Foundations of Physics

, Volume 44, Issue 12, pp 1289–1301

Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle

Article

DOI: 10.1007/s10701-014-9812-2

Cite this article as:
Page, J. & Catren, G. Found Phys (2014) 44: 1289. doi:10.1007/s10701-014-9812-2

Abstract

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.

Keywords

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

Copyright information

© 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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