Quantum Graphs : Physics of Quantum Singularity (6-1)
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Scale Anomaly and Point Interactions 1
We now ask a question why there are so many different ways to place singular points quantum mechanically in comparison to classical mechanical counterpart.
An inportant element here is the length scale L_0 which emerged "from nowhere" in the process of determining the quantum connection condition at the site of singularity. As is observed in (2.11) and (4.5), this scale paraemter L_0 morphs into the strength parameter fro delta and epsilon potentials, thus is essential in the sxistence of the quantum point interaction itself. Note that L_0 does not give a new degree of freedom and its value itself is not of any importance, since the system can be rescaled (or the unit of the length is renamed) to get arbitrary new value of L_0. It is the fact that quantization entails the introduction of a classically nonexistent length scale that is important.
As indicated earlier, classical mechanics has only one trivial type of point interaction in the form of inpenetrable bouncing barrier. (Or two, if you include "zero strength"- that is no barrier at all.) This is a direct result of the fact that the "sizeless barrier" has to be scale invariant by its nature, and therefore, there is no place to bring length scale into the problem. In quantum system however, we have to consider the connection condition of wave function, which has the wave length as a natural length scale, that "forces" the system with point interaction to "come up with its own length scale" that eventually characterizes the coupling strength.
A scale invariant system in classical mechanics can loose the invariance when it is quantized; this phenomenon is known as quantum scale anomaly, and is one particular example of quantum anomaly phenomena in which classical invariance is lost with quantization. The quantum anomaly has been regarded as an exclusive phenomenon for quantum field theory in which the neccesary intriduction of renormalization constitutes a breeding ground for the emergence of anomaly. Looking at our example of scale anomaly in the quantum mechanical system with point interaction, it appears that the contact force nature - locality of interaction, in the field theory context - is the real cause of quantum anomaly, neither infinite degree of freedom nor the renormalization.
The decisive role of the emergence of scale parameter L_0 for the existence of nontrivial quantum point interaction is evident in the separable limit of point interaction (4.3). Here, the interaction parameters (\theta_+, \theta-) is combined with the length scale L_0 to decide the portion of wave that is "lost" with the reflection by the barrier.
In hidsight, it is rather astonishing that the entire the nontrivial structures formed by quantum point interaction detailed in previous pages is generated "out of nowhere" by the quantum breaking of scale invariance, thus can be regarded as a result of a kind of quantum black magic. It certainly shows the power of quantum mechanics which lurkes beneath the seemingly mandane dayly classical life of ours. |
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