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An Alternative Perspective on Pauli's Theorem

Zhiyong Wang, Cai-Dong Xiong

Published May 11, 2006• 0 citations
OpenAlex

Abstract

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.

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Introduction

Jan Nesemann

In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.

Vieweg+Teubner eBooks•Jan 1, 2011•0 citations

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

Eric A. Galapon

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.

•Aug 11, 1999•3 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.

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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

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
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.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
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.
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.
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.