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 Consistency of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum

Eric A. Galapon

Published Aug 11, 1999• 3 citations
OpenAlex

Abstract

Pauli's well known theorem (W. Pauli, Hanbuch der Physik vol. 5/1, ed. S. Flugge, (1926) p.60) asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the time operator and the Hamitlonian posses completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a Hamiltonian with a spectrum which is a proper subspace of the real line. But we challenge this conclusion. We show rigourously the consitency of assuming a bounded, self-adjoint time operator conjugate to a Hamiltonian with an unbounded, or semibounded, or finitely countable point spectrum. Pauli implicitly assumed unconditionally that the domain of the Hamiltonian is invariant under the action of $U_\\beta=\\exp(-i\\beta T)$, where $T$ is the time operator, for arbitrary real number $\\betaA$. But we show that the $\\beta$'s are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a simibounded or finitely countable point spectrum, we show that no $U_\\beta$. We demonstrate our claim by giving an explicit example.

Related papers

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

Self-adjoint Time Operator is the Rule for Discrete Semibounded Hamiltonians

Eric A. Galapon

We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. Keywords: Time operators; Quantum canonical pairs; Pauli’s theorem 1.

•Nov 10, 2001•64 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

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

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

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
Pauli's well known theorem (W. Pauli, Hanbuch der Physik vol. 5/1, ed. S. Flugge, (1926) p.60) asserts that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian implies that the time operator and the Hamitlonian posses completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a Hamiltonian with a spectrum which is a proper subspace of the real line. But we challenge this conclusion. We show rigourously the consitency of assuming a bounded, self-adjoint time operator conjugate to a Hamiltonian with an unbounded, or semibounded, or finitely countable point spectrum. Pauli implicitly assumed unconditionally that the domain of the Hamiltonian is invariant under the action of U_\\beta=\\exp(-i\\beta T), where TTT is the time operator, for arbitrary real number betaA\\betaAbetaA. But we show that the beta\\betabeta's are at most the differences of the eigenvalues of the Hamiltonian. And this happens, under some other conditions, when the Hamiltonian has a non-empty point spectrum extending from negative to positive infinity. For a Hamiltonian with a simibounded or finitely countable point spectrum, we show that no U_\\beta. We demonstrate our claim by giving an explicit example.
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.
We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. Keywords: Time operators; Quantum canonical pairs; Pauli’s theorem 1.
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.
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.
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.
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.