On the discrete Heisenberg group and commutative modular variables in quantum mechanics: II. Synchronization of unitary actions and homological Abelianization

Since, the quantum modular variables are encoded in terms of one-parameter unitary groups, we are led to a careful re-evaluation of the Heisenberg group, from where the commutation relations emerge from. In particular, the re-evaluation pertains to the discrete Heisenberg group from the perspective of its genuine descent from a more fundamental layer of structure targeting the origin of the non-commutativity of quantum observables. Due to the fact that the modular variables commute, they give rise to an integrality condition inherent to the structure of the discrete Heisenberg group. In this manner, this group should mediate in the structural transition from non-commutativity to its integral Abelian shadow, which is qualified symplectically via the non-squeezing theorem. In particular, the integrality of symplectic area pertains to the global topology of the torus, meaning that the phase space of the 2-d Abelian shadow is topologically toroidal and isomorphic to the modular lattice $$\frac{\mathbb R^2}{\mathbb Z^2}$$ , such that it is universally covered by $$\mathbb R^2$$ , where $$\mathbb Z^2$$ is the free Abelian group in two generators acting by integer translations. The main conclusion of this work is that the structural transition from the non-Abelian-free fundamental group of based loops $$\Theta _2$$ , expressing the synchronization condition of the joint action of modular variables, to the Abelian-free homology group of cycles $$\mathbb Z^2$$ , where the entangled symplectic area pertains, factorizes though the discrete Heisenberg group. This elucidates the role and status of modular variables in quantum mechanics and constitutes a viable theoretical explanation of the nature and appearance of quantum interference phenomena underlying the significance of the notion of a geometric phase.