**1. Introduction**

It is an old classical result that the nonrelativistic quantum current algebra realizes [1–3] a representation of the Lie algebra G, related to the semidirect product *G* := Diff(R*m*) - *<sup>F</sup>*(R*<sup>m</sup>*; R) of the topological diffeomorphism group Diff(R*m*) of the real space R*m* and the space *<sup>F</sup>*(R*<sup>m</sup>*; R) of smooth Schwartz type functions on it. As it was later shown by G. Goldin, with collaborators [4–7], in fact, all nonrelativistic quantum manyparticle Hamiltonian systems allow the equivalent representation by means of the current algebra operators and their realization on some specially constructed generalized Hilbert spaces with cyclic vector structure, strongly depending on the groundstate vectors of the corresponding Hamiltonian operators. The detailed analysis of this representation [8–10] made it possible to reveal a deep connection of the specially factorized operator structure of Hamiltonian operators and the quantum complete integrability of the corresponding Heisenberg type operator dynamical systems. Moreover, studying vector field representations of the quantum current algebra, related to the semidirect product Diff(S<sup>1</sup>) - *F*(S1; R) of the topological diffeomorphism group Diff(S<sup>1</sup>) of the circle S1 and the space *F*(S2; R) of the smooth periodic functions on it, their isomorphism was stated [11–13] with all completely known and up to date Lax type integrable classical dynamical systems on spatially one-dimensional functional manifolds. Closely related algebraic aspects of representation theory of the canonical creation–annihilation operators, both on the Fock space and on related cyclic Hilbert spaces, gave rise to the construction of the effective linearizing scheme of any smooth dynamical system on functional Hilbert space. As the main analytical trick of these schemes is based on the coherent vector representation of the canonical creation– annihilation operators on the Fock space, we describe their unbelievable and impressive applications to the theory of nonlinear dynamical systems on Hilbert spaces, their linearization and integrability, previously initiated in [14,15] and continued in [16]. We briefly review the coherent vector representations of the Bargmann–Segal space H*k* of complex holomorphic functions on C*k*, and describe a general approach to constructing coherent states and their applications both to the linearization of nonlinear dynamical systems on Hilbert spaces, and to describing their complete integrability. The latter is developed using the modern Lie-algebraic approach [11,17–19] to nonlinear dynamical systems on Poissonian functional manifolds, and proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces.

Other very important applications of the current algebra representations are related both to statistical physics, classical and quantum, and to hydrodynamics. The quantum current algebra quasi-classical representations made it possible to analytically describe [20–22] the so-called collective variable approach in equilibrium statistical physics and calculate the main thermodynamical quantities at finite temperatures. The related quantum current algebra quasi-classical Wigner type representations proved to be effective in describing the kinetic theory [23,24] of many-particle systems and calculating both the corresponding evolution equations for the infinite hierarchy of many-particle distribution functions, and developing a new approach to their dynamically compatible splitting, based on the well known Dirac type reduction of Poissonian systems on functional submanifolds.

A very rich geometric structure of liquid flow in a domain Ω ⊂ R<sup>3</sup> and its properties can be deeply described by means of the corresponding diffeomorphism group Diff(Ω)

and its semi-direct products with different functional spaces on the domain Ω ⊂ R3. It is well known that the same physical system is often described using different sets of variables, related with their different physical interpretation. It was observed [25–33] that the corresponding mathematical structures used for describing the analytical properties of hydrodynamical systems are canonically related to each other. Simultaneously, mathematical properties, against a background of their analytical description, make it possible to study additional important parameters [34–50] of different hydrodynamic and magnetohydrodynamic systems. Amongst these, we will mention integral invariants, describing such internal fluid motion peculiarities as vortices, topological singularities [51] and other different instability states, strongly depending [52,53] on imposed isentropic fluid motion constraints. Being interested in their general properties and mathematical structures which are responsible for their existence and behavior, we present [54] a detailed differential geometrical approach to thermodynamically investigating quasi-stationary isentropic fluid motions, paying more attention to the analytical argumentation of tricks and techniques used during the presentation. Amongst the systems analyzed here, we mention the Hamiltonian analysis and adiabatic magneto-hydrodynamic superfluid motion, as well constructing a modified current Lie algebra and describing magneto-hydrodynamic invariants and their geometry. In particular, we studied a modified current Lie algebra symmetry on torus, its Lie-algebraic structure and related integrable heavenly type dynamical systems, describing the quasi-conformal metrics of Riemannian spaces in general relativity.

#### **2. The Fock Space, Non-Relativistic Quantum Current Algebra and Its Cyclic Representations**

#### *2.1. The Fock Space Representation*

Let a Hilbert space Φ*F* possess the standard canonical Fock space structure [5,11,55–60], that is

$$
\Phi\_F = \oplus\_{n \in \mathbb{Z}\_+} \Phi\_{\binom{\mathbb{S}^n}{\mathbb{S}}}^{\otimes \mathbb{N}},\tag{1}
$$

where subspaces Φ⊗*<sup>n</sup>* (*s*) , *n* ∈ Z+, are the symmetrized tensor products of the Hilbert space *H* - *L*(*s*) 2 (R*<sup>m</sup>*; <sup>C</sup>*<sup>k</sup>*). If a vector *ϕ* := (*ϕ*0, *ϕ*1, ..., *ϕ<sup>n</sup>*, ...) ∈ Φ*F*, its norm

$$\|\|\boldsymbol{\varrho}\|\|\_{\Phi} := \left(\sum\_{n \in \mathbb{Z}\_+} \|\|\boldsymbol{\varrho}\_n\|\|\_{n}^2\right)^{1/2},\tag{2}$$

where *ϕn* ∈ Φ⊗*<sup>n</sup>* (*s*) - *L*(*s*) 2 (( <sup>R</sup>*<sup>m</sup>*)<sup>⊗</sup>*<sup>n</sup>*; C) and ... *n* is the corresponding norm in Φ⊗*<sup>n</sup>* (*s*) for all *n* ∈ Z+. Note here that concerning the rigging structure (18), there holds the corresponding rigging for the Hilbert spaces Φ⊗*<sup>n</sup>* (*s*) , *n* ∈ Z+, that is

$$\mathcal{D}\_{(s)}^{\mathfrak{n}} \subset \Phi\_{(s),+}^{\otimes \mathfrak{n}} \subset \Phi\_{(s)}^{\otimes \mathfrak{n}} \subset \Phi\_{(s),-}^{\otimes \mathfrak{n}} \tag{3}$$

with some suitably chosen dense and separable topological spaces of symmetric functions D*n* (*s*) , *n* ∈ Z+. Concerning expansion (1), we obtain by means of projective and inductive limits [55,57,61,62] the quasi-nucleus rigging of the Fock space Φ in the form (18).

Consider now any basis vector |(*α*)*n*) ∈ Φ⊗*<sup>n</sup>* (*s*) , *n* ∈ N, which can be written [56,57,63–65] in the following canonical Dirac ket-form:

$$|(\mathfrak{a})\_{\mathfrak{n}}\rangle := |\mathfrak{a}\_1, \mathfrak{a}\_{2'}, \dots, \mathfrak{a}\_{\mathfrak{n}}\rangle,\tag{4}$$

where, by definition,

$$|\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_{\mathfrak{n}}\rangle := \frac{1}{\sqrt{n!}} \sum\_{\sigma \in S\_n} |\mathfrak{a}\_{\sigma(1)}\rangle \otimes |\mathfrak{a}\_{\sigma(2)}\rangle \dots |\mathfrak{a}\_{\sigma(n)}\rangle \tag{5}$$

and vectors |*<sup>α</sup>j*) ∈ *H*+, Φ⊗<sup>1</sup> (*s*) - *H*, *j*, *k* ∈ N, are bi-orthogonal to each other, that is (*<sup>α</sup>k*|*<sup>α</sup>j*)*H* = *<sup>δ</sup>k*,*<sup>j</sup>* for any *k*, *j* ∈ N. The corresponding scalar product of base vectors as (5) is given as follows:

$$\begin{aligned} ( (\beta)\_n | (\boldsymbol{a})\_n ) &:= (\beta\_n, \beta\_{n-1}, \dots, \beta\_2, \beta\_1 | \boldsymbol{a}\_1, \boldsymbol{a}\_2, \dots, \boldsymbol{a}\_{n-1}, \boldsymbol{a}\_n) \\ &= \sum\_{\boldsymbol{\sigma} \in S\_n} (\beta\_1 | \boldsymbol{a}\_{\boldsymbol{\sigma}(1)}) \boldsymbol{\mu} \dots (\beta\_n | \boldsymbol{a}\_{\boldsymbol{\sigma}(n)}) \boldsymbol{\mu} := \operatorname{per} \{ (\beta\_i | \boldsymbol{a}\_j)\_{\boldsymbol{\Pi}} \}\_{i, j = \overline{1, \boldsymbol{\mu}}} \end{aligned} \tag{6}$$

where "*per*" denotes the permanence of the matrix and (·|·) is the corresponding scalar product in the Hilbert space *H*. Based now on the representation (4), one can define an operator *a*+(*α*) : Φ⊗*<sup>n</sup>* (*s*) −→ Φ<sup>⊗</sup>(*n*+<sup>1</sup>) (*s*) for any |*α*) ∈ *H*− as follows:

$$a^+(\mathfrak{a})|\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n\rangle := |\mathfrak{a}, \mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n\rangle,\tag{7}$$

which is called the "*creation*" operator in the Fock space Φ*F*. The adjoint operator *a*(*β*) := (*a*+(*β*))<sup>∗</sup> : Φ<sup>⊗</sup>(*n*+<sup>1</sup>) (*s*) −→ Φ⊗*<sup>n</sup>* (*s*) with respect to the Fock space Φ*F* (1) for any |*β*) ∈ *H*−, called the "*annihilation*" operator, acts as follows:

$$a(\beta)|a\_1, a\_2, \dots, a\_{n+1}\rangle := \sum\_{\sigma \in \mathbb{S}\_n} (\beta|a\_j)|a\_1, a\_2, \dots, a\_{j-1}, \pounds\_j a\_{j+1}, \dots, a\_{n+1}\rangle,\tag{8}$$

where the hat "ˆ· over a vector denotes that it should be omitted from the sequence. It is easy to check that the commutator relationship

$$[a(\mathfrak{a}), a^+(\mathfrak{k})] = (\mathfrak{a}|\mathfrak{k})\_H \tag{9}$$

holds for any vectors |*α*) ∈ *H* and |*β*) ∈ *H*. Expression (9), owing to the rigging structure (18), can be naturally extended to the general case, when vectors |*α*) and |*β*) ∈ *H*−, conserving its form. In particular, taking |*α*) := |*α*(*y*)) = √12*π e<sup>i</sup> <sup>y</sup>*|*<sup>x</sup> k* ∈ *H*− := *<sup>L</sup>*2,−(R*<sup>m</sup>*; C*<sup>k</sup>*) for any *y* ∈ E*<sup>m</sup>*, one easily gets from (9) that

$$[a\_i(\mathbf{x}), a\_j^+(\mathbf{y})] = \delta\_{ij}\delta(\mathbf{x} - \mathbf{y})\tag{10}$$

for any *i*, *j* = 1, *k*, where we put, by definition, ·|· the usual scalar product in the *m*-dimensional Euclidean space E*m* := (R*<sup>m</sup>*; ·|· ), *a*+*j* (*y*) := *a*+*j* (*y*(*x*)) and *aj*(*y*) := *aj*(*y*(*x*)), *j* = 1, *k*, for all *x*, *y* ∈ R*m* and denoted by *<sup>δ</sup>*(·) the classical Dirac delta-function.

The construction above makes it possible to observe easily that there exists the unique vacuum vector |0) ∈ Φ⊗<sup>1</sup> (*s*), such that for any *x* ∈ R*m*

$$a\_{\!\!\!\!/}(\mathbf{x})|0\rangle = 0 \tag{11}$$

for all *j* ∈ 1, *k*, and the set of vectors

$$\left(\prod\_{j=1}^{k} \prod\_{i=1}^{n\_j} \left(a\_j^+\right) \left(\mathbf{x}\_j^{(i)}\right)\right) |0\rangle \in \Phi\_{(s)}^{\otimes n} \tag{12}$$

is total in Φ⊗*<sup>n</sup>* (*s*) , that is, their linear integral hull over the functional spaces Φ⊗*<sup>n</sup>* (*s*) is dense in the Hilbert space Φ⊗*<sup>n</sup>* (*s*) for every *n* = <sup>∑</sup>*kj*=<sup>1</sup> *nj* ∈ N. This means that for any vector *ϕ* ∈ Φ*F*, the following canonical representation

$$\varphi = \sum\_{n=\sum\_{j=1}^{k}n\_{j} \in \mathbb{Z}\_{+}}^{\oplus} \int\_{(\mathbb{R}^{m})^{n}} \rho\_{n\_{1}n\_{2}\ldots n\_{s}}^{(n)}(\mathbf{x}\_{1}^{(1)}, \mathbf{x}\_{1}^{(2)}, \ldots, \mathbf{x}\_{1}^{(n\_{1})}; \mathbf{x}\_{2}^{(1)}, \mathbf{x}\_{2}^{(2)}, \ldots, \mathbf{x}\_{2}^{(n\_{2})}; \ldots}^{(n\_{1})} \tag{13}$$
 
$$\varphi\_{k}^{(1)}, \mathbf{x}\_{k}^{(2)}, \ldots, \mathbf{x}\_{k}^{(n\_{m})}) \prod\_{j=1}^{k} \frac{1}{\sqrt{n\_{j}!}!} \prod\_{s=1}^{n\_{j}} a\_{j}^{+}(\mathbf{x}\_{j}^{(s)}) |0\rangle$$

holds with the Fourier type coefficients *ϕ*(*n*) *n*1*n*2...*ns* ∈ Φ⊗*<sup>n</sup>* (*s*) for all *n* = <sup>∑</sup>*kj*=<sup>1</sup> *nj* ∈ Z+. The latter is naturally endowed with the Gelfand type quasi-nucleus rigging, dual to

$$H\_{+} \subset H \subset H\_{-},$$

making it possible to construct a quasi-nucleous rigging of the dual Fock space Φ*F* := <sup>⊕</sup>*n*∈Z+Φ⊗*<sup>n</sup>* (*s*). Thereby, the chain (14) generates the dual Fock space quasi-nucleolus rigging

$$\mathcal{D} \subset \Phi\_{\bar{F},+} \subset \Phi\_{\bar{F}} \subset \Phi\_{\bar{F},-} \subset \mathcal{D}' \tag{15}$$

with respect to the Fock space Φ*F*, easily following from (1) and (14).

Construct now the following self-adjoint operator *ρ*(*x*) : Φ*F* → Φ*F* as

$$\rho(\mathbf{x}) := \langle a^\top(\mathbf{x}) | a(\mathbf{x}) \rangle,\tag{16}$$

called the density operator at a point *x* ∈ R*<sup>m</sup>*, satisfying the commutation properties:

$$\begin{aligned} \left[\rho(\mathbf{x}), \rho(y)\right] &= 0, \\ \left[\rho(\mathbf{x}), a(y)\right] &= -a(y)\delta(\mathbf{x} - y), \\ \left[\rho(\mathbf{x}), a^+(y)\right] &= a^+(y)\delta(\mathbf{x} - y) \end{aligned} \tag{17}$$

for any *x*, *y* ∈ R*<sup>m</sup>*.

Assume now that Φ is a separable Hilbert space, *F* is a topological real linear space and A := {*A*(f) : f ∈ *F*} is a family of commuting self-adjoint operators in Φ (i.e., these operators commute in the sense of their resolutions of the identity) with dense in Φ domain Dom*A*(f) := *DA*(f) ⊂ Φ, f ∈ *F*. Consider the corresponding Gelfand rigging [57,61,66] of the Hilbert space Φ, i.e., a chain

$$\mathcal{D} \subset \Phi\_+ \subset \Phi \subset \Phi\_- \subset \mathcal{D}' \tag{18}$$

in which Φ+ is a Hilbert space, topologically (densely and continuously) and quasi-nucleus (the inclusion operator *i* : Φ+ −→ Φ is of the Hilbert–Schmidt type) embedded into Φ, the space Φ− is the dual to Φ+ as the completion of functionals on Φ+ with respect to the norm ||f||− := sup ||*u*||+=<sup>1</sup> |(f|*u*)Φ|, *u* ∈ Φ, a linear dense in Φ+ topological space D ⊆ Φ+ is such

that D ⊂ *DA*(f) ⊂ Φ and the mapping *A*(f) : D → Φ+ is continuous for any f ∈ *F*. Then, the following structural theorem [4,5,16,57,61,62,67–69] about the cyclic representations of the family A := {*A*(f) : f ∈ *F*} of commuting self-adjoint operators in the separable Hilbert space Φ holds.

#### **Theorem 1.** *Assume that the family of operators* A *satisfies the following conditions:*

*(a) for <sup>A</sup>*(f), f ∈ *F*, *the closure of the operator A*(f) *in* Φ *coincides with A*(f) *for any* f ∈ *F*, *that is A*(f) = *A*(f) *on domain DA*(f) *in* Φ;

*(b) the Range A*(f) ⊂ Φ *for any* f ∈ *F*;

*(c) for every ϕ* ∈ D *the mapping F* f −→ *<sup>A</sup>*(f)|*ϕ*) ∈ Φ+ *is linear and continuous;*

*(d) there exists a strong cyclic vector* |Ω) ∈ <sup>f</sup>∈*<sup>F</sup> DA*(f), *such that the set of all vectors* |Ω) *and* ∏*nj*=<sup>1</sup> *<sup>A</sup>*(f*j*)|Ω), *n* ∈ Z+, *is total in* Φ+ *(i.e., their linear hull is dense in* Φ+*).*

*Then there exists a probability measure μ on* (*F*, *<sup>C</sup>σ*(*F*))*, where F is the dual of F and <sup>C</sup>σ*(*F*) *is the σ-algebra generated by cylinder sets in F such that, for μ*−*almost every η* ∈ *F there is a generalized common eigenvector ω*(*η*) ∈ Φ− *of the family* A, *corresponding to the common eigenvalue η* ∈ *F*, *that is for any ϕ* ∈D⊂ Φ+ *and A*(f) ∈ A

$$(\omega(\eta)|A(\mathbf{f})\boldsymbol{\varrho})\_{\Phi\_- \times \Phi\_+} = \eta(\mathbf{f})(\omega(\eta)|\boldsymbol{\varrho})\_{\Phi\_- \times \Phi\_+} \tag{19}$$

*with η*(f) ∈ R, *denoting here the result of the pairing between F and F*.*The mapping*

$$\mathcal{D} \ni |\varrho\rangle \longrightarrow (\omega(\eta)|\varrho)\_{\Phi\_- \times \Phi\_+} := \varrho(\eta) \in \mathbb{C} \tag{20}$$

*for any η* ∈ *F can be continuously extended to a unitary surjective operator* <sup>F</sup>*η* : Φ+ −→ *L*(*μ*) 2(*F*; C), *where*

$$\mathcal{F}\_{\eta} \left| \varrho \right> := \eta \left( \varrho \right) \tag{21}$$

*for any η* ∈ *F is a generalized Fourier transform, corresponding to the family* A. *Moreover, the image of the operator <sup>A</sup>*(f), f ∈ *F*, *under the* <sup>F</sup>*η- mapping is the operator of multiplication by the function F η* → *η*(f) ∈ R.

Now, if to construct the following self-adjoint family R := *ρ*(f) :<sup>=</sup> <sup>R</sup>*m ρ*(*x*)f(*x*)*dx* : f ∈ *F* of linear operators in the Hilbert space <sup>Φ</sup>*μ*, where *F* := S(R*<sup>m</sup>*; R) is the Schwartz functional space dense in *H*, one can derive, making use of Theorem 1, that there exists the generalized Fourier transform (21), such that

$$\Phi\_{\mu} = L\_2^{(\mu)}(F'; \mathbb{C}) \simeq \int\_{F'}^{\oplus} \Phi\_{(\eta)} d\mu(\eta) \tag{22}$$

for some Hilbert space sets <sup>Φ</sup>(*η*), *η* ∈ *F*, and a suitable measure *μ* on *F*, with respect to which the corresponding joint eigenvector *ω*(*η*) ∈ Φ− for any *η* ∈ *F* generates the Fourier transformed family {*η*(f) ∈ R : f ∈ *<sup>F</sup>*}. Moreover, if dim <sup>Φ</sup>*η* = 1 for all *η* ∈ *F*, the Fourier transformed eigenvector *ω*(*η*) := <sup>Ω</sup>(*η*) = 1 for all *η* ∈ *F*.

Now we will consider the family of self-adjoint operators *ρ*(f) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, f ∈ *F*, as generating a unitary family U := {*U*(f) : f ∈ *<sup>F</sup>*}, where the operator

$$\mathcal{U}(\mathbf{f}) := \exp[i\rho(\mathbf{f})] \tag{23}$$

is unitary, satisfying the abelian commutation condition

$$\mathcal{U}(\mathbf{f\_1})\mathcal{U}(\mathbf{f\_2}) = \mathcal{U}(\mathbf{f\_1} + \mathbf{f\_2}) \tag{24}$$

for any f1, f2 ∈ *F*. Since, in general, the unitary family U is defined in the Hilbert space <sup>Φ</sup>*μ*, not coinciding, in general with the canonical Fock type space, the important problem of describing its cyclic unitary representation spaces arises, within which the factorization jointly with relationships (17) hold for any f ∈ *F*. This problem can be treated using mathematical tools devised both within the representation theory of C∗-algebras [4,5,57,63] and the Gelfand–Vilenkin [66] approach. Below we will describe the main features of the Gelfand–Vilenkin formalism, being much more suitable for the task, providing a reasonably unified framework of constructing the corresponding representations. The next definitions will be used in our construction.

**Definition 1.** *Let F be a locally convex topological vector space, F*0 ⊂ *F be a finite dimensional subspace of F*. *Let F*<sup>0</sup> ⊆ *F be defined by*

$$F^0 := \{ \sigma \in F' \, : \, \sigma|\_{F\_0} = 0 \},$$

*and called the annihilator of F*0.

The quotient space *F*<sup>0</sup> := *F* /*F*<sup>0</sup> may be, evidently, identified with *F* 0 ⊂ *F*, the adjoint space of *F*0.

**Definition 2.** *Let Q* ⊆ *F*0; *then the subset*

$$X\_{F^0}^{(Q)} := \left\{ \sigma \in F' : \sigma + F^0 \subset Q \right\} \tag{26}$$

*is called the cylinder set with the base Q and the generating subspace F*0.

**Definition 3.** *Let n* = dim *F*0 = dim *F* 0 = dim *F*0. *One says that a cylinder set X*(*Q*) *has Borel base, if Q is a Borel set, when regarded as a subset of* R*<sup>m</sup>*.

The family of cylinder sets with Borel base forms an algebra of sets, which is a key stone for defining measurable sets in and the corresponding measures on *F* .

**Definition 4.** *The measurable sets in F are the elements of the σ-algebra generated by the cylinder sets with Borel base.*

**Definition 5.** *A cylindrical measure in F is a non-negative σ-pre-additive function μ defined on the algebra of cylinder sets with a Borel base and satisfying the conditions* 0 ≤ *μ*(*X*) ≤ 1 *for any X*, *μ*(*F*) = 1 *and μ <sup>j</sup>*∈<sup>N</sup> *Xj* = ∑*j*∈<sup>N</sup> *<sup>μ</sup>*(*Xj*), *if all sets Xj* ⊂ *F* , *j* ∈ N, *have a common generating subspace F*0 ⊂ *F*.

**Definition 6.** *A cylindrical measure μ satisfies the commutativity condition if, and only if, for any bounded continuous function, α* : R*n* −→ R *of n* ∈ N *real variables the function*

$$\mathfrak{a}\left[\mathfrak{f}\_1, \mathfrak{f}\_2, \dots, \mathfrak{f}\_n\right] := \int\_{F'} \mathfrak{a}\left(\eta(\mathfrak{f}\_1), \eta(\mathfrak{f}\_2), \dots, \eta(\mathfrak{f}\_n)\right) d\mu(\eta) \tag{27}$$

*is sequentially continuous in* f*j* ∈ *F*, *j* = 1, *m*.

**Remark 1.** *It is known [4,57,66] that in countably normalized spaces, the properties of sequential and ordinary continuity are equivalent.*

**Definition 7.** *A cylindrical measure μ is countably additive if, and only if, for any cylinder set X* = *<sup>j</sup>*∈<sup>N</sup> *Xj*, *which is the union of countably many mutually disjoints cylinder sets Xj* ⊂ *F* , *j* ∈ N, *μ*(*X*) = ∑*j*∈<sup>N</sup> *<sup>μ</sup>*(*Xj*).

The next two standard propositions [4,57,66,70,71], characterizing extensions of the measure *μ* on *X* = *<sup>j</sup>*∈<sup>N</sup> *Xj*, hold.

**Proposition 1.** *A countably additive cylindrical measure μ can be extended to a countably additive measure on the σ-algebra, generated by the cylinder sets with a Borel base. Such a measure will also be called a cylindrical measure.*

**Proposition 2.** *Let F be a nuclear space. Then, any cylindrical measure μ on F* , *satisfying the continuity condition, is countably additive.*

#### *2.2. Non-Relativistic Quantum Current Algebra and Its Cyclic Representations*

Based on the Fock space Φ*F*, defined by (18) and generated by the creation–annihilation operators (7) and (8), the current operator *J*(*x*) : Φ*F* → Φ*m F* , *x* ∈ R*<sup>m</sup>*, can be easily constructed as follows:

$$J(\mathbf{x}) = \frac{1}{2i} [a^+(\mathbf{x}) \,\nabla\_{\mathbf{x}} a(\mathbf{x}) - \nabla\_{\mathbf{x}} a^+(\mathbf{x}) \, a(\mathbf{x})],\tag{28}$$

satisfying jointly with the density operator *ρ*(*x*) : Φ*F* → Φ*F*, *x* ∈ R*<sup>m</sup>*, defined by (16), the following quantum current Lie algebra symmetry [4–7,59,68,72] relationships:

$$\left[f(\mathbf{g}\_1), f(\mathbf{g}\_2)\right] = i f(\left[\mathbf{g}\_1, \mathbf{g}\_2\right]), \ \left[\rho(\mathbf{f}\_1), \rho(\mathbf{f}\_2)\right] = 0,\tag{29}$$

$$\left[f(\mathbf{g}\_1), \rho(\mathbf{f}\_1)\right] = i\rho(\left(\mathbf{g}\_1 \middle| \nabla \mathbf{f}\_1\right)),$$

holding for all f1, f1 ∈ *F* and g1, g2 ∈ *Fm*, where we put, by definition,

$$
\langle \mathbf{g}\_1 \mathbf{g}\_2 \rangle := \langle \mathbf{g}\_1 | \nabla \rangle \ \mathbf{g}\_2 - \langle \mathbf{g}\_2 | \nabla \rangle \ \mathbf{g}\_1. \tag{30}
$$

being the usual commutator of vector fields g1|∇ and g2|∇ on the configuration space R*<sup>m</sup>*. It is easy to observe that the current algebra (29) is the Lie algebra G, corresponding to the Banach group *G* := Diff (R*m*) - *F*, the semidirect product of the Banach group of diffeomorphisms Diff (R*m*) of the *m*-dimensional space R*m* and the Abelian group *F*. As the Lie algebra Γ(R*m*) of smooth vector fields on R*m* with the Lie bracket (17) is isomorphic to the Lie algebra Diff(R*m*) of the Banach diffeomorphism group Diff (R*<sup>m</sup>*), it is natural to construct the corresponding unitary operators

$$V(\mathfrak{q}\_t^{\mathfrak{G}}) := \exp[i I(\mathfrak{g})],\tag{31}$$

on the Representation Hilbert space <sup>Φ</sup>*μ*, where for any g ∈ *Fm*, there holds *<sup>d</sup>ϕ*g*t* /*dt* = <sup>g</sup>(*ϕ*g*t* ), *ϕ*g*t* (*x*)|*<sup>t</sup>*=<sup>0</sup> = *x* ∈ R*<sup>m</sup>*, where *ϕ*g*t* ∈ Diff(R*<sup>m</sup>*), *t* ∈ R. The constructed above exponential currents (23) and (31) constitute together a unitary operator group on the Hilbert space Φ, endowed with the following composition law

$$\begin{aligned} \mathcal{U}(\mathbf{f}\_1)\mathcal{U}(\mathbf{f}\_2) &= \mathcal{U}(\mathbf{f}\_1 + \mathbf{f}\_2), \mathcal{V}(\boldsymbol{\varrho}\_1)\mathcal{V}(\boldsymbol{\varrho}\_2) = \mathcal{V}(\boldsymbol{\varrho}\_2 \circ \boldsymbol{\varrho}\_1), \\ \mathcal{V}(\boldsymbol{\varrho})\mathcal{U}(\mathbf{f}) &= \mathcal{U}(\mathbf{f} \circ \boldsymbol{\varrho})\mathcal{V}(\boldsymbol{\varrho}) \end{aligned} \tag{32}$$

for all f1, f2, f ∈ *F* and *ϕ*, *ϕ*2, *ϕ* ∈ Diff(R*<sup>m</sup>*). The operator group (32) is, evidently, isomorphic to the semidirect product group *G*, which is endowed, respectively, with the natural composition law

$$(\mathfrak{q}\_1, \mathfrak{k}\_1) \circ (\mathfrak{q}\_2, \mathfrak{k}\_2) = (\mathfrak{q}\_2 \circ \mathfrak{q}\_1, \mathfrak{k}\_1 + \mathfrak{k}\_2 \circ \mathfrak{q}\_1) \tag{33}$$

for all f1, f2 ∈ *F* and *ϕ*1, *ϕ*2 ∈ Diff(R*<sup>m</sup>*). Concerning a more adequate mathematical description of the Banach diffeomorphism group Diff(R*<sup>m</sup>*), it is useful to consider the subgroup Diff0(R*m*) of smooth diffeomorphisms of R*m* with compact supports, which is a topological space with the topology given by a counted family of the metrics ||*ϕ*1 − *ϕ*2||*n* := max|*k*|=0,*<sup>n</sup>* sup*x*∈R*<sup>m</sup>* (1 + |*x*|<sup>2</sup>)*n*|*ϕ*(*k*) 1 (*x*) − *ϕ*(*k*) 2 | for all *n* ∈ Z+ and *ϕ*1, *ϕ*2 ∈ Diff0(R*<sup>m</sup>*). So, the diffeomorphism group Diff(R*m*) can be defined as the completion of the space Diff0(R*m*) with respect to the topology introduced above. This way, the constructed group Diff(R*m*) is topological, locally linear connected and metrizable with a countable topology basis at each of its points. In particular, the group Diff(R*m*) contains diffeomorphisms with noncompact supports, ye<sup>t</sup> in the limit |*x*| → <sup>∞</sup>, *x* ∈ R*<sup>m</sup>*, they can be approximated by the identity mapping in Diff(R*<sup>m</sup>*). The latter makes it possible to state that for any g ∈ *Fm* the element *ϕ*g*t* ∈ Diff(R*m*) for all *t* ∈ R generates the uniform continuous mapping *Fm* g → *ϕ*g*t*∈ Diff(R*<sup>m</sup>*).

Proceeding now to the Banach group of currents *G* = Diff (R*m*) - *F*, we have that the separable Hilbert space <sup>Φ</sup>*μ* for every irreducible cyclic representation will be unitary equivalent to the Hilbert space (45), which in many physical applications reduces in the case dim <sup>Φ</sup>(*η*) = 1 for all *η* ∈ *F* to the following form:

$$\Phi\_{\mu} \simeq L\_2^{(\mu)}(F'; \mathbb{C}), \tag{34}$$

being the space of square integrable functions with respect to the measure *μ* on *F*.

Assume now that an element *ω* ∈ <sup>Φ</sup>*μ* is taken arbitrarily and consider the action of the Banach group of currents *G* on it:

$$\mathcal{U}(\mathbf{f})\omega(\boldsymbol{\eta}) = \exp[i(\boldsymbol{\eta}(\mathbf{f}))\omega(\boldsymbol{\eta})] \,\tag{35}$$

$$\mathcal{V}(\boldsymbol{\varrho})\omega(\boldsymbol{\eta}) = \chi\_{\boldsymbol{\varrho}}(\boldsymbol{\eta})\omega(\boldsymbol{\varrho}^\*\boldsymbol{\eta}) \left[\frac{d\mu(\boldsymbol{\varrho}^\*(\boldsymbol{\eta}))}{d\mu(\boldsymbol{\eta})}\right]^{1/2} \,\tag{36}$$

where, by definition, *ϕ*<sup>∗</sup>*η*(f) := *η*(<sup>f</sup> ◦ *ϕ*) for all f ∈ *F*, *dμ*(*ϕ*<sup>∗</sup>*η*) *dμ*(*η*) is the corresponding Radon– Nikodym derivative [19,73] of the measure *μ* ◦ *ϕ*∗ with respect to the measure *μ* on *F* and *χϕ*(*η*) is a complex-valued character of the unit norm, satisfying the relationship

$$
\chi\_{\mathfrak{P}\_2}(\eta)\chi\_{\mathfrak{P}\_1}(\mathfrak{p}\_2^\*\eta) = \chi\_{\mathfrak{P}\_1 \cap \mathfrak{P}\_2}(\eta) \tag{36}
$$

for all *ϕj* ∈ Diff(R*<sup>m</sup>*), *j* = 1, 2, *η* ∈ *F*. For the Radon–Nikodym derivative above to exist, the measure *μ* on *F* should be quasi-invariant with respect to the diffeomorphism group Diff(T*<sup>n</sup>*), that is, for any measurable set *Q* ⊂ *F* the condition *μ*(*Q*) = 0 if, and only if, *μ*(*ϕ*<sup>∗</sup>*Q*) for arbitrary *ϕ* ∈ Diff(R*<sup>m</sup>*).

In physics applications, the representation (35) is uniquely determined by the measure *μ* on *F*, which in the general case has a very complicated [20,72] structure, and its analytic construction is nontrivial. One of the fairly effective approaches to this problem is the quantum method of Bogolubov generating functionals developed in [2,6,7,20,74]. Another approach, which is of considerable interest for the theory of dynamical systems, is based on algebraic methods of constructing self-adjoint functional-operator representations of the original current Lie algebra (29). In particular, the representation (35), corresponding to a quantum-mechanical system of *N* ∈ N identical bose-particles localized at points *xj* ∈ R*<sup>m</sup>*, has a measure *μ* with supports [20,72] on Dirac delta-functions *η* := *ηN* ∈ *F* of the form:

$$\eta\_N(\mathbf{x}) = \sum\_{j \in \overline{1, N}} \delta(\mathbf{x} - \mathbf{x}\_j) \tag{37}$$

at any *x* ∈ R*m* with a measure *μ* of the form:

$$d\mu(\eta\_N) = \Omega\_N^\* \Omega\_N \prod\_{j=\overline{1,N}} d\mathbf{x}\_j \delta(\eta - \eta\_N(\mathbf{x})),\tag{38}$$

where Ω*N* ∈ Φ*N* - *L*(*s*) 2 ((R*<sup>m</sup>*)<sup>⊗</sup>*N*; C) is the corresponding symmetric ground-state wave function of the related quantum Hamiltonian system, satisfying the conditions (49) and (49), reduced on the invariant subspace Φ*N*. Moreover, the following general expressions hold: <sup>Ω</sup>(*η*) = 1 and for any *ω* ∈ *L*(*μ*) 2 (*F*; C)

$$\rho(\mathbf{x})\omega(\eta\_N) = \sum\_{j \in \overline{\mathbb{T}, \mathcal{N}}} \delta(\mathbf{x} - \mathbf{x}\_j)\omega(\eta\_N), \tag{39}$$

$$J(\mathbf{x})\omega(\eta\_N) = \frac{1}{2i} \sum\_{j \in \overline{\mathbb{T}N}} \left[ \delta(\mathbf{x} - \mathbf{x}\_j) \circ \partial / \partial \mathbf{x}\_j + \partial / \partial \mathbf{x}\_j \circ \delta(\mathbf{x} - \mathbf{x}\_j) \right] \omega(\eta\_N),$$

where, by definition, *<sup>ω</sup>*(*ηN*) ∈ Φ*N* - *L*(*s*) 2 ((R*<sup>m</sup>*)<sup>⊗</sup>*N*; C). As a simple consequence of the actions (39), one derives that

$$\mathcal{U}(f)\omega(\eta\_N) \,=\exp[i\sum\_{j\in\overline{1,N}}f(\mathbf{x}\_j)]\omega(\eta\_N),\tag{40}$$

$$V(\boldsymbol{\varrho})\boldsymbol{\omega}(\boldsymbol{\eta}\_N) = \boldsymbol{\omega}(\boldsymbol{\varrho}^\*\boldsymbol{\eta}\_N) \left[ \left| \det \left( \frac{\partial \boldsymbol{\varrho}(\boldsymbol{x})}{\partial \boldsymbol{x}} \right) \right| \right]^{1/2}.$$

where we put, for brevity, that the character *χϕ*(*ηN*) = 1 for all *ϕ* ∈ Diff(R*<sup>m</sup>*).

*2.3. The Generating Functional Equation, Cyclic Current Algebra Representation and Hamiltonian Operator Groundstate*

Concerning the Fourier transform of a cylindrical measure *μ* in *F*, we will use the following natural definitions.

**Definition 8.** *Let μ be a cylindrical measure in F*. *The Fourier transform of μ is the nonlinear functional*

$$\mathcal{L}(\mathbf{f}) := \int\_{F'} \exp[i\eta(\mathbf{f})] d\mu(\eta), \tag{41}$$

*coinciding with the characteristic functional of the measure μ*.

**Definition 9.** *The nonlinear functional* L : *F* −→ C *on F*, *defined by (41), is called positive definite, if, and only if, for all* f*j* ∈ *F and λj* ∈ C, *j* = 1, *n*, *the condition*

$$\sum\_{j,k=1}^{n} \bar{\lambda}\_j \mathcal{L}(\mathbf{f}\_k - \mathbf{f}\_j) \lambda\_k \ge 0 \tag{42}$$

*holds for any n* ∈ N.

The following important proposition, owing to Gelfand and Vilenkin [4,66], Araki [75] and Goldin [1,4], holds.

**Proposition 3.** *The functional* L : *F* −→ C *on F*, *defined by (41), is the Fourier transform of a cylindrical measure on F if, and only if, it is positive definite, sequentially continuous and satisfying the condition* L(0) = 1*. Suppose now that we have a continuous unitary representation of the unitary family* U *in a suitable Hilbert space* <sup>Φ</sup>*μ with a cyclic vector* |Ω) ∈ <sup>Φ</sup>*μ*. *Then we can put*

$$\mathcal{L}(\mathbf{f}) := (\Omega | \mathcal{U}(\mathbf{f}) | \Omega) \tag{43}$$

*for any* f ∈ *F* := S(R*<sup>n</sup>*; <sup>R</sup>), *being the Schwartz space on* R*<sup>m</sup>*, *and observe that functional (43) is continuous on F owing to the continuity of the representation. Therefore, this functional is the generalized Fourier transform of a cylindrical measure μ on F:*

$$\langle \Omega | \mathcal{U}(\mathbf{f}) | \Omega \rangle = \int\_{\mathcal{S}'} \exp[i\eta(\mathbf{f})] d\mu(\eta). \tag{44}$$

*From the spectral point of view, based on Theorem 1, there is an isomorphism between the Hilbert spaces* <sup>Φ</sup>*μ and L*(*μ*) 2 (*<sup>F</sup>*; C), *defined by* |Ω) −→ <sup>Ω</sup>(*η*) = 1 *and U*(f)|Ω) −→ exp[*<sup>i</sup>η*(f)] *and next extended by linearity upon the whole Hilbert space* Φ. *In the non-cyclic case, there exists a finite or countably infinite family of measures* {*μk* : *k* ∈ <sup>Z</sup>+} *on F*, *with* <sup>Φ</sup>*μ*- ⊕*k*∈Z+ *L*(*μ<sup>k</sup>* ) 2 (*F*; C) *and the unitary operator U*(f) :Φ*μ*−→Φ*μ for any* f ∈ *F corresponds in all L*(*μ<sup>k</sup>* ) 2 (*F*; C), *k* ∈ Z+, *to a multiplication operator on the exponent function* exp[*<sup>i</sup>η*(f)]. *This means that there exists a single cylindrical measure μ on F and a μ*− *measurable field of Hilbert spaces* <sup>Φ</sup>(*η*) *on F*, *such that*

$$\Phi\_{\mu} \simeq \int\_{F'}^{\ominus} \Phi\_{(\eta)} d\mu(\eta),\tag{45}$$

*with* U(f) : <sup>Φ</sup>*μ*−→Φ*μ*, *corresponding [66] to the operator of multiplication by* exp[*<sup>i</sup>η*(f)] *for any* f ∈ *F and η* ∈ *F*. *Thereby, having constructed the nonlinear functional (41) in an exact analytical form, one can retrieve the representation of the unitary family* U *on the corresponding Hilbert space* <sup>Φ</sup>*μ*, *as follows:* <sup>Φ</sup>*μ* = <sup>⊕</sup>*n*∈Z+Φ*<sup>n</sup>*, *where*

$$\Phi\_n = \prod\_{j=\overline{1,n}} \rho(\mathbf{x}\_j) |\Omega\rangle,\tag{46}$$

*for all n* ∈ N.

The cyclic vector |Ω) ∈ <sup>Φ</sup>*μ* can be, in particular, obtained as the ground state vector of some unbounded self-adjoint positive definite Hamiltonian operator H : <sup>Φ</sup>*μ*−→ <sup>Φ</sup>*μ*, commuting with the self-adjoint non-negative particle number operator

$$\mathcal{N} := \int\_{\mathbb{R}^m} d\mathbf{x} \rho(\mathbf{x}),\tag{47}$$

that is [H, N] = 0. Moreover, the conditions

$$\mathbb{H}|\Omega\rangle = 0\tag{48}$$

and

$$\inf\_{\varphi \in D\_{\mathcal{H}}} (\varphi | \mathcal{H} | \varphi) = (\Omega | \mathcal{H} | \Omega) = 0 \tag{49}$$

hold for the operator H : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ*, where *D*H denotes its domain of definition, dense in <sup>Φ</sup>*μ*. To find the functional (43), which is called the generating Bogolubov type functional for moment distribution functions

$$f\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) := (\Omega | : \rho(\mathbf{x}\_1)\rho(\mathbf{x}\_2)...\rho(\mathbf{x}\_n) : | \Omega \rangle,\tag{50}$$

where *xj* ∈ R*<sup>m</sup>*, *j* = 1, *n*, and the normal ordering operation: · : is defined [4,6,7,55,56,68] as

$$\rho: \rho(\mathbf{x}\_1)\rho(\mathbf{x}\_2)...\rho(\mathbf{x}\_n): = \prod\_{j=1}^n \left(\rho(\mathbf{x}\_j) - \sum\_{k=1}^{j-1} \delta(\mathbf{x}\_j - \mathbf{x}\_k)\right),\tag{51}$$

it is convenient first to choose the Hamilton operator H : Φ*F* → Φ*F* in the following secondly quantized [4,5,56] representation

$$\mathcal{H} := \frac{1}{2} \int\_{\mathbb{R}^m} \langle \nabla\_{\mathbf{x}} a^+(\mathbf{x}) | \nabla\_{\mathbf{x}} a(\mathbf{x}) \rangle d\mathbf{x} + \mathcal{V}(\rho), \tag{52}$$

on the related Fock space Φ*F* , where the sign "∇*x* means the usual gradient operation with respect to *x* ∈ R*m* in the Euclidean space E*<sup>m</sup>* - (R*<sup>m</sup>*; ·|· ). If the energy spectrum density of the Hamiltonian operator (52) on the cyclic representation Hilbert space <sup>Φ</sup>*μ* is bounded from below, in works done by Goldin G.A., Grodnik J., Menikov R. Powers R.T. and Sharp D. [4,5,76] it was stated that this Hamiltonian, modulo the ground state energy eigenvalue, can be algebraically represented on a suitably constructed *current algebra symmetry representation Hilbert space* <sup>Φ</sup>*μ*, as the positive definite gauge type factorized operator

$$\mathcal{H} = \frac{1}{2} \int\_{\mathbb{R}^m} \left\langle \left( \mathbf{K}^+(\mathbf{x}) - \mathbf{A}(\mathbf{x}; \boldsymbol{\rho}) \right) \middle| \boldsymbol{\rho}^{-1}(\mathbf{x}) \left( \mathbf{K}(\mathbf{x}) - \mathbf{A}(\mathbf{x}; \boldsymbol{\rho}) \right) \right\rangle d\mathbf{x},\tag{53}$$

satisfying conditions (48) and (49), where <sup>A</sup>(*x*; *ρ*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*<sup>m</sup>μ* , *x* ∈ R*<sup>n</sup>*, is some specially constructed [68,77] linear self-adjoint operator, satisfying the condition

$$\mathbf{K}(\mathbf{x})|\Omega\rangle = \mathbf{A}(\mathbf{x};\rho)|\Omega\rangle\tag{54}$$

= ∑*n*∈Z+ 1*n*! <sup>R</sup>*m*×*<sup>n</sup>*

with the ground state |Ω) ∈ <sup>Φ</sup>*μ*, corresponding to chosen potential operators <sup>V</sup>(*ρ*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ*. The singular structure of the operator (53) was previously analyzed in detail in [2] where, in part, its well-posedness was showed.

The "*potential*" operator <sup>V</sup>(*ρ*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ* is, in general, a polynomial (or analytical) functional of the density operator *ρ*(*x*) : <sup>Φ</sup>*μ*−→ <sup>Φ</sup>*μ* for any *x* ∈ R*<sup>m</sup>*, and the operator <sup>K</sup>(*x*) <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*<sup>m</sup>μ* is defined as

$$\mathbf{K}(\mathbf{x}) := \nabla\_{\mathbf{x}} \rho(\mathbf{x})/2 + i\mathbf{l}(\mathbf{x}),\tag{55}$$

:

where the self-adjoint "current" operator *J*(*x*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*<sup>m</sup>μ* can be naturally defined (but non-uniquely) from the continuity equality

$$\partial \rho / \partial t = \mathrm{i}[\mathrm{H}, \rho(\mathbf{x})] = -\langle \nabla | f(\mathbf{x}) \rangle,\tag{56}$$

holding for all *x* ∈ R*<sup>m</sup>*. Such an operator *J*(*x*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*<sup>m</sup>μ* , *x* ∈ R*<sup>m</sup>*, can exist owing to the commutation condition [H, N] = 0, giving rise to the continuity relationship (56), if, additionally, to take into account that supports supp *ρ* of the density operator *ρ*(*x*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ*, *x* ∈ R*<sup>m</sup>*, can be chosen arbitrarily, owing to the independence of (56) on the potential operator <sup>V</sup>(*ρ*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ*, but its strict dependence on the corresponding representation (45).

**Remark 2.** *The self-adjointness of the operator* <sup>A</sup>(g; *ρ*) : <sup>Φ</sup>*μ*<sup>→</sup> <sup>Φ</sup>*μ*, g ∈ *F*, *can be stated following schemes from works [5,68,72] under the additional existence of such a linear anti-unitary mapping* T : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ that the following invariance conditions hold:*

$$\mathbf{T}\rho(\mathbf{x})\mathbf{T}^{-1} = \rho(\mathbf{x}), \qquad \mathbf{T}\ f(\mathbf{x})\ \mathbf{T}^{-1} = -f(\mathbf{x}), \qquad \mathbf{T}|\Omega\rangle = |\Omega\rangle \tag{57}$$

*for any x* ∈ R*<sup>m</sup>*. *Thereby, owing to conditions (57), the following equalities*

$$\mathbf{K}(\mathbf{x})|\Omega\rangle = \mathbf{A}(\mathbf{x};\rho)|\Omega\rangle\tag{58}$$

*hold for any x* ∈ R*<sup>m</sup>*, *giving rise to the self-adjointness of the operator* <sup>A</sup>(g; *ρ*) :Φ*μ*−→ <sup>Φ</sup>*μ*, g ∈ *Fm*.

It is easy to observe that the time-reversal condition (57) imposes the real value relationship for the real valued ground state Ω*N* = Ω*N* ∈ Φ*N* - *L*(*s*) 2 (R*<sup>m</sup>*×*N*; C) of the canonically represented *<sup>N</sup>*-particle Hamiltonian H*N* : Φ*N* → Φ*N* for arbitrary *N* ∈ N. Moreover, taking into account the relationship (58), one can easily observe that on the invariant subspace Φ*N* ⊂ Φ*F*, the operator <sup>K</sup>(*x*) : Φ*N* −→ Φ*N* is representable as

$$\mathcal{K}\_{\mathcal{N}}(\mathbf{x}) = \sum\_{j=\overline{1,\mathcal{N}}} \delta(\mathbf{x} - \mathbf{x}\_{j}) \frac{\partial}{\partial \mathbf{x}\_{j}},\tag{59}$$

entailing the following expression for the related operator *AN*(*x*; *ρ*) : Φ*N* → Φ*N* on the subspace Φ*N* ⊂ Φ :

$$A\_N(x;\rho) = \sum\_{j=\overline{1,N}} \delta(x - x\_j) \nabla\_{x\_j} \ln|\Omega\_N(x\_1, x\_2, \dots, x\_N)|.\tag{60}$$

The latter makes it possible to derive its secondly quantized [56,57,78,79] expression as

$$\mathbf{A}(\mathbf{x};\boldsymbol{\rho}) = \int\_{\mathbb{R}^{m \times N}} d\mathbf{x}\_2 d\mathbf{x}\_3...d\mathbf{x}\_N : \boldsymbol{\rho}(\mathbf{x})\boldsymbol{\rho}(\mathbf{x}\_2)\boldsymbol{\rho}(\mathbf{x}\_3)...\boldsymbol{\rho}(\mathbf{x}\_N) : \nabla\_{\mathbf{x}}\ln|\Omega\_N(\mathbf{x},\mathbf{x}\_2,...,\mathbf{x}\_N)|, \quad \text{(61)}$$

which holds for any *x* ∈ R*m* and arbitrary *N* ∈ Z+. Being interested in the infinite particle case when *N* → <sup>∞</sup>, the expression (61) can be naturally decomposed [77,79] as

> :

$$\begin{aligned} \mathcal{A}(\mathbf{x}; \rho) &:= \rho(\mathbf{x}) \nabla \frac{\delta}{\delta \rho(\mathbf{x})} \mathcal{W}(\rho) = \\\\ \mathcal{dy}\_1 \mathcal{dy}\_2 \dots \mathcal{dy}\_n &: \rho(\mathbf{x}) \rho(\mathbf{y}\_1) \rho(\mathbf{y}) \rho(\mathbf{y}\_3) \dots \rho(\mathbf{y}\_n) : \nabla\_{\mathbf{x}} \mathcal{W}\_{n+1}(\mathbf{x}; \mathbf{y}\_1, \mathbf{y}\_2, \dots, \mathbf{y}\_n), \end{aligned} \tag{62}$$

 *y*1, *y*2, ...,

50

 :

where the corresponding real-valued coefficients *Wn* ∈ *H*(1) 2 (R*<sup>m</sup>*×*<sup>n</sup>*; R) should be such functions that the series (62) were convergen<sup>t</sup> in a suitably chosen representation Fock space Φ*F*, for which the resulting ground state lim*N*→∞ Ω*N* - |Ω) ∈ Φ*F* is necessarily cyclic and normalized.

Based now on the construction above, one easily deduces from expression (55) that the generating Bogolubov type functional (43) obeys for all *x* ∈ R*m* the following functionaldifferential equation:

$$\left[\nabla\_{\mathbf{x}} - i\nabla\_{\mathbf{x}}\mathbf{f}\right] \frac{1}{2i} \frac{\delta \mathcal{L}(\mathbf{f})}{\delta \mathbf{f}(\mathbf{x})} = \mathbf{A}\left(\mathbf{x}; \frac{1}{i} \frac{\delta}{\delta \mathbf{f}}\right) \mathcal{L}(\mathbf{f}),\tag{63}$$

whose solutions should satisfy [3,74] the Fourier transform representation (44), and which were, in part, studied in [74]. In particular, a wide class of special so-called Poissonian white noise type solutions to the functional-differential Equation (63) was obtained in [5,61,62,68,71,80] by means of functional-operator methods in the following generalized form:

$$\mathcal{L}(\mathbf{f}) = \exp\left\{2\int\_{\mathbb{R}^m} \mathcal{W}\left(\frac{1}{i}\frac{\delta}{\delta\mathbf{f}}\right) d\mathbf{x}\right\} \exp\left(\rho\int\_{\mathbb{R}^m} \{\exp[i\mathbf{f}(\mathbf{x})] - 1\} d\mathbf{x}\right),\tag{64}$$

where *ρ*¯ = (Ω|*ρ*|Ω) ∈ R+ is a suitable Poisson process parameter and the operator <sup>A</sup>(*x*; *ρ*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*<sup>m</sup>μ* , *x* ∈ R*<sup>m</sup>*, resulting from the expression (62) for some scalar operator <sup>W</sup>(*ρ*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*.

**Remark 3.** *It is worth remarking here that solutions to Equation (63) realize the suitable physically motivated representations of the abelian Banach subgroup F of the Banach group G* = Diff(R*m*) - *F*, *mentioned above. In the general case of this Banach group G one can also construct [5,6,16,81] a generalized Bogolubov type functional equation, whose solutions give rise to suitable physically motivated representations of the corresponding current Lie algebra* G.

Recalling now the Hamiltonian operator representation (53), one can readily deduce that the following weak representation Hilbert space <sup>Φ</sup>*μ* weak relationship

$$\left( \left< \mathbf{A} \middle| \boldsymbol{\rho}^{-1} \mathbf{A} \right> - \left< \mathbf{K}^\* \middle| \boldsymbol{\rho}^{-1} \mathbf{A} \right> - \left< \mathbf{A} \middle| \boldsymbol{\rho}^{-1} \mathbf{K} \right> \right) / 2 - \mathbf{V}(\boldsymbol{\rho}) = \left. \varepsilon \mathbf{0} \right> \tag{65}$$

where 0 ∈ R is the corresponding ground state energy density value. Thus, the main analytical problem is now reduced to constructing the expansion (62) corresponding to a suitable cyclic representation Hilbert space <sup>Φ</sup>*μ* of the quantum current algebra (29), compatible with the Hamiltonian operator structure (52).

**Remark 4.** *Here we mention that the operator* <sup>K</sup>(*x*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*<sup>m</sup>μ* , *x* ∈ R*<sup>m</sup>*, *defined by (55), relates to that from the work [4,5,76] via scaling* <sup>K</sup>(*x*) → <sup>K</sup>(*x*)/2, *x* ∈ R*<sup>m</sup>*.

#### *2.4. The Hamiltonian Operator Reconstruction and the Cyclic Current Algebra Representation*

We will assume that we are given a Banach current group *G* = Diff(R*m*) - *F* cyclic representation in a Hilbert space <sup>Φ</sup>*μ* with respect to *F* with a cyclic vector |Ω) ∈ Φ+ ⊂ <sup>Φ</sup>*μ*. Based on the well known Araki reconstruction theorem [5,75] for the canonical Weyl commutation relations, we can first readily obtain from (56) that

$$\left[\mathbf{H}, \mathcal{U}(\mathbf{f})\right] = f(\nabla \mathbf{f})\mathcal{U}(\mathbf{f}) - 1/2\rho(\langle \nabla \mathbf{f}\_1 | \nabla \mathbf{f}\_2 \rangle)\mathcal{U}(\mathbf{f}),\tag{66}$$

where *U*(f) = exp[*<sup>i</sup>ρ*(f)], f ∈ *F*, is an element of the unitary family U. The expression (66) makes it possible to calculate the bilinear form

$$\begin{array}{c} (\mathcal{U}(\mathbf{f\_1})\boldsymbol{\Omega}|\mathbf{H}|\mathcal{U}(\mathbf{f\_2})\boldsymbol{\Omega}) = (\mathcal{U}(\mathbf{f\_1})\boldsymbol{\Omega}|\mathcal{J}(\nabla \mathbf{f\_1})|\mathcal{U}(\mathbf{f\_2})\boldsymbol{\Omega}) - \\ -1/2(\mathcal{U}(\mathbf{f\_1})\boldsymbol{\Omega}|\rho(\langle \nabla \mathbf{f\_1}|\nabla \mathbf{f\_2}))|\mathcal{U}(\mathbf{f\_2})\boldsymbol{\Omega}) \end{array} \tag{67}$$

for any f1, f2 ∈ *F*. Taking into account the symmetry properties (57), we finally deduce from (67) that for arbitrary functions f1, f2 ∈ *F*

$$\mathcal{I}\left(\mathcal{U}(\mathbf{f}\_1)\Omega|\mathcal{H}|\mathcal{U}(\mathbf{f}\_2)|\Omega\right) = 1/2(\mathcal{U}(\mathbf{f}\_1)\Omega|\rho(\langle\nabla\mathbf{f}\_1|\nabla\mathbf{f}\_2\rangle))\mathcal{U}(\mathbf{f}\_2)\Omega).\tag{68}$$

The standard reasonings make it possible to state that the bilinear symmetric form (68) determines on <sup>Φ</sup>*μ* a self-adjoint non-negative definite Hamiltonian operator H : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, densely defined on the domain *D*H := *span* f ∈*F* { exp[*<sup>i</sup>ρ*(f)]|Ω) ∈ <sup>Φ</sup>*μ*}. Really, for any set of

functions f*j* ∈ *F*, *j* = 1, *n*, the following inequalities

$$\sum\_{j,k=\overline{1,n}} \mathbb{s}\_{j} \mathbf{s}\_{k} \langle \nabla \mathbf{f}\_{j} | \nabla \mathbf{f}\_{k} \rangle \ge 0, \quad \sum\_{j,k=\overline{1,n}} \mathbb{s}\_{j} \mathbf{s}\_{k} (\mathbf{U}(\mathbf{f}\_{j}) \Omega | \rho(\mathbf{x}) | \mathbf{U}(\mathbf{f}\_{k}) \Omega) \ge 0 \tag{69}$$

hold for any complex numbers *sj* ∈ C, *j* = 1, *n*, and arbitrary *n* ∈ N. Since, for any non-negative definite complex matrices *A*, *B* ∈ End R*<sup>n</sup>*, the matrix *C* := {*AjkBjk* : *j*, *k* = 1, *n*} ∈ End C*n* proves to be non-negative definite [75,82] too, one ensures that the bilinear form (69) is also non-negative definite. Then, as follows from the classical Friedrichs' theorem [69,83–85], there exists a self-adjoint densely defined and non-negative definite operator H : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*.

*2.5. Current Algebra Representations, Generating Functional Method and Ergodicity of the Hilbert Space Representation Measure*

In view of the importance of the current algebra representations of the Banach group *G* = Diff (R*m*) - *F* for physics applications, we consider their construction by means of the generating functional method [6,7,20,86]. From the very beginning, let us introduce a governing definition in connection with this method.

**Definition 10.** *A generating functional on a group G is a complex-valued function E on G with the following conditions: (1) E*(1) = 1, 1 ∈ *G*; *(2) <sup>E</sup>*(*<sup>a</sup>*1 exp(*tA*)*<sup>a</sup>*2) *is a continuous function of the parameter t* ∈ R *for all A* ∈ G *and a*1, *a*2 ∈ *G; (3) the matrix <sup>E</sup>*(*a*<sup>−</sup><sup>1</sup> *k aj*), *k*, *j* = 1, *N*, *is positive definite for any N* ∈ N;

The following theorem [75] holds.

**Theorem 2.** *The function E is a generating functional on G if, and only if, there exists a continuous unitary representation π* : *G* → Aut(<sup>Φ</sup>*μ*) *on a separable Hilbert space* (<sup>Φ</sup>*μ*;(·|·)) *with a cyclic vector* Ω ∈ <sup>Φ</sup>*μ*, *such that*

$$E(a) = \left(\Omega \middle| \pi(a)\Omega\right) \tag{70}$$

*holds for all a* ∈ *G*.

The vector Ω ∈ <sup>Φ</sup>*μ* is said to be cyclic with respect to the representation *π* : *G* → *Aut*(<sup>Φ</sup>*μ*), if the set {*π*(*a*)<sup>Ω</sup> : *a* ∈ *G*} is complete in <sup>Φ</sup>*μ*, i.e., is dense in <sup>Φ</sup>*μ*, if taken together with its linear combinations over C. The significance of this theorem is that one can implicitly construct unitary representations of the Banach current group *G* = Diff(R*m*) - *F* and, thus, the current Lie algebra G by means of an appropriately defined generating functional on *G*. This is important, since frequently the latter problem is much simpler than the initial problem.

We now consider the representation *π* : *G* → Aut(<sup>Φ</sup>*μ*), restricted to the Abelian subgroup *F* in the group *G* = Diff (R*m*) - *F* and its corresponding generating functional L(f), f ∈ *F*, in the form

$$\mathcal{L}(f) := (\Omega | \exp[i\rho(\mathbf{f})] \Omega) = \int\_{F'} d\mu(\eta) \exp[i\eta(f)],\tag{71}$$

where the cyclic vector Ω ∈ <sup>Φ</sup>*μ* is normalized to unity: (Ω|Ω) = 1. In many physically interesting cases [6,7,20,86] the expression (71) can be replaced by means of the following equivalent trace-representation:

$$\mathcal{L}(\mathbf{f}) = \text{Tr}(P \exp[i\rho(\mathbf{f})]),\tag{72}$$

where *P* : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* is the corresponding so called statistical operator, depending on the Hamiltonian operator H : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*. The constructed above generating functional (71) should possess the following necessary properties: (1) L(f) = L(−f) for all f ∈ *F*; (2) L(0) = 1; (3) |L(f)| ≤ 1 for all f ∈ *F*; (4) L(f) is a positive definite functional on *F* : the inequality ∑*j*,*k*=1,*<sup>N</sup> <sup>c</sup>*¯*k*L(f*k* − <sup>f</sup>*j*)*cj* ≥ 0 holds for all *cj* ∈ C, *j* = 1, *N*, and arbitrary *N* ∈ N. As one can show, a generating functional L : *F* → C, satisfying the properties (1)–(4) always defines [66] a measure *μ* on *F*, defining the searched for unitary representation *π* : *G* → Aut(<sup>Φ</sup>*μ*) of the Abelian subgroup *F* of the current Banach group *G* = Diff(R*<sup>m</sup>*)- *F* . If the measure *μ* is in addition quasi-invariant and the factors *χϕ*(*η*) in (35) are known for all *ϕ* ∈ Diff(R*<sup>m</sup>*), *η* ∈ *F*, the corresponding representation of the current Lie algebra G is completely determined. Yet, if being interested only by irreducible representations of the current Banach algebra G, it is well known [66] that the corresponding measure *μ* on *F* is ergodic for the diffeomorphism subgroup Diff(R*<sup>m</sup>*), that is for any measurable and invariant subset *Q* ⊂ *F* either *μ*(*Q*) = 0, or *μ*(*F*\*Q*) = 0. Moreover, an arbitrary invariant set is in the general case a nondenumerable union of a family of mutually non-intersecting orbits. Assuming that the orbits containing an invariant subset *Q* ⊂ *F* are measurable, we obtain that there exist only two possibilities for ergodicity of the cylindrical measure *μ* on *F* : either it is concentrated on one orbit, or each orbit has zero measure, and these two possibilities really occur in applications. For instance, the case when the measure is concentrated on functionals of the form (37) leads to irreducibility of the generating functional representation on the Hilbert space *<sup>L</sup>*2(R*mN*; C) for any finite *N* ∈ N.

#### *2.6. The Creation–Annihilation Heisenberg Algebra, Its Coherent State Representations and Linearization of Nonlinear Dynamical Systems on Hilbert Spaces*

It is well known [87,88] that the representation theory of the quantum current algebra in a separable Hilbert space <sup>Φ</sup>*μ* is very close to the cyclic Hilbert space representations of the canonical creation–annihilation operator Heisenberg algebra H family {*a*+(f), *a*(f) : Φ*F* → Φ*F* : f ∈ *<sup>F</sup>*}, defined on the Fock space Φ*F*. The coherent states, being venerable objects in physics, were invented by Schrëdinger [89], as far back as in 1926, in the context of the quantum harmonic oscillator, they seemed to have lapsed into oblivion for some obscure reasons. About thirty-five years later, they were rediscovered, almost simultaneously, by Glauber [90], Klauder [91,92] and Sudarshan [93], in the context of a quantum optical description of coherent light beams emitted by lasers. Since then, coherent states have pervaded nearly all branches of quantum physics—including, of course, quantum optics in the study of lasers, nuclear, atomic and solid state physics, quantum electrodynamics, quantization and dequantization problems and path integrals, to mention just a few. For original references, the reader is referred to the review [88] and reprint volume of Klauder and Skagerstam [94]. In many of these applications, the question naturally poses itself as to whether it might not be possible to find other families of states, sharing some properties of the original or canonical coherent states, emanating from the quantum oscillator and which could possibly be useful to ye<sup>t</sup> other areas of physics.

Already, in 1926, Schrëdinger had tried unsuccessfully to construct coherent states appropriate to the hydrogen atom problem. This was motivated by the quasi-classical character of the canonical coherent states which made them very desirable for studying the quantization of classical dynamical systems, a point which we discuss in some detail below. The key to the generalization of the notion of a coherent state was the observation by Perelomov [95] and independently by Gilmore [96,97], that the construction of the oscillator coherent states could be reformulated as a problem in group representation theory: the canonical coherent states could be obtained by acting on the oscillator ground state with the operators of a unitary representation of the group generated by the creation and annihilation operators, namely the Weyl–Heisenberg group.

The link between the Schrëdinger and the Perelomov approaches is the uniqueness theorem [98,99] of von Neumann for the quantum mechanics of a system with finitely many degrees of freedom. In addition, a unitary representation *T* : *G* → *U*(*G*) of a compact symmetry group *G* in a separable Hilbert space Φ, used for building up the system of canonical coherent states, has the property of square integrability with respect to the left (or right) invariant Haar measure on *G*. Furthermore, the physical states, associated with the coherent states, are not indexed by elements of *G* itself, but by points in the coset space *G*/*Gc*, where *Gc* is the Cartan subgroup of *G* and is isomorphic to the torus.

Since its introduction in 1972, the concept of coherent states was widely exploited [14–16,87,100,101] in many fields of mathematical physics, whose leading idea consisting of considering the translates of a fixed cyclic vector under a group action is as old as the celebrated Gel'fand-Raikov theorem [66] on locally bicompact groups. Their common properties, namely that the related homogeneous space has a complex homogeneous structure, and the corresponding representation Hilbert space can be identified in the coherent state basis with a space of holomorphic functions on the homogeneous space. As it was stated in [87], a homogeneous complex structure is actually present quite generally, and on the basis of the homogeneous complex structure, the related homogeneous manifolds are just the classical phase spaces on which the group acts through canonical transformations. From this point of view, coherent states can be interpreted just as probability wave packets over the classical phase space, that is a well-known result for the harmonic oscillator coherent states. The converse problem, i.e., the construction of irreducible unitary representations of the group, starting from its phase space realization, was considered in [102] and found a definite mathematical setting.

To look at the coherent vector representation problem within the Fock type space, its main idea becomes very transparent and motivative owing to the classical Bargmann– Segal [103] construction. Namely, there is considered the Hilbert space

$$\mathcal{H}\_k := \{ f \in H(\mathbb{C}^k) : \int\_{\mathbb{C}^n} |f(z)|^2 d\mu(z) \tag{73}$$

of holomorphic functions *<sup>H</sup>*(C*<sup>k</sup>*), *k* ∈ N, with the scalar product (*f* |*g*) := C*<sup>k</sup> f*(*z*)*g*(*z*)*dμ*(*z*) for arbitrary *f* , *g* ∈ H*k* with respect to the measure *dμ*(*z*) = *π*<sup>−</sup>*<sup>k</sup>* exp(− *z*|*z* ) *dz*¯∧*dz* (2*i*) *k* for *z* ∈ C*k*. It is easy to observe that the Hilbert space H*k* is the direct sum of the symmetric polynomial subspaces H(*s*) *k*,*s* ∈ Z+:

$$\mathcal{H}\_k = \ominus\_{s=0}^{\infty} \mathcal{H}\_k^{(s)},\tag{74}$$

where, by definition,

$$\mathcal{H}\_k^{(s)} := \{ \sum\_{s=n\_1+n\_2+\ldots,n\_k} c\_{n\_1 n\_2 \ldots n\_k} z\_1^{n\_1} z\_2^{n\_2} \ldots z\_k^{n\_k} : c\_{n\_1 n\_2 \ldots n\_k} z\_j \in \mathbb{C}, j = \overline{1,k} \}. \tag{75}$$

Moreover, it is easy to check that the polynomials

$$x\_{n\_1 n\_2 \dots n\_k}(z) = \prod\_{j=\overline{1,k}} \frac{z\_j^{n\_j}}{\sqrt{n\_j!}} \tag{76}$$

form a complete and orthogonal base in H*k*, that is (*en*1*n*2...*nk* |*em*1*m*2...*mk* ) = ∏ *j*=1,*k <sup>δ</sup>njmj* . The next important observation, made by V. Bargmann, was the point boundedness of any function *f* ∈ H*k* : | *f*(*z*)| ≤ || *f* || exp( *z*|*z* /2), *z* ∈ C*<sup>n</sup>*. Really, for any *f* ∈ H*k* there holds the expansion

$$f(z) = \sum\_{s \in \mathbb{Z}\_+} \sum\_{s=n\_1+n\_2+\ldots n\_k} (c\_{n\_1 n\_2 \ldots n\_k} | f) \prod\_{j=1,k} \frac{z\_j^{n\_j}}{\sqrt{n\_j!}} \tag{77}$$

from which one easily ensues, owing to the closedness property and Schwartz inequality on *l*2(C), that

$$|f(z)| \le \left(\sum\_{s \in \mathbb{Z}\_+} \sum\_{s=n\_1+n\_2+\ldots, n\_k} |(c\_{n\_1 n\_2 \ldots n\_k} | f)|^2 \right)^{1/2} \times$$

$$\times \left(\sum\_{s \in \mathbb{Z}\_+} \sum\_{s=n\_1+n\_2+\ldots, n\_k} \prod\_{j=1,k} \frac{|z\_j|^{2n\_j}}{n\_j!} \right)^{1/2} =:= ||f||\exp(\langle z|z\rangle/2)$$

The latter makes it possible to define for any *u* ∈ C*n* the following dual to (78) bounded functional

*u*

*u* ˆ(*f*) := *f*(*u*), ||*u*<sup>ˆ</sup>|| ≤ exp( *u*|*u* /2) (79)

on H*k*, whose Riesz representation

$$
\Psi(f) = (h\_u|f) \tag{80}
$$

defines the unique element *hu* ∈ H*k*, or equivalently

$$f(u) = \int\_{\mathbb{C}^k} \overline{h\_{\mu}(\xi^x\_{\sharp})} f(\xi^x\_{\sharp}) \exp\left(-\langle \xi^x\_{\sharp} | \xi^x\_{\sharp} \rangle \right) \frac{d\xi^x \wedge d\xi^x\_{\sharp}}{\left(2i\right)^k}.\tag{81}$$

Taking into account the orthogonality of the base vectors (76) in H*k*, it is easy to calculate that the vector *hu*(*ξ*) = exp( *u*|*z* ) ∈ H*k*, whose norm ||*hu*|| = exp( *u*|*u* /2) for any *u* ∈ C*n* and which is called the "*coherent vector*". It is worth remarking here that the function representation (81) is well known in the operator theory [104,105] and is called the "reproducing kernel" representation with the kernel *hu* ∈ H*k*, *u* ∈ C*k*.

The Hilbert space H*k*, as the direct sum (74) of symmetrical polynomial subspaces, possesses the Fock space structure, allowing the introduction of the creation operators *a*+*j* : H(*s*) *k* → H(*s*+<sup>1</sup>) *k* for any *j* = 1, *k* and all *s* ∈ Z+ as multiplication operators: for any *f* ∈ H(*s*) *k a*+*j f*(*z*) := *zj f*(*z*) for any *j* = 1, *k* and all *s* ∈ Z+. The corresponding adjoint expressions *a*+*j* ∗ := *aj* : H(*s*) *k* → H(*<sup>s</sup>*−<sup>1</sup>) *k* act as *aj f*(*z*) = *<sup>∂</sup>*/*∂zj f*(*z*) on arbitrary *f* ∈ H(*s*) *k* for any *j* = 1, *k* and all *s* ∈ Z+, where, by definition, H(0) *k* - C. Now one can easily check that the coherent vector *hu* = exp( *u*|· ) ∈ H*k* for any *u* ∈ C*<sup>k</sup>* is a common eigenvector of the annihilation operators *aj* : H*k* → H*k*, *j* = 1, *k*:

$$
a\_j h\_u(z) = 
u\_j h\_u(z) \tag{82}$$

with the eigenvalues *uj* ∈ C, *j* = 1, *k*. It is important also to mention here that the creation– annihilation operators defined above satisfy the canonical commutation relationships:

$$[a\_{\dot{\prime}}, a\_n] = 0 = [a\_{\dot{\prime}}^+, a\_n^+]\_{\prime} \ [a\_{\dot{\prime}}, a\_n^+] = \delta\_{\dot{\prime}, n} \tag{83}$$

for all *j*, *n* = 1, *k*.

The coherent vector representation scheme described above can be respectively generalized to arbitrary symmetric Fock space Φ that will be effectively used in the sections proceeding below. Returning back to the algebraic properties of coherent states, we proceed to describing their unbelievable and impressive applications to theory of nonlinear dynamical systems on Hilbert spaces, their linearization and integrability, previously initiated

in [14,15] and continued in [16]. We briefly reviewed the cyclic Hilbert space representations of the quantum Heisenberg algebra and presented a general approach to constructing the coherent states and their applications both to the linearization of nonlinear dynamical systems on Hilbert spaces, and to describing their complete integrability. The latter is developed using the modern Lie-algebraic approach [11,17–19] to nonlinear dynamical systems on Poissonian functional manifolds, and proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces.

Jointly with the cyclic Hilbert space representations of the Heisenberg algebra H, we briefly reviewed the closely related cyclic Hilbert space density representations [4,6,87,88] of the canonical quantum current algebra G on the circle S1, whose vector field representations on smooth spatially one-dimensional functional manifolds coincide exactly with the related symmetry algebra of completely integrable nonlinear Hamiltonian systems on these manifolds. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on smooth spatially one-dimensional functional manifolds. The algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are also described.

#### *2.7. The Canonical Heisenberg Algebra and Its Cyclic Hilbert Space Representations*

Let (<sup>Φ</sup>;(·|·)) be a separable Hilbert space, *F* be a topological real linear space and A := {*A*(f) : f ∈ *F*} a family of commuting self-adjoint operators in Φ (i.e., these operators commute in the sense of their resolutions of the identity) with dense in Φ domain *Dom A*(f) := *DA*(f), f ∈ *F*. Consider the Gelfand rigging [57,61,66] of the Hilbert space Φ, i.e., a chain

$$\mathcal{D} \subset \Phi\_+ \subset \Phi \subset \Phi\_- \subset \mathcal{D}' \tag{84}$$

in which Φ+ is a Hilbert space, topologically (densely and continuously) and quasi-nucleus (the inclusion operator *i* : Φ+ −→ Φ is of the Hilbert-Schmidt type) embedded into Φ, the space Φ− is the dual to Φ+ as the completion of functionals on Φ+ with respect to the norm ||f||− := sup ||*u*||+=<sup>1</sup> |(f|*u*)Φ|, *u* ∈ Φ, a linear dense in Φ+ topological space D ⊆ Φ+ is such that

D ⊂ *DA*(f) ⊂ Φ and the mapping *A*(f) : D → Φ+ is continuous for any f ∈ *F*. Then, owing to the structural theorem (1) there exists a cyclic representation of the canonical creation– annihilation Heisenberg operator algebra H family {*a*+(f), *a*(f) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* : f ∈ *F*} on the separable Hilbert space <sup>Φ</sup>*μ*, whose generalized Fourier transform is given by the expression

$$\Phi\_{\mu} = L\_2^{(\mu)}(F'; \mathbb{C}) \simeq \int\_{F'}^{\oplus} \Phi\_{(\eta)} d\mu(\eta) \tag{85}$$

for some Hilbert space sets <sup>Φ</sup>*η*, *η* ∈ *F*, and a suitable measure *μ* on *F*, with respect to which the corresponding joint eigenvector *ω*(*η*) ∈ Φ− for any *η* ∈ *F* generates the Fourier transformed family {*η*(f) ∈ R : f ∈ *<sup>F</sup>*}. Moreover, if dim <sup>Φ</sup>*η* = 1 for all *η* ∈ *F*, the Fourier transformed eigenvector *ω*(*η*) := <sup>Ω</sup>(*η*) = 1 for all *η* ∈ *F* .

Next, we will consider the family of self-adjoint operators *<sup>P</sup>*(f), *Q*(g) : <sup>Φ</sup>*η* → <sup>Φ</sup>*η* : f, g ∈ *F* , as generating a unitary Heisenberg group

$$\mathfrak{H} := \{ \exp(i P(\mathbf{f})), V(\mathbf{g}) = \exp(i Q(\mathbf{g})) : \\ \tag{86} $$

$$P := \left( a^+ + a \right) / 2, \mathbf{Q} := i \left( a - a^+ \right) / 2, \mathbf{f}, \mathbf{g} \in F, \}$$

satisfying the commutation conditions

$$\mathcal{U}(\mathbf{f})V(\mathbf{g}) = \exp(-i(f|\mathbf{g}))V(\mathbf{g})\mathcal{U}(\mathbf{f}),\tag{87}$$

$$\mathcal{U}(\mathbf{f}')\mathcal{U}(\mathbf{g}) = \mathcal{U}(\mathbf{f}+\mathbf{g}),\ V(\mathbf{f}')V(\mathbf{g}) = V(\mathbf{f}+\mathbf{g}'),$$

for any f, g ∈ *F*. Since, in general, the unitary Heisenberg group H is defined on a representation Hilbert space <sup>Φ</sup>*μ*, not coinciding, in general, with the canonical Fock type space

Φ*F*, the important problem of describing its cyclic unitary representation spaces arises, within which the factorization (86) jointly with relationships (87) should hold. Below, we will briefly describe only the main features of the Gelfand–Vilenkin formalism, being much more suitable for the task, providing a reasonably unified framework of constructing the corresponding cyclic representations of the family A := {*Q*(f) : f ∈ *F*} of commuting self-adjoint operators in a separable Hilbert space Φ.

Proceeding now to the Heisenberg group H, the separable Hilbert space <sup>Φ</sup>*μ* for its every irreducible representation will be unitary equivalent to the Hilbert space (45), which in many physical applications reduces in the case dim <sup>Φ</sup>(*η*) = 1 for all *η* ∈ *F* to the following form:

$$\Phi\_{\mu} \simeq L\_2^{(\mu)}(F'; \mathbb{C}), \tag{88}$$

.

being the space of square integrable functions with respect to the measure *μ* on *F* 

Assume now that an element *ω* ∈ <sup>Φ</sup>*μ* is taken arbitrarily and consider [75] the action of the Heisenberg group H on it:

$$\mathcal{U}(\mathbf{f})\omega(\boldsymbol{\eta}) = \exp[i(\boldsymbol{\eta}(\mathbf{f}))\omega(\boldsymbol{\eta})] \,\tag{89}$$

$$\mathcal{V}(\mathbf{g})\omega(\boldsymbol{\eta}) = \chi\_{\mathbb{g}}(\boldsymbol{\eta})\omega(\boldsymbol{\eta} + \mathbf{g}) \left[\frac{d\mu(\boldsymbol{\eta} + \mathbf{g})}{d\mu(\boldsymbol{\eta})}\right]^{1/2} \,\tag{90}$$

where, by definition, for any f ∈ *F* the expression *<sup>d</sup>μ*(*η*+g) *dμ*(*η*) at *η* ∈ *F* means the corresponding Radon–Nikodym derivative [19,73] of the measure *μ*(◦ + g) with respect to the measure *μ* on *F* and *<sup>χ</sup>*g(*η*) is a complex-valued character of the unit norm, satisfying the relationship

$$
\chi\_{\mathbf{f}}\left(\boldsymbol{\eta}\right)\chi\_{\mathbb{R}}(\boldsymbol{\eta}+\mathbf{f}) = \chi\_{\mathbf{f}+\mathbf{g}}(\boldsymbol{\eta})\tag{90}
$$

for all f, g ∈ *F* ⊂ *H* and arbitrary *η* ∈ *F*. For the Radon–Nikodym derivative above to exist, the measure *μ* on *F* should be quasi-invariant with respect to the shift group elements {*F η* → *η* + g ∈ *F* }, that is, for any measurable set *Q* ⊂ *F* the condition *μ*(*Q*) = 0 if, and only if, *μ*(*Q* + g) for arbitrary g ∈ *F* ⊂ *F* .

**Definition 11.** *A vector* |*u*) ∈ <sup>Φ</sup>*μ is called a coherent vector state in the representation Hilbert space* <sup>Φ</sup>*μ with respect to an element u* ∈ *H* - *<sup>L</sup>*2(R*<sup>m</sup>*; <sup>R</sup>*<sup>k</sup>*), *if it satisfies the eigenfunction condition*

$$u\_j(\mathbf{x})|\mu\rangle = u\_j(\mathbf{x})|\mu\rangle\tag{91}$$

*for each j* = 1, *k and all x* ∈ R*<sup>m</sup>*.

It is easy to check that for any *u* ∈ *H* the coherent ket-vector |*u*) ∈ <sup>Φ</sup>*μ* exists: really, the following vector expression

$$|u\rangle := \exp[(u|a^+)\_H]|\Omega\rangle\tag{92}$$

where Ω ∈ Φ+ ⊂ <sup>Φ</sup>*μ* is a cyclic vector for the creation–annihilation operator algebra family {*a*+(f), *a*(f) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* : f ∈ *F*} and satisfies the defining condition (91), where the operator *a*+(*u*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ*, *u* ∈ *H*, action ensues from the determining condition (19): for any *ϕ* ∈ <sup>Φ</sup>*μ* there exists a unique vector *<sup>ω</sup>*(*ηa*) ∈ <sup>Φ</sup>*μ* for which

$$(\omega(\eta\_a)|(a^+(\mu)\varrho)\_\mu = \eta\_a(\mu)\left(\omega(\eta\_a)|\varrho\right)\_\mu\tag{93}$$

for all *u* ∈ *H*. Moreover, as the Hilbert space *H* ⊂ *F*, the eigenvalue *ηa*(*u*) ∈ R is bounded jointly with the Hilbert space <sup>Φ</sup>*μ* norm

$$||u|| := (u|u)^{1/2} = \exp(\frac{1}{2}||u||\_H^2) < \infty,\tag{94}$$

since *u* ∈ *H* and its Hilbert space norm *uH* is *a priori* bounded.

Consider now any function *u* ∈ *H* and observe that the Hilbert spaces embedding mapping

$$\emptyset \,:\, H \ni u \longrightarrow |u\rangle \in \Phi\_{\mu\prime} \tag{95}$$

defined by means of the coherent vector expression (92), realizes a smooth isomorphism between the Hilbert spaces *H* and the image *ξ*(*H*) ⊂ <sup>Φ</sup>*μ*. The inverse mapping *ξ*−<sup>1</sup> : *ξ*(*H*) ⊂ <sup>Φ</sup>*μ* −→ *H* is given by the following exact expression:

$$(u|\eta)\_H = (\Omega|a(\eta)|u)/(\Omega|u),\tag{96}$$

holding for any *η* ∈ *H*. Owing to condition (94), one finds from (96) and the classical Riesz type theorem [85,106] that the corresponding function *u* ∈ *H*.

Let now define on the Hilbert space *H* a nonlinear in general dynamical system (which can, in general, be non-autonomous) in partial derivatives

$$d\mathfrak{u}/dt = \mathbb{K}[\mathfrak{u}]\_\prime \tag{97}$$

where *t* ∈ R+ is the corresponding evolution parameter, [*u*] := (*x*; *u*, *ux*, *uxx*, ...,) ∈ *J*(*k*)(R*<sup>m</sup>*; R*s*) belongs to the jet-space *J*(*k*)(R*<sup>m</sup>*, R*n*) of the order *k* ∈ Z+, and, in general, a nonlinear mapping *K* : *H* −→ *H* is Frechet smooth. Assume also that the corresponding Cauchy problem

$$
u|\_{t=\!+0} = \mathfrak{u}\_0\tag{98}$$

for the nonlinear dynamical system (97) is solvable in the Hilbert space *H* for any *u*0 ∈ *H* on an interval [0, *T*) ⊂ <sup>R</sup>1+ for some *T* > 0. Thus, there is determined a smooth evolution mapping

$$T\_l: H \ni \mathfrak{u}\_0 \longrightarrow \mathfrak{u}(t|\mathfrak{u}\_0) \in H,\tag{99}$$

for all *t* ∈ [0, *<sup>T</sup>*). Now, it is natural to consider the following commuting diagram:

$$\begin{array}{ccccc} H & \stackrel{\xi}{\longrightarrow} & \Phi\_{\mu} \\ T\_{t} \downarrow & & \downarrow \stackrel{\tau}{\mathcal{T}\_{t}} \\ H & \stackrel{\overline{\xi}}{\longrightarrow} & \Phi\_{\mu\nu} \end{array} \tag{100}$$

where the mapping T*t* : <sup>Φ</sup>*μ* −→ <sup>Φ</sup>*μ*, *t* ∈ [0, *<sup>T</sup>*), is defined from the conjugation relationship on the image *ξ*(*H*) ⊂ <sup>Φ</sup>*μ* of the mapping (95):

$$
\xi \circ T\_t = \mathcal{T}\_t \circ \xi \tag{101}
$$

Now take coherent vector |*<sup>u</sup>*0) ∈ <sup>Φ</sup>*μ*, corresponding to the Cauchy data *u*0 ∈ *H*, and construct the vector

$$|\mathfrak{u}\rangle := \mathcal{T}\_t \cdot |\mathfrak{u}\_0\rangle \in \Phi\_\mu \tag{102}$$

for all *t* ∈ [0, *<sup>T</sup>*). Since the vector (102) is, by construction, coherent, that is

$$a\_j(\mathbf{x})|\mu\rangle := u\_j(\mathbf{x}, t|\mu\_0\rangle|\mu\rangle\tag{103}$$

for each *j* = 1, *k*, *t* ∈ [0, *T*) and almost all *x* ∈ R*<sup>m</sup>*, owing to the smoothness of the mapping *ξ* : *H* −→ <sup>Φ</sup>*μ* with respect to the corresponding norms in the Hilbert spaces *H* and <sup>Φ</sup>*μ*, we derive that the coherent vector (102) is differentiable with respect to the evolution parameter *t* ∈ [0, *<sup>T</sup>*). Thus, one can easily find [14,15] that

$$\frac{d}{dt}|u\rangle = \mathbb{K}(a^+,a)|u\rangle,\tag{104}$$

where

$$|u\rangle|\_{t=\bar{\phantom{e}}+0} = |u\_0\rangle\tag{105}$$

and an operator mapping <sup>K</sup>(*a*+, *a*) : <sup>Φ</sup>*μ* −→ <sup>Φ</sup>*μ* is defined by means of the exact analytical expression

$$\mathbb{K}(a^+,a) := (a^+|\mathbb{K}[a])\_H. \tag{106}$$

As a result of the consideration above we obtain the following theorem.

**Theorem 3.** *Any smooth nonlinear dynamical system (97) in Hilbert space H is representable by means of the Hilbert spaces embedding isomorphism ξ* : *H* −→ <sup>Φ</sup>*μ via the completely linear form (104).*

We now make some comments concerning the solution to the linear Equation (104) under the Cauchy condition (105) in the case of the Fock representation space Φ*F*. Since any vector |*ω*) ∈ Φ*F* allows the series representation

$$\begin{split} |\omega\rangle = \bigoplus\_{n=\sum\_{j=1}^{k}n\_{j}\in\mathbb{Z}\_{+}} \int\_{(\mathbb{R}^{m})^{n}} \omega\_{n\_{1}n\_{2}\dots n\_{k}}^{(n)}(\mathbf{x}\_{1}^{(1)}, \mathbf{x}\_{1}^{(2)}, \dots, \mathbf{x}\_{1}^{(n\_{1})}) ; \\ \mathbf{x}\_{2}^{(1)}, \mathbf{x}\_{2}^{(2)}, \dots, \mathbf{x}\_{2}^{(n\_{2})}; \dots; \mathbf{x}\_{k}^{(1)}, \mathbf{x}\_{k}^{(2)}, \dots, \mathbf{x}\_{k}^{(n\_{m})}) \prod\_{j=1}^{k} \left( \frac{1}{\sqrt{n\_{j}!}} \prod\_{k=1}^{n\_{j}} d\mathbf{x}\_{k}^{(j)} a\_{j}^{+}(\mathbf{x}\_{j}^{(k)}) \right) |\Omega\rangle, \end{split} \tag{107}$$

where for any *n* = <sup>∑</sup>*kj*=<sup>1</sup> *nj* ∈ N functions

$$\omega\_{n\_1 n\_2 \ldots n\_k}^{(n)} \in \bigotimes\_{j=1}^k L\_2^{(s)}((\mathbb{R}^m)^{n\_j}; \mathbb{C}) \simeq L\_2^{(s)}(\mathbb{R}^{mn\_1} \times \mathbb{R}^{mn\_2} \times \ldots \mathbb{R}^{mn\_k}; \mathbb{C}),\tag{108}$$

its Fock space norm is easily calculated as

$$\left\|\omega\right\|^2 = \sum\_{n=\sum\_{j=1}^k n\_j \in \mathbb{N}} \left\|\omega\_{n\_1 n\_2 \dots n\_k}^{(n)}\right\|\_2^2. \tag{109}$$

For the case of the coherent vector |*u*) ∈ Φ*F* its norm is easily obtained as ||*u*|| = exp(*u*<sup>2</sup>*H*/2), coinciding with the result (94). Moreover, substituting (107) into Equation (104), reduces (104) to an infinite recurrent set of linear evolution equations in partial derivatives on coefficient functions (108). The latter can often be solved [14] step by step analytically in exact form, thereby, making it possible to obtain, owing to representation (96), the exact solution *u* ∈ *H* to the Cauchy problem (98) for our nonlinear dynamical system in partial derivatives (97).

Concerning possible applications of nonlinear dynamical systems like (95) in mathematical physics, it is very important to construct their so called conservation laws or smooth invariant functionals *γ* : *H* −→ R on the Hilbert space *H*. Making use of the quantum mathematics technique described above, one can sugges<sup>t</sup> an effective algorithm for constructing these conservation laws in exact form.

Indeed, consider a vector |*γ*) ∈ <sup>Φ</sup>*μ*, satisfying the linear equation:

$$\frac{\partial}{\partial t}|\gamma\rangle + \mathcal{K}^\*(a^+, a)|\gamma\rangle = 0. \tag{110}$$

Then, the following proposition [14,15] holds.

**Proposition 4.** *The functional*

$$\gamma := (\mathfrak{u}|\gamma) \tag{111}$$

*is a conservation law for dynamical system (95), that is*

$$d\gamma/dt|\_{K} = 0\tag{112}$$

*along all orbits of the evolution mapping (99).*

It is interesting to reanalyze the dynamical system (104) from the Lie-algebraic point of view [11,19] and represent it as a coadjoint canonical Hamiltonian flow on the corresponding adjoint space to the Hilbert space <sup>Φ</sup>*μ*, considered as a Lie algebra over the field C. To do this, it is necessary to define the related Lie commutator on the Hilbert space <sup>Φ</sup>*μ* : for any vectors |*<sup>K</sup>α*) := <sup>K</sup>*α*(*a*+, *a*)∗|*ω*) ∈ <sup>Φ</sup>*μ* and |*<sup>K</sup>β*) := <sup>K</sup>*β*(*a*+, *a*)∗|*ω*) ∈ <sup>Φ</sup>*μ*, where <sup>K</sup>*α*(*a*+, *a*)<sup>∗</sup> and <sup>K</sup>*β*(*a*+, *a*)<sup>∗</sup> ∈ End <sup>Φ</sup>*μ* are smooth mappings and a central vector |*ω*) ∈ <sup>Φ</sup>*μ* is chosen to be fixed, their commutator, defined as

$$[(\mathbb{K}\_{\emptyset}), (\mathbb{K}\_{\emptyset})] := [\mathbb{K}\_{\emptyset}(a^+, a)^\*, \mathbb{K}\_{\emptyset}(a^+, a)^\* | \omega), \tag{113}$$

allows the construction of the related co-adjoint action <sup>Φ</sup><sup>∗</sup>*μ* (*l*| → *ad*∗|*<sup>K</sup>α*)(*u*| ∈ <sup>Φ</sup><sup>∗</sup>*μ* of a vector |*<sup>K</sup>α*) ∈ <sup>Φ</sup>*μ* on a fixed element (*l*| ∈ <sup>Φ</sup><sup>∗</sup>*μ*, where for any vector |*η*) ∈ <sup>Φ</sup>*μ* there holds the following identity:

$$(ad^\*\_{|\mathbb{K}\_a})(l|\eta) = -(l|(\mathbb{K}\_a), \eta|). \tag{114}$$

The latter makes it possible to define on the adjoint space <sup>Φ</sup><sup>∗</sup>*μ* the classical Lie–Poisson bracket for any smooth functionals *α* := (*l*|*α*) and *β* := (*l*|*β*) ∈ <sup>D</sup>(Φ<sup>∗</sup>*μ*):

$$\{a, \beta\} := (l | [\text{grad } a(l), \text{grad } \beta(l)]\_{\theta} | \omega) = (l | \text{grad } a(l) \theta(a^+) \text{grad } \beta(l) | \omega), \tag{115}$$

where, by definition, *<sup>ϑ</sup>*(*a*+) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* is some skew-symmetric Poisson operator, the element (*l*| := (*l*(*u*)| ∈ <sup>Φ</sup><sup>∗</sup>*μ* for any *u* ∈ *H* is interpreted as the corresponding momentum mapping *H u l* → (*l*(*u*)| : = (*u*| ∈ <sup>Φ</sup><sup>∗</sup>*μ* for the Poissonian action of the Lie algebra <sup>Φ</sup>*μ* on the Hilbert space *H*:

$$\Phi\_{\mu} \times H \ni (\text{grad } \gamma(u) \times u) \to \ K\_{\gamma}(u) \in \Phi\_{\mu} \tag{116}$$

with <sup>K</sup>*γ* := (*a*<sup>+</sup>| *<sup>K</sup>γ*[*a*])*<sup>H</sup>* ∈ End <sup>Φ</sup>*μ*, *<sup>K</sup>γ*[*a*]<sup>∗</sup> = −*<sup>ϑ</sup>*(*a*+) grad *γ*(*l*) : <sup>Φ</sup>*μ* → <sup>Φ</sup>*μ* for arbitrary *γ* ∈ <sup>D</sup>(Φ<sup>∗</sup>*μ*). The related action

$$(\operatorname{grad}\gamma(u)\times u) \, = K\_{\gamma}[u] \tag{117}$$

is a Hamiltonian vector field on *H*, generated by the corresponding Hamiltonian vector field *<sup>K</sup>γ* : *H* → *H* on the Hilbert space *H* commonly with the invariant Hamiltonian function *γ* = (*u*|*γ*) ∈ <sup>D</sup>(*H*). Simultaneously, the flow (117) for any *γ* ∈ <sup>D</sup>(Φ<sup>∗</sup>*μ*) naturally generates the linear flow on the adjoint space <sup>Φ</sup><sup>∗</sup>*μ* - <sup>Φ</sup>*μ* in the form +

$$nd^\*\_{\vartheta(a^+)|\gamma}(\mu| = (\mu|\mathbb{K}^\*\_{\gamma}. \tag{118})$$

Moreover, one easily checks that the commutator of vector fields <sup>K</sup>*α*|*u*) and <sup>K</sup>*β*|*u*) ∈ <sup>Φ</sup>*μ* equals

$$[\mathbb{K}\_{\mathfrak{a}}(\mu), \mathbb{K}\_{\mathfrak{k}}(\mu)] := [\mathbb{K}\_{\mathfrak{a}}, \mathbb{K}\_{\mathfrak{k}}](\mu) \tag{119}$$

for any smooth conservation laws *α* = (*u*|*α*) and *β* = (*u*|*β*) ∈ D(*H*) of the dynamical system (104), easily following from the evident conditions <sup>K</sup>∗|*α*) = 0 and K∗ |*μ*) = 0. Consider now the following representations of the gradient vectors grad *α*(*u*) = |*α*(*a*+)|*ω*) and grad *β*(*u*) = |*β*(*a*+)|*ω*) for some fixed central element |*ω*) ∈ <sup>Φ</sup>*μ*. Then, the Poisson bracket (115) for any *α*, *β* ∈ D(*H*) is representable as

$$\begin{split} \{\mathfrak{a},\mathfrak{\beta}\} &= -(\mathfrak{u}|[\mathbf{K}\_{\mathfrak{a}}^{\*}|\omega), \mathbf{K}\_{\mathfrak{\beta}}^{\*}|\omega)|\_{\mathfrak{\beta}}) = (\mathfrak{u}|[\mathbf{K}\_{\mathfrak{a}}, \mathbf{K}\_{\mathfrak{\beta}}]^{\*}|\omega) = \\ &= (\mathfrak{u}|\mathbf{K}\_{\{\mathfrak{a},\mathfrak{\beta}\}}^{\*}|\omega) = -(\mathfrak{u}|\mathfrak{\theta}\,\mathbf{grad}\{\mathfrak{a},\mathfrak{\beta}\}), \end{split} \tag{120}$$

being completely compatible with the Poissonian action of the Lie algebra <sup>Φ</sup>*μ* on the Hilbert space *H*. The obtained result can be summarized as the following theorem.

**Theorem 4.** *If the momentum mapping H u l* → (*l*(*u*)| : = (*u*| ∈ Φ∗ *μ*, *related with the nonlinear dynamical system (*(97) *on the Hilbert space H, is Poissonian, then all its symmetries (117), generated by smooth invariants γ* ∈ D(Φ∗ *μ*), *are represented as linear Hamiltonian flows (118) on the adjoint Hilbert space* Φ∗ *μ* - <sup>Φ</sup>*μ with respect to the canonical Lie–Poisson bracket (120).*

The theorem above plays a decisive role in constructing within the suitably modified Adler–Kostant–Souriau [11,19] scheme integrable Hamiltonian flows on the adjoint space Φ∗ *μ*, equivalent to nonlinear integrable Hamiltonian systems on the functional Hilbert space *H*.
