Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrödinger Type Dynamical Systems

Bogolubov, N. N.; Prorok, Dominik; Prykarpatski, Anatolij K.

doi:10.1134/s1063779620040152

Cited by 1 publication

(5 citation statements)

References 23 publications

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“…The expression (257) coincides exactly with that of (251), thereby proving the validity of our expressions (240) and (243) for the N.N. Bogolubov type generating functional L(f), f ∈ F, satisfying the functional Equation (242) of Proposition (8).…”

Section: The Bogolubov-zubarev "Collective" Variables Transformsupporting

confidence: 76%

“…−→ Φ ⊗n (s) with respect to the Fock space Φ F (1) for any |β) ∈ H − , called the "annihilation" operator, acts as follows: a(β)|α 1 , α 2 , ..., α n+1 ) := ∑ σ∈S n (β|α j )|α 1 , α 2 , ..., α j−1 , αj , α j+1 , ..., α n+1 ), (8) where the hat "• over a vector denotes that it should be omitted from the sequence.…”

Section: The Fock Space Non-relativistic Quantum Current Algebra and ...mentioning

confidence: 99%

“…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 m , can be easily constructed as follows:…”

Section: Non-relativistic Quantum Current Algebra and Its Cyclic Repr...mentioning

confidence: 99%

“…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][9][10] that for any integer p ∈ Z + , the suitably symmetrized Hamiltonian operator densities h(x) := D + (x)ρ(x) −1 D(x) : Φ → Φ, x ∈ R/{[0, l]Z}, commute to each other and with the particle number operator N : Φ → Φ, that is…”

Section: The Current Algebra Representation and Integrability Of The ...mentioning

confidence: 99%

“…As it was later shown by G. Goldin, with collaborators [4][5][6][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][9][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 1 ) F(S 1 ; R) of the topological diffeomorphism group Diff(S 1 ) of the circle S 1 and the space F(S 2 ; R) of the smooth periodic functions on it, their isomorphism was stated [11][12][13] with all completely known and up to date Lax type integrable classical dynamical systems on spatially one-dimensional functional manifolds.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

Prykarpatski

2022

Universe

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This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. 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 functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

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Section: The Bogolubov-zubarev "Collective" Variables Transformsupporting

confidence: 76%

Section: The Fock Space Non-relativistic Quantum Current Algebra and ...mentioning

confidence: 99%

Section: Non-relativistic Quantum Current Algebra and Its Cyclic Repr...mentioning

confidence: 99%

Section: The Current Algebra Representation and Integrability Of The ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

Prykarpatski

2022

Universe

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show abstract

Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrödinger Type Dynamical Systems

Cited by 1 publication

References 23 publications

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

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