Effective approach to open systems with probability currents and the Grothendieck formalism

An effective approach to open systems and irreversible phenomena is presented, where an open system Σ(d) with d-dimensional Hilbert space, is a subsystem of a larger isolated system Σ(2d) (the `full universe') with 2d-dimensional Hilbert space. A family of Bargmann-like representations (called z-Bargmann representations) introduces naturally the larger space. The z-Bargmann representations are defined through semi-unitary matrices (which are a coherent states formalism in disguise). The `openness' of the system is quantified with the probability current that flows from the system to the external world. The Grothendieck quantity Q is shown to be related to the probability current, and is used as a figure of merit for the `openness' of a system. Q is expressed in terms of `rescaling transformations' which change not only the phase but also the absolute value of the wavefunction, and are intimately linked to irreversible phenomena (e.g., damping/amplification). It is shown that unitary transformations in the isolated system Σ(2d) (full universe), reduce to rescaling transformations when projected to its open subsystem Σ(d). The values of the Grothendieck Q for various quantum states in an open system, are compared with those for their counterpart states in an isolated system.