On the discrete Heisenberg group and commutative modular variables in quantum mechanics: I. The Abelian symplectic shadow and integrality of area

Abstract

Global interference phenomena start to manifest in quantum transitions involving at least two projection filters which cannot be realized simultaneously, like in the case of the double-slit experiment. In terms of the 3-vertex invariants of the projective space of rays, thought of as a symplectic phase space, we encounter products involving the complex-valued ${\mathbf {I}}_3 (\varPsi _{in}, \varPsi _b, \varPsi _{fin})$ with the complex conjugate $\mathbf {I_3}^*(\varPsi _{in}, \varPsi _b', \varPsi _{fin})$, where the projection operators $|\psi _b \rangle \langle \psi _b |$ and $|\psi _b' \rangle \langle \psi _b' |$ are not simultaneously realizable. In this way, interference can be expressed in terms of products of this form defined over squares in the space of rays, which can be triangulated. Triangles in this space encode the invariant information of the geometric phase factor. From this perspective, we qualify the proposal of Aharonov and collaborators pertaining to the consideration of commutative modular variables evaluated in $\mathbb R/ {\mathbb {Z}}$ in deciphering the quantum interference pattern of the double-slit experiment. The quantum modular variables pertaining to conjugate observables are encoded in terms of one-parameter unitary groups acting jointly on the phase space, thus modeled through the continuous group action of ${\mathbb {R}}^2$. We show that $\frac{h}{2}$ expresses the minimal indistinguishable invariant area of the 2D symplectic Abelian shadow of the symplectic ball of radius $R=\sqrt{\hbar }$ in the 2n-phase space of the conjugate position and momenta. The above conclusion leads to a re-estimation of Weyl’s view of the quantum kinematical space in terms of an Abelian group of unitary ray rotations, and in particular the role that the discrete Heisenberg group plays in this conundrum.