Quantum mechanics on profinite groups and partial order

Abstract

Inverse limits and profinite groups are used in a quantum mechanical context. Two cases are considered: a quantum system with positions in the profinite group and momenta in the group , and a quantum system with positions in the profinite group and momenta in the group . The corresponding Schwatz–Bruhat spaces of wavefunctions and the Heisenberg–Weyl groups are discussed. The sets of subsystems of these systems are studied from the point of view of partial order theory. It is shown that they are directed-complete partial orders. It is also shown that they are topological spaces with T0-topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and T0-topologies, in a quantum mechanical context, is discussed.