General Properties of Irreducible Vectors and Operators

K. T. Hecht

The introduction of a set of intrinsic coordinates to give an explicit construction of the intrinsic states of vector coherent state theory has greatly simplified earlier attempts to generalize this theory to include the construction of vector coherent state realizations of operators other than the group generators. The group U(3)⊇ U(2)×U(1) is used as a prototype. The construction of irreducible tensor operators with specific shift properties is illustrated with a number of examples. These show how the Wigner calculus for a higher symmetry group can be expressed solely in terms of the recoupling coefficients of the core subgroup and the simple K‐matrix elements of vector coherent state theory.

Edgard Elbaz

Manu Mathur, Hamdi Mani

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

Felix M. Lev

Franco Fagnola, Damiano Poletti

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with [Formula: see text] is fully non-commutative.

Lev Deych