Foundations of Physics 47 (12):1597-1608 (2017)

Authors
Juan Leon
Consejo Superior De Investigaciones Científicas
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. Here we show how the Pauli argument fails when one uses an external system to track time, so that time arises as correlations between the system and the clock. 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.
Keywords Time  Foundations of quantum mechanics  Time operator
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DOI 10.1007/s10701-017-0115-2
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References found in this work BETA

Relational Quantum Mechanics.Carlo Rovelli - 1996 - International Journal of Theoretical Physics 35 (8):1637--1678.
Time in Quantum Gravity: An Hypothesis.Carlo Rovelli - 1991 - Physical Review D 43 (2):451–456.
Time in Quantum Mechanics: A Story of Confusion.Jan Hilgevoord - 2005 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36 (1):29-60.
Time in Quantum Mechanics: A Story of Confusion.Jan Hilgevoord - 2005 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36 (1):29-60.

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