Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

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

Tomer Shushi

Published Jul 24, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1209/0295-5075/adf3e0

Related papers

On the problem of time in quantum mechanics

M Bauer

Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.

arXiv:1606.02618•European Journal of Physics•Mar 1, 2017•18 citations

Quantum Gravity: A Self-Adjoint Time Operator as a Quantum Observable

Ilija Barukčić

Background:In quantum mechanics energy is an observable, whereas time is a parameter. Pauli's theorem and the problem of time in canonical quantum gravity reflect this asymmetry.Material and methods:We define the time operator $\hat{t}=\hat{x}/c$ and form, with the relativistic energy $\hat{E}_R$, Einstein's constant $\kappa$ and the time $\hat{t}_0$ in the rest system, operators of gravitational-field and curvature type. Commutation relations, wave equations and spectra are derived in the time and in the energy representation.Results:The commutator $[\hat{t},\hat{E}_R]=\tfrac{i\hbar}{c}\hat{v}_x$ is bounded with norm $\hbar$ and is not a multiple of the identity, so Pauli's theorem is not contradicted. The sum $c\,\hat{t}+\kappa\hat{E}_R$ is unitarily equivalent to the time operator and has a continuous spectrum with explicit eigenfunctions. A postulated field equation, $\hat{G}_0+\hat{\Lambda}=\kappa\hat{E}_R$, has the structure of Einstein's equation; the term $\hat{\Lambda}$ is an operator with continuous spectrum, the Ricci scalar operator contains the energy, and for small velocities the equation becomes the Schr\"odinger equation of a particle in a homogeneous field.Conclusion:Based on the postulated field equation, a quantization of the gravitational-field and curvature terms is derived: they become operators with explicit commutation relations and spectra. The metric tensor of Einstein's field equation has not been quantized.\par

•Oct 21, 2026•0 citations

On the Problem of Time(s) in Quantum Mechanics and Quantum Gravity: recent integrating developments and outlook

M. Bauer, Cesar Augusto Aguillon

Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.

arXiv:2104.10151•Apr 20, 2021•0 citations

An Alternative Perspective on Pauli's Theorem

Zhiyong Wang, Cai-Dong Xiong

From the point of view of purely mathematics, coordinates and momenta can be imaginary, and then it is not necessary for a Hamiltonian operator to have a semi-bounded spectrum, only if one imposes some physical conditions, a Hilbert space is defined and the spectrum of the Hamiltonian operator is determined. Pauli's Theorem just shows us a mathematical truth: via a time operator, one can find all the possible eigenstates of the Hamiltonian operator, but only a part of them satisfy the given physical conditions and represent the physical solutions, while the remainders represent the redundant solutions, they do not satisfy the given physical conditions and should be discarded. The transition probabilities between the physical solutions and the redundant solutions vanish. Therefore, the existence of a self-adjoint time operator actually presents no problem.

•May 11, 2006•0 citations

The Pauli Objection Addressed in a Logical Way

Espen Gaarder Haug

One of the greatest unsolved problems in quantum mechanics is related to time operators. Since the Pauli objection was first raised in 1933, time has only been considered a parameter in quantum mechanics and not as an operator. The Pauli objection basically asserts that a time operator must be Hermitian and self-adjoint, something the Pauli objection points out is actually not possible. Some theorists have gone so far as to claim that time between events does not exist in the quantum world. Others have explored various ideas to establish an acceptable type of time operator, such as a dynamic time operator, or an external clock that stands just outside the framework of the Pauli objection. However, none of these methods seem to be completely sound. We think that a better approach is to develop a deeper understanding of how elementary particles can be seen, themselves, as ticking clocks, and to examine more broadly how they relate to time.

•Jun 1, 2018•1 citation

Time and energy operators in the canonical quantization of special relativity

C A Aguillón, M Bauer, G E García

Abstract Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of special relativity is shown to provide a unified origin for (i) the complex vector space formulation of quantum mechanics; (ii) the momentum and space commutation relations and the corresponding representations; (iii) the Dirac Hamiltonian in the formulation of relativistic quantum mechanics; (iv) the existence of a self adjoint time operator that circumvents Pauli’s objection.

arXiv:1911.03371•European Journal of Physics•Jan 6, 2020•6 citations
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.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
Background:In quantum mechanics energy is an observable, whereas time is a parameter. Pauli's theorem and the problem of time in canonical quantum gravity reflect this asymmetry.Material and methods:We define the time operator t^=x^/c\hat{t}=\hat{x}/ct^=x^/c and form, with the relativistic energy E^R\hat{E}_RE^R​, Einstein's constant κ\kappaκ and the time t^0\hat{t}_0t^0​ in the rest system, operators of gravitational-field and curvature type. Commutation relations, wave equations and spectra are derived in the time and in the energy representation.Results:The commutator [t^,E^R]=iℏcv^x[\hat{t},\hat{E}_R]=\tfrac{i\hbar}{c}\hat{v}_x[t^,E^R​]=ciℏ​v^x​ is bounded with norm ℏ\hbarℏ and is not a multiple of the identity, so Pauli's theorem is not contradicted. The sum c t^+κE^Rc\,\hat{t}+\kappa\hat{E}_Rct^+κE^R​ is unitarily equivalent to the time operator and has a continuous spectrum with explicit eigenfunctions. A postulated field equation, G^0+Λ^=κE^R\hat{G}_0+\hat{\Lambda}=\kappa\hat{E}_RG^0​+Λ^=κE^R​, has the structure of Einstein's equation; the term Λ^\hat{\Lambda}Λ^ is an operator with continuous spectrum, the Ricci scalar operator contains the energy, and for small velocities the equation becomes the Schr\"odinger equation of a particle in a homogeneous field.Conclusion:Based on the postulated field equation, a quantization of the gravitational-field and curvature terms is derived: they become operators with explicit commutation relations and spectra. The metric tensor of Einstein's field equation has not been quantized.\par
Canonical quantization applied to closed systems leads to static equations, the Wheeler-deWitt equation in Quantum Gravity and the time independent Schrödinger equation in Quantum Mechanics. How to restore time is the Problem of Time(s). Integrating developments are: a) entanglement of a microscopic system with its classical environment accords it a time evolution description, the time dependent Schrödinger equation, where t is the laboratory time measured by clocks; b) canonical quantization of Special Relativity yields both the Dirac Hamiltonian and a self adjoint "time" operator, restoring to position and time the equivalent footing accorded to energy and momentum in Relativistic Quantum Mechanics. It introduces an intrinsic time property τ associated with the mass of the system, and a basis additional to the usual configuration, momentum and energy basis. As a generator of momentum displacements and consequently of energy, it invalidates Pauli's objection to the existence of a time operator. It furthermore complies with the requirements to condition the other observables in the conditional interpretation of QG. As Pauli's objection figures explicit or implicitly in most current developments of QM and QG, its invalidation opens to research the effect of this new two times perspective on such developments.
From the point of view of purely mathematics, coordinates and momenta can be imaginary, and then it is not necessary for a Hamiltonian operator to have a semi-bounded spectrum, only if one imposes some physical conditions, a Hilbert space is defined and the spectrum of the Hamiltonian operator is determined. Pauli's Theorem just shows us a mathematical truth: via a time operator, one can find all the possible eigenstates of the Hamiltonian operator, but only a part of them satisfy the given physical conditions and represent the physical solutions, while the remainders represent the redundant solutions, they do not satisfy the given physical conditions and should be discarded. The transition probabilities between the physical solutions and the redundant solutions vanish. Therefore, the existence of a self-adjoint time operator actually presents no problem.
One of the greatest unsolved problems in quantum mechanics is related to time operators. Since the Pauli objection was first raised in 1933, time has only been considered a parameter in quantum mechanics and not as an operator. The Pauli objection basically asserts that a time operator must be Hermitian and self-adjoint, something the Pauli objection points out is actually not possible. Some theorists have gone so far as to claim that time between events does not exist in the quantum world. Others have explored various ideas to establish an acceptable type of time operator, such as a dynamic time operator, or an external clock that stands just outside the framework of the Pauli objection. However, none of these methods seem to be completely sound. We think that a better approach is to develop a deeper understanding of how elementary particles can be seen, themselves, as ticking clocks, and to examine more broadly how they relate to time.