Solution to Infinity Problem of Scattering Matrix Using Time-Evolution Operators Without Needing Renormalization

The current situation of research challenging the demanding tasks of renormalization implies that the present framework of quantum scattering theory does not offer good prospect and therefore in parallel with the development of renormalization, it is desirable to attempt to formulate a new theory able to solve the infinity problem fundamentally in a general way. Our purpose is to construct an alternative mathematical formulation capable of ensuring the convergence of the scattering matrix without relying on renormalization theory, thus preventing overlapping divergences of the scattering matrix in principle. We present alternative representations of the scattering matrix in terms of the local and global time-evolution operators which replace the Dyson series and do not need the Feynman diagram. Importantly, the obtained results clarify that substantially, there does not exist the infinity problem of the scattering matrix within the framework of our formulation. Ultimately, we draw the successful conclusion that it is possible to conceive of an alternative to the conventional scattering theory and our formalism as a new proposal can contribute to formulating a consistent theory without infinity and renormalization.