:0) ∂ ˆ ξ ∗ rµ,n (cid:1) (cid:0) ∂ ˆ g Trµ,r ′ ν (cid:1) ! αβ ; (4.11) (cid:0) ∂ ˆ Y αβ (cid:1) st = − (cid:0) ∂ ˆ Y αβ (cid:1) = − (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) ,stαβ = (cid:16) ˆ P ˆ T − (cid:0) ∂ ˆ T (cid:1) ˆ P − (cid:17) αβ . (4.12)Appropriately to sections 3.2 and 3.3, we can also relate the listed density terms (4.9-4.12) to theanomalous, Euclidean integration variables with time contour path dependent phases ϕ m (˘ ~x, ˘ t p ), φ r (˘ ~x, ˘ t p ) ı (cid:0) ∂ ˆ d mn (cid:1) = 12 tan (cid:18) | ˆ b mm | + | ˆ b nn | (cid:19) h e − ı ϕ n (cid:0) ∂ ˆ b mn (cid:1) − e ı ϕ m (cid:0) ∂ ˆ b ∗ mn (cid:1)i + (4.13) −
12 tan (cid:18) | ˆ b mm | − | ˆ b nn | (cid:19) h e − ı ϕ n (cid:0) ∂ ˆ b mn (cid:1) + e ı ϕ m (cid:0) ∂ ˆ b ∗ mn (cid:1)i ı (cid:0) ∂ ˆ g rµ,r ′ ν (cid:1) = − X k =0 (cid:0) τ k τ (cid:1) µν (cid:26) tanh (cid:18) | ˆ a (2) rr | + | ˆ a (2) r ′ r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) − (cid:0) − (cid:1) k e ı φ r (cid:0) ∂ ˆ a ( k )+ rr ′ (cid:1)i + − tanh (cid:18) | ˆ a (2) rr | − | ˆ a (2) r ′ r ′ | (cid:19) h e − ı φ r ′ (cid:0) ∂ ˆ a ( k ) rr ′ (cid:1) + (cid:0) − (cid:1) k e ı φ r (cid:0) ∂ ˆ a ( k )+ rr ′ (cid:1)i(cid:27) ; (4.14)ˆ a (2) rr = 0 ; ˆ a ( k ) rr ≡ k = 0 , , ı (cid:0) ∂ ˆ ξ rµ,n (cid:1) = 12 tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) − ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ rκ,n (cid:1)i + (4.16)+ 12 tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ ∗ rκ,n (cid:1) + ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) ∂ ˆ ζ rκ,n (cid:1)i ; phases ϕ m ( ˘ ~x, ˘ t p ), φ r ( ˘ ~x, ˘ t p ) of coset eigenvalues in ˆ Y DD , ˆ X DD : ϕ m (˘ ~x, ˘ t p ) = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | ˆ b mm (˘ ~x, ˘ t ′ q ) | sin (cid:0) | ˆ b mm (˘ ~x, ˘ t ′ q ) | (cid:1) ∂β m (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q ; (4.17)0 φ r (˘ ~x, ˘ t p ) = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | ˆ a (2) rr (˘ ~x, ˘ t ′ q ) | sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ′ q ) | (cid:1) ∂α r (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q ; (4.18)ˆ Y DD (˘ ~x, ˘ t p ) = X DD (˘ ~x, ˘ t p ) e κ ˆ X + DD (˘ ~x, ˘ t p ) 0 ! ; (4.19)ˆ X DD (˘ ~x, ˘ t p ) = −| ˆ b mm (˘ ~x, ˘ t p ) | e ı ϕ m (˘ ~x, ˘ t p ) (cid:0) τ (cid:1) µν | ˆ a (2) rr (˘ ~x, ˘ t p ) | e ı φ r (˘ ~x, ˘ t p ) ! . (4.20)According to the listed Eqs. (4.3-4.20), the Lagrangian L ( d ) [ˆ Z ; ˘ J ψ ] (4.4) is outlined for its various boson-boson, fermion-fermion and odd fermion-boson, boson-fermion parts of the super-matrices. As alreadymentioned in Ref. [6], the fermion-fermion density parts are always accompanied by a phase factor of e ± ı π = − L ( d ) [ˆ Z ; ˘ J ψ ] = 2 (cid:16) ˘ c ij + 12 δ ij (cid:17) tr h(cid:0) ˘ ∂ i ˆ b + mn (cid:1) (cid:0) ˘ ∂ j ˆ b nm (cid:1) + (cid:0) ˘ ∂ i ˆ a + rµ,r ′ ν (cid:1) (cid:0) ˘ ∂ j ˆ a r ′ ν,rµ (cid:1) + 2 (cid:0) ˘ ∂ i ˆ ζ + m,r ′ ν (cid:1) (cid:0) ˘ ∂ j ˆ ζ r ′ ν,m (cid:1)i ++ tr h(cid:0) ˘ ∂ i ˆ d mn (cid:1) (cid:0) ˘ ∂ i ˆ d nm (cid:1) − (cid:0) ˘ ∂ i ˆ g rµ,r ′ ν (cid:1) (cid:0) ˘ ∂ i ˆ g r ′ ν,rµ (cid:1) + 2 (cid:0) ˘ ∂ i ˆ ξ + m,r ′ ν (cid:1) (cid:0) ˘ ∂ ij ˆ ξ r ′ ν,m (cid:1)i + (4.21) − (cid:16) ˘ u (˘ ~x ) − ˘ µ − ı ˘ ε p + (cid:10) ˘ σ (0) D (˘ ~x, ˘ t p ) (cid:11) ˆ σ (0) D (cid:17)(cid:16) L X m =1 h cos (cid:0) | ˆ b mm | (cid:1) − i − S/ X r =1 h cosh (cid:0) | ˆ a (2) rr | (cid:1) − i(cid:17) ++ (cid:20) L X m =1 cos (cid:0) | ˆ b mm | (cid:1) (cid:0) ˘ ∂ ˘ t p ˆ d mm (cid:1) − S/ X r =1 cosh (cid:0) | ˆ a (2) rr | (cid:1) X µ =1 , (cid:0) ˘ ∂ ˘ t p ˆ g rµ,rµ (cid:1)(cid:21) + − ı (cid:26) L X m =1 sin (cid:0) | ˆ b mm | (cid:1) h e ı ϕ m (cid:0) ˘ ∂ ˘ t p ˆ b + mm (cid:1) − e − ı ϕ m (cid:0) ˘ ∂ ˘ t p ˆ b mm (cid:1)i ++ S/ X r =1 sinh (cid:0) | ˆ a (2) rr | (cid:1) h e ı φ r tr (cid:2)(cid:0) τ (cid:1) µν (cid:0) ˘ ∂ ˘ t p ˆ a + rν,rµ (cid:1)(cid:3) − e − ı φ r tr (cid:2)(cid:0) τ (cid:1) νµ (cid:0) ˘ ∂ ˘ t p ˆ a rµ,rν (cid:1)(cid:3)i(cid:27) ++ 2 ı (cid:16) ˘ d ij − δ ij (cid:17) (cid:26) ˘ j +ˆ P ψ ; B h(cid:0) ˘ ∂ i ˆ b (cid:1) (cid:0) ˘ ∂ j ˆ b + (cid:1) + (cid:0) ˘ ∂ i ˆ d (cid:1) (cid:0) ˘ ∂ j ˆ d (cid:1) − (cid:0) ˘ ∂ i ˆ ζ T (cid:1) (cid:0) ˘ ∂ j ˆ ζ ∗ (cid:1) + (cid:0) ˘ ∂ i ˆ ξ + (cid:1) (cid:0) ˘ ∂ j ˆ ξ (cid:1)i ˘ j ˆ P ψ ; B + − ˘ j +ˆ P ψ ; F h(cid:0) ˘ ∂ i ˆ a (cid:1) (cid:0) ˘ ∂ j ˆ a + (cid:1) − (cid:0) ˘ ∂ i ˆ g (cid:1) (cid:0) ˘ ∂ j ˆ g (cid:1) − (cid:0) ˘ ∂ i ˆ ζ (cid:1) (cid:0) ˘ ∂ j ˆ ζ + (cid:1) − (cid:0) ˘ ∂ i ˆ ξ (cid:1) (cid:0) ˘ ∂ j ˆ ξ + (cid:1)i ˘ j ˆ P ψ ; F ++ ˘ j +ˆ P ψ ; F h(cid:0) ˘ ∂ i ˆ ξ (cid:1) (cid:0) ˘ ∂ j ˆ d (cid:1) + (cid:0) ˘ ∂ i ˆ g (cid:1) (cid:0) ˘ ∂ j ˆ ξ (cid:1) + (cid:0) ˘ ∂ i ˆ ζ (cid:1) (cid:0) ˘ ∂ j ˆ b + (cid:1) − (cid:0) ˘ ∂ i ˆ a (cid:1) (cid:0) ˘ ∂ j ˆ ζ ∗ (cid:1)i ˘ j ˆ P ψ ; B ++ ˘ j +ˆ P ψ ; B h(cid:0) ˘ ∂ i ˆ d (cid:1) (cid:0) ˘ ∂ j ˆ ξ + (cid:1) + (cid:0) ˘ ∂ i ˆ ξ + (cid:1) (cid:0) ˘ ∂ j ˆ g (cid:1) + (cid:0) ˘ ∂ i ˆ b (cid:1) (cid:0) ˘ ∂ j ˆ ζ + (cid:1) − (cid:0) ˘ ∂ i ˆ ζ T (cid:1) (cid:0) ˘ ∂ j ˆ a + (cid:1)i ˘ j ˆ P ψ ; F (cid:27) + − ˘ j +ˆ P ψ ˘ j T ˆ P ψ e κ ı ! T (cid:0) e − Y DD (cid:1) (cid:0) e − Y DD (cid:1) (cid:0) e − Y DD (cid:1) (cid:0) e − Y DD (cid:1) ! (cid:0) ˘ ∂ ˘ t p ˆ Y (cid:1) − (cid:0) ˘ ∂ ˘ t p ˆ X (cid:1)e κ (cid:0) ˘ ∂ ˘ t p ˆ X + (cid:1) − (cid:0) ˘ ∂ ˘ t p ˆ Y (cid:1) ! ˘ j ˆ P ψ ı ˘ j ∗ ˆ P ψ ! + Although the given relation (4.21) for L ( d ) [ˆ Z ; ˘ J ψ ] obscures the underlying super-symmetries and has a complicatedappearance, we represent the Lagrangian in its expanded version with split boson-boson, fermion-fermion, odd fermion-boson and boson-fermion parts in order to verify the phase jump between the boson-boson and fermion-fermion densities,respectively. .1 Variation for classical field equations with Euclidean integration variables η p (cid:26) L X m =1 h cos (cid:0) | ˆ b mm (˘ ~x, ˘ t p ) | (cid:1) − i − S/ X r =1 h cosh (cid:0) | ˆ a (2) rr | (cid:1) − i(cid:27) ;˘ j ˆ P ψ ( ~x, t p ) = ˆ P ( ~x, t p ) ˘ j ψ ( ~x, t p ) . (4.22)It remains to perform the variation of L ( d ) [ˆ Z ; ˘ J ψ ] with respect to the independent, Euclidean fields in δ ˆ Z αβ , δ ˆ Z αβ for classical equations and quadratic fluctuations and we thus tabulate the variations of thedifferent terms occurring in (4.4) or its expanded version (4.21)ˆ Z abαβ (˘ ~x, ˘ t p = ± ) = ˆ Z abαβ (˘ ~x, ˘ t ) ± δ ˆ Z abαβ (˘ ~x, ˘ t ) (4.23) δ ˆ Z abαβ (˘ ~x, ˘ t ) = δ ˆ Z a = bαβ (˘ ~x, ˘ t ) + δ ˆ Z a = bαβ (˘ ~x, ˘ t ) (4.24) δ ˆ Z αβ (˘ ~x, ˘ t ) = − δ ˆ b mn (˘ ~x, ˘ t ) δ ˆ ζ Tm,r ′ ν (˘ ~x, ˘ t ) − δ ˆ ζ rµ,n (˘ ~x, ˘ t ) (cid:0) τ k (cid:1) µν δ ˆ a ( k ) rr ′ (˘ ~x, ˘ t ) ! (4.25) δ ˆ Z αβ (˘ ~x, ˘ t ) = e κ (cid:0) δ ˆ Z αβ (˘ ~x, ˘ t ) (cid:1) + = δ ˆ b ∗ mn (˘ ~x, ˘ t ) δ ˆ ζ + m,r ′ ν (˘ ~x, ˘ t ) δ ˆ ζ ∗ rµ,n (˘ ~x, ˘ t ) (cid:0) τ k (cid:1) µν δ ˆ a ( k )+ rr ′ (˘ ~x, ˘ t ) ! (4.26) δ ˆ Z αβ (˘ ~x, ˘ t ) = ı δ ˆ d mn (˘ ~x, ˘ t ) δ ˆ ξ + m,r ′ ν (˘ ~x, ˘ t ) δ ˆ ξ rµ,n (˘ ~x, ˘ t ) δ ˆ g rµ,r ′ ν (˘ ~x, ˘ t ) ! ; (4.27) δ ˆ Z αβ (˘ ~x, ˘ t ) = − ı δ ˆ d Tmn (˘ ~x, ˘ t ) − δ ˆ ξ Tm,r ′ ν (˘ ~x, ˘ t ) δ ˆ ξ ∗ rµ,n (˘ ~x, ˘ t ) δ ˆ g Trµ,r ′ ν (˘ ~x, ˘ t ) ! ; (cid:0) δ ˆ Z αβ (˘ ~x, ˘ t ) (cid:1) st = − δ ˆ Z αβ (˘ ~x, ˘ t ) (4.28) ı (cid:0) δ ˆ d mm (cid:1) = 12 tan (cid:0) | ˆ b mm | (cid:1) h e − ı ϕ m (cid:0) δ ˆ b mm (cid:1) − e ı ϕ m (cid:0) δ ˆ b ∗ mm (cid:1)i ; (4.29) ı (cid:0) δ ˆ d mn (cid:1) m = n = 12 tan (cid:18) | ˆ b mm | + | ˆ b nn | (cid:19) h e − ı ϕ n (cid:0) δ ˆ b mn (cid:1) − e ı ϕ m (cid:0) δ ˆ b ∗ mn (cid:1)i + (4.30) −
12 tan (cid:18) | ˆ b mm | − | ˆ b nn | (cid:19) h e − ı ϕ n (cid:0) δ ˆ b mn (cid:1) + e ı ϕ m (cid:0) δ ˆ b ∗ mn (cid:1)i ; ı (cid:0) δ ˆ g rµ,rν (cid:1) = ı (cid:0) τ (cid:1) µν (cid:0) δ ˆ g (0) rr (cid:1) = (4.31)= − δ µν tanh (cid:0) | ˆ a (2) rr | (cid:1) h e − ı φ r (cid:0) δ ˆ a (2) rr (cid:1) − e ı φ r (cid:0) δ ˆ a (2) ∗ rr (cid:1)i ; ı (cid:0) δ ˆ g rµ,r ′ ν (cid:1) r = r ′ = − X k =0 (cid:0) τ k τ (cid:1) µν (cid:26) tanh (cid:18) | ˆ a (2) rr | + | ˆ a (2) r ′ r ′ | (cid:19)h e − ı φ r ′ (cid:0) δ ˆ a ( k ) rr ′ (cid:1) − (cid:0) − (cid:1) k e ı φ r (cid:0) δ ˆ a ( k )+ rr ′ (cid:1)i + − tanh (cid:18) | ˆ a (2) rr | − | ˆ a (2) r ′ r ′ | (cid:19)h e − ı φ r ′ (cid:0) δ ˆ a ( k ) rr ′ (cid:1) + (cid:0) − (cid:1) k e ı φ r (cid:0) δ ˆ a ( k )+ rr ′ (cid:1)i(cid:27) ; (4.32)ˆ a (2) rr = 0 ; ˆ a ( k ) rr ≡ k = 0 , , ı (cid:0) δ ˆ ξ rµ,n (cid:1) = 12 tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) δ ˆ ζ ∗ rκ,n (cid:1) − ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) δ ˆ ζ rκ,n (cid:1)i + (4.33)2 + 12 tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19) h e ı φ r (cid:0) τ (cid:1) µκ (cid:0) δ ˆ ζ ∗ rκ,n (cid:1) + ı e − ı ϕ n (cid:0) τ (cid:1) µκ (cid:0) δ ˆ ζ rκ,n (cid:1)i . The variation δ (exp { Y DD } ) of the coset eigenvalues also involves a variation of the phases δϕ m (˘ ~x, ˘ t p ), δφ r (˘ ~x, ˘ t p ) with respect to δβ m (˘ ~x, ˘ t p ), δα r (˘ ~x, ˘ t p ) apart from the variation of absolute values δ | ˆ b mm (˘ ~x, ˘ t p ) | , δ | ˆ a (2) rr (˘ ~x, ˘ t p ) | . The first order variation of the last term in (4.4,4.21) vanishes completely, due to theadditional contour metric η p . This term begins to contribute from second and all higher even or-der variations with universal fluctuations which are entirely determined by the coset decompositionOsp( S, S | L ) / U( L | S ) ⊗ U( L | S ). In the following relations (4.34-4.37) we arrange the various diagonalparts of the coset eigenvalue matrix (exp { Y DD } ) abαβ and also point out the non-Markovian, path de-pendent phases ϕ m ( ~x, t p ), φ r ( ~x, t p ) determined by the contour time history of the anomalous, Euclideanfields ˆ b mm ( ~x, t p ), ˆ a (2) rr ( ~x, t p ) (cid:16) exp { Y DD } (cid:17) αβ = (cid:16) exp { Y DD } (cid:17) αβ = (4.34)= cos (cid:0) | ˆ b mm (˘ ~x, ˘ t p ) | (cid:1) δ mn
00 cosh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t p ) | (cid:1) δ rr ′ δ µν ! abαβ ; (cid:16) exp { Y DD } (cid:17) αβ = (4.35)= − sin (cid:0) | ˆ b mm (˘ ~x, ˘ t p ) | (cid:1) e ı ϕ m (˘ ~x, ˘ t p ) δ mn (cid:0) τ (cid:1) µν sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t p ) | (cid:1) e ı φ r (˘ ~x, ˘ t p ) δ rr ′ ! αβ ; (cid:16) exp { Y DD } (cid:17) αβ = (4.36)= sin (cid:0) | ˆ b mm (˘ ~x, ˘ t p ) | (cid:1) e − ı ϕ m (˘ ~x, ˘ t p ) δ mn (cid:0) τ (cid:1) µν sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t p ) | (cid:1) e − ı φ r (˘ ~x, ˘ t p ) δ rr ′ ! αβ ; ϕ m (˘ ~x, ˘ t p ) = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | ˆ b mm (˘ ~x, ˘ t ′ q ) | sin (cid:0) | ˆ b mm (˘ ~x, ˘ t ′ q ) | (cid:1) ∂β m (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q ; φ r (˘ ~x, ˘ t p ) = Z ˘ t p − ˘ ∞ + d ˘ t ′ q | ˆ a (2) rr (˘ ~x, ˘ t ′ q ) | sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ′ q ) | (cid:1) ∂α r (˘ ~x, ˘ t ′ q ) ∂ ˘ t ′ q . (4.37)After partial integrations in (4.4,4.21), we obtain the first order variation δ L ( d ) [ˆ Z ; ˘ J ψ ] of the Lagrangianin terms of the matrices (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ and also list the part which starts to contributeafter second order variations for universal fluctuations around the classical solutions within the cosetdecomposition Osp( S, S | L ) / U( L | S ) ⊗ U( L | S ) δ L ( d ) [ˆ Z ; ˘ J ψ ] = STR a,α ; b,β "(cid:0) δ ˆ Z abαβ (cid:1) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.38)+ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − × .1 Variation for classical field equations with Euclidean integration variables × (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + 12 exp (cid:8) − Y DD (cid:9)(cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21) − (cid:19)(cid:21) baβα + −
12 STR a,α ; b,β "(cid:0) δ exp { Y DD } (cid:1) abαβ (cid:18)h ı ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) + ˆ1 N × N (cid:16) ˘ u (˘ ~x ) − ˘ µ + ℜ (cid:0) h ˘ σ (0) D (˘ ~x, ˘ t ) i ˆ σ (0) D (cid:1)(cid:17)i ++ h ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − i(cid:19) baβα + η p a,α ; b,β h(cid:0) δ exp { Y DD } (cid:1)i| {z } ≡ ++ η p (cid:16) ı (cid:0) ε + + ℑ (cid:0) h ˘ σ (0) D (˘ ~x, ˘ t ) i (cid:1)(cid:17) (cid:18) δ STR a,α ; b,β h(cid:0) δ exp { Y DD } (cid:1)i(cid:19) . The following steps seem to be involved and complicated, but are straightforward in order to attainthe first order variations for the classical equations with the independent, anomalous Euclidean fieldsˆ b m ≥ n ( ~x, t p ), ˆ a (2) rr ( ~x, t p ), ˆ a ( k ) r>r ′ ( ~x, t p ) ( k = 0 , , , ζ rµ,n ( ~x, t p ), (+c.c.). One has to relate the variations ofthe matrices (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ in (4.38) to these independent, anomalous, Euclidean fieldswhere each of these finally defines a classical equation. At first we specify the variations of the coseteigenvalue matrix (cid:0) δ exp { Y DD } (cid:1) abαβ in terms of ˆ b mm ( ~x, t p ) and ˆ a (2) rr ( ~x, t p ) which introduce the Sine(h)-and Cos(h)-functions of these diagonal elements into the first order variations to classical field equations (cid:0) δ exp { Y DD } (cid:1) aaαβ = (cid:0) δ exp { Y DD } (cid:1) aaBB ; mm δ mn (cid:0) δ exp { Y DD } (cid:1) aaF F ; rµ,rν δ rr ′ δ µν ! aaαβ ;(4.39) (cid:0) δ exp { Y DD } (cid:1) aaBB ; mm = − sin (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) × (4.40) × h e − ı β m (˘ ~x, ˘ t ) (cid:0) δ ˆ b mm (˘ ~x, ˘ t ) (cid:1) + e ı β m (˘ ~x, ˘ t ) (cid:0) δ ˆ b ∗ mm (˘ ~x, ˘ t ) (cid:1)i ; (cid:0) δ exp { Y DD } (cid:1) aaF F ; rµ,rν = δ µν sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) × (4.41) × h e − ı α r (˘ ~x, ˘ t ) (cid:0) δ ˆ a (2) rr (˘ ~x, ˘ t ) (cid:1) + e ı α r (˘ ~x, ˘ t ) (cid:0) δ ˆ a (2) ∗ rr (˘ ~x, ˘ t ) (cid:1)i ; (cid:0) δ exp { Y DD } (cid:1) αβ = (cid:0) δ exp { Y DD } (cid:1) BB ; mm δ mn (cid:0) δ exp { Y DD } (cid:1) F F ; rµ,rν δ rr ′ ! αβ ; (4.42) (cid:0) δ exp { Y DD } (cid:1) αβ = (cid:0) δ exp { Y DD } (cid:1) BB ; mm δ mn (cid:0) δ exp { Y DD } (cid:1) F F ; rµ,rν δ rr ′ ! αβ ; (4.43) (cid:0) δ exp { Y DD } (cid:1) BB ; mm = − (cid:0) δ exp { Y DD } (cid:1) , ∗ BB ; mm = (4.44)= − (cid:16) e ı ϕ m (˘ ~x, ˘ t ) cos (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) + 1 (cid:17) e − ı β m (˘ ~x, ˘ t ) (cid:0) δ ˆ b mm (˘ ~x, ˘ t ) (cid:1) + − (cid:16) e ı ϕ m (˘ ~x, ˘ t ) cos (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) − (cid:17) e ı β m (˘ ~x, ˘ t ) (cid:0) δ ˆ b ∗ mm (˘ ~x, ˘ t ) (cid:1) ; (cid:0) δ exp { Y DD } (cid:1) F F ; rµ,rν = (cid:0) δ exp { Y DD } (cid:1) , + F F ; rµ,rν = (4.45)4 = (cid:0) τ (cid:1) µν (cid:20)(cid:16) e ı φ r (˘ ~x, ˘ t ) cosh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) + 1 (cid:17) e − ı α r (˘ ~x, ˘ t ) (cid:0) δ ˆ a (2) rr (˘ ~x, ˘ t ) (cid:1) ++ (cid:16) e ı φ r (˘ ~x, ˘ t ) cosh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) − (cid:17) e ı α r (˘ ~x, ˘ t ) (cid:0) δ ˆ a (2) ∗ rr (˘ ~x, ˘ t ) (cid:1) . At first we specialize on the variation with diagonal (quaternion diagonal) matrix elements δ ˆ b ∗ mm (˘ ~x, ˘ t ),( δ ˆ a (2) ∗ rr (˘ ~x, ˘ t )) and have to extract these Euclidean fields from the variation within the matrices (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ which appear in δ L ( d ) [ˆ Z ; ˘ J ψ ] (4.38). As these fields δ ˆ b ∗ mm (˘ ~x, ˘ t ), ( δ ˆ a (2) ∗ rr (˘ ~x, ˘ t )) areseparated from the various parts of the variations of the matrices (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ , one hasto include coefficients B a ≥ bmm (˘ ~x, ˘ t ), Y a ≥ bmm (˘ ~x, ˘ t ) (derived from (cid:0) δ ˆ Z abαβ (cid:1) , (cid:0) δ exp { Y DD } (cid:1) abαβ ) into the resultingfield equation Variation with (cid:0) δ ˆ b ∗ mm (˘ ~x, ˘ t ) (cid:1) B mm (˘ ~x, ˘ t ) = 1 ; (4.46) B mm (˘ ~x, ˘ t ) = − B mm (˘ ~x, ˘ t ) = −
12 tan (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) e ı ϕ m (˘ ~x, ˘ t ) ; (4.47) Y mm (˘ ~x, ˘ t ) = (cid:16) e − ı ϕ m (˘ ~x, ˘ t ) cos (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) + 1 (cid:17) e ı β m (˘ ~x, ˘ t ) ; (4.48) Y mm (˘ ~x, ˘ t ) = Y mm (˘ ~x, ˘ t ) = − sin (cid:0) | ˆ b mm (˘ ~x, ˘ t ) | (cid:1) e ı β m (˘ ~x, ˘ t ) . (4.49)According to the above coefficients B a ≥ bmm (˘ ~x, ˘ t ), Y a ≥ bmm (˘ ~x, ˘ t ) of (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ , we cansimplify the resulting equation of the diagonal pair condensate fields in the boson-boson part0 ≡ ( a ≥ b ) X a,b =1 , B a ≥ bmm (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.50)+ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + 12 exp (cid:8) − Y DD (cid:9)(cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21)(cid:19)(cid:21) b ≤ aBB ; mm + − ( a ≥ b ) X a,b =1 , Y a ≥ bmm (˘ ~x, ˘ t ) (cid:18)h ı ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) + ˆ1 N × N (cid:16) ˘ u (˘ ~x ) − ˘ µ + ℜ (cid:0) h ˘ σ (0) D (˘ ~x, ˘ t ) i ˆ σ (0) D (cid:1)(cid:17)i ++ h ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − i(cid:19) b ≤ aBB ; mm . Similarly, we specify coefficients F a ≥ brµ,rν (˘ ~x, ˘ t ), Y a ≥ brµ,rν (˘ ~x, ˘ t ) from varying (cid:0) δ ˆ Z abαβ (cid:1) and (cid:0) δ exp { Y DD } (cid:1) abαβ for the quaternion diagonal elements of the BCS pair condensates within the fermion-fermion section. .1 Variation for classical field equations with Euclidean integration variables δ ˆ a (2) ∗ rr (˘ ~x, ˘ t )Variation with (cid:0) δ ˆ a (2) ∗ rr (˘ ~x, ˘ t ) (cid:1) or (cid:0) δ ˆ a rµ,rν (˘ ~x, ˘ t ) (cid:1) = − (cid:0) δ ˆ a rν,rµ (˘ ~x, ˘ t ) (cid:1) = (cid:0) τ (cid:1) µν (cid:0) δ ˆ a (2) ∗ rr (˘ ~x, ˘ t ) (cid:1) F rµ,rν (˘ ~x, ˘ t ) = − F rν,rµ (˘ ~x, ˘ t ) = (cid:0) τ (cid:1) µν ; (4.51) F rµ,rν (˘ ~x, ˘ t ) = − F rν,rµ (˘ ~x, ˘ t ) = (cid:0) τ (cid:1) µν
12 tanh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) e ı φ r (˘ ~x, ˘ t ) ; (4.52) Y rµ,rν (˘ ~x, ˘ t ) = (cid:0) τ (cid:1) µν (cid:16) e − ı φ r (˘ ~x, ˘ t ) cosh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) + 1 (cid:17) e ı α r (˘ ~x, ˘ t ) ; (4.53) Y rµ,rν (˘ ~x, ˘ t ) = Y rν,rµ (˘ ~x, ˘ t ) = (cid:0) τ (cid:1) µν sinh (cid:0) | ˆ a (2) rr (˘ ~x, ˘ t ) | (cid:1) e ı α r (˘ ~x, ˘ t ) . (4.54)These coefficients contain Sinh-, Cosh- and Tanh-functions instead of their trigonometric correspondentsfor the diagonal pair condensates within the boson-boson part and allow to disentangle the resultingmatrix equation for the quaternion diagonal fermion-fermion section. Since one has to consider quater-nion elements in the case of fermion-fermion parts, we have to perform a trace ’tr µν ’ over the 2 × ≡ − ( a ≥ b ) X a,b =1 , tr µ,ν " F a ≥ brµ,rν (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.55)+ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + e − Y DD (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21) − (cid:19)(cid:21) b ≤ aF F ; rν,rµ ++ 12 ( a ≥ b ) X a,b =1 , tr µ,ν " Y a ≥ brµ,rν (˘ ~x, ˘ t ) (cid:18)h ı ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) + ˆ1 N × N (cid:16) ˘ u (˘ ~x ) − ˘ µ + ℜ (cid:0) h ˘ σ (0) D (˘ ~x, ˘ t ) i ˆ σ (0) D (cid:1)(cid:17)i ++ h ˆ S (cid:0) ˘ ∂ ˘ t ˆ Z (cid:1) ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − i(cid:19) b ≤ aF F ; rν,rµ In the case of off-diagonal variations δ ˆ b ∗ m = n (˘ ~x, ˘ t ), δ ˆ a ( k )+ rr ′ (˘ ~x, ˘ t ) in the anomalous boson-boson or fermion-fermion part, we can neglect variations (cid:0) δ exp { Y DD } (cid:1) abαβ of the coset eigenvalue matrix because theseonly consist of diagonal (quaternion diagonal) elements ˆ b ∗ mm (˘ ~x, ˘ t ) (ˆ a (2) ∗ rr (˘ ~x, ˘ t )) apart from the non-Markovian phases ϕ m (˘ ~x, ˘ t ), φ r (˘ ~x, ˘ t ). Therefore, we have only to take into account coefficients B a ≥ bmn (˘ ~x, ˘ t )arising from the variation with the matrix (cid:0) δ ˆ Z abαβ (cid:1) Variation with (cid:0) δ ˆ b ∗ m = n (˘ ~x, ˘ t ) (cid:1) Mind the symmetry : (cid:0) δ ˆ b ∗ m = n (˘ ~x, ˘ t ) (cid:1) = (cid:0) δ ˆ b ∗ n = m (˘ ~x, ˘ t ) (cid:1) B mn (˘ ~x, ˘ t ) = 1 ; (4.56) B mn (˘ ~x, ˘ t ) = − B nm (˘ ~x, ˘ t ) = − e ı ϕ m (˘ ~x, ˘ t ) (cid:20) tan (cid:18) | ˆ b mm | + | ˆ b nn | (cid:19) + tan (cid:18) | ˆ b mm | − | ˆ b nn | (cid:19)(cid:21) . (4.57)Application of the above coefficients B a ≥ bmn (˘ ~x, ˘ t ) finally allows to give the classical field equations for theoff-diagonal anomalous, boson-boson part in abbreviated from which also includes the trigonometricfunctions0 ≡ ( ( a ≥ b ) X a,b =1 , B a ≥ bmn (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.58)+ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + e − Y DD (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21)(cid:19)(cid:21) b ≤ aBB ; nm ) ++ ( entire above terms with m ↔ n ) . In analogy we catalogue the coefficients F , ( k ) rµ,r ′ ν (˘ ~x, ˘ t ) of the variations (cid:0) δ ˆ Z abαβ (cid:1) for the off-diagonal quater-nion matrix elements within the fermion-fermion section. Since the variation with (cid:0) δ exp { Y DD } (cid:1) abαβ is restricted to the quaternion diagonal elements, we have only to introduce coefficients F , ( k ) rµ,r ′ ν (˘ ~x, ˘ t ) forthe variations within (cid:0) δ ˆ Z abαβ (cid:1) Variation with (cid:0) δ ˆ a ( k )+ rr ′ (˘ ~x, ˘ t ) (cid:1) in (cid:0) δ ˆ a + rµ,r ′ ν (˘ ~x, ˘ t ) (cid:1) = X k =0 (cid:0) τ k (cid:1) µν (cid:0) δ ˆ a ( k )+ rr ′ (˘ ~x, ˘ t ) (cid:1) ;Mind the symmetries : (cid:0) δ ˆ a + rµ,r ′ ν (˘ ~x, ˘ t ) (cid:1) = − (cid:0) δ ˆ a + r ′ ν,rµ (˘ ~x, ˘ t ) (cid:1) ; (cid:0) δ ˆ a (2)+ rr ′ (cid:1) = + (cid:0) δ ˆ a (2)+ r ′ r (cid:1) ; but (cid:0) δ ˆ a ( k )+ rr ′ (cid:1) = − (cid:0) δ ˆ a ( k )+ r ′ r (cid:1) for k = 0 , , F , ( k ) rµ,r ′ ν (˘ ~x, ˘ t ) = (cid:0) τ k (cid:1) µν ; (4.59) F , ( k ) rµ,r ′ ν (˘ ~x, ˘ t ) = − F , ( k ) r ′ ν,rµ (˘ ~x, ˘ t ) = (cid:0) − (cid:1) k (cid:0) τ k τ (cid:1) µν e ı φ r (˘ ~x, ˘ t ) × (4.60) × (cid:20) tanh (cid:18) | ˆ a (2) rr | + | ˆ a (2) r ′ r ′ | (cid:19) + tanh (cid:18) | ˆ a (2) rr | − | ˆ a r ′ r ′ | (cid:19)(cid:21) The above coefficients with hyperbolic trigonometric functions again reduce the field equations to acompact form which includes traces over the 2 × .1 Variation for classical field equations with Euclidean integration variables ≡ ( − ( a ≥ b ) X a,b =1 , tr µ,ν " F a ≥ b, ( k ) rµ,r ′ ν (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.61)+ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + e − Y DD (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21) − (cid:19)(cid:21) b ≤ aF F ; r ′ ν,rµ ++ ( entire above terms with r ↔ r ′ for k = 2 ) or − ( entire upper terms with r ↔ r ′ for k = 0 , , ) Finally, we approach the variation with respect to the anti-commuting, anomalous fields δ ˆ ζ ∗ rκ,n (˘ ~x, ˘ t ) andextract corresponding coefficients Z a ≥ b, ( κ ) F B ; rµ,n (˘ ~x, ˘ t ), Z a ≥ b, ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) for the fermion-boson and boson-fermionparts which are derived from the matrix (cid:0) δ ˆ Z abαβ (cid:1) Variation with (cid:0) δ ˆ ζ ∗ rκ,n (˘ ~x, ˘ t ) (cid:1) ; (cid:0) δ ˆ ζ + n,rκ (˘ ~x, ˘ t ) (cid:1) Z , ( κ ) F B ; rµ,n (˘ ~x, ˘ t ) = 1 ; (4.62) Z , ( κ ) F B ; rµ,n (˘ ~x, ˘ t ) = e ı φ r (˘ ~x, ˘ t ) (cid:20) tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19) + tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19)(cid:21) (cid:0) τ (cid:1) µκ ; (4.63) Z , ( κ ) F B ; rµ,n (˘ ~x, ˘ t ) = ı e ı ϕ n (˘ ~x, ˘ t ) (cid:20) tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19) − tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19)(cid:21) (cid:0) τ (cid:1) µκ ; (4.64) Z , ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) = 1 ; (4.65) Z , ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) = − ı e ı ϕ n (˘ ~x, ˘ t ) (cid:20) tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19) − tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19)(cid:21) (cid:0) τ (cid:1) κµ ; (4.66) Z , ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) = − e ı φ r (˘ ~x, ˘ t ) (cid:20) tanh (cid:18) | ˆ a (2) rr | + ı | ˆ b nn | (cid:19) + tanh (cid:18) | ˆ a (2) rr | − ı | ˆ b nn | (cid:19)(cid:21) (cid:0) τ (cid:1) κµ . (4.67)The coefficients Z a ≥ b, ( κ ) F B ; rµ,n (˘ ~x, ˘ t ), Z a ≥ b, ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) are partially composed of compact and non-compact (hy-perbolic) trigonometric functions and have to be summed over the two spin degrees of freedom with therest of the field equation which finally takes values within the Grassmann sector of the super-symmetricmatrices0 ≡ − ( a ≥ b ) X a,b =1 , X µ =1 , " Z a ≥ b, ( κ ) F B ; rµ,n (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) + (4.68)8 + (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + e − Y DD (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21) − (cid:19)(cid:21) b ≤ aBF ; n,rµ ++ ( a ≥ b ) X a,b =1 , X µ =1 , " Z a ≥ b, ( κ ) BF ; n,rµ (˘ ~x, ˘ t ) (cid:20) ˘ ∂ i (cid:18) c ij (˘ ~x, ˘ t ) (cid:0) ˘ ∂ j ˆ Z b = aβα (cid:1) (cid:0) − δ ab (cid:1) + (cid:0) ˘ ∂ i ˆ Z baβα (cid:1) ++ (cid:26) ı (cid:16) ˘ d ij (˘ ~x, ˘ t ) − δ ij (cid:17) (cid:0) ˘ ∂ j ˆ Z (cid:1) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:27) + (cid:19) ++ 12 ˘ ∂ ˘ t (cid:18)(cid:16) ı ˆ1 N × N + ˆ P ˆ I − ˘ J ψ ⊗ ˘ J + ψ ˆ I ˆ K ˆ P − (cid:17) exp { Y DD } ˆ S (cid:19) + (cid:18)(cid:18) −→ ∂ ˆ P −→ ∂ ˆ Z (cid:19) ˆ P − ×× (cid:20)(cid:16)(cid:0) ˘ ∂ i ˆ Z (cid:1)(cid:0) ˘ ∂ j ˆ Z (cid:1) ı (cid:16) ˘ d ij − δ ij (cid:17) + e − Y DD (cid:0) ˘ ∂ ˘ t p ˆ Z (cid:1) ˆ S (cid:17) , ˆ P ˆ I ˘ J ψ ⊗ ˘ J + ψ ˆ I e K ˆ P − (cid:21) − (cid:19)(cid:21) b ≤ aF B ; rµ,n We have classified in relations (4.46-4.68) the various classical field equations following from firstorder variations of the independent, Euclidean pair condensate fields. Since we have considered generalangular momentum degrees of freedom of the boson-boson, fermion-fermion and the odd fermion-boson,boson-fermion parts, various coefficients [cf. Eqs. (4.46-4.49), (4.51-4.54), (4.56-4.57), (4.59-4.60), (4.62-4.67)] have to be used as part of the variation within (cid:0) δ ˆ Z abαβ (cid:1) or (cid:0) δ exp { Y DD } (cid:1) abαβ . One achieves coupledsuper-symmetric matrix equations which are composed of Sine(h)-, Cos(h)- or Tan(h)-functions of thediagonal, Euclidean pair condensate fields so that these illustrate modifications of the well-known,integrable Sine-Gordon equations in 1+1 or 2+1 dimensions. These matrix equations correspond tothe Gross-Pitaevskii equation in a transferred sense if one regards the coherent super-symmetric paircondensates in analogy to the coherent BEC-wavefunctions. Apart from the gradient expansion with (4.69), we have also to take into account the generating sourcefield e J ( ˆ T − , ˆ T ) (4.70) whose second order expansion of the effective actions is listed in relation (4.71).This generating source field e J ( ˆ T − , ˆ T ) (4.70) can be replaced by derivatives with respect to the paircondensate ’seeds’ ı ˆ J a = bψψ ; αβ ( ~x, t p ) e K (2.63,2.64) of the action A ˆ J ψψ [ ˆ T ] (2.65-2.69) for observables whichgo beyond the second order expansion of e J ( ˆ T − , ˆ T ) in relation (4.71). However, the pair condensate’seed’ fields ı ˆ J a = bψψ ; αβ ( ~x, t p ) e K (2.63,2.64) of the action A ˆ J ψψ [ ˆ T ] (2.65-2.69) do not allow for generatingdensity terms as the source field e J ( ˆ T − , ˆ T ) (4.70). In correspondence to chapter 4 of Ref. [6], one canperform the gradient expansion with δ ˆ H ( ˆ T − , ˆ T ) (4.69) and with the source term e J ( ˆ T − , ˆ T ) (4.70) in .3 Outlook for relations between chaotic and integrable systems with modified r-s matrices δ ˆ H ( ˆ T − , ˆ T ) = − ˆ η (cid:16) ˆ T − ˆ S (cid:0) E p ˆ T (cid:1) + ˆ T − (cid:0) e ∂ i e ∂ i ˆ T (cid:1) + (cid:0) ˆ T − ˆ S ˆ T − ˆ S (cid:1) ˆ E p + 2 ˆ T − (cid:0) e ∂ i ˆ T (cid:1) e ∂ i (cid:17) (4.69) e J ab~x,α ; ~x ′ ,β ( ˆ T − ( t p ) , ˆ T ( t ′ q )) = ˆ T − aa ′ αα ′ ( ~x, t p ) ˆ I ˆ K η p ˆ J a ′ b ′ ~x,α ′ ; ~x ′ ,β ′ ( t p , t ′ q ) N x η q ˆ K ˆ I e K ˆ T b ′ bβ ′ β ( ~x ′ , t ′ q ) (4.70) A ′ (cid:2) ˆ T ; ˆ J (cid:3) = − (cid:28) Tr STR (cid:20)e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] (cid:21)(cid:29) ˆ σ (0) D + (4.71) − ı N [ h J bψ ; β | ˆ η (cid:16) ˆ I e K ˆ T ˆ G (0) [ˆ σ (0) D ] (cid:16)e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] (cid:17) ˆ T − ˆ I (cid:17) baβα ˆ η | [ J aψ ; α i (cid:29) ˆ σ (0) D + − (cid:28) Tr STR (cid:20) δ ˆ H ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] (cid:21)(cid:29) ˆ σ (0) D + − ı N (cid:28) [ h J bψ ; β | ˆ η (cid:18) ˆ I e K ˆ T ˆ G (0) [ˆ σ (0) D ] (cid:16) δ ˆ H ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] e J ( ˆ T − , ˆ T ) ++ e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] δ ˆ H ( ˆ T − , ˆ T ) (cid:17) ˆ G (0) [ˆ σ (0) D ] ˆ T − ˆ I (cid:19) baβα ˆ η | [ J aψ ; α i (cid:29) ˆ σ (0) D ++ 12 Tr STR (cid:20)e J ( ˆ T − , ˆ T ) D ˆ G (0) [ˆ σ (0) D ] E ˆ σ (0) D (cid:21) ++ ı N (cid:28) [ h J bψ ; β | ˆ η (cid:16) ˆ I e K ˆ T ˆ G (0) [ˆ σ (0) D ] e J ( ˆ T − , ˆ T ) ˆ G (0) [ˆ σ (0) D ] ˆ T − ˆ I (cid:17) baβα ˆ η | [ J aψ ; α i (cid:29) ˆ σ (0) D . The action A ′ [ ˆ T ; ˆ J ] also involves the averaging h . . . i ˆ σ (0) D over the background density field σ (0) D ( ~x, t p ) withgenerating function Z [ j ψ ; ˆ σ (0) D ] (2.74) (compare (2.70-2.75)). However, we can simplify this averagingprocess by taking the classical field value which results from the saddle point equation outlined in (2.76). A particular property of the nonlinear sigma-model equations (4.46-4.68) is the integrability for specialdimensions, as 1+1 or even 2+1 [53]-[60]. These properties of integrability are determined by r-s matrixproperties which can be investigated as quantum groups [57]-[60]. However, as already suggested in [6],one can also try to classify chaotic systems as extensions of these r-s matrix bi-algebras in analogy ofextensions of group or algebraic properties if one adds symmetry breaking generators or group elementsto the classical equations. 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