Initial-state definition and evolution under topological regularization

Determine how initial states and their time evolution are defined, preserved, and transformed within the topological regularization framework for quantum field theory, and ascertain whether configurations produced by topological mappings correspond to empirically observable physical states.

Background

The paper proposes topological regularization as a geometric framework to handle ultraviolet divergences by embedding quantum field theories into manifolds with controlled defects and emphasizing homotopy equivalence of regularization schemes. The Physical Equivalence Theorem posits that homotopy-equivalent regularizations produce identical renormalized observables.

Despite these formal developments, the authors explicitly acknowledge a gap between the mathematical equivalences achieved via topological mappings and experimental realities. In particular, they highlight uncertainty about how initial states and their evolution behave under such mappings and whether the resulting configurations correspond to physically realizable states. This connects the theoretical construction to empirical validation and the operational definition of states within the proposed framework.