1301.0258 (Hai-Jhun Wanng)

Scaling Transformation for Nonlocal Interactions    [PDF]

Hai-Jhun Wanng

More often than not, previous investigations on nonlocal interaction tried to fit certain results to those of renormalization. Reversely, in this paper we extract a scaling transformation for nonlocal interaction in the light of renormalization group method (RGM). By RGM the one-loop renormalized $S$% -matrix keeps exactly its tree-level form except for the running of physical parameters (charge and mass etc.) with respect to a scale parameter. The symmetric method here falls into the stream of using local method to study nonlocality. Foremost the similarity between the dispensable local-form of nonlocal interaction in momentum representation and the Hamiltonian in one-loop renormalized $S$-matrix suggests that the scale transformation may underly the nonlocal interaction. Then by referencing the process of Lorentz transformation applied to Dirac equation, we apply the scale transformation to a scale-invariant interaction vertex. Consequently the scale transformation $e^{\frac u2\gamma ^5}$ in spinor representation and the scale-invariant vertex $\gamma ^\mu (1\pm\gamma ^5)$ are obtained simultaneously. Make the transformation $e^{\frac u2\gamma ^5}$ performed repeatedly on the vector vertex $\gamma ^\mu$ one obtains a varying vertex $% \gamma ^\mu (a\ + b\gamma ^5)$, i.e. normal vector vertex $\gamma ^\mu$ would evolve to include the part of $\gamma ^\mu (1\pm\gamma ^5) $ due to scale(size)-variation of extended particle with different energies in scattering. In this sense, the invariant vertex $\gamma ^\mu (1\pm\gamma ^5)$ can be viewed as the extremes of vector interaction under scaling transformations, and the varying coefficient $a$ before $\gamma ^\mu$ is assumed relating compatibly to running charges.

View original: http://arxiv.org/abs/1301.0258