Quantum computations are easily represented in the graphical notation known as the zx-calculus, a.k.a. the red-green calculus. We demonstrate its use in reasoning about measurement-based quantum computing, where the graphical syntax directly captures the structure of the entangled states usesd to represent computations, and show that the notion of information flow within the entangled states gi...

We review the present status of gauge theories built on various quantum space–times described by noncommutative space–times. The mathematical tools and notions underlying their construction are given. Different formulations theory models Moyal spaces as well whose coordinates form a Lie algebra covered, with particular emphasis some explored properties. Recent attempts aiming to include gravity...

When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the rst part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justiication to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner produc...