Counting functional freedom for quantum theories

Develop a rigorous, generalizable procedure to define and compute functional freedom (FF), and in particular dynamical functional freedom (DFF), for quantum theories such as perturbative string theory, without relying on classical Hamiltonian formulations and while accommodating cases lacking unique classical limits due to dualities.

Background

The paper reconstructs Penrose’s notion of functional freedom (FF) and proposes a formalization using the machinery of constrained Hamiltonian systems, which successfully counts dynamical functional freedom (DFF) for classical field theories admitting a Hamiltonian description.

However, this counting method is limited to classical theories; the authors note that extending it to quantum theories is non-trivial. Many quantum theories are not straightforward quantizations of classical field theories, and strong–weak dualities suggest multiple classical counterparts, potentially yielding inconsistent DFF counts.

Since string theory is primarily a quantum theory, the lack of a rigorous quantum FF/DFF counting method critically undermines Penrose’s FF-based arguments against string theory and raises a general methodological gap in applying FF to quantum theories.