The new formalism

Ary W. Espinosa Müller, Adelio Matamala-V�squez

A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his formalism. In this regard, it has been introduced a new set of superoperators featured by their algebraic structure. Also, two lemmas and one theorem have been developed in order to algebraically reformulate the theory on more rigorous grounds. Finally, we have been able to build a coherent and self-contained formalism independent on any matricial representation , removing in this way the degeneracy problem .

Hiroyasu Koizumi

Abstract Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U (1) phase neglected by Dirac in the Schrödinger representation of quantum mechanics. The neglected phase gives rise to a U (1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected U (1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the U (1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does not agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exist in the new one in accordance with a recent experiment (Murani et al 2020 Phys. Rev. X 10 021003).

Riccardo Franco

In this article we describe a cognitive heuristic known as the unpacking effect by using a mathematical model, based on the quantum formalism, already introduced for the conjunction fallacy. We present the basic postulates of such quantum-like model and we show that the presence of interference terms is responsible of the unpacking effect. In particular, the sign of the interference and its functional form are able to describe the experimental results about subadditivity, superadditivity and additivity. A comparison with previous models is presented, as well as new experimental predictions, allowing to conclude that this new formalism and the basic concepts of quantum information processing provide a new promising way to describe and understand human judgement and categorization.