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The Pauli Objection

Abstract

Schrödinger’s equation says that the Hamiltonian is the generator of time translations. This seems to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian. Then both time and energy must have the same spectrum since conjugate operators are unitarily equivalent. Clearly this is not always true: normal Hamiltonians have lower bounded spectrum and often only have discrete eigenvalues, whereas we typically desire that time can take any real value. Pauli concluded that constructing a general a time operator is impossible (although clearly it can be done in specific cases). Here we show how the Pauli argument fails when one uses an external system (a “clock”) to track time, so that time arises as correlations between the system and the clock (conditional probability amplitudes framework). In this case, the time operator is conjugate to the clock Hamiltonian and not to the system Hamiltonian, but its eigenvalues still satisfy the Schrödinger equation for arbitrary system Hamiltonians.

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Acknowledgements

We acknowledge the FQXi foundation for financial support in the “Physics of what happens” program. JL acknowledges support from MINECO/FEDER Project FIS2015-70856-P and CAM PRICYT Project QUITEMAD+ S2013/ICE-2801 and Giacomo Mauro D’Ariano for the kind hospitality at Pavia University. LM acknowledges very useful feedback from Vittorio Giovannetti and Seth Lloyd.

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Correspondence to Lorenzo Maccone.

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Leon, J., Maccone, L. The Pauli Objection. Found Phys 47, 1597–1608 (2017). https://doi.org/10.1007/s10701-017-0115-2

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  • DOI: https://doi.org/10.1007/s10701-017-0115-2

Keywords

  • Time
  • Quantum mechanics
  • Foundations of quantum mechanics
  • Time operator