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Streszczenie:
The one-dimensional system of particles with a 1/x2 repulsive potential is known as the Calogero–Moser system. Its classical version can be generalized by substituting the coupling constants with additional degrees of freedom, which span the so(N) or su(N) algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and tΣhe reduced free quantum Hamiltonian manifests this trait through an additional attractive term i<j−¯h2(xi−xj)2 , which is absent when one performs the straightforward canonical Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero–Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles N = 2, 3, and we diagonalize the Hamiltonian partially for a general value of N.

Słowa kluczowe:
Calogero–Moser models, Quantum many-body systems, Inverse-square interactions

Streszczenie:
Many-body one-dimensional systems with 1/x2 interactions are known as Calogero–Moser systems. The so called ordinary system with a common coupling constant for all interacting pairs can be generalized into models with couplings which evolve as additional degrees of freedom. This can be done via unitary reduction of a linear matrix model or, alternatively, by assigning a vectorial degree of freedom to each particle. We briefly review the matrix and vector formulations in the context of Hamiltonian mechanics. Various models of this type were investigated separately in several previous studies. In this paper we focus on relations among them. In particular we prove that the seemingly two-particle variables governing the repulsion of each pair are in fact functions of one-particle variables. Identification of the ordinary Calogero–Moser system as an isolated stationary point in the space of generalized models is a starting point to classifying the generalized models according to the rank of the matrix of couplings or, equivalently, the effective number of independent non-spacial degrees of freedom. Finally, we present two new examples of generalized Calogero–Moser models. The first one combines the matrix and vector degrees of freedom and results in additional interaction terms. The second one stems from a Hamilton function which mimics the Hamiltonian of a quantum model with spin exchange terms and leads to matrix anticommutator equations of motion.

Słowa kluczowe:
Calogero–Moser models, Matrix dynamics, Hamiltonian systems, Many-body systems, One-dimensional systems

Streszczenie:
We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as an identification of states with ‘the same correlations properties’, i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question ‘what can be transformed into what via available means?’. Exactly such an interpretation, i.e. in terms of mutual transformability, can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.

Słowa kluczowe:
quantum entanglement, reduced density matrices, momentum map, entanglement polytopes, symplectic reduction, invariant polynomials