Quantum Graphs : Physics of Quantum Singularity (1-1)

Transformation of Quantum Mechanics and Quantum Graph

Naturally exing objects ijn microscopic world usually have high simmetry in three dimensional space. Traditional courses of quantum mechanics that were honed to tackle the problem in atomic, nuclear and particle physics used to start dealing with rotational group O(3) early on. You just have to recall the names like Clebsch and Racah.

However, in recent decades, human technology finally has caught up with the theory, and we can now directly controll the objects living in microscopic scales. This is the advent of "quantum technologyg".

At the bottom of quantum theory, there is an unintelligible characteristics, even a mystery that directly confronts our basic intuition. The most prominent example of that mystery is the breaking of local causality as displayed in the breaking of Bell inequalities. Traditionally, researches on that mysterious aspects of quantum mechanics used to be viewed with some suspicion as could be observed from the term "quantum metaphysics". The recent flowering of quantum information science is nothing but the resurgence of quantum metaphysics in a shining costume of high-technology.

Imagine, for a moment a laptop computer in a future, 50 years from now, say, that utilizes the quantum computing algorithm. The central processing unit would be made up of quantum wires connected at nodes on which there are quantum mechanisms to control the flow of electrons.

In the road connecting current computers to that quantum computing Nirvana, we should expect a consideraqble amount of obstacles. One of such obstacle is undoubtedly of theoretical sort: We need to develope a theory of quantum wires and node in full generality and with all the physical intricacies, even before having a specific theory for each type of devices.

In the past, one-dimensional system has been treated as a simple test ground for the full treatment of complicated three-dimensional system which has been the real targets of research. The current interest in one-dimensional system as a direct model of reality is a quite recent phenomena.

A general basic theory of quantum device that is made from connected quantum wires - that is exactly the thery of quantum graphs.

Among the quantum graphs, the simplest is a line with a point interaction, which we might think as something we know very well as the "Dirac's delta function" potential. The readers will discover, in this eLecture, that even in this simplest limit, there are unfamiliar and intriguing variants to the delta potentials.