Evolution of quantum observables: from non-commutativity to commutativity

  • S. Fortin
  • M. Gadella
  • F. Holik
  • M. LosadaEmail author


A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infinite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.


Foundations of quantum mechanics Gamow vectors Rigged Hilbert space Classical limit Quantum resonances 



M. Gadella acknowledges partial financial support to the Spanish Government Grant MTM2014-57129-C2-1-P, the Junta de Castilla y León Grants BU229P18, VA137G18. S. Fortin, F. Holik and M. Losada wish to acknowledge the financial support of the Universidad de Buenos Aires, the Grant PICT-2014-2812 from the Consejo Nacional de Investigaciones Científicas y Técnicas of Argentina.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CONICETUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Física Teórica, Atómica y Óptica and IMUVAUniversidad de ValladolidValladolidSpain
  3. 3.Instituto de Física La PlataConsejo Nacional de Investigaciones Científicas y TécnicasLa PlataArgentina

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