Quantum Graphs : Physics of Quantum Singularity (1-2)

Vista from Quantum Graph Theory

On top of the utility as the general theory of quantum device, the theory of quantum graphs has other aspects of potential utilities and interests. As a phsical syste, the target of quantum graph theory is a very simple, apparently trivial system. Recent researches, however, have revealed that there are rather profound physics beneath its simplistic appearence.

There are, in life, something too profound and arcane to be useful, and something too useful to be a subject of serious research. Full life demands something which is bothy profound and useful at the same time, which is, in fact, a rather rare comodity. It could just be that quantum graph is one of them.

At any rate, if we idealize the motion of a particle on a quantum lines as a free quantum motion in one dimension, all physics occurs at the joint of the lines, or the graph nodes. There, at the nodes, we find a quantum monster lurking.

The name of the monster turns out to be the quantum anomaly. This is a phenomenon in which classical symmetry is lost after the quantization. It is related to the peculiar aspect of quantum mechanics. While in classical mechanics, observables are physical quantity that appears in the equation motion, in quantum physics, they are "operators" residing in the Hilbert space that has to be supplemented by the "vectors" in that space to produce physically observed quantities.

Traditionally, the quantum anomaly are thought to be a rare spices found only in quantum field theories that describe high energy elementary particle phenomena. Here we have quantum anomaly residing at the node of quantum devices which is expected to be in your every-day electronic device in near future.

Just as the existence of the entanglement at the core of quantum information, there is the quantum anomaly at the heart of quantum graph and quantum device. This is the reason that the theory of quantum graph is intereting in its own right as a subject of theoretical and mathematical physics. We shall explore some selective aspects of it with the minimal use of equations in this eLecture.