Exactly solvable 2D model for photon propagation in curved space: loss of interference and Bell inequality violation

Classified as probably not a substantial claim.

In quantum physics, observables described by operators with eigenvalues ±1 can violate the CHSH inequality.

“In quantum physics, the observables are described by operators Â, Â , B, B with eigenvalues ±1, for which the CHSH inequality can be violated [32].”

• cites [32] Aspect, 1982, Phys. Rev. Lett. sourcePartly supported

  The passage on the experimental setup (indexed under Abstract; page unavailable) describes four polarizer orientations and an inequality similar in form to the Clauser–Horne–Shimony–Holt inequality. The results report a violation consistent with quantum-mechanical predictions. Our copy does not describe the observables as operators with eigenvalues ±1 or establish a violation of the CHSH inequality in that specific formulation. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

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

Time-bin encoding has experimentally demonstrated Bell inequality violation by Marcikic et al.

“To generate entanglement, we use time-bin encoding, for which BIV has been experimentally demonstrated by Marcikic et al [39].”

• cites [39] Marcikic, 2004, Phys. Rev. Lett. sourceSupported

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

The example supports the idea that the concept of a particle is linked to simultaneity.

“The example provided here also supports the idea that the concept of a particle is linked to simultaneity [29].”

• cites [29] Dolby, 2006, Simultaneity and the concept of ‘particle’ sourceSupported

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

For continuous variables like position and momentum, these operators may include parity and two other variables that test even and odd wavefunctions.

“For continuous variables like position and momentum, these operators may include parity and two other variables that test even and odd wavefunctions [34][35][36][37][38].”

• cites [34] Banaszek, 1998, Phys. Rev. A sourcePartly supported

  On p. 1, the paper analyzes position–momentum correlations using displaced parity operators. On p. 2, it describes parity measurements as having outcomes of +1 or −1. It does not describe two additional variables that test even and odd wavefunctions. (LLM, conf=0.87)

  Experimental — accuracy not yet measured
• cites [35] Banaszek, 1999, Phys. Rev. Lett. sourcePartly supported

  On p. 2, the paper describes a pair of projection operators that distinguish an even from an odd number of detected photons. On p. 3, it identifies the resulting correlation with a displaced parity operator. It does not describe those operators as testing even and odd wavefunctions; its measurement concerns photon-number parity in an optical setup. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [36] Chen, 2002, Phys. Rev. Lett. sourcePartly supported

  On p. 2, the paper defines three pseudospin operators for continuous-variable light fields: one is related to photon-number parity, and the others act in photon parity space. It describes the latter as parity-flip operators, however, not as variables that test even and odd wavefunctions. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

Classified as probably not a substantial claim.

In time-bin encoding, a qubit is encoded into two identical wavepackets, which at a given time t are spatially displaced relative to each other.

“In time-bin encoding, a qubit is encoded into two identical wavepackets, which at a given time t are spatially displaced relative to each other.”

• cites [39] Marcikic, 2004, Phys. Rev. Lett. sourcePartly supported

  On p. 1, the paper defines the qubit states as photons in a first and a second time bin, produced using an unbalanced interferometer. This supports the two temporally separated components, but our copy does not specify that their wavepackets are identical or describe them as spatially displaced at a given time. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

May combine two claims.

Far inside the warp drive, the metric is also flat, although t is not a time-like coordinate in that region anymore.

“Far inside the warp drive, the metric is also flat, although t is not a time-like coordinate in that region anymore.”

May combine two claims.

f(x) is differentiable, but its second-order derivative is discontinuous at x=0, which results in a discontinuity of spacetime curvature.

“f (x) is differentiable, but its secondorder derivative is discontinuous at x = 0, which results in a discontinuity of spacetime curvature.”

May combine two claims.

For positive wavenumber k, right-moving wavepackets correspond to η=-1, and for this choice the wavepacket can be normalized by setting N = 1/√(-2kη√πΔX).

“Right-moving wavepackets would correspond to the choice η = -1, and for this choice, the wavepacket can be normalized by setting N = 1/ -2kη √ πΔX.”

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

The Alcubierre metric does not possess a time-like Killing vector.

“However, the Alcubierre metric does not possess a time-like Killing vector.”

May combine two claims.

The rightmost pulse starts well within the warp drive, in a region that has very little spacetime curvature, and its shape is undisturbed since it propagates through a flat portion of spacetime.

“The rightmost pulse in figure 4 starts well within the warp drive, in a region that has very little spacetime curvature.The shape of this pulse is also undisturbed since it propagates through a flat portion of spacetime.”

May combine two claims.

In conformal Cartesian coordinates, the phase gradient is ∂T α = kη, which results in a positive mean frequency for kη < 0.

“In conformal Cartesian coordinates, its gradient is given by ∂ T α = kη, which results in a positive mean frequency for kη < 0.”

May combine two claims.

The leftmost pulse starts in a region well outside of the warp drive, is hardly disturbed by its motion, and propagates like in a flat spacetime.

“The leftmost pulse starts in a region well outside of the warp drive.It is hardly disturbed by its motion and propagates like in a flat spacetime.”

May combine two claims.

Photons co-moving with the warp drive (η=-1) always have the same sign of ∂t α, but their apparent frequency increases.

“Photons co-moving with the warp drive (η = -1) always have the same sign of ∂ t α, but their apparent frequency increases.”

The quote only partly states this claim.

As a result, the center pulse gets stretched.

“As a result, the pulse gets stretched.”

May combine two claims.

The center pulse passes through the region where the warp drive generates a large amount of curvature; curvature stretches the distance between light rays, tearing the pulse apart: parts in the tail fall behind, while parts further inside are dragged along with the warp drive.

“The center pulse in figure 4 passes through the region where the warp drive generates a large amount of curvature.Since curvature stretches the distance between light rays, it tears the pulse apart: parts of the pulse in the tail of the drive fall behind, while parts further inside are dragged along with the warp drive.”

May combine two claims.

It is possible to construct local positive-frequency wavepackets and to expand the field operator in terms of these states.

“In the following, we will show that it is possible to construct local positive-frequency wavepackets and to expand the field operator in terms of these states.”

May combine two claims.

The wave equation in light ray coordinates takes the form ∂x+∂x-ψ=0, which has the general solution ψ(t,x)=ψ+(x+)+ψ-(x-).

“The wave equation in light ray coordinates takes the form ∂ x + ∂ x -ψ = 0, which has the general solution”

May combine two claims.

In the left region, the Ricci scalar changes sign, with a maximum/minimum of about ±3v^2 σ/z_f.

“In the left region, R changes sign, with a maximum/minimum of about ±3v 2 σ/z f .”

May combine two claims.

Warp drive curvature generates a delay (or speedup) in arrival times of light rays, and this delay persists.

“The reason is that warp drive curvature generates a delay (or speedup) in arrival times of light rays, and this delay persists.”

May combine two claims.

The wavepacket appears stretched and distorted on a hypersurface of constant t, while it preserves its shape on lines of constant T.

“As a result, the wavepacket appears stretched and distorted on a hypersurface of constant t, as shown in figure 4. On the other hand, the wavepacket preserves its shape on the lines of constant T.”

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

The last result is valid if the trip distance z_f is much larger than the width of the curvature area, z_f σ >> 1.

“The last result is valid if the trip distance z f is much larger than the width of the curvature area, z f σ 1.”

May combine two claims.

Curvature also induces a phase shift Δθ, but the phase shift is only large for very small fringe visibility.

“Curvature also induces a phase shift Δθ, but the phase shift is only large for very small fringe visibility.”

May combine two claims.

When wavepacket ψ+ starts sufficiently far to the left, it only travels through flat space; consequently, the overlap with the detector is perfect and no phase shift is induced.

“When wavepacket ψ + starts sufficiently far to the left, it only travels through flat space.As a consequence, the overlap with the detector is perfect and no phase shift is induced.”

May combine two claims.

In the right region R, the Ricci scalar is strictly positive and reaches a maximum of about 4v^2 σ^2.

“In the right region R, the Ricci scalar is strictly positive and reaches a maximum of about 4v 2 σ 2 .”

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

Trajectory (19) gives very good control over the regions in spacetime with non-zero curvature.

“Trajectory (19) gives us very good control over the regions in spacetime with non-zero curvature.”

May combine two claims.

Using inner product (36), the wavepacket is normalized to unity for N = 1/(2√(Ẽ(k)ΔX√π)), and the overlap between two localized wavepackets is given by equation (64).

“Using inner product (36), the wavepacket is then normalized to unity for N = 1/ 2 Ẽ(k)ΔX √ π, and the overlap between two localized wavepackets is given by Ψ X 0 ,k |Ψ X 1 ,k 1 = E(k) + E(k 1 ) 2 √ E(k)E(k 1 ) exp - (k -k 1 ) 2 4 ΔX 2 - (X 0 -X 1 ) 2 4ΔX 2 + i 2 (X 0 -X 1 )(k + k 1 ) . (64”

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

For a conformally flat metric (21), the retarded Green's function for massless particles in curved space is related to the Green's function in Minkowski space via G_ret(x,y)=√g_Σ(y)G_Mink_ret(x,y).

“For completeness, we also mention that for a conformally flat metric (21), the retarded Green's function for massless particles in curved space is related to the Green's function in Minkowski space via G ret (x, y) = g Σ (y)G Mink ret (x, y).(B13)”

Classified as probably not a substantial claim.

The expansion (66) of the field operator in terms of annihilation operators for localized wavepackets with positive frequency was derived.

“Furthermore, we derived an expansion (66) of the field operator in terms of annihilation operators for localized wavepackets with positive frequency.”

May combine two claims.

Equation (64) demonstrates that localized wavepackets are not orthonormal, so calculations that employ expansion (66) will generally involve integrals over mean position and wavenumber of the wavepackets.

“Equation (64) demonstrates that localized wavepackets are not orthonormal, so that calculations that employ expansion (66) will generally involve integrals over mean position and wavenumber of the wavepackets.”

May combine two claims.

The overall shape of the curve results from the fact that for σx0 ∈ (-8, -5) only the right (R) wavepacket passes through the region of curvature, and for σx0 ∈ (-3, 0) only the left wavepacket passes through it; between these two intervals, both wavepackets pass through warp-induced curvature.

“The overall shape of the curve results from the fact that for values σx 0 ∈ (-8, -5) only the right (R) wavepacket passes through the region of curvature, and for values σx 0 ∈ (-3, 0) only the left wavepacket passes through it.Between these two intervals, both wavepackets pass through warp-induced curvature.”

May combine two claims.

The theory was applied to describe the reduction of fringe visibility in a single-photon Mach-Zehnder interferometer, and the non-violation of the CHSH inequality when wavepackets pass through a region with spacetime curvature.

“We have applied this theory to describe the reduction of fringe visibility in a single-photon Mach-Zehnder interferometer, and the non-violation of the CHSH inequality when wavepackets pass through a region with spacetime curvature.”

Classified as probably not a substantial claim.

Numerical evaluation of expression (61) with the same wavepacket parameters as in section 6 and displacement ±5ΔX for wavepackets R and L relative to center x0 yields the result shown in figure 9.

“We have evaluated expression (61) numerically with the same numerical parameters for wavepackets as in section 6, and with a displacement of ±5ΔX for wavepackets R and L relative to the center point x 0 between them.The result is shown in figure 9.”

May combine two claims.

Expansion (66) is based on an over-completeness relation and does not utilize a set of basis vectors.

“Second, it is based on an over-completeness relation and does not utilize a set of basis vectors.”

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

For values σx0 > 1, the photon is dragged along with the warp drive but does not pass through curvature, and the Bell inequality remains maximally violated.

“This is also the case for values σx 0 > 1, when the photon is dragged along with the warp drive but does not pass through curvature.”

May combine two claims.

For values σx0 < -8, the right-moving photon will not enter the region of curvature and the Bell inequality remains maximally violated.

“For values σx 0 < -8, the right-moving photon will not enter the region of curvature and the Bell inequality remains maximally violated.”

Classified as probably not a substantial claim.

Evaluating the mean value of the Bell operator B = √2(σx+σx- + σz+σz-) in state (56) yields equation (61).

“Evaluating the mean value of the Bell operator B = √ 2(σ x+ σx-+ σz+ σz-) in state (56) amounts to a repeated application of equations (39) and (60).The result is lengthy but can be greatly simplified if one takes into account that ψ+R |ψ +R = ψ+L |ψ +L = 1 and ψ+R |ψ +L = ψ+L |ψ +R = 0. We then arrive at ψ| B|ψ = 1 √ 2 ψ-R |ψ -R + ψ-L |ψ -L 2 -ψ-R |ψ -L -ψ-L |ψ -R 2 . (61”

• AssumptionThe authors state this as an assumption: Since detection happens after the warp drive stopped, annihilation operators can be defined as in flat space and âψ±R|0 = âψ±L|0 = 0.
• AssumptionThe authors state this as an assumption: The detector wavepackets are identical to the initial photon wavepackets, but displaced so that their peak matches that of the propagating wavepackets at the time of measurement.

Probably not a substantial claim: may not pass S2 (checkable).

A warp drive that only moves temporarily can dramatically simplify the physical interpretation.

“In addition, from a theoretical perspective, a warp drive that only moves temporarily can dramatically simplify the physical interpretation.”

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

The quantization procedure also applies to massive and charged scalar fields in an n-dimensional globally hyperbolic spacetime.

“The quantization procedure also applies to massive and charged scalar fields in an n-dimensional globally hyperbolic spacetime.”

Classified as probably not a substantial claim.

A 2D model cannot display all features that a quantized electromagnetic field in a 4D spacetime would exhibit.

“A 2D model cannot display all features that a quantized electromagnetic field in a 4D spacetime would exhibit.”

May combine two claims.

Since there are no transverse spatial degrees of freedom in a 2D spacetime, polarization of light and the dispersion of wavepackets will be absent.

“Since there are no transverse spatial degrees of freedom in a 2D spacetime, polarization of light and the dispersion of wavepackets will be absent.”

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

Bell inequality violation can be obtained for many different systems; a simple way is to replace the Pauli matrices by three other operators that obey the same commutation relations.

“Bell inequality violation (BIV) can be obtained for many different systems.A simple way to achieve this is to replace the Pauli matrices by three other operators σi that obey the same commutation relations.”

Classified as probably not a substantial claim.

In curved space, the interpretation of these operators is generally ambiguous: what appears to be a particle to one observer may appear to be an antiparticle to another.

“In curved space, the interpretation of these operators is generally ambiguous: what appears to be a particle (positive-energy solution) to one observer may appear to be an antiparticle (negative-energy solution) to another one.”

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

In flat space, the boundary condition ψ(xB,t)=0 can be fulfilled by the method of mirror images.

“In flat space, this boundary condition can be fulfilled by the method of mirror images.”

May combine two claims.

Using approximation (33), ∂t α ≈ k(η - ż). Assuming ż > 0, this implies that the phase gradient changes its sign for left-moving waves (η=1) for superluminal warp speed because they are dragged to the right by the warp drive.

“To understand this, we use approximation (33) for photons deep inside the warp drive, which yields ∂ t α ≈ k(ηż).Assuming that ż > 0, this implies that the phase gradient changes its sign for left-moving waves (η = 1) for superluminal warp speed because they are dragged to the right by the warp drive.”

• AssumptionThe authors state this as an assumption: ż > 0 for the phase-gradient sign-change analysis.

May combine two claims.

An observer who moves along with the warp drive would likely consider a line of constant conformal time T as a spacelike hypersurface; in this coordinate system, photons would appear undistorted and with unchanged frequency.

“On the other hand, an observer who moves along with the warp drive would likely consider a line of constant conformal time T (see figure 3) as a spacelike hypersurface.In this coordinate system, photons would appear undistorted and with unchanged frequency.”

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

In flat space, the interpretation of these operators is independent of an observer's reference frame.

“In flat space, the interpretation of these operators is independent of an observer's reference frame: if a wavepacket is normalized to 1, then âψ can be interpreted as a bosonic annihilation operator for a positive-energy solution (a particle, for charged quantum fields) in mode ψ(x), and b † ψ * as the creation operator for a corresponding negative-energy solution (an antiparticle).”

Classified as probably not a substantial claim.

An unambiguous definition of a vacuum state is possible if spacetime possesses a timelike Killing vector.

“An unambiguous definition of a vacuum state is possible if spacetime possesses a timelike Killing vector.”

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

The piecewise definition (2) of the metric leads to a discontinuity in curvature, but this effect has no significant consequences for the results.

“As discussed before, the piecewise definition (2) of our metric leads to a discontinuity in curvature, but this effect has no significant consequences for our results.”

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

Lichnerowicz has shown that both Green's functions exist in an open neighborhood of a point.

“Lichnerowicz [22] has shown that both Green's functions exist in an open neighborhood of a point.”

• cites [22] Lichnerowicz, 1961, Publ. Math. Inst. Hautes Sci. sourceNot checked — no full text of the source

May combine two claims.

For the chosen parameters, the mean wavelength 2π/k is much smaller than the width ΔX of the wavepacket; consequently its width in momentum space is narrow and a gravitationally induced change in mean momentum will quickly reduce the overlap with the detector mode in momentum space.

“The reason is that for our choice of parameters the mean wavelength 2π/k is much smaller than the width ΔX of the wavepacket.Consequently, its width in momentum space is narrow and a gravitationally induced change in mean momentum will quickly reduce the overlap with the detector mode in momentum space.”

May combine two claims.

An observer who uses radar coordinates would likely use constant t to define simultaneity; in this coordinate system, photons that traveled through a region with curvature appear stretched and their frequency is red-shifted.

“An observer who uses radar coordinates would likely use constant t to define simultaneity.In this coordinate system, photons that traveled through a region with curvature appear stretched as in figure 4, and their frequency is red-shifted, as shown by the light-blue areas in figure 5.”

Classified as probably not a substantial claim.

The scalar product is preserved in the sense that it does not depend on the choice of the hypersurface if ψ, φ are solutions to the Klein-Gordon equation.

“This scalar product is preserved in the sense that it does not depend on the choice of the hypersurface if ψ, φ are solutions to the Klein-Gordon equation [24].”

• cites [24] Parker, 2009 sourceNot checked — no full text of the source

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

The method using the Pauli-Jordan propagator and the conserved scalar product can be used in a more general setting, including a higher-dimensional spacetime, massive scalar fields, and non-Hermitian scalar fields.

“The method can be used in a more general setting, including a higher-dimensional spacetime, massive scalar fields, and non-Hermitian scalar fields.”

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

The left-moving '+' photon starts in a superposition of ψ+L(x) and ψ+R(x), corresponding to two wavepackets displaced to the left and right; similarly, the right-moving photon is in a superposition of ψ-L(x) and ψ-R(x).

“The left-moving '+' photon starts in a superposition of ψ +L (x) and ψ +R (x), corresponding to two wavepackets that are displaced to the left (L) and to the right (R), respectively.For perfect encoding, the overlap between the two wavepackets would vanish, ψ +R |ψ +L = 0. Similarly, the right-moving photon is in a superposition of ψ -L (x) and ψ -R (x).”

May combine two claims.

Since the left-moving photon only travels through flat space, this implies ψ+R = ψ+R and ψ+L = ψ+L.

“Since the left-moving photon only travels through flat space, this implies that ψ+R = ψ +R and ψ+L = ψ +L .”

• AssumptionThe authors state this as an assumption: The left-moving photon will always travel through a flat portion of spacetime, so that it will not change its shape.

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

Up to the normalization factor N, equation (62) is equivalent to the overcompleteness relation for coherent states.

“Up to the hitherto unspecified normalization factor N, this is equivalent to the overcompleteness relation for coherent states.”

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

This also happens in flat regions of spacetime, where different observers may come to different conclusions.

“This also happens in flat regions of spacetime, where different observers may come to different conclusions.”

May combine two claims.

The derivation of result (66) only made use of the properties of a conformally flat metric (21), not the specific form of the Alcubierre metric; therefore it holds for any two-dimensional spacetime.

“The derivation of result (66) has only made use of the properties of a conformally flat metric (21), not of the specific form of the Alcubierre metric.It therefore holds for any two-dimensional spacetime.”