Quantum Theory of Circuit Systems – I. Aspects of Symmetry Principles and a Generalization of the Nonlinear Schr？dinger Equation in Charge Space

The quantization of circuits finds many interesting applications, e.g. quantum computer, molecular and biophysics. The method can be extended to nuclear physics, if the exchange interactions between nuclear particles are described by currents. A system of mutually coupled circuits can be treated by the linear Schr？dinger equation yielding symmetries such as SU₂, SU₃, SU₄, etc. A generalization of the principles to a nonlinear/nonlocal Schr？dinger is presented; the nonlocal exchange between elementary particles is mediated by a Gaussian kernel (integral transform) in the charge space. The SU₃ (or SU₄, if four charges Q₁,…,Q₄ are accounted for) are used as the basic structure of the circuits. In nonlocal fields these symmetries hold, in the same fashion, too, but the energy levels are not equidistant as it the case at circuit quantization, equivalent to the 3D harmonic oscillator. The Gaussian kernel assumes zero value in the positive half-plane with E > 0. To avoid this hindrance a kernel is proposed, which only uses terms up to the second order of the Gaussian kernel, and a self-interacting 3D oscillator keeps excitations of arbitrary order to indemnify the confinement of quarks