The Representation Aspect of the Generalized Hydrogen Atoms

Let $D\ge 1$ be an integer. In the Enright-Howe-Wallach classification list of the unitary highest weight modules of \widetilde{\mr{Spin}}(2, D+1), the (nontrivial) Wallach representations in Case II, Case III, and the mirror of Case III are special in the sense that they are precisely the ones that can be realized by the Hilbert space of bound states for a generalized hydrogen atom in dimension D. It has been shown recently that each of these special Wallach representations can be realized as the space of L^2-sections of a canonical hermitian bundle over the punctured {\bb R}^D. Here a simple algebraic characterization of these special Wallach representations is found.Comment: 18 pages, simplified proo

Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons

Due to their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications. They carry G_2-structures and admit the G_2-instanton equations, which are natural BPS equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons-type functional. We consider the Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces, which is equivalent to Spin(7)-instantons on a cylinder over them. For a general SU(3)-equivariant gauge connection, the generalized instanton equations turn into gradient-flow equations on C^3 x R^2, with a particular cubic superpotential. For the simplest member of the Aloff-Wallach family (with 3-Sasakian structure) we present an explicit instanton solution of tanh-like shape.Comment: 1+17 pages, 1 figur

An observable measure of entanglement for pure states of multi-qubit systems

Recently, Meyer and Wallach [D.A. Meyer and N.R. Wallach (2002), J. of Math. Phys., 43, pp. 4273] proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.Comment: 8 pages, 2 figure

Letter to Kate Wallach regarding SEAALL projects, June 23, 1956

A letter from Corinne Bass to Kate Wallach thanking Wallach for a materials exchange and for Wallach\u27s offer to help SEAALL with projects