The current algebra representations of quantum many-particle Schrödinger type Hamiltonian models, their factorized structure and integrability

Abstract

There is developed a current algebra representation scheme for reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space and its application to quantum Hamiltonian and symmetry operators in case of quantum integrable spatially many-and one-dimensional dynamical systems. As examples, we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially many-and one-dimensional models as generalized oscillatory, Calogero-Sutherland, Coulomb type and nonlinear Schrödinger dynamical systems of spinless bose-particles.group Diff(R 3 ) of the space R 3 and the Schwarz space of smooth real valued functions on R 3 . Moreover, the corresponding quantum Hamiltonian operators of the Schrödinger type in the Hilbert space Φ, as it appeared to be very surprising, possess a very nice factorized structure, completely determined by this groundstate vector |Ω) ∈ Φ. This fact posed a very interesting and important problem of studying the related mathematical structure of these factorized operators and the correspondence to the classical Hamiltonian many-particle non-relativistic systems generating them, specified by some kinetic and interparticle potential energy.Analytical studies in modern mathematical physics are strongly based on the exactly solvable physical models which are of great help in understanding their mathematical and frequently hidden physical nature. Especially, the solvable models are of great importance in quantum many-particle physics, amongst which one can single out the oscillatory systems and Coulomb systems, modelling phenomena in plasma physics, the well known Calogero-Moser and Calogero-Moser-Sutherland models, describing a system of many particles on an axis, interacting pair-wise through long range potentials, modelling both some quantum-gravity and fractional statistics effects. In this work we developed investigations of local quantum current algebra symmetry representations in suitably renormalized representation Hilbert spaces, suggested and devised before by G.A. Goldin with his collaborators, having further applied their results to constructing the related factorized operator representations for secondly-quantized many-particle integrable Hamiltonian systems. The main technical ingredient of the current algebra symmetry representation approach consists in the weak equivalence of the initial many-particle quantum Hamiltonian operator to a suitably constructed quantum Hamiltonian operator in the factorized form, strictly depending only on its ground state vector. The latter makes it possible to reconstruct the initial quantum Hamiltonian operator in the case of its strong equivalence to the related factorized Hamiltonian operator form, thereby constructing, as a by-product, the corresponding N-particle groundstate vector for arbitrary N ∈ Z + . Being uniquely defined by means of the Bethe groundstate vector representation in the Hilbert space, the analyzed factorized operator structure of quantum compl...

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.…”