On quantum systems with a self-adjoint time operator without violating Pauli's theorem

Center for Quantum Science and Technology & Department of Business Administration, Guilford Glazer Faculty of Business and Management, Ben-Gurion University of the Negev - Beer-Sheva, Israel

Received: 8 April 2025
Accepted: 24 July 2025

Abstract

Pauli's theorem asserts that no self-adjoint time operator can exist that is canonically conjugate to a Hamiltonian with a spectrum bounded from below, preventing time and energy from both being operators in standard quantum mechanics. In this paper, we propose a new reformulation that circumvents this constraint by inverting the roles of time and energy: time becomes a self-adjoint operator, while energy is treated as an external parameter. This dual perspective yields a mathematically equivalent yet physically distinct formulation, where quantum systems evolve according to internal clocks. The model accounts for the measurement problem without invoking collapse, offering an observer-relative interpretation tied to temporal misalignment. The proposed model also suggests a fresh perspective on the relationship between quantum systems and spacetime geometry.