Quantum Graphs : Physics of Quantum Singularity (3-1)

Geometry of Parameter Space 1

Now that we have learnt the existence of family of quantum interactions which are not described by Dirac's delta function, natural questions to be asked are how the entire four-parameter family of point interactions can be characterized, and what kinds of secrets are hidden there. To answer this dawnting question seems to require some sophisticated mathematics. However, it turns out that we can actually approach this problem with nothing more than an elementary topology of introductory kind. All we need is a detailed analysis of the topological structure of "nontrivial" four-dimensional manifold S_4 that is synonimous to the parameter space U(4). We begin by pointing out the fundamental role of Pauli spin matrices in this 4-sphare S_4. Let us consider the following three transformations acting on the wave function of a particle on a line with a point interaction;

(3.1)

The first of thse, P_1, is nothing but the parity transformation that exchanges right and left. The realtions P_i² = 1 betrays the fact that, in fact, all three of P_i are of similar type, which we might call a set of "generalized parity transformations". A very intriguing relations can be confirmed by direct calculation. The reason behind them can be found by observing the fact that the trasformation P_i acts as spin mtrices \sigma_i,

(3.2)

when actetd upon the vectors \Phi and \Phi".

If we paraphrase what we have found up to now, a spin algebra SU(2) is realizable on an entire set of complex function on a real axis which is smooth everywhere except at one point x = 0, at which point a discontunuity is allowed with the condition of "quantum probability flux conservation" - a previously unnoticed remarkable fact.

We now consider three real numbers c1, c2, c3 (with the constraint ) and define the transformation . If we consider a parallel spin rotation and consider the rotation of U

(3.3)

along with the transformtions , , this tranformed U' also represents a point interaction since the condition (2.6) is kept intact with this rotation. Point interaction specified by U' is , in general, different from the one specified by the original U. But U and U' are isospectral, meaning they ashare the same quantum spectra. To see this, we first note taht the transformation between the Hamiltonian H_U specified by U and the Hamiltonian H_{U'} specified by U' are related by . If we now write the eigenvalue of H_U as E and its eigenfunction as |\psi>, we have the relation H_U' P|\psi>= P H_U |\psi> = E P |\psi>, which obviously means that the state P |\psi> is an eigenfunction of H_U' with the same eigenvalue E.