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arXiv:quant-ph/0211047 · Journal of Physics A Mathematical and General

Time in quantum mechanics and quantum field theory

Z Y Wang, B Chen, Chenxi Xiong

Published Apr 24, 2003• 11 citations
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Abstract

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.

DOI
10.1088/0305-4470/36/18/317

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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

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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.
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.
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.
Following homogeneous canonical formalism where the time is treated on an equal footing with the space coordinates, we propose an alternative formulation of quantum mechanics in which the time is considered as a dynamical variable. The basic equation corresponding to and being equivalent to the ordinary Schrödinger equation is obtained when we consider the space coordinates x as a parameter corresponding to the time t in the ordinary formulation. The formal solution of this equation is given and examined. By making use of this solution the scattering theory is formulated in our formalism. It is shown that observables corresponding to the time and space coordinates can be defined. A short comment on the energy-time uncertainty relation is also given.
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.