*3.4. Conclusions*

In this Section, we analyzed the algebraic structure of symmetries of nonlinear integrable infinite-dimensional integrable Hamiltonian dynamical systems. It was stated that the Banach group of currents Di*f f*(S<sup>1</sup>) - *C*∞(S1; R) on the circle S1 is a universal symmetry group of all completely integrable bi-Hamiltonian systems. Applications of this phenomenon to the problem of constructing effective criteria of integrability of nonlinear dynamical systems of theoretical and mathematical physics are presented.

#### **4. The Current Algebra Representations and the Factorized Structure of Quantum Integrable Many-Particle Hamiltonian Systems**

*4.1. The Current Algebra Representation and the Hamiltonian Reconstruction of the Calogero–Moser–Sutherland Quantum Model*

The periodic Calogero–Moser–Sutherland quantum bosonic model on the finite interval [0, *l*] - R/{[0, *l*]Z} is governed by the *<sup>N</sup>*-particle Hamiltonian

$$H\_N := -\sum\_{j=\overline{1,N}} \frac{\partial^2}{\partial \mathbf{x}\_j^2} + \sum\_{j \neq k=\overline{1,N}} \frac{\pi^2 \beta(\beta - 1)}{l^2 \sin^2[\frac{\pi}{l}(\mathbf{x}\_j - \mathbf{x}\_k)]} \tag{165}$$

on the symmetric Hilbert space *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C), where *N* ∈ Z+ and *β* ∈ R is an interaction parameter. As it was stated in very interesting and highly speculative works [132,133], there exists linear differential operators

$$\mathcal{D}\_{\dot{I}} := \frac{\partial}{\partial \mathbf{x}\_{\dot{I}}} - \frac{\pi \beta}{l} \sum\_{k=\overline{1,N}, k \neq \dot{I}} c t \mathbf{g} \left[ \frac{\pi}{l} (\mathbf{x}\_{\dot{I}} - \mathbf{x}\_{k}) \right] \tag{166}$$

for *j* = 1, *N*, such that the Hamiltonian (165) is factorized as the bounded from below symmetric operator

$$H\_N = \sum\_{j=\overline{1,N}} \mathcal{D}\_j^+ \mathcal{D}\_j + E\_{n\_\prime} \tag{167}$$

where

$$E\_N = \frac{1}{3} \left(\frac{\pi \beta}{l}\right)^2 N(N^2 - 1) \tag{168}$$

is the ground state energy of of the Hamiltonian operator (165), that there exists such a vector |<sup>Ω</sup>*N*) ∈ *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C), satisfying for any *N* ∈ Z+ the eigenfunction condition

$$H\_N|\Omega\_N\rangle = E\_N|\Omega\_N\rangle\tag{169}$$

and equals

$$\left(\left|\Omega\_N\right>\right) = \prod\_{j$$

coinciding with the corresponding Bethe anzatz representation [134,135] for the groundstate of the quantum Calogero–Moser-Sutherland model (165).

Being additionally interested in proving the quantum integrability of the Calogero–Moser–Sutherland model (165), we will proceed to its second quantized representation [9,10,13,56,57,60,68,135,136] and studying it by means of the density operator representation approach to the current algebra, described above in Section 2 and devised previously in [1,4–7,76,77].

The secondly quantized form of the Calogero–Moser–Sutherland Hamiltonian operator (165) looks as

$$\mathcal{H} = \int\_0^l dx \psi\_x^+(\mathbf{x}) \psi\_x(\mathbf{x}) + \left(\frac{\pi}{l}\right)^2 \beta(\beta - 1) \int\_0^l dx \int\_0^l dy \frac{\psi^+(\mathbf{x}) \psi^+(y) \psi(y) \psi(\mathbf{x})}{\sin^2[\frac{\pi}{l}(\mathbf{x} - y)]},\tag{171}$$

acting on the corresponding Fock space Φ*F* := <sup>⊕</sup>*n*∈Z+Φ(*s*) *n* , Φ(*s*) *n* - *L*(*s*) 2 ([0, *l*]*<sup>n</sup>*; C), *n* ∈ Z+. To proceed to the current algebra representation of the Hamiltonian operator (171), it would

useful to recall the factorized representation (167) and construct preliminarily the following singular Dunkl type [132,133,137,138] symmetrized differential operator

$$D\_{N}(\mathbf{x}) := \sum\_{j=1,\overline{\lambda}} \delta(\mathbf{x} - \mathbf{x}\_{j}) \frac{\partial}{\partial \mathbf{x}\_{j}} - \tag{172}$$

$$- \frac{\pi \mathcal{E}}{2l} \sum\_{j \neq k=\overline{1,N}} \Big( \delta(\mathbf{x} - \mathbf{x}\_{j}) \text{ctg} \Big[ \frac{\pi}{l} (\mathbf{x}\_{j} - \mathbf{x}\_{k}) \Big] + \delta(\mathbf{x} - \mathbf{x}\_{k}) \text{ctg} \Big[ \frac{\pi}{l} (\mathbf{x}\_{k} - \mathbf{x}\_{j}) \Big] \Big)$$

on the Hilbert space *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C), *N* ∈ Z+, parametrized by a running point *x* ∈ R/{[0, *l*]Z}. The corresponding secondly quantized representation of the operator (172) looks as

$$\mathbf{D}(\mathbf{x}) = \boldsymbol{\psi}^{+}(\mathbf{x})\boldsymbol{\psi}\_{\mathbf{}}(\mathbf{x}) - \frac{\pi\beta}{l} \int\_{0}^{l} d\mathbf{y} \,\mathrm{c}\mathbf{t} \mathbf{g} [\frac{\pi}{l}(\mathbf{x} - \mathbf{y})] : \boldsymbol{\psi}^{+}(\mathbf{x})\boldsymbol{\psi}^{+}(\mathbf{y})\boldsymbol{\psi}(\mathbf{y})\boldsymbol{\psi}(\mathbf{x}):\tag{173}$$

for any *x* ∈ R/[0, *l*]<sup>Z</sup>, or on the density operator *ρ* : Φ*F* → Φ*F* representation form, as

$$\begin{aligned} \mathbf{D}(\mathbf{x}) &= \nabla\_{\mathbf{x}} \rho(\mathbf{x})/2 + i\mathbf{j}(\mathbf{x}) - \\\\ -\frac{\pi \mathfrak{k}}{2\mathbb{I}} \int\_0^l dy \left[ \mathrm{c}t g[\frac{\pi}{\mathsf{I}}(\mathbf{x} - \mathbf{y})] : \rho(\mathbf{x})\rho(\mathbf{y}) : - \mathrm{c}t \mathrm{g}[\frac{\pi}{\mathsf{I}}(\mathbf{y} - \mathbf{x})] : \rho(\mathbf{y})\rho(\mathbf{x}) : \right], \end{aligned} \tag{174}$$

which is equivalently representable in a suitable current algebra symmetry representation Hilbert space Φ, as

$$\begin{aligned} \mathbf{D}(\mathbf{x}) &= \mathbf{K}(\mathbf{x}) - \\\\ -\frac{\pi\mathfrak{k}}{2\mathsf{I}} \int\_{0}^{l} dy \Big[ \mathrm{c}t \mathrm{g} \big[ \mp(\mathbf{x} - \mathbf{y}) \big] : \rho(\mathbf{x})\rho(\mathbf{y}) : - \mathrm{c}t \mathrm{g} \big[ \mp(\mathbf{y} - \mathbf{x}) \big] : \rho(\mathbf{y})\rho(\mathbf{x}) : \Big]. \end{aligned} \tag{175}$$

Now, based on the operator (174), one can formulate [10] the following proposition.

**Proposition 7.** *The secondly quantized Calogero–Moser–Sutherland Hamiltonian operator (171) in a suitable current algebra symmetry representation Hilbert space* Φ *is weakly equivalent to the factorized Hamiltonian operator*

$$
\hat{\mathbf{H}} = \int\_0^l d\mathbf{x} \mathbf{D}^+(\mathbf{x}) \rho(\mathbf{x})^{-1} \mathbf{D}(\mathbf{x}) \tag{176}
$$

*modulo the ground state energy operator* E : Φ → Φ, *where*

$$\mathbf{E} = \frac{1}{3} \left( \frac{\pi \beta}{l} \right)^2 : \mathbf{N}^3 : + \left( \frac{\pi \beta}{l} \right)^2 : \mathbf{N}^2 :,\tag{177}$$

*where, as before,*

$$\mathbf{N} := \int\_0^l \rho(\mathbf{x})d\mathbf{x} \tag{178}$$

*is the particle number operator, and satisfies the determining conditions*

$$(\mathbf{H} - \mathbf{E})|\Omega\rangle = 0, \ \mathbf{D}(\mathbf{x})|\Omega\rangle = 0 \tag{179}$$

*on the suitably renormalized groundstate vector* |Ω) ∈ Φ *for all x* ∈ R/[0, *l*]<sup>Z</sup>. *Moreover, for any integer N* ∈ Z+ *the corresponding projected vector* |<sup>Ω</sup>*N*) := |Ω)|<sup>Φ</sup>*N exactly coincides with the* *related Bethe groundstate vector for the N-particle Calogero–Moser–Sutherland model (165) and satisfies the following eigenfunction relationships:*

$$\begin{aligned} \mathrm{N}|\Omega\_{N}\rangle &= N|\Omega\_{N}\rangle, \; \mathrm{E}|\Omega\_{N}\rangle = \left(\frac{1}{3}\left(\frac{\pi\beta}{l}\right)^{2} : \mathrm{N}^{3} : + \left(\frac{\pi\beta}{l}\right)^{2} : \mathrm{N}^{2} :\right)|\Omega\_{N}\rangle = \\\\ &= \left[\frac{1}{3}\left(\frac{\pi\beta}{l}\right)^{2}(N^{3} - 3N^{2} + 2N) + \left(\frac{\pi\beta}{l}\right)^{2}N(N-1)\right]|\Omega\_{N}\rangle = \\\\ &= \left[\frac{1}{3}\left(\frac{\pi\beta}{l}\right)^{2}(N^{3} - 3N^{2} + 2N + 3N^{2} - 3N)\right]|\Omega\_{N}\rangle = \\\\ &= \left[\frac{1}{3}\left(\frac{\pi\beta}{l}\right)^{2}N(N^{2}-1)\right]|\Omega\_{N}\rangle := E\_{N}|\Omega\_{N}\rangle, \end{aligned}$$

*exactly ensuing the result (168).*

**Remark 5.** *When deriving the expression (180), we have used the identities*

$$
\rho(\mathbf{x})\rho(\mathbf{y}) = \, :
\rho(\mathbf{x})\rho(\mathbf{y}) : + \rho(\mathbf{y})\delta(\mathbf{x} - \mathbf{y}),
$$

$$
\rho(\mathbf{x})\rho(\mathbf{y})\rho(\mathbf{z}) = \, :
\rho(\mathbf{x})\rho(\mathbf{y})\rho(\mathbf{z}) : + :
\rho(\mathbf{x})\rho(\mathbf{y}) : \delta(\mathbf{y} - \mathbf{z}) + \tag{181}
$$

$$
\rho: \rho(\mathbf{y})\rho(\mathbf{z}) : \delta(\mathbf{z} - \mathbf{x}) + :
\rho(\mathbf{z})\rho(\mathbf{x}) : \delta(\mathbf{x} - \mathbf{y}) + :
\rho(\mathbf{x})\delta(\mathbf{y} - \mathbf{z})\delta(\mathbf{z} - \mathbf{x}),
$$

*which hold [5,55,56,77] for the density operator ρ* : Φ → Φ *at any points x*, *y*, *z* ∈ R/{[0, *l*]Z}.

Observe now that the operator (173) can be rewritten down in Φ as

$$\mathbf{D}(\mathbf{x}) = \mathbf{K}(\mathbf{x}) - \mathbf{A}(\mathbf{x}),\tag{182}$$

where, by definition,

+

$$\mathcal{K}(\mathbf{x}) := \nabla\_{\mathbf{x}} \rho(\mathbf{x}) / 2 + i \mathbf{J}(\mathbf{x}), \ \mathcal{A}(\mathbf{x}) := \frac{\pi \beta}{l} \int\_0^l dy \, \text{tg}[\frac{\pi}{l}(\mathbf{x} - \mathbf{y})] : \rho(\mathbf{x}) \rho(\mathbf{y}) : \tag{183}$$

for all *x* ∈ R/{[0, *l*]Z}. Recalling now the second condition of (179), one can rewrite it equivalently as

$$\mathbf{K}(\mathbf{x})|\Omega\rangle = \mathbf{A}(\mathbf{x})|\Omega\rangle\tag{184}$$

on the renormalized ground state vector |Ω) ∈ Φ for all *x* ∈ R/{[0, *l*]Z}. On the other hand, owing to the expression (176), we obtain the searched for current algebra representation

$$\hat{\mathbf{H}} = \int\_0^l dx (\mathbf{K}^+(\mathbf{x}) - \mathbf{A}(\mathbf{x})) \rho(\mathbf{x})^{-1} (\mathbf{K}(\mathbf{x}) - \mathbf{A}(\mathbf{x})) \tag{185}$$

of the Calogero–Moser–Sutherland Hamiltonian operator (165) on the suitably renormalized Hilbert space Φ, as it was already demonstrated in the work [76,77], using the condition (184) in the form (61).

#### *4.2. The Current Algebra Representation and Integrability of the Calogero–Moser–Sutherland Quantum Model*

We now briefly discuss the quantum integrability of the Calogero–Moser–Sutherland model (165). Owing to the factorized representation (185), one can easily observe [8–10]

that for any integer *p* ∈ Z+, the suitably symmetrized Hamiltonian operator densities <sup>h</sup>(*x*) := <sup>D</sup>+(*x*)*ρ*(*x*)−1D(*x*) : Φ → Φ, *x* ∈ R/{[0, *l*]Z}, commute to each other and with the particle number operator N : Φ → Φ, that is

$$\left[\mathbf{h}(\mathbf{x}), \mathbf{h}(\mathbf{y})\right] = 0, \ \left[\mathbf{h}(\mathbf{x}), \mathbf{N}\right] = 0 \tag{186}$$

for any *x*, *y* ∈ R/[0, *l*]<sup>Z</sup>. As a result of the commutation property (186), one easily obtains that for any integer *p* ∈ Z+ the symmetric operators

$$\mathbf{\hat{H}}^{(p)} := \int\_0^l d\mathbf{x} \mathbf{h}(\mathbf{x})^p \tag{187}$$

also commute to each other

$$[\hat{\mathcal{H}}^{(p)}, \hat{\mathcal{H}}^{(q)}] = 0 \tag{188}$$

for all integers *p*, *q* ∈ Z+, and in particular, commute to the Calogero–Moser–Sutherland Hamiltonian operator (176):

$$[\hat{\mathbb{H}}^{(p)}, \mathbb{H}] = 0.\tag{189}$$

Concerning the related *<sup>N</sup>*-particle differential expressions for the operators (187), it is enough to calculate their projections on the *<sup>N</sup>*-particle Fock subspace Φ(*s*) *N* ⊂ Φ*F*, *N* ∈ N. Namely, let an arbitrary vector |*ϕN*) ∈ Φ(*s*) *N* be representable as

$$|\varphi\_N\rangle := \int\_{[0,l]^N} \varphi\_N(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_N) \prod\_{j=\overline{1,N}} d\mathbf{x}\_j \psi^+(\mathbf{x}\_j) |0\rangle \tag{190}$$

for some coefficient function *ϕN* ∈ *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C). Then, by definition,

$$\mathbf{\hat{H}}^{(p)}|\varphi\_N\rangle := |\varphi\_N^{(p)}\rangle,\tag{191}$$

where

$$\left|\boldsymbol{\varrho}\_{N}^{(p)}\right> = \int\_{[0J]^N} (H\_N^{(p)}\boldsymbol{\varrho}\_N) (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_N) \prod\_{j=1,N} d\mathbf{x}\_j \boldsymbol{\psi}^+(\mathbf{x}\_j) |0\rangle \tag{192}$$

for a given *p* ∈ Z+ any *N* ∈ Z+. In particular, for *p* = 2, when H ˆ (2) + E = H : Φ*F* → Φ*F*, one easily retrieves the shifted Calogero–Moser–Sutherland Hamiltonian operator (165):

$$H\_N^{(2)} = -\sum\_{j=\overline{1,N}} \frac{\hat{\sigma}^2}{\partial \mathbf{x}\_j^2} + \sum\_{j \neq k=\overline{1,N}} \frac{\pi^2 \beta(\beta - 1)}{l^2 \sin^2[\frac{\pi}{l}(\mathbf{x}\_j - \mathbf{x}\_k)]} - \left(\frac{\pi \beta}{l}\right)^2 \frac{N(N^2 - 1)}{3}.\tag{193}$$

Respectively for higher integers *p* > 2 the resulting *<sup>N</sup>*-particle differential operator expressions *H*(*p*) *N* : *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C) → *L*(*s*) 2 ([0, *<sup>l</sup>*]*<sup>N</sup>*; C), *N* ∈ Z+, can be obtained the described above way by means of simple ye<sup>t</sup> cumbersome calculations, and which will prove to be completely equivalent to those calculated previously in good work [132].

**Remark 6.** *In the thermodynamical limit, when* lim*N*→∞,*l*→<sup>∞</sup> *N*/*πl* := *ρ*¯ > 0, *the structural operator* <sup>D</sup>(*x*) : Φ → Φ, *x* ∈ R/{[0, *l*]Z}, *reduces to*

$$\vec{D}(\mathbf{x}) := \lim\_{N/l \to \boldsymbol{\rho}} \mathcal{D}(\mathbf{x}) = \nabla\_{\mathbf{x}} \rho(\mathbf{x})/2 + i \mathcal{I}(\mathbf{x}) - \beta \int\_{\mathbb{R}} dy \, \frac{\colon \rho(y)\rho(\mathbf{x}) : \mathbf{y}}{\mathbf{x} - \mathbf{y}} \,, \tag{194}$$

*and, respectively, the operator (165) reduces to*

$$H\_N = -\sum\_{j=\overline{1,N}} \frac{\partial^2}{\partial x\_j^2} + \beta(\beta - 1) \sum\_{j \neq k=\overline{1,N}} \frac{1}{(x\_j - x\_k)^2} \tag{195}$$

*on the Hilbert space L*(*s*) 2 (R *N*; C) *for any N* ∈ Z+, *whose density operator representation in a suitable Hilbert space* Φ, *respectively, equals*

$$\mathbf{H} = \int\_{\mathbb{R}} d\mathbf{x} \left( \mathbf{D}^+(\mathbf{x}) \rho(\mathbf{x})^{-1} \mathbf{D}(\mathbf{x}) + \epsilon\_0 \right), \tag{196}$$

*where* 0 := lim *<sup>N</sup>*/*l*→*ρ*¯ *EN l* = *ρ*¯3/3 *denotes the average energy density of the reduced Calogero– Moser–Sutherland Hamiltonian operator (195) as N* → <sup>∞</sup>, *exactly coinciding with the before obtained results in [77,133].*

#### **5. The Dual Current Algebra Density Representation and the Factorized Structure of Quantum Integrable Many-Particle Hamiltonian Systems**

*5.1. The Current Algebra Density Representation*

We are now interested in constructing a special density functional representation of the local current algebra (29) on the corresponding representation Hilbert space <sup>Φ</sup>*μ* - ⊕*ρ* with the cyclic vector |Ω) = 1 ∈ <sup>Φ</sup>*ρ*. To do this, let us first consider the *creation ψ*+(*x*) and annihilation operators *ψ*(*x*), *x* ∈ R*<sup>m</sup>*, defined via (56) on the canonical Fock space Φ*F*, which can be formally represented as

$$\psi^{+}(\mathbf{x}) = \sqrt{\rho(\mathbf{x})} \exp[-i\theta(\mathbf{x})], \psi(\mathbf{x}) = \exp[i\theta(\mathbf{x})] \sqrt{\rho(\mathbf{x})},\tag{197}$$

where *ρ*(*x*) : Φ*F* → Φ*F* is our density operator and *<sup>ϑ</sup>*(*x*) : Φ*F* → Φ*F*,*<sup>x</sup>* ∈ R*<sup>m</sup>*, is some self-adjoint operator. What is important is the operators *ρ*(*x*) and *<sup>ϑ</sup>*(*x*) : Φ*F*<sup>→</sup> Φ*F* realize the canonical [55,56,60,63,85] commutation relationships

$$[\rho(\mathbf{x}), \rho(y)] = 0 = [\theta(\mathbf{x}), \theta(y)],\tag{198}$$

$$[\rho(y), \theta(\mathbf{x})] = i\delta(\mathbf{x} - y)$$

for any *x*, *y* ∈ R*<sup>m</sup>*. Concerning the current operator *J*(*x*) : Φ*F* → Φ*m F* ,*x* ∈ R*<sup>m</sup>*, one can easily obtain its equivalent expression

$$J(\mathbf{x}) = \rho(\mathbf{x}) \nabla \theta(\mathbf{x}). \tag{199}$$

Based on the canonical relationships (198) one can easily obtain, following [72,85,139], that

$$\vartheta(\mathbf{x}) = \frac{1}{i} \frac{\delta}{\delta \rho(\mathbf{x})} + i\sigma[\rho(\mathbf{x})]\_{\prime} \tag{200}$$

where *σ*[*ρ*(*x*)] : <sup>Φ</sup>*ρ* → <sup>Φ</sup>*ρ* acts on the corresponding Hilbert representation space <sup>Φ</sup>*ρ* and is some function of the density operator *ρ*(*x*) : <sup>Φ</sup>*ρ* → <sup>Φ</sup>*ρ*,*<sup>x</sup>* ∈ R*<sup>m</sup>*. Then, respectively, the current operator (199) is representable in <sup>Φ</sup>*ρ* as

$$J(\mathbf{x}) = -i\rho(\mathbf{x})\nabla \frac{\delta}{\delta\rho(\mathbf{x})} + \rho(\mathbf{x})\nabla\sigma[\rho(\mathbf{x})].\tag{201}$$

The functional-operator expression (201) proves to make sense [5,72,75,139] as operators on the Hilbert space <sup>Φ</sup>*ρ* of functional valued complex-functions on the manifold M, coordinated by the density functional parameter *ρ* : <sup>Φ</sup>*ρ* → <sup>Φ</sup>*ρ* and endowed with the scalar product (*a*|*b*)<sup>Φ</sup>*ρ* := M *a*(*ρ*)*b*(*ρ*)*dμ*(*ρ*) subject to some measure *μ* on M. To calculate this measure *μ* on M, we will present an explicit isomorphism between this Hilbert space <sup>Φ</sup>*ρ* and the corresponding Fock space Φ of spinless bosonic particles in R*<sup>m</sup>*. First, we determine the support *supp μ* ⊂ M of the measure *μ*, having assumed that the manifold

$$\mathcal{M} = \cup\_{n \in \mathbb{Z}\_+} \mathcal{M}\_{n\prime} \tag{202}$$

where M*n* := {*a*(*ρ*) : *ρ*(*x*) := ∑*nj*=<sup>1</sup> *<sup>δ</sup>*(*x* − *cj*) : *a* ∈ *<sup>C</sup>*<sup>∞</sup>(*F*; *End* <sup>Φ</sup>*ρ*)}, where *cj* ∈ R*<sup>m</sup>*, *j* = 1, *n*, *n* ∈ N, are arbitrary vector parameters. The restriction *dμn* of the measure *μ* on the submanifold M*n* can be presented [4,5,57,68,71] as

$$d\mu\_n = \gamma\_n(c\_1, c\_2, \dots, c\_n) \prod\_{j=\overline{1,n}} d c\_{j\prime} \tag{203}$$

where functions *γn* : R*m*×*n* → R+, *n* ∈ N, should be determined from the condition (201). In accordance with the manifold structure (202), we can decompose the Hilbert space <sup>Φ</sup>*ρ* as

$$
\Phi\_{\rho} = \ominus\_{n \in \mathbb{N}} \Phi\_{n\prime} \tag{204}
$$

where the space Φ*n* depends on the mapping *σ* : M <sup>→</sup>End(<sup>Φ</sup>*ρ*) and consists of functionals that are bounded on M*<sup>n</sup>*, in particular, for any *a*(*ρ*) ∈ M the restrictions *<sup>a</sup>*(*ρ*)|<sup>Φ</sup>*n* , *n* ∈ N, consist of functions of vectors (*<sup>c</sup>*1, *c*2, ..., *cn*) ∈ R*<sup>m</sup>*×*n*, *n* ∈ N, respectively. The scalar product in Φ*<sup>n</sup>*, *n* ∈ N, is suitably defined by means of the expressions (203). Now we can construct the isomorphism between the Hilbert spaces Φ*<sup>n</sup>*, *n* ∈ N, and the corresponding components Φ*<sup>n</sup>*, *n* ∈ N, of the corresponding Fock space Φ, representing spinless bosonic particles in R*<sup>m</sup>*. In the Hilbert space Φ*n* := Φ(*σ*) *n* , *n* ∈ N, one can easily calculate the eigenfunctions *ϕ*(*σ*) *p*1, *p*2,...,*pn* (*ρ*) ∈ Φ(*σ*) *n* of the free Hamiltonian

$$\mathcal{H}\_0^{(\sigma)} := \frac{1}{2} \int\_{\mathbb{R}^m} d\mathbf{x} \langle K^+(\mathbf{x}) | \rho^{-1}(\mathbf{x}) K(\mathbf{x}) \rangle \tag{205}$$

with structural

$$\mathcal{K}(\mathbf{x}) := \frac{1}{2}\nabla\rho(\mathbf{x}) + i l^{(\sigma)}(\mathbf{x}), \ \mathcal{K}^+(\mathbf{x}) := \frac{1}{2}\nabla\rho(\mathbf{x}) - i l^{(\sigma)}(\mathbf{x}) \tag{206}$$

and the momentum

$$\mathcal{P}^{(\sigma)} := \int\_{\mathbb{R}^m} d\mathbf{x} f^{(\sigma)}(\mathbf{x}) \tag{207}$$

operators:

$$\mathbb{H}\_0^{(\sigma)} \varrho\_{p\_{1,\mathcal{P}\_2,\ldots,\mathcal{P}\_n}^{(\sigma)}}(\rho) = (\sum\_{j=\overline{1,n}} E\_j) \varrho\_{p\_{1,\mathcal{P}\_2,\ldots,\mathcal{P}\_n}^{(\sigma)}}^{(\sigma)}(\rho),\tag{208}$$

$$\mathbf{P}^{(\sigma)} \boldsymbol{q}\_{p\_{1,\mathcal{P}\_{\bullet}}^{(\sigma)}, \dots, p\_{n}}^{(\sigma)}(\boldsymbol{\rho}) = (\sum\_{j=\overline{1,n}} p\_j) \boldsymbol{q}\_{p\_{1,\mathcal{P}\_{\bullet}}^{(\sigma)}, \dots, p\_n}^{(\sigma)}(\boldsymbol{\rho})\_{\boldsymbol{\rho}}$$

where *pj* ∈ R*m* , *j* = 1, *n*, are momentums of bose-particles in R*<sup>m</sup>*, the operator H(*σ*) 0 : <sup>Φ</sup>*ρ*<sup>→</sup> <sup>Φ</sup>*ρ* is given by the expressions (55), (201) and (205) and the operator P(*σ*) : <sup>Φ</sup>*ρ*<sup>→</sup> <sup>Φ</sup>*ρ* is given by the expressions (201) and (206), respectively, within which the current operator *J*(*σ*)(*x*) : <sup>Φ</sup>*ρ*<sup>→</sup> <sup>Φ</sup>*ρ* is realized under the condition ∇*σ*[*ρ*(*x*)] := *σρ*(*x*)−1∇*ρ*(*x*) as

$$J^{(\sigma)}(\mathbf{x}) = -i\rho(\mathbf{x})\nabla \frac{\delta}{\delta\rho(\mathbf{x})} + i\sigma\nabla\rho(\mathbf{x}) \tag{209}$$

where *σ* ∈ R is a fixed real-valued parameter. In this case, the eigenfunctions *ϕ*(*σ*) *p*1, *p*2,...,*pn* (*ρ*) ∈ Φ(*σ*) *n* , *n* ∈ N, can be expressed [72,139] as

$$\left(\mathfrak{p}\_{p\_1, p\_2, \ldots, p\_n}^{(\sigma)}(\rho)\right) = \frac{1}{n!} \mathfrak{p}\_0^{(\sigma)}(\rho) \left(\prod\_{j=\overline{1,n}} B\_{p\_j}(\rho) \cdot 1\right),\tag{210}$$

where

$$\phi\_0^{(\sigma)}(\rho) := \exp\left[ (\sigma - 1/2) \int\_{\mathbb{R}^m} d\mathbf{x} \rho(\mathbf{x}) \ln \rho(\mathbf{x}) \right],\tag{211}$$

$$B\_{p\_j}(\rho) := \int\_{\mathbb{R}^m} d\mathbf{x} \exp(i \langle p|\mathbf{x}\rangle \ ) \rho(\mathbf{x}) \exp(-\frac{\delta}{\delta \rho(\mathbf{x})}) \dots$$

The corresponding *n*-particle Fock subspaces Φ(*σ*) *n* , *n* ∈ N, can be naturally represented by means of the vectors

$$\mathbb{P}\left(\boldsymbol{\varrho}\_{n}^{(\sigma)}\right) := \frac{1}{\sqrt{n!}} \int\_{\mathbb{R}^{m \times n}} \prod\_{j=1,n} d\boldsymbol{p}\_{j} \, q\_{n}^{(\sigma)}(\boldsymbol{p}\_{1}, \boldsymbol{p}\_{2}, \dots, \boldsymbol{p}\_{n}) a^{+}(\boldsymbol{p}\_{1}) a^{+}(\boldsymbol{p}\_{2}) \dots a^{+}(\boldsymbol{p}\_{n}) |0\rangle \tag{212}$$

with functions *ϕ*(*σ*) *n* ∈ *L*(*s*) 2 (R*<sup>m</sup>*×*<sup>n</sup>*; C), *n* ∈ N, where

$$a^+(p) := \frac{1}{(2\pi)^{m/2}} \int\_{\mathbb{R}^m} d\mathbf{x} \, \exp(i \langle \mathbf{x} | p \rangle) a^+(\mathbf{x}) \tag{213}$$

denotes the momentum creation operator for any *p* ∈ R*<sup>m</sup>*.

Moreover, any functional *ϕ*(*σ*) *n* (*ρ*) ∈ Φ(*σ*) *n* , *n* ∈ N, can be uniquely represented as

$$\mathfrak{gl}\_n^{(\sigma)}(\rho) := \int\_{\mathbb{R}^{m \times n}} \prod\_{j=1,n} d\boldsymbol{p}\_j \Phi\_n^{(\sigma)}(\boldsymbol{p}\_1, \boldsymbol{p}\_2, \dots, \boldsymbol{p}\_n) \, \mathfrak{gl}\_{\boldsymbol{p}\_1, \mathbf{p}\_2, \dots, \mathbf{p}\_n}^{(\sigma)}(\rho) \tag{214}$$

for *ϕ*˜(*σ*) *n* ∈ *L*(*s*) 2 (R*<sup>m</sup>*×*<sup>n</sup>*; C), since the following condition

$$\left(B\_{p\_{n+1}}(\rho)\prod\_{j=\overline{1,n}}B\_{p\_j}(\rho)\cdot 1\right)\bigg|\_{\rho=a^+(x)a(x)}|\rho\_n^{(\sigma)}\rangle = 0\tag{215}$$

holds identically for all *pj* ∈ R*<sup>m</sup>*, *j* = 1, *n* + 1, and arbitrary state |*ϕ*(*σ*) *n* ) ∈ Φ*F*, *n* ∈ N.

**Remark 7.** *The condition (215) jointly with the constraint* <sup>R</sup>*m ρ*(*x*)*dx* = *n in* Φ(*σ*) *n* , *n* ∈ N, *should be, in general, naturally satisfied for any current algebra representation space* <sup>Φ</sup>*ρ*, *if and only if ρ*(*x*) = ∑*j*=1,*<sup>n</sup> <sup>δ</sup>*(*x* − *cj*) ∈ M*n for arbitrary n* ∈ N.

As a result of the construction above, we can state that the Hilbert spaces Φ(*σ*) *n* , *n* ∈ N, embed, respectively, isomorphically into the related Fock subspaces Φ(*σ*) *n* ,*n* ∈ N. As a consequence, we derive that the Hilbert space <sup>Φ</sup>*ρ* allows an isomorphic embedding into the related Fock space Φ*F*.

Consider now, following [5,72,139], the action of the current operator (209) on the basic vectors *ϕ*(*σ*) *n* (*ρ*) ∈ Φ(*σ*) *n* , *n* ∈ N:

$$J^{(\sigma)}(\mathbf{x})\boldsymbol{q}\_{\boldsymbol{n}}^{(\sigma)}(\boldsymbol{\rho}) = \boldsymbol{\phi}\_{0}^{(\sigma)}(\boldsymbol{\rho})[-i\boldsymbol{\rho}(\mathbf{x})\nabla\frac{\delta}{\delta\boldsymbol{\rho}(\mathbf{x})} + i\sigma\nabla\boldsymbol{\rho}(\mathbf{x})]\boldsymbol{q}\_{\boldsymbol{n}}^{(\sigma)}(\boldsymbol{\rho}),\tag{216}$$

from which one ensues easily at *σ* = 1/2 its *n*-particle representation on the functional manifold M*n*:

$$\begin{aligned} \left. f^{(1/2)}(\mathbf{x}) \varphi\_n^{(1/2)}(\boldsymbol{\rho}) \right|\_{\boldsymbol{\rho}(\mathbf{y}) = \sum\_{j=1,2} \delta(\mathbf{y} - \boldsymbol{\varepsilon}\_j)} &= \\ = \sum\_{j=1,2} \frac{1}{2} [-i\delta(\mathbf{x} - \boldsymbol{\varepsilon}\_j) \nabla\_{\boldsymbol{\varepsilon}\_j} + i \nabla\_{\boldsymbol{\varepsilon}\_j} \diamond \delta(\mathbf{x} - \boldsymbol{\varepsilon}\_j)] \tilde{f}\_n^{(1/2)}(\mathbf{c}\_1, \mathbf{c}\_2, \dots, \mathbf{c}\_n), \end{aligned} \tag{217}$$

where we took into account that *ϕ*¯(1/2) 0 (*ρ*) = 1 for all densities *ρ* : Φ → Φ and have put, by definition, the Fourier transform

$$f\_n^{(1/2)}(c\_1, c\_2, \dots, c\_n) := \int\_{\mathbb{R}^{m \times n}} \prod\_{j=1,n} d p\_j f\_n^{(1/2)}(p\_1, p\_2, \dots, p\_n) \exp(i \sum\_{j=\overline{1,n}} \langle p\_j | c\_j \rangle) \tag{218}$$

for any fixed particle position vectors *cj* ∈ R*<sup>n</sup>*, *j* = 1, *n*, and for arbitrary *n* ∈ N. The expression (217), in particular, means that the current operator *J*(1/2)(*x*) : <sup>Φ</sup>*ρ* → <sup>Φ</sup>*ρ* is symmetric with respect to the measure *<sup>d</sup>μ*(1/2) *n* := *βn* ∏ *j*=1,*<sup>n</sup> dcj* on each functional submanifold M*n* for all *n* ∈ N, where the constants *βn* ∈ R+, *n* ∈ N, can be determined from the normalization condition ||*ϕ*(1/2) *n* (*ρ*)||Φ(1/2) *n* = (*ϕ*(1/2) *n* |*ϕ*(1/2) *n* )1/2 Φ(1/2) *n* , *n* ∈ N. The latter gives rise [1,4,57,68,71,72,139] to the following symbolic measure expression

$$d\mu\_n^{(1/2)} := \prod\_{\mathbf{x} \in \mathbb{R}^m} \delta\left(\rho(\mathbf{x}) - \sum\_{j=\overline{1,n}} \delta(\mathbf{x} - \mathbf{c}\_j)\right) \prod\_{j=\overline{1,n}} \frac{d\mathbf{c}\_j}{(2\pi)^m} \tag{219}$$

for all *cj* ∈ R*<sup>n</sup>*, *j* = 1, *n*, and arbitrary *n* ∈ N.

**Remark 8.** *As was aptly observed in [72], the choice σ* = 1/2 *makes it possible to realize the current algebra representation on the space* M *of analytic functions, which will be a priori assumed for further, that is the corresponding measure can be symbolically expressed as*

$$d\mu\_n := \prod\_{\mathbf{x} \in \mathbb{R}^m} \delta\left(\rho(\mathbf{x}) - \sum\_{j=\overline{1,n}} \delta(\mathbf{x} - c\_j)\right) \prod\_{j=\overline{1,n}} \frac{dc\_j}{(2\pi)^m} \tag{220}$$

*on the subspace* M*n for any n* ∈ N.

*5.2. The Current Algebra Representation and Hamiltonian Reconstruction: A Many-Dimensional Quantum Oscillator Model*

As a classical application of the construction above, one can consider a density current algebra representation of the quantum Hamiltonian operator

$$\mathcal{H}^{(\omega)} = \frac{1}{2} \int\_{\mathbb{R}^m} \langle \mathcal{K}(\mathbf{x})^+ | \rho(\mathbf{x})^{-1} \mathcal{K}(\mathbf{x}) \rangle d\mathbf{x} + \frac{1}{2} \int\_{\mathbb{R}^m} \langle \omega \mathbf{x} | \omega \mathbf{x} \rangle \rho(\mathbf{x}) d\mathbf{x} \tag{221}$$

on the corresponding representation Hilbert space <sup>Φ</sup>*ρ* of the generalized quantum *N*particle oscillatory Hamiltonian

$$H\_N^{(\omega)} = \frac{1}{2} \sum\_{j=\overline{1,N}} \left( \langle \nabla\_{x\_j} | \nabla\_{x\_j} \rangle + \langle \omega \mathbf{x}\_j | \omega \mathbf{x}\_j \rangle \right) \tag{222}$$

for *N* ∈ Z+ bose-particles in the *m*-dimensional space R*m* under the external oscillatory potential, parametrized by the positive definite frequency matrix *ω* ∈ *End* R*<sup>m</sup>*.

Having shifted the representation Hilbert space <sup>Φ</sup>*ρ* by the functional *ϕ*¯(1/2) 0 (*ρ*) := exp[−12 <sup>R</sup>*m x*|*ωx ρ*(*x*)*dx*] ∈ <sup>Φ</sup>*ρ*, the corresponding current operator (209) becomes

$$J^{(\omega)}(\mathbf{x}) = -i\rho(\mathbf{x})\nabla \frac{\delta}{\delta\rho(\mathbf{x})} + \frac{i}{2}\nabla\rho(\mathbf{x}) - i\omega\mathbf{x}\rho(\mathbf{x}),\tag{223}$$

simultaneously entailing the related *K*-operator changing

$$K(\mathbf{x}) = \rho(\mathbf{x}) \nabla \frac{\delta}{\delta \rho(\mathbf{x})} \to K^{(\omega)}(\mathbf{x}) = \rho(\mathbf{x}) \nabla \frac{\delta}{\delta \rho(\mathbf{x})} + \omega \mathbf{x} \rho(\mathbf{x}) \tag{224}$$

for any *x* ∈ R*<sup>m</sup>*. The latter gives rise, respectively, to the following equivalent current algebra functional representation of the oscillatory Hamiltonian (221):

$$\mathcal{H}^{(\omega)} = \frac{1}{2} \int\_{\mathbb{R}^m} \langle \mathcal{K}^{(\omega)}(\mathbf{x})^+ | \rho(\mathbf{x})^{-1} \mathcal{K}^{(\omega)}(\mathbf{x}) \rangle d\mathbf{x} + \frac{1}{2} \text{tr}\omega \int\_{\mathbb{R}^m} \rho(\mathbf{x}) d\mathbf{x} \tag{225}$$

for any positive defined matrix *ω* ∈ *End* R*<sup>m</sup>*. The shifted current operator (223) makes it possible to construct the suitably deformed free particle measure

$$d\mu\_1^{(\omega)}(\rho) := \exp\left(-\int\_{\mathbb{R}^m} d\mathbf{x} \rho(\mathbf{x}) \langle \mathbf{x} | \omega \mathbf{x} \rangle \right) d\mu\_1^{(1/2)}(\rho) \tag{226}$$

on the one-particle functional manifold M1, for which the following expression

$$\langle \Omega | \mathcal{H}^{(\omega)} | \mathcal{U}(\mathbf{f}) | \Omega \rangle = \int\_{\mathcal{M}} \exp[i\rho(\mathbf{f})] d\mu\_1^{(\omega)}(\rho) \tag{227}$$

holds for any test function f ∈ *F*. The latter, jointly with the related ground state condition |Ω) = 1 ∈ <sup>Φ</sup>*ρ*, makes it possible to easily calculate the scalar product elements

$$\langle \langle \mathcal{U}(\mathfrak{f}\_1) \Omega | \mathcal{H}^{(\omega)} | \mathcal{U}(\mathfrak{f}\_2) | \Omega \rangle = \int\_{\mathbb{R}^m} \exp[i \mathfrak{f}\_1(\mathfrak{c}) + i \mathfrak{f}\_2(\mathfrak{c})] \exp(-\langle \mathfrak{c} | \omega \mathfrak{c} \rangle) \frac{d\mathfrak{c}}{(2\pi)^m} \tag{228}$$

for any test functions f1, f2 ∈ *F*. The expression (228) makes it possible to successfully calculate the matrix elements (*ρ*(f*p*1 )Ω|H(*ω*)|*ρ*(f*p*2 )|Ω) of the Hamiltonian H(*ω*) : <sup>Φ</sup>*ρ* → <sup>Φ</sup>*ρ* on the corresponding eigenvectors *<sup>ρ</sup>*(f*p*)|Ω) ∈ <sup>Φ</sup>*ρ* for arbitrary *p* = *p*1, *p*2 ∈ N and, therefore, to find its spectrum.

Consider now the operator (55), taking into account the analytical current representation (216) at *σ* = 1/2:

$$\begin{split} K(\mathbf{x}) \boldsymbol{\varrho}\_{\mathrm{n}}^{(1/2)}(\boldsymbol{\rho}) &= \, \left[ \boldsymbol{\rho}(\mathbf{x}) \nabla \frac{\boldsymbol{\delta}}{\delta \boldsymbol{\rho}(\mathbf{x})} - 1/2 \nabla \boldsymbol{\rho}(\mathbf{x}) \right] \boldsymbol{\varrho}\_{\mathrm{n}}^{(1/2)}(\boldsymbol{\rho}) + \\ &+ 1/2 \nabla \boldsymbol{\rho}(\mathbf{x}) \boldsymbol{\varrho}\_{\mathrm{n}}^{(1/2)}(\boldsymbol{\rho}) = \boldsymbol{\rho}(\mathbf{x}) \nabla \frac{\boldsymbol{\delta}}{\delta \boldsymbol{\rho}(\mathbf{x})} \boldsymbol{\varrho}\_{\mathrm{n}}^{(1/2)}(\boldsymbol{\rho}) \end{split} \tag{229}$$

for any *n* ∈ N. Having substituted instead of *ϕ*(1/2) *n* (*ρ*) ∈ Φ(*n*) *ρ* , *n* ∈ N, the ground state eigenfunction <sup>Ω</sup>(*ρ*) = 1 ∈ <sup>Φ</sup>*ρ*, we can easily retrieve the before derived expression (61). Moreover, based on the representation (224) and the definition (54), one can calculate that

$$\begin{split} K^{(\omega)}(\mathbf{x}) \, \phi\_0^{(1/2)}(\boldsymbol{\rho}) &= \left[ \boldsymbol{\rho}(\mathbf{x}) \nabla \frac{\boldsymbol{\delta}}{\delta \boldsymbol{\rho}(\mathbf{x})} + \omega \mathbf{x} \boldsymbol{\rho}(\mathbf{x}) \right] \phi\_0^{(1/2)}(\boldsymbol{\rho}) = \mathbf{0} = \\ &= \mathbf{A}^{(\omega)}(\mathbf{x}; \boldsymbol{\rho}) \boldsymbol{\rho}^{(1/2)}(\boldsymbol{\rho}), \end{split} \tag{230}$$

where *ϕ*¯(1/2) 0 (*ρ*) = exp[−12 <sup>R</sup>*m x*|*ωx ρ*(*x*)*dx*] ∈ Φ(1/2) *ρ* - <sup>Φ</sup>*ρ*. The latter means, in particular, that the corresponding multiplication operator <sup>A</sup>(*ω*)(*x*; *ρ*) = 0, or, respectively,

$$\mathbf{A}(\mathbf{x})\boldsymbol{\phi}\_{0}^{(1/2)}(\boldsymbol{\rho}) := \mathbf{A}(\mathbf{x};\boldsymbol{\rho})\boldsymbol{\phi}\_{0}^{(1/2)}(\boldsymbol{\rho}) = -\omega \mathbf{x}\boldsymbol{\rho}(\mathbf{x})\boldsymbol{\phi}\_{0}^{(1/2)}(\boldsymbol{\rho}),\tag{231}$$

where *ϕ*¯(1/2) 0 (*ρ*) := |Ω(*ρ*)) ∈ <sup>Φ</sup>*ρ* is the corresponding ground state vector in <sup>Φ</sup>*ρ* for the oscillatory Hamiltonian operator (222). Making use of the operator (226), based on expression (64), one can present a special solution to the functional Equation (63) in the form

$$\mathcal{L}(\mathbf{f}) = \exp\left(-\int\_{\mathbb{R}^m} d\mathbf{x} \langle \omega \mathbf{x} | \mathbf{x} \rangle \frac{1}{2i} \frac{\delta}{\delta \mathbf{f}(\mathbf{x})}\right) \exp\left(\bar{\rho} \int\_{\mathbb{R}^m} \{\exp[i\mathbf{f}(\mathbf{x})] - 1\} d\mathbf{x}\right),\tag{232}$$

confirming similar statements from [5,6,77].
