Quantum Graphs : Physics of Quantum Singularity (7-1)

Scale Invariant Subfamily and Macroscopic Quantum Chaos 1

This is how a set of quantum point interaction emerge from the quantum mechanical breaking of classical systemmries. It must be added in rush that not all quantum point interactions are gerenated because of the quantum scale anomaly. That is evident in the existence of scale invariant quantum point interactions.

Consider the family of point interaction spanned by (\mu, \nu) with fixed values \theta_+ = 0 and \theta_- = \pi in (3.4), (3.5) and (3.6). This, we recall, is nothing but the "spectral invariant sphere" in the previous chapters. Explicitely, U is given by

(7.1)

We see one particular characteristics of this family of U, which is that the scale parameter L_0 does not appear anywhere. We can also show converse quite easily. Namely all scale invaiant point interaction is described (7.1). Therfore, the sphere representing the parameter space can also be called "scale invariant sphere". If we translate this U into the transfer matrix form (2.11), we have

(7.2)

Here we have defined \alpha = \cos\mu. This show that the point interaction belonging to the scale invariant sphere casues the wave function discontinuity that enlarge (shrink) the wave function by factor \alpha and shrink(enlarge) its derivative by factor (1/\alpha) when moving from one side of the singularity to the other. A crucial aspect of this family of point interaction can be found in its scattering matrix, which is obtained straight from (3.11) as

、

(7.3)

These expressions show that the transmission and the reflection probabilities are energy independent constants, as they are expected to be from the scale invariant interactions.

Strangely, this fact does not change at any high energy limit, signifying its persistence to the scale we can handle macroscpically. Therefore, it must mean that this quantum obstacle act as a stochastic semi-transparent barrier for a "classical" experimenter-observer. The trick is of course in thge fact that this high energy limit is not really a classical limit in any usual sense. But this does have macroscopically observable consequence, classical or not, and it illustrates the mystery of quantumpoint interaction in very vivid way.