Interferometry with relativistic electrons

May combine two claims.

For non-relativistic velocities, the Dirac equation can be solved using a Foldy-Wouthuysen transformation, and the analysis of Ref. [13] can therefore be applied to the first two laser pulses.

“For non-relativistic velocities the Dirac equation can be solved using a Foldy-Wouthuysen transformation [15]. The large components of the Dirac spinor are then of the form ψ = exp( -imc 2 t/ ℏ ) ˜ ψ , where ˜ ψ is a solution to the Schr¨ odinger equation with the same vector potential. The analysis of Ref. [13] can therefore be applied to the first two laser pulses.”

• cites [15] J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics (McGraw-Hill, 1998).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [13] K.-P. Marzlin, Phys. Rev. A 88 , 043621 (2013), URL http://arxiv.org/abs/1307.4020 . sourceSupported

May combine two claims.

The distance between wavepackets A and B in Fig. 2 is not exactly equal to the shift (γ-1)∆z derived in Sec. II, but is given by Eq. (25), because Eq. (25) accounts for the finite laser pulse duration t_L while the discussion in Secs. II and III assumes instantaneous momentum change.

“We remark that the distance between wavepackets A and B in Fig. 2 is not exactly equal to the shift ( γ -1)∆ z derived in Sec. II, but is rather given by”

• Also stated in: § V. NUMERICAL RESULTS · p. 5 · plain

  “The reason is that Eq. (25) takes into account the finite duration t L of the laser pulses, while the discussion in Secs. II and III assumes that the momentum of the electrons changes instantaneously.”

• AssumptionThe authors state this as an assumption: The momentum of the electrons changes instantaneously (as assumed in Secs. II and III).

May combine two claims.

For the parameters used in Sec. V, the total photon emission probability P_em(u) varies very slowly with u and is less than 5% for u > 10^-4.

“For the parameters used in Sec. V, P em ( u ) varies very slowly with u and is less than 5% for u > 10 -4 .”

May combine two claims.

For constant η and E(p), the maximal transition probability between positive- and negative-energy solutions is ħ2η2/(E(p)2+ħ2η2), significant only for accelerations a≈10^30 m/s2 or larger.

“For constant values of η and E ( p ) we find that the maximal transition probability is given by ℏ 2 η 2 / ( E ( p ) 2 + ℏ 2 η 2 ), which is only significant for extreme accelerations of a ≈ 10 30 m/s 2 or larger.”

May combine two claims.

The loss of coherence from photon emission is not a fundamental problem for the proposed experiment, but the associated change in electron momentum may nevertheless affect the displacement between partial beams A and B.

“While point (i) implies that this loss of coherence is not a fundamental problem, the associated change in the electron momentum may nevertheless affect the displacement between partial beams A and B.”

Classified as probably not a substantial claim. May combine two claims.

The theoretical analysis is based on simplifying assumptions including a homogeneous electric field, laser pulses switched on and off at specific times, and a one-dimensional analysis that ignores forces in the x- and y-direction.

“The theoretical analysis that we have presented above is based on several simplifying assumptions, including a homogeneous electric field, laser pulses that are switched on and off at specific times, and a one-dimensional analysis that ignores forces in the x - and y -direction.”

• AssumptionThe authors state this as an assumption: The electric field is homogeneous.
• AssumptionThe authors state this as an assumption: The laser pulses are switched on and off at specific times.
• AssumptionThe authors state this as an assumption: The analysis is one-dimensional and ignores forces in the x- and y-direction.

May combine two claims.

A disadvantage of velocity-selection is that runs in which an electron has the wrong velocity will not contribute to the measurement, so the total number of experimental runs needed will be increased.

“A disadvantage of velocity-selection is that runs in which an electron has the wrong velocity will not contribute to the measurement. The total number of experimental runs needed will therefore be increased.”

May combine two claims.

Once electrons move at speed v, the proper distance ∆z′ in their rest frame is γ∆z, where γ=1/√(1−β2), because ∆z is the Lorentz contraction of ∆z′.

“We then have ∆ z ′ = γ ∆ z > ∆ z , where γ = 1 / √ 1 -β 2 , because ∆ z is the Lorentz contraction of ∆ z ′ .”

Probably not a substantial claim: may not pass S4 (based).

Temporal fluctuations in the electric field could significantly change the electron dynamics between different runs and thus make it impossible to measure the beam displacement.

“Temporal fluctuations in the electric field could significantly change the dynamics of the electrons between different runs and thus make it impossible to measure the beam displacement.”

May combine two claims.

If the two partial beams are partially overlapping, the fringe visibility can provide information about the displacement; coherence is therefore helpful but not essential.

“However, if the two partial beams are partially overlapping, the fringe visibility can provide information about the displacement. Coherence is therefore helpful but not essential.”

May combine two claims. The quote only partly states this claim.

The resonant interaction with laser pulses selects those electrons which have obtained the proper velocity and position to contribute to the measured observable, overcoming the problem of temporal field fluctuations.

“Thus, the resonant interaction with laser pulses selects those electrons which have obtained the proper velocity and position to contribute to the measured observable.”

May combine two claims.

The measured observable is a relative displacement of two partial beams along the direction of the laser pulses; effects on motion in other directions (such as x- or y-forces) do not affect the result, and z-forces affecting both partial beams equally do not affect the displacement.

“(ii) The measured observable is a relative displacement of two partial beams along the direction of the laser pulses. Any effects that affect the motion in other directions, such as forces in the x - or y -direction, do therefore not affect the result. Similarly, forces in the z -direction that affect both partial beams in the same way will not affect the displacement.”

May combine two claims.

For the acceleration considered, rule (iii) provides an excellent approximation and can be evaluated directly.

“For the acceleration of 1 . 8 × 10 16 m/s 2 that we considered, rule (iii) provides an excellent approximation and can also be evaluated directly.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Lorentz contraction has generally only been confirmed indirectly, for instance through the compressed charge distribution in high-energy ion collisions and the wavelength of free-electron lasers.

“Lorentz contraction generally has only been confirmed indirectly, for instance through the compressed charge distribution in high-energy ion collisions [3] and the wavelength of free-electron lasers [4].”

• cites [3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [4]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

Most authors who addressed whether Lorentz contraction causes stress in the string concluded that the string in Bell's paradox would break.

“Most authors came to the conclusion that the string would break.”

• cites [25] J. S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge University Press, 2004). sourceNot checked — no full text of the source
• cites [26] E. Dewan and M. Beran, Am. J. Phys. 27 , 517 (1959), URL http://link.aip.org/link/?AJP/27/517/1 .Not checked yet

  Not checked in the free run.

• cites [31] P. J. Nawrocki, Am. J. Phys. 30 , 771 (1962), URL http: //link.aip.org/link/?AJP/30/771/1 .Not checked yet

  Not checked in the free run.

• cites [32] E. M. Dewan, Am. J. Phys. 31 , 383 (1963), URL http: //link.aip.org/link/?AJP/31/383/1 .Not checked yet

  Not checked in the free run.

• cites [33] D. T. Cornwell, Europhys. Lett. 71 , 699 (2005), URL http://stacks.iop.org/0295-5075/71/i=5/a=699 .Not checked yet

  Not checked in the free run.

• cites [34] J. Franklin, European Journal of Physics 31 , 291 (2010), URL http://stacks.iop.org/0143-0807/31/i= 2/a=006 .Not checked yet

  Not checked in the free run.

• cites [35] A. Tartaglia and M. L. Ruggiero, Europ. J. Phys. 24 , 215 (2003), URL http://stacks.iop.org/0143-0807/24/i= 2/a=361 .Not checked yet

  Not checked in the free run.

• cites [36] D. F. Styer, Am. J. Phys. 75 , 805 (2007), URL http: //link.aip.org/link/?AJP/75/805/1 .Not checked yet

  Not checked in the free run.

May combine two claims.

In both Bell's paradox and the proposed experiment, it is Lorentz contraction of the final separation ∆z′ that is responsible for a physical effect: the mismatch between the final positions of beams A and B in the interferometer, and the breaking of the string in Bell's paradox.

“In both cases, it is Lorentz contraction of the final separation ∆ z ′ that is responsible for a physical effect: the mismatch between the final positions of beam A and B in the electron interferometer, and the breaking of the string in Bell's paradox.”

May combine two claims.

Spatial homogeneity of the electric field is not a critical assumption, because it only matters that both partial beams achieve the same Lorentz factor γ at the end of the acceleration phase.

“Spatial homogeneity is not a critical assumption because it only matters that both partial beams achieve the same Lorentz factor γ at the end of the acceleration phase.”

May combine two claims.

Because both partial beams essentially follow the same path and are separated only along the z-axis, they undergo the same (non-constant) acceleration before recombination; hence field fluctuations along the z-axis would not affect the experimental outcome.

“Because both partial beams essentially follow the same path and are only separated along the z -axis, both beams would undergo the same (non-constant) acceleration before they are recombined. Hence, field fluctuations along the z -axis would not affect the experimental outcome.”

May combine two claims.

In the proposed setup, pulse timing is not a problem because only partial beam A actually interacts with the pulses, while partial beam B does not obtain a momentum transfer.

“In the proposed setup this is not a problem because only partial beam A actually interacts with the pulses, while partial beam B does not obtain a momentum transfer.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The numerical results were checked by verifying that the wavefunction is normalized and that the mean position of all partial beams agrees with the position (in the rest frame of beam B) of a classical relativistic point particle receiving a specific momentum kick at each laser pulse.

“To check the numerical results we have verified that the wavefunction is normalized and the mean position of all partial beams agrees with the position (in the rest frame of beam B) of a classical relativistic point particle that receives a specific momentum kick at each laser pulse.”

May combine two claims.

Bell's spaceship paradox and the proposed interference experiment are closely related; except for the parts in which electron beams A and B are split and recombined, the two space-time diagrams coincide.

“Comparing Figs. 1b) and 3 one can see that Bell's paradox and the proposed interference experiment are closely related. Except for the parts in which electron beam A and B are split and recombined, the two space-time diagrams coincide.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Rules (i) and (ii) can be evaluated exactly in momentum space.

“Rules (i) and (ii) can be evaluated exactly in momentum space.”

May combine two claims.

Strictly speaking, the proposed experiment is not an interference experiment; the measured quantity is a displacement between two partial beams, which would also be produced for incoherent electron beams.

“(i) Strictly speaking, the proposed experiment is not an interference experiment. The measured quantity is a displacement between to partial beams, which would also be produced for incoherent electron beams.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The beam splitting process is velocity-selective: the electron beam is only split or recombined for electrons within a specific velocity range.

“(iii) The beam splitting process is velocity-selective, i.e., the electron beam is only split or recombined for electrons within a specific velocity range.”