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 ${\mathbb{Z}}_p$ and momenta in the group ${\mathbb{Q}}_p/{\mathbb{Z}}_p$; and a quantum system with positions in the profinite group $\hat{\mathbb{Z}}$ and momenta in the group $\mathbb{Q}/\mathbb{Z}$. 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 $T_0$ topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and $T_0$ topologies, in a quantum mechanical context, is discussed.

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Cited by in corpus (5)

• The complete Heyting algebra of subsystems and contextuality
• Trace class operators and states in p-adic quantum mechanics
• Transition probabilities and transition rates in discrete phase space
• A quantum prediction as a collection of epistemically restricted classical predictions
• The tensor product of p-adic Hilbert spaces