Symmetry of Hamiltonian Systems: Classical and Quantum Aspects

Dear Colleagues,

Symmetry serves for an exact mathematical notion known as group, and in general, in fact, Hamiltonian systems are related with some subgroup orbits on the naturally related to them spaces, as their cotangent spaces, possessing always a priori Hamiltonian structures. In fact, this statement is very old, and was invented still by Sophus Lie in 1887, and is called today the Lie-Poisson structure. This structure was many times later in past century reinvented by such mathematicians as Arnold, Adler, Berezin, Kirillov, Kostant, and others, and today plays a leading role in Hamiltonian systems studies. From another point of view, this deep observation from practical point of view lasts up to date as an art, insomuch as retrieving this hidden group structure needs very deep and speculative efforts, and within which the main modern investigations are centered.

From application point of view the insight on Hamiltonian systems as mathematical objects with hidden symmetries should be unifying all possible efforts of researchers on the field. As demonstrate examples, in many cases the governing symmetry can be restored if considered to be based on the general reduction method by symmetry jointly with Hamilton's principle. This symmetry-reduction theme is widely used in geometric mechanics from the Jacobi, Lagrange, D'Alembert and Euler-Poincare viewpoints jointly with a fundamental Lie symmetry approach. The Lie symmetries in Hamilton's systems deserve to be mentioned separately as very often owing to them one can derive basic symmetry-reduced equations of motion, analyze their solutions and resume their decisive role in the related integrability theory. As this research scheme is deeply founded on searching for the complete set of invariants to a Hamiltonian system under regard, the mathematical structures, describing different forms of the Legendre transformation, providing the Hamiltonian formulation of these equations in terms of Lie-Poisson brackets, are looking especially important for further understanding their symmetry nature. If for example, one considers Lagrangian picture, the problem is described on the tangent space T(M) to some configuration manifold M; and recovering its symmetry group G makes it possible often to proceed to studying the related vector fields on this group, naturally forming its Lie algebra g: Using an analogy with the Legendre transformation, reducing the problem to its Hamiltonian description by means of the corresponding Poisson bracket on the cotangent space T^*(M); one can reduce the studying of the problem under regard to that on the adjoint space g^* possessing its own canonical Lie-Poisson structure, completely equivalent[4–6,19–21,35,37,38] to the canonical symplectic structure on the coadjoint space T^*(G) to the symmetry group G, suitably reduced on some invariant subgroup.

As demonstrated past century comprehensive studies [1,8,9,11,25–27,30,31] on the integrability theory of both finite and infinite-dimensional Hamiltonian systems, almost all of them were interpreted as the corresponding Hamiltonian flows on the adjoint spaces to their hidden group symmetries, that makes the investigation of symmetry properties of nonlinear Hamiltonian systems crucially important both for their mathematical theory and wide applications in modern geometry, mechanics, field theory of classical and quantum physics and in biological sciences. Especially there are of worth mentioning the symmetry analysis of a wide class of space distributed Hamiltonian systems, describing continuous flows in aero- and hydrodynamic, in plasma media, nuclear matter and some others, and having important and wide applications in modern science and technology.

Given any classical Hamiltonian many-particle non-relativistic system with the standard cotangent phase space T*(ℝ³)^⊗N, where the quantity of particles N∈ℤ₊ is fixed, there is a standard recipe for producing a quantum system by a method known as “canonical quantization", assigning to the system a suitably constructed [34] self-adjoint Hamiltonian operator, acting in the related Hilbert space ℋ=L₂(ℝ^3⊗N;ℂ). In case when all the particles are equivalent to each other and the particle number N∈ℤ₊ can vary within the system, there is applied to this system another recipe, called the”second quantization”, producing the corresponding [7,10] quantum self-adjoint Hamiltonian operator, acting already in a specially constructed Fock space Φ_F, whose basis vectors are generated by means of actions of additional so called “creation” and “annihilation” operators on a uniquely defined “vacuum” zero-particle vector state |0>∈Φ_F; whose structure in most practical cases is hidden. Even though this method appeared to be very effective for studying many quantum many-particle Hamiltonian systems, some important problems related with the a priori non-self-adjointness of the “creation” and” annihilation” operators in the Fock space Φ_F; stimulated researchers to suggest a dual quantization scheme, based strictly only on physically “observable” operators in a suitably constructed cyclic Hilbert space Φ, generated by means of the so called "groundstate” vector |Ω∈Φ, and being completely different from the Fock space Φ_F.

Several authors have been developing this idea of quantizing nonrelativistic models, making use of the local current algebra operators [3, 12, 13, 14, 15, 16, 29] as the basic dynamical variables, that is the density ρ(x): Φ→Φ and current J(x): Φ→Φ operators at spatial point x∈ℝ³, representing, as is well known, generators of the fundamental physical symmetry group Diff(ℝ³)⋉S(ℝ³;ℝ), the semidirect product of the diffeomorphism group Diff (ℝ³) of the space ℝ³ and the Schwarz space of smooth real valued functions on ℝ³. Moreover, the corresponding quantum Hamiltonian operators of the Schrödinger type in the Hilbert space Φ, as there appeared to be very surprising, possess a very nice factorized structure, completely determined by this groundstate vector |Ω>∈Φ. This fact posed today a very interesting and important problem of studying the related mathematical structures of these factorized operators and the correspondence to the generating them classical Hamiltonian many-particle non-relativistic systems, specified by some kinetic and inter-particle potential energy.

Analytical studies in modern mathematical physics are strongly based on the exactly solvable physical models which are of great help in the understanding of their mathematical and often hidden physical nature. Especially the solvable models are of great importance in quantum many particle physics, amongst which one can single out such as 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, modeling both some quantum-gravity and fractional statistics effects. One needs here to stress on classical investigations of local quantum current algebra symmetry representations in suitably renormalized representation Hilbert spaces, suggested and developed by G. A. Goldin with his collaborators [12–16,23,24] and which have a great importance for constructing the related factorized operator representations of secondly-quantized many-particle integrable Hamiltonian systems. As 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, classification of such integrable models became also so challenging for modern mathematical physics and quantum field theory specialists. Moreover, their study makes it possible to reconstruct the initial quantum Hamiltonian operators 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₊. In particular, being uniquely defined by means of the Bethe groundstate vector representation in the Hilbert space, the analyzed factorized operator structure of quantum completely integrable many-particle Hamiltonian systems on the axis proves to be closely related to their quantum integrability by means of the quantum inverse scattering transform [32–34], being a new and very fruitful field of quantum symmetry studies.

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Prof. Dr. Alexander A. Balinsky
Prof. Dr. Anatolij K. Prykarpatski
Guest Editors

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• Hamiltonian systems
• Lagrangian analysis
• Legendre transformations
• Hamiltonian group action invariance
• Poisson and Lie-Poisson structures
• symmetry analysis and reduction structures
• conditional symmetry analysis
• Lie and Lie-Backlund symmetry
• symplectic and canonical transformations
• classical integrability and symmetry analysis
• quantum integrability and symmetry analysis
• geometry of cotangent space and Hamiltonian analysis
• diffeomorphism group symmetry in continuous media dynamics
• Hamiltonian approach in gravity theory
• quantization and quantum Hamiltonian systems
• quantum integrability and related factorization structures
• quantum Hamiltonian systems and quantum inverse scattering transform