The Pauli objection

Abstract

Schroedinger'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 not conjugate to the system Hamiltonian, but its eigenvalues still satisfy the Schroedinger equation for arbitrary Hamiltonians.

6 pages

References in corpus (2)

• Conditional probabilities with Dirac observables and the problem of time in quantum gravity
• Time as a quantum observable

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• Quantum Space, Quantum Time, and Relativistic Quantum Mechanics
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• Finite Mathematics, Finite Quantum Theory And A Conjecture On The Nature Of Time
• Emergent dynamics from entangled mixed states
• Dynamical symmetrization of the state of identical particles
• Relational Dynamics with Periodic Clocks
• Proposal for a solid-state magnetoresistive Larmor quantum clock