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

The Pauli Objection Addressed in a Logical Way

Espen Gaarder Haug

Published Jun 1, 2018• 1 citation
OpenAlex

Abstract

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.

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

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

Tomer Shushi

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.

Europhysics Letters (EPL)•Jul 24, 2025•1 citation

Time as a Dynamical Variable

Z. Y. Wang, B. Chen, C. D. Xiong

There are enough reasons for us to consider time as a dynamical variable or operator; but according to Pauli's argument the existence of a self-adjoint time operator is incompatible with the semi-boundedness of Hamiltonian spectrum. In this article, we study the expressions of time operator and the definitions of mean time from a general and new viewpoint, make a new comment on Pauli's objections, and clarify some possible confusion in existing theories of time in quantum mechanics. From which we try to provide a new gateway for the conundrum of time in quantum mechanics and reconstruct a unified foundation for some issues of time.

arXiv:quant-ph/0311188•Nov 27, 2003•0 citations

What could have we been missing while Pauli's Theorem was in force?

Eric A. Galapon

Pauli's theorem asserts that the canonical commutation relation $[T,H]=iI$ only admits Hilbert space solutions that form a system of imprimitivities on the real line, so that only non-self-adjoint time operators exist in single Hilbert quantum mechanics. This, however, is contrary to the fact that there is a large class of solutions to $[T,H]=iI$, including self-adjoint time operator solutions for semibounded and discrete Hamiltonians. Consequently the theorem has brushed aside and downplayed the rest of the solution set of the time-energy canonical commutation relation.

arXiv:quant-ph/0303106•Mar 18, 2003•7 citations

Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

Eric Galapon

In single Hilbert space, Pauli's well–known theorem implies that the existence of a self–adjoint time operator canonically conjugate to a given Hamiltonian requires the Hamiltonian to possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self–adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well–known illustrative, implicit counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues surrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.

arXiv:quant-ph/9908033•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Jan 8, 2002•131 citations

Time in quantum mechanics and quantum field theory

Z Y Wang, B Chen, Chenxi Xiong

W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥ μ (with Ĥ 0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂ μ ψ = Ĥ μ ψ and Heisenberg equation d /d x μ = ∂ μ + i[Ĥ μ , ] are obtained, from which we have: (1) t is to Ĥ 0 as x j is to Ĥ j ( j = 1, 2, 3); likewise, t is to i∂ 0 as x j is to i∂ j ; (2) the proposed time operator is canonically conjugate to i∂ 0 rather than to Ĥ 0 , therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δ x μ Δ p μ ≥ 1/2 and the Mandelstam–Tamm treatment Δ x μ Δ H μ ≥ 1/2, have been formulated.

arXiv:quant-ph/0211047•Journal of Physics A Mathematical and General•Apr 24, 2003•11 citations
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.
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.
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.
There are enough reasons for us to consider time as a dynamical variable or operator; but according to Pauli's argument the existence of a self-adjoint time operator is incompatible with the semi-boundedness of Hamiltonian spectrum. In this article, we study the expressions of time operator and the definitions of mean time from a general and new viewpoint, make a new comment on Pauli's objections, and clarify some possible confusion in existing theories of time in quantum mechanics. From which we try to provide a new gateway for the conundrum of time in quantum mechanics and reconstruct a unified foundation for some issues of time.
Pauli's theorem asserts that the canonical commutation relation [T,H]=iI[T,H]=iI[T,H]=iI only admits Hilbert space solutions that form a system of imprimitivities on the real line, so that only non-self-adjoint time operators exist in single Hilbert quantum mechanics. This, however, is contrary to the fact that there is a large class of solutions to [T,H]=iI[T,H]=iI[T,H]=iI, including self-adjoint time operator solutions for semibounded and discrete Hamiltonians. Consequently the theorem has brushed aside and downplayed the rest of the solution set of the time-energy canonical commutation relation.
In single Hilbert space, Pauli's well–known theorem implies that the existence of a self–adjoint time operator canonically conjugate to a given Hamiltonian requires the Hamiltonian to possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self–adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well–known illustrative, implicit counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of assuming a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with an unbounded, or semibounded, or finite point spectrum. We point out Pauli's implicit assumptions and show that they are not consistent. We demonstrate our analysis by giving two explicit examples. Moreover, we clarify issues surrounding the different solutions to the canonical commutation relations, and, consequently, expand the class of acceptable canonical pairs beyond the solutions required by Pauli's theorem.
W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥ μ (with Ĥ 0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂ μ ψ = Ĥ μ ψ and Heisenberg equation d /d x μ = ∂ μ + i[Ĥ μ , ] are obtained, from which we have: (1) t is to Ĥ 0 as x j is to Ĥ j ( j = 1, 2, 3); likewise, t is to i∂ 0 as x j is to i∂ j ; (2) the proposed time operator is canonically conjugate to i∂ 0 rather than to Ĥ 0 , therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δ x μ Δ p μ ≥ 1/2 and the Mandelstam–Tamm treatment Δ x μ Δ H μ ≥ 1/2, have been formulated.