Quantum Current Algebra Symmetries and Integrable Many-Particle Schrödinger Type Quantum Hamiltonian Operators

Abstract

Based on the G. Goldin’s quantum current algebra symmetry representation theory, have succeeded in explaining a hidden relationship between the quantum many-particle Hamiltonian operators, defined in the Fock space, their factorized structure and integrability. Interesting for applications quantum oscillatory Hamiltonian operators are considered, the quantum symmetries of the integrable quantum Calogero-Sutherland model are analyzed in detail.

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

“…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)) = 1 √ 2π e i y|x k ∈ H − := L 2,− (R m ; C k ) for any y ∈ E m , one easily gets from (9) that [a i (x), a + j (y)] = δ ij δ(x − y) (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 m ; •|• ), a + j (y) := a + j (y(x)) and a j (y) := a j (y(x)), j = 1, k, for all x, y ∈ R m and denoted by δ(•) the classical Dirac delta-function.…”

“…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][5][6][7]76,77].…”

“…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…”

“…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.…”