SU(2) Yang-Mills quantum mechanics of spatially constant fields

As a first step towards a strong coupling expansion of Yang-Mills theory, the SU(2) Yang-Mills quantum mechanics of spatially constant gauge fields is investigated in the symmetric gauge, with the six physical fields represented in terms of a positive definite symmetric (3 x 3) matrix S. Representing the eigenvalues of S in terms of elementary symmetric polynomials, the eigenstates of the corresponding harmonic oscillator problem can be calculated analytically and used as orthonormal basis of trial states for a variational calculation of the Yang-Mills quantum mechanics. In this way high precision results are obtained in a very effective way for the lowest eigenstates in the spin-0 sector as well as for higher spin. Furthermore I find, that practically all excitation energy of the eigenstates, independently of whether it is a vibrational or a rotational excitation, leads to an increase of the expectation value of the largest eigenvalue , whereas the expectation values of the other two eigenvalues, and , and also the component = g of the magnetic field, remain at their vacuum values.Comment: 7 pages, 4 figure

On the generation of random ensembles of qubits and qutrits Computing separability probabilities for fixed rank states

The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank

On the stratifications of 2-qubits X-state space

The 7-dimensional family $\mathfrak{P}_X$ of so-called mixed X-states of 2-qubits is considered. Two types of stratification of 2-qubits $X-$state space, i.e., partitions of $\mathfrak{P}_X\,$ into orbit types with respect to the adjoint group actions, one of the global unitary group $G_X \subset SU(4)$ and another one under the action of the local unitary group $LG_X \subset G_X$, is described. The equations and inequalities in the invariants of the corresponding groups, determining each stratification component, are given

On the family of Wigner functions for N-level quantum system

The family of unitary non-equivalent Weyl-Stratonovich kernels determining the Wigner probability distribution function of an arbitrary N-level quantum system is constructed

Generalized Calogero-Moser-Sutherland models from geodesic motion on GL(n, R) group manifold

It is shown that geodesic motion on the GL(n, R) group manifold endowed with the bi-invariant metric d s^2 = tr(g^{-1} d g)^2 corresponds to a generalization of the hyperbolic n-particle Calogero-Moser-Sutherland model. In particular, considering the motion on Principal orbit stratum of the SO(n, R) group action, we arrive at dynamics of a generalized n-particle Calogero-Moser-Sutherland system with two types of internal degrees of freedom obeying SO(n, R) \bigoplus SO(n, R) algebra. For the Singular orbit strata of SO(n, R) group action the geodesic motion corresponds to certain deformations of the Calogero-Moser-Sutherland model in a sense of description of particles with different masses. The mass ratios depend on the type of Singular orbit stratum and are determined by its degeneracy. Using reduction due to discrete and continuous symmetries of the system a relation to II A_n Euler-Calogero-Moser-Sutherland model is demonstrated.Comment: 16 pages, LaTeX, no figures. V2: Typos corrected, two references added. V3: Abstract changed, typos corrected, a few formulas and references added. The presentation in the last section has been clarified and it was restricted to the case of GL(3, R) group, the analysis of GL(4, R) will be given elsewhere. V4: Minor corrections in the whole text, more formulas and references added, accepted for publication in PL