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arXiv · 1808.08812

Quantum Holonomies and the Heisenberg Group

J. E. Nelson·R. F. Picken
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Abstract

Quantum holonomies of closed paths on the torus $T^2$ are interpreted as elements of the Heisenberg group $H_1$. Group composition in $H_1$ corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group $\pi_1$ of $T^2$, making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of $H_1$ adjust these signed areas, and the discrete symplectic transformations of $H_1$ generate the modular group of $T^2$.

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J. E. Nelson, R. F. Picken. 2018-08-27. Quantum Holonomies and the Heisenberg Group. https://doi.org/10.1142/s0217732319502560

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