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Towards a Galoisian lnterpretation of Heisenberg lndeterminacy principle. (English) Zbl 1312.81102

Summary: We revisit Heisenberg indeterminacy principle in the light of the Galois-Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois-Grothendieck duality between finite \(K\)-algebras split by a Galois extension \(L\) and finite \(\mathrm{Gal}(L{:}K)\)-sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle (formulated in terms of the notion of entropic indeterminacy) can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \(H\subseteq G\) of the states in a \(G\)-set \(\mathcal {O}\simeq G/H\), the smaller the “conjugate” algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \(H\) can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations.

MSC:

81S05 Commutation relations and statistics as related to quantum mechanics (general)
81P05 General and philosophical questions in quantum theory
14F20 Étale and other Grothendieck topologies and (co)homologies
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
00A79 Physics
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