Foundations of Physics

, Volume 44, Issue 12, pp 1289–1301 | Cite as

Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle

  • Julien Page
  • Gabriel Catren


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 \(Gal(L{:}K)\)-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.


Galois–Grothendieck theory Quantum mechanics Heisenberg indeterminacy principle Symmetries-invariants 



The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013 Grant Agreement n\(^{\circ }\) 263523). We also thank Daniel Bennequin, Mathieu Anel, and Alexandre Afgoustidis for helpful discussions.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Laboratoire SPHERE (UMR 7219)Université Paris Diderot - CNRSParisFrance
  2. 2.Facultad de Filosofía y LetrasUniversidad de Buenos Aires - CONICETBuenos AiresArgentina

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