Algebras, States, Representations

L. E. Ballentine

Hans Plesner Jakobsen, C.-W. H. Lee

There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way. Moreover, they are the only states with a finite number of non-zero quantum numbers with respect to a certain set of maximally commuting linearly independent quantum observables. Any other state is a tensor product of a multiple meson state and a state coming from a representation of a quotient algebra that extends and generalizes the Virasoro algebra. We expect the representation theory of this quotient algebra to describe physical systems at the thermodynamic limit.

Harry Paul

Daniel A. Dubin, Mark A. Hennings

Part 1 Basic quantum mechanics: elementary quantum systems states and observables representations of the ccr rapidly decreasing sequences topological structure of s(d) uniqueness of s-Class representations the Schrodiger representation angular momentum and spin compound systems - the exclusion principle tensor products and symmetry quantum canonical transformations. Part 2 Topological algebras: algebras locally convex algebras states, representations and order. Part 3 The algebra of observables: order properties complete sets of commuting observables ideals of L+(D). Part 4 The states of the system: states as positive functionals states as density matrices states and representations algebraic and topological properties of A' positivity preserving maps generalized eigenvectors and Dirac's notation. Part 5 Dynamics and symmetries: the Hamiltonian smoothed coulomb potentials dynamical automorphisms time invariant states automorphisms and symmetries implementibility ergodicity. Part 6 Quantum measurements: orientation operations and instruments continuity and regularity composition and conditioning a class of instruments on S(R) the axioms.

Lev Deych

Yong Zhang, Naihuan Jing, Mo-Lin Ge

The antisymmetric solution of the braided Yang--Baxter equation called the Bell matrix becomes interesting in quantum information theory because it can generate all Bell states from product states. In this paper, we study the quantum algebra through the FRT construction of the Bell matrix. In its four dimensional representations via the coproduct of its two dimensional representations, we find algebraic structures including a composition series and a direct sum of its two dimensional representations to characterize this quantum algebra. We also present the quantum algebra using the FRT construction of Yang--Baxterization of the Bell matrix.