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 $\mathrm{\Sigma }(d)$ with $d$-dimensional Hilbert space, is a subsystem of a larger isolated system $\mathrm{\Sigma }(2d)$ (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 $\mathcal{Q}$ is shown to be related to the probability current, and is used as a figure of merit for the openness' of a system. ${\cal Q}$ 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 $\mathrm{\Sigma }(2d)$ (full universe), reduce to rescaling transformations when projected to its open subsystem $\mathrm{\Sigma }(d)$. The values of the Grothendieck $\mathcal{Q}$ for various quantum states in an open system, are compared with those for their counterpart states in an isolated system.