We consider the motion of a classical particle in (1+1)-dimensional space-time. Four probability distributions govern the trajectory of the particle; these give the probability of moving to the left or right in space while moving backwards or forwards in time. If these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner, then the master equations for these probabilities give rise to the Dirac equation without recourse to direct analytic continuation. In contrast, when a particle always moves forward in time, an analytic continuation is required to recover the Dirac equation.We consider the motion of a classical particle in (1+1)-dimensional space-time. Four probability distributions govern the trajectory of the particle; these give the probability of moving to the left or right in space while moving backwards or forwards in time. If these probabilities are randomly distributed and if the probability of moving backwards in time is related to the probability of moving forwards in time in a prescribed manner, then the master equations for these probabilities give rise to the Dirac equation without recourse to direct analytic continuation. In contrast, when a particle always moves forward in time, an analytic continuation is required to recover the Dirac equation.

Physical Review LettersJul 6, 199278 citations

Moving mirrors and time-varying dielectrics

Harry Johnston, Sarben Sarkar

The spectral distribution of light generated by a finite cavity with one moving mirror is compared with that produced by a fixed cavity containing a time-varying dielectric. In both cases a motion over a finite time interval is considered. Although the moving mirror is usually considered to be an idealization for the time-varying dielectric, there are qualitative differences in the spectra produced. The spectral distribution for the moving mirror case behaves as 1/ ${\mathit{n}}^{3}$ , while that for the time-varying dielectric behaves on average as 1/ ${\mathit{n}}^{4}$ but is rapidly oscillating.The spectral distribution of light generated by a finite cavity with one moving mirror is compared with that produced by a fixed cavity containing a time-varying dielectric. In both cases a motion over a finite time interval is considered. Although the moving mirror is usually considered to be an idealization for the time-varying dielectric, there are qualitative differences in the spectra produced. The spectral distribution for the moving mirror case behaves as 1/${\mathit{n}}^{3}$, while that for the time-varying dielectric behaves on average as 1/${\mathit{n}}^{4}$ but is rapidly oscillating.

Physical Review AMay 1, 199537 citations

Time to think

Mark Buchanan

Nature PhysicsDec 1, 20130 citations

Just a matter of time

David Gevaux

Nature PhysicsNov 1, 20100 citations

Propagator and multiple scattering approach to the time of arrival problem

Victor F. Los, Andrei V. Los

The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t0) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V0, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V0, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V0 → 0, t − t0 ~ /2V0 → ∞ (2V0(t − t0)/ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V0 including the Zeno case (V0 → ∞) and the normalized TOA rate can be introduced for all values of V0 as a probability distribution. The latter is illustrated for the Gaussian wave packet.The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t0) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V0, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V0, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V0 → 0, t − t0 ~ /2V0 → ∞ (2V0(t − t0)/ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V0 including the Zeno case (V0 → ∞) and the normalized TOA rate can be introduced for all values of V0 as a probability distribution. The latter is illustrated for the Gaussian wave packet.

Journal of Physics A Mathematical and TheoreticalApr 21, 20117 citations

{\mathit{n}}^{3}

, while that for the time-varying dielectric behaves on average as 1/

{\mathit{n}}^{4}

but is rapidly oscillating.