A typical cover letter I sent to the editors of the most recent journals I submitted my Conflict paper to

Charles Michael Fox

Editors of ____________________________________________,

Please note that my paper does not claim that I have or that there is a method of controllable faster-than-light (FTL) communication or signaling, or that special relativity is false. (If somewhere in the paper I refer to "my FTL method", it is only because I got tired of listing all the following qualifications.) My paper does say that if quantum mechanics is entirely true, and in addition an entangled particle pair described by one particular (specified) kind of wavefunction similar to the one discussed near the end of the EPR paper is constructible, and that so-far-unobserved restrictions on human action do not occur, then controllable FTL signaling is possible. Therefore, at least one of the above underlined conditions must not occur, unless there is a violation of special relativity, as explained in this letter's last paragraph. That is the point of my paper, which suggests that the second condition, constructability of the required type of entangled particle-pair state, is the one which doesn't occur, but my paper doesn't give nor do I know any strong argument for that being the one.

The key to the main conclusion of my paper "A Possible Severe Conflict between Quantum Mechanics and Special Relativity" is the existence of continuous EPR (the 1935 Einstein-Podolsky-Rosen paper)-type variables, position and its conjugate variable momentum, whose measurement uncertainties are related by the Heisenberg Uncertainty Principle, of two spatially separated particles entangled in a certain way. EPR said that locality was true, so measurement of either variable of one particle would not affect the distant particle, so through the entanglement correlation the distant particle's corresponding variable could be measured without disturbing that particle's other, conjugate variable, thus violating the Uncertainty Principle (and so demonstrating that quantum mechanics is not complete). Modern physics says, however, that locality does not hold in this case, but that the Uncertainty Principle does, so that measuring nearly precisely, e.g., the position of one particle, and so the position of the other particle, would indeed affect the momentum uncertainty of the other particle, greatly and measurably increasing it. Voila, controllable instant communication between the particles' separate locations! This possibility seems to have somehow been overlooked by the physics community. (Determining at the distant [receiving] location whether the momentum uncertainty of its particle had increased, which would communicate to that distant location a "one" bit, of perhaps a long message, would require measurement of the momentum of its member of each of an ensemble of many identically prepared entangled pairs and, for the sending location to send a "one" instead of a "zero", the sending location to measure many of its [the sending location's] particles' positions nearly precisely.)

Of course, such controllable FTL, indeed instant, signaling would violate the microcausality postulate of relativistic quantum field theory and, according to special relativity, using a well-known procedure enable controllable signaling into the past light cone of the signaler, giving the possibility of thereby preventing that signaling (the causality-violating grandfather paradox of time travel into the past, and so of FTL signaling). This obviously cannot occur, so something in quantum mechanics or special relativity, the constructibility of the required entangled state, or the freedom of human action, must not be true. An actual advantage for physics of this problem occurring is suggested.

Charles Michael Fox

A Possible Severe Conflict between Quantum Mechanics and Special Relativity

Charles Michael Fox

Abstract

The position and momentum variables of two spatially separated particles in an entangled joint state similar to the entangled state described in the 1935 Einstein- Podolsky-Rosen (EPR) paper can, if quantum mechanics is entirely true, and, in addition, such a pair in a state described by any one of a certain class of quantum wavefunctions similar to the one discussed in EPR is constructible, do what the polarization or spin variables of two entangled photons or massive particles cannot do-- be used to provide faster-than-light communication of information decided on by a human experimenter -- how? -- by nearly precisely measuring the position of the local particle of each of an ensemble of (perhaps, but not necessarily, identically prepared) entangled pairs of such particles, causing its position standard deviation (s. d.) to become smaller, thus through the entanglement correlation causing the position s. d. of the distant particle of each pair to become smaller, thus causing its momentum s. d. to become larger, thus causing the momentum s. d. of the ensemble to become larger, which change is measurable at the distant location, thus communicating to it FTL that the position measurements on the local particles of the ensemble have been performed. This possibility, however, conflicts with the microcausality postulate of relativistic quantum field theory and, even more seriously, with the Lorentz transformations of special relativity. One possible solution to this problem is offered.

Keywords: EPR, faster-than-light, no-signaling, causality, special relativity, quantum mechanics

I. Introduction

This paper describes a near contradiction between (1) basic aspects of quantum mechanics (QM) and (2) special relativity (SR), which together provide for certain types of correlations between the properties of two separated particles which are entangled (in general but, importantly, not in detail, in ways similar to the spin or polarization entanglements much discussed currently) which could be used to produce controllable faster-than-light communication, and thus communication into the past, and with that the contradiction of the grandfather paradox. A possible way in which such a conflict might not occur is that no joint state in which the two particles are entangled in the required way is constructible. If this is not the case, then either a fundamental conflict between the two theories must exist, or there would exist a hitherto unobserved type of restriction on human action involved with such communication into the past.

The Einstein-Podolsky-Rosen (EPR) paper [1] argued that if (a) the quantum-mechanical description of reality provided by the wavefunction is complete and (b) two systems separated by a non-trivial spatial interval do not interact, then the two physical quantities of a system corresponding to two noncommuting operators, e.g., the position and momentum of a particle, could both exist simultaneously, contradicting the Heisenberg Uncertainty Principle (HUP) of QM and showing that QM’s description of reality by the wave function is actually not complete.

Perhaps no practical experiment to test the locality axiom (b) was known in 1935, but in 1951 David Bohm suggested [2] a type of entanglement which involved two particles in what is termed the "singlet" state, in which, according to QM, the probability of their spins being measured to be in various directions varied with the angle between those directions in ways that did not depend on their spatial separation, and in 1964 J. S. Bell published a paper [3] showing that if the universe obeyed a local theory, either deterministic or probabilistic, satisfying assumption (b) of EPR, those correlations could not occur. Numerous experiments since then have (almost) conclusively confirmed that those spin direction correlations predicted by QM, or similar correlations involving the measured polarizations of a pair of entangled photons, do occur, showing that the universe does not obey any entirely local theory. Some of these experiments have ruled out, specifically, theories which are merely Einsteinian local, i.e., which forbid causal interaction between space-time points which are outside of each other's forward and backward light-cones, i.e., are separated by a space-like interval.

Although these correlations include predictions with a 100% probability of being true of the outcome of certain spin or polarization measurements of a distant particle or photon, based on the simultaneously (in some inertial coordinate frame) measured spin or polarization direction of a local particle or photon, that local spin or polarization direction cannot be controlled by the local measuring experimenter, so these correlations cannot be used to send an arbitrary message, decided on by the local experimenter, faster than light (FTL).

However, this paper gives in III a way, if (importantly!) certain major principles of QM are true and, in addition, entangled pairs of spatially separated massive particles or photons with a joint wavefunction similar to that described in the EPR paper are constructible, for each inertial coordinate frame I to send to any distant point instantly, in I, any message encodable in a finite number of binary bits, using not the correlated spin or polarization directions of the particles of such pairs but rather the original EPR correlated positions and momenta of the particles -- in principle, even if not currently in practice.

The contradictions of the possibility of FTL communication with the no-signaling theorem(s) and with the microcausality postulate (MC) of relativistic quantum field theory (RQFT) are discussed in V and VI. However, the most serious conflict with established physics of this prediction of the possibility of instant, even just FTL, signaling in each inertial frame is, I think, with the well-tested and confirmed Lorentz transformations of SR, which imply that, without some improbably severe restrictions on human or human mechanisms' actions, such signaling could be used to send, by a well-known procedure, a signal into the past of the apparatus which emitted the signal, and thereby, possibly through some prearranged feature of that apparatus, prevent the sending of the signal (the causality-violating problem-- the grandfather paradox-- of time travel into the past, and thus of FTL signaling). This is discussed in VI and VII. I have no probable solution to this problem other than that no entangled 2-particle state similar to the state in the EPR paper and such as I specify, suitable for my FTL communication method, is constructible.

II. Measurements Involving Correlated Spins of Entangled Wavefunctions

Correlations predicted by QM between the measured spin directions of two massive particles (or the polarization direction of two photons) in the singlet state include the prediction that if the spin of one particle of such a pair of massive particles is measured in direction a or -a (the setting of the measurement apparatus) and found, e.g., to be in direction a (the outcome of the experiment), the spin of the other, if measured in direction a or -a, will be found to be in direction -a. (A similar statement holds, with appropriate modifications, for photon polarization.) Although whether either one of the two's spin is measured to be in direction a rather than -a cannot be controlled by the (local) experimenter (Alice) or anyone else, the fact that the distant particle's spin must measure to be in direction -a if the spin of the local particle measures to be in direction a, if the distant one's is measured (by Bob) in the same or opposite direction as the local one, might seem to show that the direction of the first's spin is communicated to the location of the second arbitrarily, perhaps infinitely, rapidly. However, as shown in [3], that correlation can be explained as a function of some hidden variable originally encoded in each of the singlet state pair before they separated, and doesn't require for its explanation FTL communication between them after they are separated. However, the detailed dependence of the probability of the measurement of the distant particle's spin being in direction b if measured in that direction when the spin of the local particle has been measured as being in direction a, given by P(a,b) = sin²(θ/2), where θ is the angle between a and b, cannot be explained by a local theory, as also shown in [3].

This apparently suggests to some that the measurement of the spin of particle 1 does not transmit any information to the location of distant particle 2, since it is only the probabilistic correlation between the measurements of spin, in various directions, of the two particles that shows nonlocality. However, I think this is not so, since, according to QM, after the singlet state has been established and the particles have separated, if the spin of particle 1 is not measured, for every direction b the probability of measuring the spin of particle 2 to be in that direction, if measured in direction b, is 1/2, regardless of any information known by Alice, Bob, or anyone else, whereas if Alice measures the spin of particle 1 to be in direction -a, the spin of particle 2, if measured by Bob in direction a or -a, is certain to measure to be in direction a, so if Alice and Bob measure in direction a or -a and Bob knows this, and measures his particle's spin to be in direction -a, he will know that Alice's measured to be in direction a, even though he will not know whether she measured in direction a or measured in direction -a, since it is only that if Alice's particle's spin had measured to be in direction -a, not whether she had measured in direction a or measured in direction -a, which determines that his would have measured to be in direction a. Thus Alice cannot send information to Bob by choosing whether to measure in direction a or -a. However, whether Alice's measurement result was a or -a is communicated to Bob; this is a message decided upon by Nature, rather than Alice, sent FTL from Alice's location to Bob's location. Alice's Shannon entropy E, if she doesn't measure her particle's spin direction, of whether the spin direction measured by Bob will be a, or instead will be -a, if he measures it, is -[(1/2)(logbase2((1/2) + (1/2)(logbase2(1/2)] = -[(1/2)(-1 -1)] = 1, while if she measures her particle's spin, E = (1)[logbase2(1)] = 0. There is a corresponding entropy decrease for Bob. Additionally, does a supernova explosion 10,000 light-years from Earth fail to send a message (that the star exploded, & many other things eagerly awaited by astronomers) to us because no one in the universe can control whether it explodes (if in fact no one can)? Abner Shimony wrote a paper approximately titled "Controllable and noncontrollable signaling" for a Japanese conference, which I have not been able to obtain, which probably discussed this.

III. Faster-than-light Signaling

The (only maybe feasible) FTL communication method described in this paper does not depend on spin or polarization correlations of two particles, such as in the spin or polarization singlet state; instead, it uses the position and linear momentum correlations of two particles of the same type (both electrons, both photons, etc.) in (approximately) the entangled state described in EPR, or a similar state described below, together with HUP, which states a fundamental relationship between the uncertainties (Heisenberg's original term "indeterminacy", "Unbestimmtheit ", may actually be better than "uncertainty") of, e.g., those position and linear momentum variables of each particle of such a state, which is essentially used in the FTL communication method whose possibility this paper shows is implied by HUP together with another basic principle of QM, with the addition of the somewhat less certain possibility of constructing at least one system with a particular type of quantum state; no such relationship as HUP is available in the spin or polarization cases. EPR denied non-locality, assuming that no action on one particle of even an entangled pair could influence the state of the other distant particle of the pair, at least until a signal had traveled between the spatial location of the first particle to that of the second-- assumption (b). That this signal could not travel faster than light was not mentioned in EPR, but clearly would have been believed by at least one of its authors-- Albert Einstein.

While EPR denied the existence of non-local influences, the experiments mentioned above which were intended to test QM predictions for the singlet state, needed to confirm the hypothesis of Bell's Theorem (or perhaps that of its logically equivalent contrapositive, depending on which of the two is considered to be Bell's Theorem) have clearly shown, in conjunction with that theorem, that non-local influences can occur. Therefore, this paper will assume that the refutation of HUP attempted by EPR, which depends on the assumption of locality, is invalid, and that HUP is probably true.

EPR discusses a system S composed of two particles whose (spatially one-dimensional) wavefunctions are known at time t = 0, next interact till t = T, and after that do not interact. S's wavefunction at t = T is 𝛹(x₁ , x₂ ), which EPR chooses to be such that the sum p₁ + p₂ of the momenta of the two particles is zero, and the difference x₂ - x₁ of their positions is x₀ (at time T). EPR gives this as

[The integral symbol wouldn't reproduce here, so I show the relevant equation as:] 𝛹(x₁ , x₂ ) = Integral between - infinity and + infinity of {exp[(2𝜋i/h)(x₁- x₂ + x₀ )p]dp}.

According to EPR, if p is measured as the momentum of particle 1, -p would be measured as the momentum of particle 2, and if x is measured as the position of particle 1, x + x₀ would be measured as the position of particle 2 ([1], p. 779-80).

That this is almost an achievable joint state (not quite achievable, since the integral doesn't exist. For it to exist, and so the wavefunction possibly to be actually achievable, the integrand might be attenuated at large |p| so that it's norm was square integrable; see other considerations for 𝛹 below.) is made plausible by the fact that the sum P₁ + P₂ of the two momentum operators of the two particles commutes with the difference Q₂ - Q₁ of the two position operators, so the sum of the momenta and the difference of the positions may be measured without the measurements interfering with each other. Also, P₂ - P₁ commutes with Q₁ + Q₂. Call a wavefunction similar to but having eigenvalues of these two operators as its eigenvalues instead "𝛷".

EPR says that x₁ may be precisely measured without affecting particle 2, so x₂ may thus be precisely determined without any effect on particle 2. On the other hand, one could have precisely measured p₁ without affecting particle 2, so p₂ could also have thus been precisely determined without affecting particle 2. Thus, both x₂ and p₂ must have simultaneously existed, contradicting HUP ([1], p. 780).

However, according to current belief, since the precise position measurement on particle 1, through the entanglement of the two particles that enabled the difference of their positions to have a precise value, would also localize particle 2 precisely, the momentum of particle 2 would, because of HUP, be rendered completely uncertain, with an infinite expectation value of the spread (standard deviation--s.d., often symbolized by "∆") in measured momentum values of an ensemble of such particles, if their momenta were measured.

That is the crux of this paper's (possibly unworkable) FTL communication procedure that QM almost implies is workable, which is to send a signal FTL from particle 1's location to particle 2's location, which may in principle be arbitrarily far from particle 1, by first putting S into a state having wavefunction 𝛹' (or 𝛷' ) like the EPR one 𝛹 (or 𝛷) except with the momentum sum and position difference only nearly precisely determined (to avoid assuming S to have a wavefunction defined in terms of non-existing integrals), and with the individual position uncertainties ∆(x₁) and ∆(x₂) of particles 1 and 2 both >> ∆(x₁-x₂), and such that ∆(p₁) and ∆(p₂)are fairly small (the requirement is more nearly exactly stated in (A) below; it will be satisfied if, e.g., the ∆(x)'s are large and the x's and p's have an approximately Gaussian probability distribution), next measuring the position of particle 1 nearly precisely, thereby fixing the position of particle 2 nearly precisely, thereby making the spread (uncertainty) in momentum of particle 2's wavefunction considerably greater than it would have been if the positions of particle 1, and as a consequence of particle 2, had not been measured (this initial momentum spread is ∆(p_2,) which is, of course, known to Bob), which increase in momentum uncertainty of particle 2 is in principle measurable by Bob at his location by measuring the momentum of each of his particles in an ensemble of identically prepared particle pairs, and by this increase indicating at particle 2's location that particle 1's position has been measured (which can serve, e.g., to change some predecided binary bit from 0 to 1), the information indicating that the measurement of particle 1's position has been made thus having been transmitted from 1's location to 2's location FTL, in fact instantaneously in Alice's reference frame. This description of particle 1 being localized, which causes particle 2 to be localized, due to the difference of their positions being almost precisely determined, which in turn causes particle 2's momentum to be considerably more uncertain than it was, is consistent with the alternate description which says that particle 1's being localized causes its momentum to be considerably more uncertain than it was, which causes particle 2's momentum to be correspondingly more uncertain than it was, due to the sum of the two momenta being almost precisely determined. (Actually, it is not clear that it is necessary, for my FTL method to work, that P₁ + P₂ be almost precisely determined, even though the EPR argument did require this; it seems to be enough that Q₁ - Q₂ be almost precisely determined [for the case in which Alice measures her particles' positions nearly precisely], even though in some types of experimental/signaling arrangements this would require that P₁ + P₂ also be almost precisely determined.)

Some considerations for practical FTL communication using this method are:

(1) Whether 𝛹′ or 𝛷′ would be better to use would depend on what equipment was available at what cost, and how far apart Alice and Bob would be when communicating.

(2) The momentum of particle 1 (Alice's) could be nearly precisely measured instead of its position, thereby increasing the position spread of particle 2's (Bob's) wavefunction instead of its momentum spread.

(3) The spread of the momentum or position measurements of Bob's particle if Alice never made a measurement on her particle would depend on the details of S's wavefunction, which would be determined by the exact procedure used to prepare it. Whatever it was, under the assumed conditions, by a sufficiently precise position or momentum measurement on her particle, Alice could make the spread of momentum or position measurements of Bob's particle considerably and measurably greater than the amount it would have been if she had not measured. A large spread increase in either would make this increase easier for Bob to detect, if his apparatus could find and measure his particle at all, but would make that finding and measuring more difficult. If Alice and Bob were light-years apart, this finding and measuring might be very difficult. A few tens of meters might be the most that would be practical for an experiment currently.

(4) To measure the spread of momentum or position implied by his particle's wavefunction, Bob would need to measure the momentum (or position) of his particle in each member of an ensemble of many instances of such a system S identically prepared in state 𝛹 ' or 𝛷', and after that with Alice either measuring or not measuring the position (or momentum) of her particle, depending on whether she intended to send a 1 or a 0 to Bob. The more copies measured, the more reliable the bit transmission.

That encoding scheme uses a signal's being sent by Alice (and received by Bob) to indicate a binary 1, and the absence of such a signal to indicate a 0. An alternate, and perhaps more reliable, scheme uses one type of signal to indicate a 0, with Alice causing, e.g., the spread in momentum measurements of Bob's particle to be ≈ some 𝜎 >> the spread in momentum measurements of his particle in the original entangled state 𝛹 ' or 𝛷' (i.e., the case in which she makes no measurement on her particle) in order to send a 0, and another type of signal, for which Alice causes the spread in momentum measurements of Bob's particle to be, say, > 3𝜎, to send a 1.

If Alice is sending a 1 and so measures nearly exactly her particle's, e.g., position, Bob's particle will, in the many different instances in the ensemble, generally have many different wavefunctions just after the measurement, depending on what small region Alice's measurement shows her particle to be in, but each of Bob's wavefunctions will have a spread in momentum > some 𝛽 > 0, e.g., 3𝜎 in the second encoding scheme, and so the spread in momentum of the entire ensemble will be > 𝛽, by an elementary statistical theorem. Since for each bit of the transmitted message it is not necessary to know, for any of Bob's particles in the ensemble used to transmit that bit, which of Alice's particles is in the same pair as that one of Bob's, all of Alice's measurements on her particles of that ensemble could be made simultaneously, and similarly for Bob's measurements, without having or needing any method of determining which two particles belong to the same pair.

(5) After the entangled particle pair is created and the two particles begin to move away from each other (if their states cause them to do so), in free space p₁ and p₂ will not change, so ∆(p₁ ± p₂) will not change, but assuming realistic non-zero ∆p₁ and ∆p₂, for particles with non-zero rest mass, ∆(x₁ ± x₂) will increase with time, so if Alice uses position measurement of her particle to signal, its effect will decrease with time, since even an exact determination of x₁ can’t decrease ∆x₂ to less than ∆(x₁ ± x₂), so ∆p₂ can’t be increased by HUP beyond h/[4π∆(x₁ ± x₂)], which will eventually become less than the initial (before Alice measures) ∆p_2,, so will be ineffective for signaling that Alice has performed a measurement; if she uses momentum measurement to signal, the resulting increase in ∆x₂ will progressively become smaller and harder to reliably determine as the initial ∆x₂ increases. Thus, there will be a maximum distance, depending on the entangled pair characteristics, at which FTL communication can be reliably achieved. However, for particles, e.g., photons, with zero rest mass, this seems not to be a problem, since they all have the same speed, C, regardless of their momenta, so ∆(x₁ ± x₂) won’t increase with time. I am a little uncertain of this conclusion.

I am not familiar enough with present particle momentum and position measurement abilities to know whether, assuming QM is entirely correct and that some entangled state satisfying (A) below is creatable, signals could presently be sent FTL by this method. However, the theoretical possibility of this is, I think, more important than its practical use in communication. It would, of course, be desirable to have an experimental test of this method's ability to communicate FTL.

This technique of FTL, even instant in Alice's inertial frame, communication requires the following:

(A) the createability of particle pairs in entangled states 𝛹′ or 𝛷′, with nearly precisely determined differences or sums of the two particles' positions (momenta) which are considerably more precisely determined than the individual particles' positions (momenta), with the conjugate variable's (momentum or position, respectively) uncertainty of one of the particles being fairly small (more nearly exactly: measurably smaller than the maximum uncertainty which could be produced in it by a achievably nearly precise measurement of the other variable of the other particle),

(B) that such a state’s wavefunction "collapses" (changes) instantly, or nearly so, in Alice's inertial reference frame, when Alice performs her position (momentum) measurement, to separate states in which Alice's, and so Bob's, particle has a nearly precise position (momentum), as (at least seemingly) guaranteed by (A), and

(C) that HUP is true, so that the momentum (position) spread of Bob's particle's wavefunction is thus considerably increased.

Nearly instantaneous action at a distance, required by (B) and predicted by (nonrelativistic) QM, has been shown to occur in experiments demonstrating correlations between two distant entangled particles' spins, and between two entangled photons' polarizations, due to a nearly instant change, transmitted FTL, everywhere in the entangling joint wavefunction upon measurement at one of the particle's locations, of a kind which would not be possible if the locality hypothesis of EPR were true, but such FTL change for a joint wavefunction involving two particles' positions and momenta entangled in the way proposed by EPR and required by my FTL signaling method, although also predicted by QM, as far as I know has not been experimentally shown to occur.

I don't know whether FTL signaling of the type contemplated here might, in some circumstances, transmit energy, linear, or angular momentum FTL, or violate conservation of one of them.

IV. An Experimental Paper Describing an Entangled Photon Pair Almost Satisfying (A)

Several experimental papers involving entangled position and momentum variables have been published, and one, [4], involved an entangled state between two photons, with the transverse position and momentum variables of the photons being correlated with each other by the entanglement, produced by spontaneous parametric down conversion, which seemed as if it might satisfy my requirements. However, not all the information necessary to determine whether requirement (A) was satisfied by the entangled photon states produced in this experiment was included in the paper. In addition, the combination of the small final separation between the two photons of a pair (less than one meter) with the very slow method of measuring the (transverse) momentum spread of the photons would not allow actual verification of FTL communication between the photons' location even if it occurred. (The combination of nearly precise localization of one photon and measurement of the resulting increase in momentum uncertainty of the other photon, measured using an ensemble of such photons and required by my FTL signaling method was, as far as I know, not attempted in [4]'s experiment.)

One of [4]'s authors confirmed that there could easily be a much faster method of measuring the photons' momentum spread, and while I didn't inquire about the feasibility of a similar experiment with larger separation between the two photons, that seems obviously possible using the basic technique of that paper. However, this coauthor also indicated that, while my requirement that ∆x, the transverse position uncertainty of each photon in the entangled state, be much greater than ∆(x₁-x₂), the uncertainty in the photons' position difference in that state, which is one part of (A), was satisfied, since it was about 8.5 times greater, the momentum uncertainty ∆p of the photons in the entangled state, instead of being about h/(4𝜋∆x), the minimum allowed by HUP, which it would have been if the variable distributions were Gaussian, was instead h/[4𝜋∆ (x₁-x₂)], about 8.5 times bigger than that. Since Bob's photon can't be localized any more precisely than ∆(x₁-x₂) by even a precise measurement of the position of Alice's photon, Bob's photon's momentum uncertainty ∆p isn't guaranteed by HUP to be any greater than h/[4𝜋∆ (x₁-x₂)] after a position measurement by Alice. Since this would be the initial ∆p of Bob's photon before measurement by Alice of the position of her photon, that measurement wouldn't be guaranteed by HUP to increase ∆p of Bob's photon, so Alice might not be able to cause such a measurable increase to send a signal to its location. Of course, some measurements by Alice of the position of her photon would change its position probability distribution, and so would, in some cases possibly, change the position probability distribution of Bob's photon, even if it didn't decrease its ∆x₂, and so perhaps would change its momentum probability distribution, even if it didn't increase ∆p₂, and this change would in principle be measurable, but determining whether it had occurred could require momentum measurements on many more photons than a measurement of ∆p₂ which would give the same statistical certainty as in the case in which ∆p₂ could be made to increase considerably.

I have been unable to determine from the authors of [4] the reason for the large initial uncertainty in the photons' momenta, or whether it can be reduced by a modification of their experiment's entangled state preparation procedure. [4]’s entangled photon state turned out to be not suitable for this paper's FTL communication method, and I don't know whether its unsuitability is related to some general impossibility of creating an entangled photon or other pair suitable for that FTL method (which I think may actually be infeasible, because of such non-constructability of a suitable entangled state, or for other reasons now unknown- see VI and VII.)

V. Some Objections to Signaling by Entanglement

I have not read all the so-called proofs of all the variations of the no-signaling theorem, which roughly says that quantum entanglement between two spatially separated systems cannot be used to send a signal between them. Those I have read that can claim to apply generally beg the question (i.e., assume what they are trying to prove) either:

(1) by assuming that the state of the system S composed of the two supposedly entangled systems A and B is given by the wavefunction 𝛹 = 𝛹_A ⊗ 𝛹_B , or mixtures--not superpositions--of such wavefunctions, where 𝛹_A and 𝛹_B are the wavefunctions of A and B. For such an S, by definition of "entangled", A and B are not entangled, and thus, of course, any change brought about in A has no effect on B (until they, perhaps, later interact in the normal way); or

(2) by just assuming that the actions, e.g., measurements, performed on A have no effect on B by, e.g., assuming that the measurement apparatus performing a measurement on A does not interact with B. It is not surprising that in such systems the entanglement cannot be used to send a signal from A to B, since sending a signal from A to B means that some action by someone or something at A affects, or could have affected but intentionally was made not to, as in the sending of a binary 0 in the first scheme, someone or something at B. Thus these supposed general no-signaling proofs, by assuming that A doesn't interact with B, assume what they are said to prove. (According to my paper, if QM is correct, for its EPR-type wavefunctions the assumption of a measurement on A having no effect on B is, of course, false.)

Those would-be proofs of a no-signaling theorem which get the idea of entanglement right and do not beg the question restrict themselves to two systems in which only the spin (or polarization) states are entangled, as in the spin correlation tests to check the QM predictions relevant to J. S. Bell's theorem, as described above. For every spatial direction and its opposite , the wavefunction for the entangled singlet spin state of S is (1/√2) )(|A↑⟩|B↓⟩ − |A↓⟩|B↑⟩), with |A↑⟩ representing the state of A having spin in direction ↑, and |A↓⟩ in direction ↓, and similarly for B. In this state, the A particle and the B particle have no individual (pure, rather than mixed) states or wavefunctions, only the total system S has a state or wavefunction. (This does not imply, as is sometimes claimed, that the two particles do not have locations. A is at Alice's location, since she can measure its spin there, and similarly B is at Bob's location. The projection of the joint entangled state's wavefunction onto A's Hilbert space is localized at Alice's location, and its projection onto B's Hilbert space is localized at Bob's location.) When S is in this entangled state, for each direction a and b, a measurement on A of its spin in direction a does not change the probability of measuring the spin of B to be in direction b, as these restricted no-signaling theorems correctly claim. A spin measurement on A in direction a breaks the entanglement, changing the state of S from that shown to either |A↑⟩|B↓⟩ or |A↓⟩|B↑⟩ (letting ↑ = a), each occurring with probability 1/2. That the probability of measuring the spin of B to be in direction b does not change when the spin of A is measured in direction a is easily shown as follows: With S having been created in the singlet state, and without any measurement of A, for each b the probability of measuring spin of B to be in direction b is 1/2, since S is spherically symmetric. For each a and b, after a spin measurement of A in direction a, the probability of measuring the spin of B to be in direction b is

P = (probability of spin of A being measured to be in direction a) (probability in that case of spin of B being measured to be in direction b) + (probability of spin of A being measured to be in direction -a) (probability in that case of spin of B being measured to be in direction b)

= (1/2)[sin²(𝜃/2)] + (1/2)[sin²(𝜋/2- 𝜃/2)]

= (1/2)[sin²(𝜃/2) + cos²(𝜃/2)]

= (1/2)(1) = 1/2.

It might be asked, then, why this paper's FTR signaling scheme does not suffer from the same defect, with Alice not being able to control any possible signaling variable enough to be able to send a signal to Bob. In the case, e.g., of her reducing the position probability spread of her particle to increase the momentum probability spread of Bob's particle, there is indeed one variable that she cannot control--the measured position or position range of her particle. It could be in any location allowed by the original entangled waveform 𝛹′ or 𝛷′. If, in each measurement, she is certain to localizes her particle to some interval of width ∆x (which measured interval may, of course, be different for different particles), and after the measurement the particle's wavefunction is its wavefunction just before the measurement projected onto that interval (really its subspace) and normalized, that after-measurement wavefunction's probability distribution may be quite different than its before-measurement distribution. However, the total position probability distribution of the measurement, which is the sum over all the possible intervals of the normalized probability distributions for those intervals weighted by the probability of finding the particle in that interval, is just the before-measurement probability distribution, which would probably be close to the distribution of a long series of measurements, and, by an extension of this argument, Bob's position measurement's distribution, after the measurements by Alice, would also be his before-measurement distribution, so Alice couldn't send Bob a controllable signal with both of them using just position measurements. (Admittedly, I don't know a proof of this for all possible types of position measurements.) The position of Alice's particle, however, is not used to signal, it is the position probability spread of its wavefunction after measurement, which she can control, or at least establish an upper bound to, that sends the signal. Even though Alice cannot know precisely where her particle will be measured to be, she can, assuming III (A), with some conceivable apparatus sufficiently limit its position uncertainty after measurement to sufficiently increase Bob's particle's momentum uncertainty to effect communication with him (with the qualification at the end of IV). The statistical considerations discussed in III (4) are relevant to this. For example, a set of short enough wavelength lasers could probe enough space that there would be at least a 99% probability of them hitting her particle, and thereby, for small enough laser beam diameters and extensive enough detectors to almost certainly detect the reflection, limit its position spread to an adequately small region. If this scheme actually is impractical, some other, perhaps using photons instead of massive particles and an array of photodiodes or charge coupled devices to measure their (transverse) positions, might even now be doable. Similarly, Bob will not know exactly where his particle will be found, but with some conceivable apparatus will be able almost certainly to find it and measure its momentum, and so the increased momentum spread, if Alice creates one, of the particles in his ensemble of particles.

VI. More Serious Objections to Faster-than-light Signaling

This brings us to the conflict of FTL signaling with MC. My knowledge of RQFT is not very extensive, but my understanding of MC is that it says that two field operators, and so two measurements, at spacelike separated locations commute with each other, so that neither can affect the probability of any outcome of the other, which effect my scheme, of course, requires. MC is not violated by the singlet state spin measurements, as shown above, even though the non-relativistic QM theory which leads to this conclusion also indicates that a measurement on one of the pair of particles in that state instantly affects the total wavefunction 𝛹' or 𝛷' .

Digression on infinitely fast waveform change propagation:

Non-relativistic QM has both infinitely fast wavefunction propagation in the non-measurement regime governed by the Schrodinger equation, and infinitely fast "collapse" of the wave packet in the measurement regime not governed by that equation. The infinitely fast propagation of wavefunction changes in the non-measurement regime which is characteristic of the Schrodinger equation may be an error of non-relativistic QM, but the infinitely fast "collapse" of the wavefunction upon measurement is not totally in error, as shown by Bell's theorem and the experiments confirming the relevant correlation predictions of QM.

Eugene Wigner, in a brief talk at a QM conference I attended, discussed the problem that"infinitely fast collapse" isn't a relativistically invariant concept, since an infinitely rapidly propagated signal in one inertial reference frame will not be infinitely rapidly propagated in some others, because a constant-time slice of space-time in one frame is not a constant- time slice in some others in ways that lead to noninfinite rapidity of propagation in those frames. He said that he had tried to formulate a QM theory in which the collapse of the wave packet occurred on the backward (past) light cone of the observation causing the collapse, which would be relativistically invariant, but could not make such a theory work, for reasons he did not explain. (The backward light cone instead of the forward, which might seem more appropriate, because the collapse's occurring instantly in every inertial frame makes the forward light cone unsuitable.) I am unable to do what Wigner could not do in this case, which is related to the causality violation possibility involved in instant signaling discussed below.

End of digression

Although the infinitely fast propagation of changes of the wavefunction in a singlet spin state observation does not lead to a violation of MC, since it does not lead to a change in (overall, rather than conditional) distant observation probabilities, the establishment FTL of the QM spin correlations over a space-like interval seems to require some effect to be transmitted FTL. The RQFT references I have seen say little other than MC about the act of observation, and I am uncertain exactly how RQFT treats that, in particular, how it explains the QM-EPR-Bohm-Bell spin correlations and consequent non-locality conclusion.

A more serious problem, however, is that the infinitely fast propagation of changes in the wavefunction 𝛹′ or 𝛷′ which is required for my signaling method and which I have claimed would occur if QM is entirely true, definitely violates MC, since it does lead to observable changes in probabilities of distant measurements, upon which changes my technique depends. As stated in III, the possibility of such infinitely fast propagation requires (only) three things:

(A') that at least one of the specified 2-particle entangled joint states is constructible,

(B') the nearly precise measurement of the position (or momentum) of one of the two particles being instantly reflected in a large decrease of the probability distribution spread of the position (or momentum) of the other particle, as required by the correlations of the positions and momenta in the join wavefunction, and

(C') HUP, one of the basic tenets of quantum theory.

(B') and (C') seem to be sound according to current QM, and (A') to be at least allowed [although it may not actually be true], but their combination violates MC of RQFT.

Perhaps (depending on what parts of SR are implied by RQFT, of which I am uncertain) more serious than even this is, I think, the conflict with SR shown by the causality violation described next, which makes me question whether instant signaling by the method described in this paper is possible. (It is assumed that the space-time points involved in the following thought experiment are contained in a region of space-time that is causally iseomorphic (bi-continuously isomorphic) to a convex, open region of Minkowski space-time.)

If, for each inertial frame I, signaling from every (x,t) instantaneously in I to each space-like related (y,t) were possible, signaling from each (x,t) into the past of (x,t) to prevent that signaling could be arranged as follows: Send a signal instantaneously, in frame I, from (x,t) to some (y,t) with y ≠x, from (y,t) send, in frame I' moving away from x, a signal instantaneously, in I', to some point (z,t') in the past light cone of (x,t), which is possible since a simultaneity slice in I' through (y,t) intersects that past light cone, next from (z,t'), by slower-than-light signaling, send a signal to a mechanism that will prevent the initial sending of the signal from (x,t) to (y,t). A similar but slightly more complex argument shows that signaling into the past of an (x,t) can be done as long as signaling at a speed at least as great as some fixed speed above light speed can be done from every s-t point. Also, the possibility of such signaling into the past of the s-t point sending the signal follows from the existence of FTL signaling in one inertial frame, the fact that FTL signaling in any inertial frame is instantaneous signaling in some other inertial frame, and the Principle of Special Relativity.

Because the standard deviation (or some other suitable measure of spread) of the set of actual measurements of Bob's particles' momenta can be, although improbably so, quite different from the s.d. computed from the QM probabilities, my FTL communication method is not 100% reliable. Even if QM and SR are entirely correct, the sender might send a signal but the receiver not indicate that one had been received, and the receiver might indicate that one had been received even though the sender had not sent one. However, if enough entangled pairs are used that each attempt at communication into the past of the sender is very likely, according to QM + SR, to succeed, and many attempts to do so are made without a signal both being sent and not being sent (which last conjunction almost everyone will agree could not occur), standard logic of scientific inference for probabilistic theories would conclude that QM + SR is very probably in some way incorrect, in a way not generally recognized.

VII. Conclusion

The problem that the combination of (A), (B), and (C) [or (A'), (B'), and (C')], all three members implied by QM (almost-- (A) and (A') are not quite implied), is in conflict with SR generally, and specifically implies, in conjunction with SR, the possibility of the causality-violating, with near certainty, thought experiment described above, seems to be unavoidable, so something in one of the two almost has to be incorrect, but I do not know what that could be, other than, perhaps, the constructibility of an entangled particle pair in a joint state suitable for this paper's FTL communication method.

As discussed near the end of IV, even if large changes in ∆p₂ couldn’t be caused by Alice through her measurements of x₁, small changes in the probability distribution of p₂ might be caused by measurements of x₁, and these could be used, with extensive enough measurements of the p₂s, for controllable FTL signaling from Alice to Bob. Also, measurements of momenta by Alice might be used to cause measurable changes in the probability distribution of the position of Bob’s particle. Finally, other conjugate pairs than position-momentum might be used.

A proof, using just current QM theory, that it is not possible to construct an entangled pair of particles with a joint wavefunction which would allow measurement or other modification of one variable of a conjugate pair of variables of Alice’s particle to change the probability distribution of the other variable of the conjugate pair, of Bob’s particle, would eliminate the possibility of controllable FTL communication using a technique similar to my suggested FTL method, and so eliminate the possibility of this paper’s possible conflict between QM and SR.

However, evidence that an entangled pair satisfying (A), (B), and (C), or equivalently (A’), (B’), and (C’), could be constructed, and so according to QM controllable FTL communication was possible, and so according to QM together with SR controllable signaling into the past, and so the grandfather paradox, was also possible (assuming no improbable so-far-unobserved types of restrictions on human action would interfere), so some major defect in QM or SR existed, should not, and probably would not, be immediately considered to be a disaster for physics. It is known that there are significant problems with current physics, such as those with the Standard Model and with quantizing general relativity (GR), probably requiring changes to QM and GR, and changes to QM or SR, and so GR, to prevent the grandfather paradox might also be, or point the way to, the changes required to fix those already recognized problems with the combination of the Standard Model and GR.

Two final notes: (1) Even if this paper's controllable FTL communication method, which allows the sender to send FTL a message decided upon by him/her, proves to be unworkable, the distant EPR correlations predicted by non-relativistic QM and actually observed in many experiments constitute an uncontrollable FTL communication method which effectively sends a message about the physical local state to a distant location, even though the sender cannot choose exactly what message is sent (as discussed at the end of II above). This is an important characteristic of the universe which, as far as I have been able to determine, is not explained by, and may even be forbidden by, current quantum field theory, making that theory significantly incomplete or even incorrect. (2) I recently ran across a proposal, "Exploiting the Heisenberg Uncertainty Principle as a means to communicate", essentially identical to my FTL method, which had been posted on Physics Stack Exchange about 7 & 1/2 years ago (2013) by someone who signed himself QEntanglement. In response to his "What is flawed in this proposal?" there were 3 comments and 1 more extended answer, all critical of his proposal and all indicating almost complete lack of understanding of it by the person criticizing. His proposal received 2 up votes and the critical extended answer received 2 up votes. I also recently ran across another briefly expressed but apparently similar proposal by someone whose name I don't remember at a location on the Internet I don't remember; it received several comments, all of which were critical, and all of which showed almost complete lack of understanding of what the proposal was, as with the PSE one, including the commenters thinking that both the sending and receiving persons measured the same variable, position, instead of the sending person measuring position but the receiving person measuring momentum, as was actually proposed in this and in my proposal.

References

[1] Einstein, A., Podolsky, B., and Rosen, N., “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, Phys. Rev. 47, 777 (1935).

https://journals.aps.org/pr/pdf/10.1103/PhysRev.47.777

[2] Bohm, D., Quantum Theory, pp. 614-9, Prentice-Hall, Englewood Cliffs, NJ (1951).

[3] Bell, J. S., “On the Einstein Podolsky Rosen Paradox”, Physics 1, 195 (1964). https://journals.aps.org/ppf/pdf/10.1103/PhysicsPhysiqueFizika.1.195

[4] Howell, J. C., Bennink, R. S., Bentley, S. J., and Boyd, R. W., “Realization of the Einstein-Podolsky-Rosen Paradox Using Momentum- and Position-Entangled Photons from Spontaneous

Parametric Down Conversion”, Phys. Rev. Lett. 92, 210403 (2004).

http://www.boydnlo.ca/wp-content/uploads/2017/10/Realization%20of%20the%20Einstein-Podolsky-Rosen%20Paradox%20Using%20Momentum-and%20Position-

Entangled%20Photons%20from%20Spontaneous%20Parametric%20Down%20Conversion.pdf

Report by Dr. Pieter Kok of the Editorial Board of Physical Review A upholding the rejection by them of my "A Possible Severe Conflict...", then titled "Faster-than-light...", paper

CMF's preliminary comments on Dr. Kok's report:

My view on revealing publicly the rejection reports of journals has been stated on the Home page. I nevertheless asked Physical Review A for permission to post here Dr. Kok's report below before doing so, but that journal declined to give me that permission. Dr. Kok's upholding of Phys. Rev. A's rejection was the most nearly sensible rejection report I got from any of the seven journals I submitted versions of my paper to. Nevertheless, it's only technical criticism of my paper, that it assumes that it is possible to measure the uncertainty in an observable such as position or momentum directly, although such a measurement is not possible in a single shot measurement, and therefore my scheme fails at FTL signaling, is wrong both in its statement that my paper assumes that single shot measurements of momentum uncertainty are possible, and in its conclusion that the inability to be able to measure momentum uncertainty in a single shot measurement makes my FTL scheme fail. The version of my paper that Kok read was similar to the present version above in clearly stating the need for Alice, if she intended to signal to Bob that she had measured her particle's position, to measure the position of her one of the two particles of each of many identically prepared entangled pairs, and for Bob to measure the momentum of his one of the two particles of each of those many pairs. If Kok had read III, Faster-than-light Signaling, subsection (4), of my paper, he would have understood this. Even if he had missed this, it should have been obvious to anyone familiar with basic quantum mechanics and of Kok's apparent intelligence that there was no necessity for Bob to be able to measure the momentum uncertainty of his particle by measuring the momentum of his particle of just one pair. As many copies of the entangled pair as needed to get the required statistics could have been prepared, with Alice measuring the position of her particle of each pair, or not, depending on the signal she wanted to send, and Bob measuring the momentum of his particle of each of the pairs, to determine whether there had been an increase in the momentum spread of his particle(s), as of course was stated in my paper.

In addition, Dr. Kok's more general criticism of my paper, that in it I claim that FTL signaling is possible, even though it almost certainly is not, is also clearly false. While at certain points in it I claim that QM implies that FTL signaling is possible, in the Abstract and I, Introduction, through V, Conclusion, I make it clear that this is only to demonstrate that there is a conflict between QM and SR (assuming the constructability of entangled pairs of the required type). To demonstrate this was clearly the point of my paper. The current version makes this even clearer than the version Dr. Kok had, but no reading with understanding of even Kok's version could have led the reader to believe that the paper was claiming that FTL signaling is actually possible, or that to show this was the paper's purpose.

I realize Phys. Rev. A probably receives many incorrect faster-than-light signaling papers, and can't review them all in detail. However, Dr. Kok nevertheless obviously did read much of my paper, but must have skipped important parts of it, and either failed to seriously think about his criticism of it, or was woefully ignorant of elementary quantum mechanics, knowledge of which was obviously necessary for understanding the paper. Therefore, as stated on the Home page, I consider that Phys. Rev did not fulfill its reviewer's duty to me, and so I have no moral duty to respect its wishes about my not publicizing Dr. Kok's report, which I never agreed not to do, any supposed duty on my part not to do so being Phys. Rev.'s invention.

Dr. Kok's report:

Re: AE11792 Faster-than-light communication using original EPR entangled wavefunctions by Charles Michael Fox

Report from the Editorial Board

In this paper, the author claims that Faster Than Light (FTL) signalling is possible using position and momentum entangled states. The error in this paper is the assumption (page 4, first column) that it is possible to measure the uncertainty in an observable such as position or momentum directly. Such a measurement is not possible in a single shot measurement, and therefore this scheme, as many before it, fails at FTL signalling. The behaviour of entanglement, both for discrete variables such as spin or polarisation, and for continuous variables such as position and momentum, has been studied at length over the past nine decades. The impossibility of FTL signalling, like perpetual motion machines, has become a corner stone of physical reasoning because its resulting theoretical predictions are borne out in experiments again and again. FTL signalling would break important parts of our physical theories, as the author himself points out. This allows us to say with confidence that FTL signalling is not possible. Any genuine claim of an FTL protocol would require an exceedingly high proof threshold, which has never been met so far. Physical Review A aims to publish only science that is correct and significant. The overwhelming evidence that FTL signalling is impossible, together with the constant stream of submissions that purport to show FTL signalling, makes it reasonable that any paper claiming FTL signalling will be rejected without review. Having pointed out the flaw in the paper explicitly, I recommend that the rejection is upheld.

Sincerely, Dr. Pieter Kok Member of the Editorial Board for Physical Review A.