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

arXiv:1911.03371 · European Journal of Physics

Time and energy operators in the canonical quantization of special relativity

C A Aguillón, M Bauer, G E García

Published Jan 6, 2020• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1361-6404/ab67d6

Related papers

Second quantization of time and energy in Relativistic Quantum Mechanics

M. Bauer, Cesar Augusto Aguillon

Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) has been shown to provide a unified origin for the existence of Dirac's Hamiltonian and a self adjoint time operator that circumvents Pauli's objection. As such, this approach restores to Quantum Mechanics (QM) the treatment of space and time on an equivalent footing as that of momentum and energy. Second quantization of the time operator field follows step by step that of the Dirac Hamiltonian field. It introduces the concept of time quanta, in a similar way to the energy quanta in Quantum Field Theory (QFT). An early connection is found allready in Feshbach's unified theory of nuclear reactions. Its possible relevance in current developments such as Feshbach resonances in the fields of cold atom systems, of Bose-Einstein condensates and in the problem of time in Quantum Gravity is noted. .

arXiv:2102.01042•Feb 1, 2021•2 citations

A dynamical time operator in relativistic quantum mechanics

M. Bauer

A self-adjoint dynamical time operator is introduced in Dirac’s relativistic formulation of quantum mechanics and shown to satisfy with the Hamiltonian a commutation relation analogous to that of the position and momentum operators. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie’s hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli’s objection is shown to be resolved or circumvented. Possible relevance to current developments in interference in time and in cosmology is noted.

•Aug 19, 2009•1 citation

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

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

Canonical commutation relations and special relativity

F. R. Tangherlini

As a way of studying the consequences of a possible breakdown of Lorentz invariance, the canonical commutation relations are enlarged to include time and energy as additional conjugate variables, and with this enlargement, they are shown to be invariant under the Lorentz transformation under the assumption that commutators of non-conjugate operators vanish. Although time does not commute with the energy operator, it does commute with the Hamiltonian. It is also shown that the invariance of the enlarged commutation relations holds for arbitrary non-singular linear transformations that describe a velocity boost to another uniformly moving frame, under the same assumptions about non-conjugate commutators. A covariant enlargement of the Poisson brackets, together with Dirac's interpretation, that would predict this invariance of the commutators is presented. There is a discussion of improper coordinate transformations, and a canonical time-reversal operator is introduced, and compared with the standard time-reversal operator T. It is concluded that even if the Lorentz transformation were to fail, and were to be replaced by another linear transformation, the enlarged canonical commutation relations could still hold.

Physica Scripta•Jun 1, 2008•1 citation

A dynamical time operator in Dirac's relativistic quantum mechanics

M. Bauer

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

arXiv:0908.2789•International Journal of Modern Physics A•Mar 4, 2014•38 citations
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.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) has been shown to provide a unified origin for the existence of Dirac's Hamiltonian and a self adjoint time operator that circumvents Pauli's objection. As such, this approach restores to Quantum Mechanics (QM) the treatment of space and time on an equivalent footing as that of momentum and energy. Second quantization of the time operator field follows step by step that of the Dirac Hamiltonian field. It introduces the concept of time quanta, in a similar way to the energy quanta in Quantum Field Theory (QFT). An early connection is found allready in Feshbach's unified theory of nuclear reactions. Its possible relevance in current developments such as Feshbach resonances in the fields of cold atom systems, of Bose-Einstein condensates and in the problem of time in Quantum Gravity is noted. .
A self-adjoint dynamical time operator is introduced in Dirac’s relativistic formulation of quantum mechanics and shown to satisfy with the Hamiltonian a commutation relation analogous to that of the position and momentum operators. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie’s hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli’s objection is shown to be resolved or circumvented. Possible relevance to current developments in interference in time and in cosmology is noted.
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.
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.
As a way of studying the consequences of a possible breakdown of Lorentz invariance, the canonical commutation relations are enlarged to include time and energy as additional conjugate variables, and with this enlargement, they are shown to be invariant under the Lorentz transformation under the assumption that commutators of non-conjugate operators vanish. Although time does not commute with the energy operator, it does commute with the Hamiltonian. It is also shown that the invariance of the enlarged commutation relations holds for arbitrary non-singular linear transformations that describe a velocity boost to another uniformly moving frame, under the same assumptions about non-conjugate commutators. A covariant enlargement of the Poisson brackets, together with Dirac's interpretation, that would predict this invariance of the commutators is presented. There is a discussion of improper coordinate transformations, and a canonical time-reversal operator is introduced, and compared with the standard time-reversal operator T. It is concluded that even if the Lorentz transformation were to fail, and were to be replaced by another linear transformation, the enlarged canonical commutation relations could still hold.
A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.