A dynamical time operator in Dirac's relativistic quantum mechanics

Bauer, Mariano

doi:10.1142/s0217751x14500365

Cited by 37 publications

(67 citation statements)

References 33 publications

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“…However one condition to be satisfied is local concordance with SR, i.e., any acceptable theory of Quantum Gravity (QG) must allow to recover the classical spacetime in the appropiate limit [28]. It follows that a venue to be explored is whether this bottom up completion of Dirac's RQM with a time operator as derived above helps to resolve some of the issues noted [22,23]. To represent observables the operatorsx µ andp µ are selfadjoint (x µ =x † µ , p µ =p † µ ) , which insures real eigenvalues.…”

Section: Resultsmentioning

confidence: 99%

“…Here T = α.r/c + βτ 0 is the time operator introduced earlier by analogy to the Dirac Hamiltonian [15].…”

Section: Lorentz and Reciprocity Invariants In The Canonical Quantizamentioning

confidence: 99%

“…This proposed principle arises from noting that the Hamiltonian formulation of classical mechanics is invariant under the transformations x i → p i , p i → −x i and from the equivalence of the configuration and momentum representations in QM. Although Born acknowledges to be unsuccessful in his intended applications 1 , the reciprocity principle is currently receiving a renewed interest [8,9,10].In the present paper it is shown that it complements the required Lorentz invariance in the canonical quantization of Special Relativity (SR) to provide a unified origin for: i) the complex vector space formulation of QM; ii) the momentum and position operators' commutation relations and their corresponding representations; iii) the Hamiltonian in Dirac's formulation of Relativistic Quantum Mechanics (RQM) [1,11,12,13]; iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection [14,15,16].…”

Section: Introductionmentioning

confidence: 99%

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Time and energy operators in the canonical quantization of special relativity

2020

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Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.

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Section: Resultsmentioning

confidence: 99%

“…Here T = α.r/c + βτ 0 is the time operator introduced earlier by analogy to the Dirac Hamiltonian [15].…”

Section: Lorentz and Reciprocity Invariants In The Canonical Quantizamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time and energy operators in the canonical quantization of special relativity

2020

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“…On the basis of the present picture, the TDSE simulation do not make a distinction between tunneling (Ttime) and ionization (ionization time), which is qualitatively a crucial point for two reasons. The way the time variable interning in TDSE, where some authors claim that only this time is a parameter and not generally the time in quantum mechanics [38,39]. And second, in the model used in the TDSE simulation by Saindadh et al [22] and others, e.g.…”

Section: The Hydrogen Atommentioning

confidence: 99%

Tunneling time in attosecond experiment for hydrogen atom

Kullie

2018

J. Phys. Commun.

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Tunneling and tunneling time are hot debated and very interesting issues because of their fundamental role in the quantum mechanics. The measurement of the tunneling time in today's attosecond and strong field (low-frequency) experiments, despite its controversial discussion offers a fruitful opportunity to understand the time measurement and the role of time in quantum mechanics. In previous work Kullie (2015 Phys. Rev. A 92, 052118), we suggested a model and derived a simple relation to calculate the tunneling time, which showed a good agreement with the experimental result for He-atom. In the present work we analyze and discuss our model considering the experimental result for H-atom, which is obtained recently by Satya Sainadh et al (2017 arXiv:1707.05445). In the tunneling region, we find that our model shows a good agreement with their experimental result (similar to the He-atom in our previous work), and with the accompanied time-dependent Schrödiger equation simulations. However, Sainadh et al use a different picture of the tunneling, in which tunneling time becomes an imaginary quantity. Whereas our model represents a real tunneling time picture, precisely a delay time with respect to the ionization time at the atomic field strength. Moreover, even though the above-threshold-ionization is beyond the tunneling regime, we still see that the actual ionization time is related to our model. However, crucial points arise and keep some questions open especially on the experimental side.

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“…The question of existence of an algebraic form for a self-adjoint time operator of Dirac's equation has been unsolved for a long time, and mostly people have come up with approximate solutions [15,16]. Wang et al [17,18] arXiv:1610.00005v6 [quant-ph]…”

Section: Time Operatormentioning

confidence: 99%

Time Operator in Relativistic Quantum Mechanics

Khorasani¹

2017

Commun. Theor. Phys.

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It is first shown that the Dirac's equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4 × 4 relativistic time operator for spin-1 2 particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but non-zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli's objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found. TIME SPECTRAIn their recent paper, Faizal et al.[1] suggest that an operator for time could be defined and introduced to the non-relativistic Schrödinger's equation, which would recast it in the formwhere α with the dimension of inverse energy is introduced through appropriate deformed algebra. The resulting feature of (1) is that there will result two distinct energy eigenvalues, and thus the the time-coordinate would acquire a minimum time-step feature. While the subject of time has been quite controversial, as nicely summarized by Muller [2][3][4], the existence of a Hermitian timeoperator has been a subject of long debate. Pauli [5,6] was the first to point out that it was impossible to have such an operator of time. He did not present a well-founded proof, however, and his reason was that the time needed to be a continuous variable while energy eigenstates must be bounded from below [7]. This viewpoint has been followed in many of the later studies which have been discussed in several recent reviews of this subject [8-10]. However, it has been rigorously shown [11,12] that a self-adjoint Hermitian Timeof-Arrival operator can be indeed accurately defined and used. Also, in his comment [13] and earlier works [14], Sidharth also points out the possibility of discreteness at the Compton Scale to develop a consistent theoretical framework for cosmology. It is here shown that this is in fact quite possible in the fully relativistic picture of quantum mechanics, without retaining to such a non-standard algebra.The 4 × 4 Dirac's equation subject to a potential readswhere [V ± ] are appropriate perturbing spin-matrix potentials for particles and antiparticles, and |ψ ± are spinors. The vector σ is defined as σ = σ xx + σ yŷ + σ zẑ where σ j with j = x, y, z are 2 × 2 Pauli's spin matrices [5,7]. The 2 × 2 identity matrix [I] = σ 0 has not been shown for the convenience of notation. The above equation may be rearranged by taking time-derivatives from each row of (2) and straightforward substitution, to yield the modified 4 × 4 Schrödinger's equation of the typesimilar to (1) with α = 1/mc 2 . Curiously, this value of α satisfies the expected upper bound of 7.2 × 10 23 J −1 [1]...

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A dynamical time operator in Dirac's relativistic quantum mechanics

Cited by 37 publications

References 33 publications

Time and energy operators in the canonical quantization of special relativity

Time and energy operators in the canonical quantization of special relativity

Tunneling time in attosecond experiment for hydrogen atom

Time Operator in Relativistic Quantum Mechanics

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