  In the light of their relationships with renormalization, in this paper we
associate the scaling transformation with nonlocal interactions. On one hand,
the association leads us to interpret the nonlocality with locally symmetric
method. On the other hand, we find that the nonlocal interaction between
hadrons could be test ground for scaling transformation if ascribing the
running effects in renormalization to scaling transformation. First we derive
directly from group theory the operator/coordinate representation and
unitary/spinor representation for scaling transformation, then link them
together by inquiring a scaling-invariant interaction vertex mimicking the
similar process of Lorentz transformation applied to Dirac equation. The main
feature of this paper is that we discuss both the representations in a sole
physical frame. The representations correspond respectively to the spatial
freedom and the intrinsic freedom of the same quantum system. And the latter is
recognized to contribute to spin angular momentum that in literature has never
been considered seriously. The nonlocal interaction Lagrangian turns out to
vary under scaling transformation, analogous to running cases in
renormalization. And the total Lagrangian becomes scale invariant only under
some extreme conditions. The conservation law of this extreme Lagrangian is
discussed and a contribution named scalum appears to the spin angular momentum.
Finally a mechanism is designed to test the scaling effect on nonlocal
interaction.
