Abstract
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.
Highlights
Nowadays, on account of the developments of string theory [1] [2], Lattice QCD [3] and the necessity to describe nonperturbatively the intermediate strong interaction between extended hadrons [4], the construction of a consistent nonlocal theory is still called for [5]-[20]
The pioneering study of nonlocal interaction dates back to the 1930’s [21] when quantum field theory was in its infancy
The phenomenology of nonlocal interaction commenced with the primary attempts to describe the interaction between extended particles, whilst to cope with the divergence appearing in local quantum field theories (LQFT)
Summary
On account of the developments of string theory [1] [2], Lattice QCD [3] and the necessity to describe nonperturbatively the intermediate strong interaction between extended hadrons [4], the construction of a consistent nonlocal theory is still called for [5]-[20]. Enlightened by Lorentz transformation, we try to link physically the spatial form of scaling transformation with its spinor/unitary form, the former representing the realistic expansions and contractions of space-time (dilatation and shrinkage means the same), the latter representing the intrinsic freedom very like spin angular momentum Considering both representations in a sole frame is the main feature of this paper. Let’s first find the invariant vertex Γμ under the scaling transformation by mimicking the method of utilizing Lorentz transformation to Dirac equation In this way we link its spatial form with its spinor form. The weak interaction between neutrinos and leptons belongs to such category
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