Introductive Backgrounds to Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems

Prykarpatsky, Anatoliy K.; Bogoli︠u︡bov, N. N.; Golenia, Jolanta; Taneri, Ufuk

doi:10.1007/s10773-008-9721-2

Cited by 10 publications

(22 citation statements)

References 24 publications

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“…satisfying conditions (22) and (23), where A(x; ρ) : Φ → Φ m , x ∈ R m , is some specially constructed linear self-adjoint operator, satisfying the condition…”

Section: The Unitary Family and Generating Functional Equationsmentioning

confidence: 99%

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

Prorok

Prykarpatski

2019

Symmetry

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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.

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“…satisfying conditions (22) and (23), where A(x; ρ) : Φ → Φ m , x ∈ R m , is some specially constructed linear self-adjoint operator, satisfying the condition…”

Section: The Unitary Family and Generating Functional Equationsmentioning

confidence: 99%

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

Prorok

Prykarpatski

2019

Symmetry

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“…linear dense in ⊕ + topological space D ⊆ ⊕ + is such that D ⊂ D A( f ) ⊂ Φ and the mapping A( f ) : D → Φ + is continuous for any f ∈ F. Then, the following structural theorem [5,7,11,12,[14][15][16][17] holds.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is worth to remark here that solutions to equation (2.54) 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[7,8,16] 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 (2.41), one can readily deduce that the following weak representation Hilbert space Φ is a weak relationship…”

mentioning

confidence: 99%

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

Prorok¹,

Prykarpatski²

2019

Condens. Matter Phys.

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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...

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“…That this is what happens is shown by experiment. Here, we can also recall an analogy from the modern quantum physics of infinite particle systems described using the second quantization [29], [31]- [33] suggested by Fock in 1932. In this case, particle creation and annihilation effects are also realized, which occur as a result of the interparticle interaction forces.…”

Section: Special Relativity Theory and Dynamical Field Equationsmentioning

confidence: 99%

The vacuum structure, special relativity theory, and quantum mechanics: A return to the field theory approach without geometry

2009

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We formulate the main fundamental principles characterizing the vacuum field structure and also analyze the model of the related vacuum medium and charged point particle dynamics using the developed field theory methods. We consider a new approach to Maxwell's theory of electrodynamics, newly deriving the basic equations of that theory from the suggested vacuum field structure principles; we obtain the classical special relativity theory relation between the energy and the corresponding point particle mass. We reconsider and analyze the expression for the Lorentz force in arbitrary noninertial reference frames. We also present some new interpretations of the relations between special relativity theory and quantum mechanics. We obtain the famous quantum mechanical Schrödinger-type equations for a relativistic point particle in external potential and magnetic fields in the semiclassical approximation as the Planck constant → 0 and the speed of light c → ∞.

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Introductive Backgrounds to Modern Quantum Mathematics with Application to Nonlinear Dynamical Systems

Cited by 10 publications

References 24 publications

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

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

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

The vacuum structure, special relativity theory, and quantum mechanics: A return to the field theory approach without geometry

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