Superluminal rates of separation and EPR photons

Transcription

1 ENSEÑANZA REVISTA MEXICANA DE FÍSICA 48 () ABRIL 00 Superluminal rates of separation and EPR photons J. Karles 1, H. Perez Rojas 1,,3 1 Departamento de Fisia, Universidad Naional, Sede Medellin, Apartado Aereo 3840, Medellin, Colombia jkarles@perseus.unalmed.edu.o Helsinki Institute of Physis, Siltavuorenpenger 0C, FIN-00014, Helsinki, Finland, 3 Grupo de Fisia Teória, Instituto de Cibernétia Matemátia y Físia, Calle E No. 309, Vedado, La Habana 4, Cuba hugo@idet.imf.inf.u Reibido el 9 de mayo de 001; aeptado el 9 de otubre de 001 We onsider rates of separation between two partiles greater than (whih is not in ontradition with speial relativity) in understanding some onflit between speial relativity and quantum mehanis found by moving observers of orrelated EPR pairs of photons. The photon frequenies observed in the moving frame have opposite shifts than those found when the detetors are fixed in that frame. By observing from two different frames the arrival of a photon from an EPR pair to a given detetor, it is illustrated how the measurement of the simultaneous arrival of the other photon of the EPR pair in one frame disturbs the measurement in the other frame. Keywords: Superluminal veloities; orrelated EPR photon pairs. Consideramos veloidades de separaión entre dos partíulas mayores que (lo que no está en ontradiión on la relatividad espeial) en la búsqueda de la omprensión del onflito entre la relatividad espeial y la meánia uántia, enontrado por observadores móviles de pares orrelaionados de fotones EPR en movimiento. Las freuenias de los fotones observadas en el maro de referenia móvil tiene desplazamiento opuesto al que se enuentra uando los detetores están fijos en tal maro. Mediante la observaión, desde dos maros diferentes, del arribo de uno de los fotones de un par EPR a un detetor dado, se ilustra ómo la mediión del arribo simultáneo del fotón ontraparte en uno de los maros, perturba la mediión del arribo simultáneo de diho fotón en ualquier otro maro. Desriptores: Veloidades superiores a la de la luz; par de fotones orrelaionados en EPR. PACS: Bz 1. Introdution In his well-known book Speakable and unspeakable in quantum mehanis, the late John S. Bell [1] states...for me then this is the real problem with quantum theory: the apparently onflit between any sharp formulation and fundamental relativity. That is to say, we have an apparent inompatibility, at the deepest level, between the two fundamental pillars of ontemporary theory...and of our meeting...it may be that a real synthesis of quantum and relativity theories requires not just tehnial developments but radial oneptual renewal. Perhaps the radial oneptual renewal suggested by Bell would ome from the onsideration of fundamental extended objets, as is string theory. But keeping us in the framework of standard relativity and quantum theory we would like, however, to have a loser look at the onflit whih arises from the unompatibity among the ideas of instantaneous redution of the wavepaket and the relative simultaneity of events. We start by disussing briefly the problem of superluminal rates of separation between two objets in Se., and in Se. 3 we apply suh ideas to pairs of photons and detetors in a EPR experiment, as a way for the observer in a moving frame to understand his observations and to make them ompatible with those of the observer at rest, from the quantum mehanial and speial relativity points of view. In Se. 4 the shift in frequenies measured by the moving observer is found, and ompared to those obtained by him with detetors fixed to his frame. Then by onsidering the event of arrival of a photon from an EPR pair to a detetor, as observed from two different frames of referene, it is disussed the simultaneous measurement of the other member of the EPR pair as made by both observers. Se. 5 deals with the onlusions.. Superluminal veloities and rates of separation It is usually believed that speial relativity forbids any veloity greater than, whih is not right. Some readers would be felt worried even in reading suh idea. What is established by speial relativity is the onstany of the veloity of light for all observers, or in other words, that the maximum veloity of propagation of a signal in vauum annot exeed in any frame of referene. To be more preise, this point must be related to transport of energy and ausality. Then the statement must be made in terms of bounded veloity of transport of energy ( ) to preserve Einstein ausality. Thus, it is possible to oneive systems in whih superluminal veloities appear without violating Einstein ausality [, 3], but as pointed out before we would like to onsider the problem in the present letter from the kinematial side, i.e., by onsidering the rate

2 SUPERLUMINALRATESOFSEPARATIONANDEPRPHOTONS 151 of spatial separation between two objets with regard to a third body with an observer O, whih an be produed at a veloity greater than without violating the postulate of onstany of the veloity of light. Even Einstein in his original paper [4] introdued formally rates of separation ± V, and Feynman [5] did the same in disussing the Mihelson- Morley experiment. On the opposite side is Fok [6], who defined the relative veloity between two moving objets as the veloity of the seond with regard to the referene system fixed on the first. In this way he obtained a Lorentz-invariant expression for the relative veloity. However, we will onern with the rate of separation understood in the first sense, whih is in general not Lorentz-invariant, but is, however intuitively lear. We name it the Einstein-Feynman (EF) rate of separation As an extreme ase of the EF rate of separation (on whih we will deal in the rest of our paper), we give the elementary example whih shows that it an be greater than. For light oming from a point soure, we have a spherial wave front propagating with veloity. If one onsider the wavefronts at the extremes of a diameter, we have obviously a relative veloity v =, i.e., photons separating at a rate with regard to an observer in the system where the soure is at rest. Or looking at the problem in another form, the surfae joining the wavefronts in the light one is parallel to the four-dimensional hyperplane perpendiular to the x 4 axis and moves upwards along the t axis at the speed, and it is obviously spaelike. Opposite photons along a diameter, lying on the one surfae, separate at the speed with regard to the observer at rest. Any observer would find that eah photon moves at veloity. With regard to one of the photons, the other photon reedes with veloity, (this is the Fok relative veloity) and it is impossible for it to inform the seond photon, say, about its state of polarization by using a signal propagating at veloity v. For an observer at rest, the spaelike interval grows at two times the veloity of light and is Lorentz-invariant. For partiles moving at veloities smaller than, however, this rate of separation is not Lorentzinvariant (see below). This is of espeial interest in onsidering EPR pairs of orrelated photons. The onept might be useful also in inflationary theories [7] whih are based on the idea of exponential bubble expansion in whih spaelike separated regions may expand at a speed larger than. We want to write a general expression for the rate of separation u 1 of two points 1 and, moving in the rest system respetively at veloities u 1, u, aording to a previous report of one of the present authors [8]. We write, thus, u 1 = u u 1.If u 1 = ±, one would have u 1 = u, and it is not in ontradition with speial relativity at all. In a frame moving at veloity V, the respetive veloities are w 1, w, where w i = u i + V 1 + u i V/. (1) One an write then w 1 = (u 1 )(1 β ) (1 + β β)(1 + β 1 β), () where β i = u i /, β = V/ and u 1 = u u 1. It is partiularly interesting the ase in whih u 1 = u, whih does not mean that w 1 = w (exept if u = ). We have w 1 = u 1(1 β ) 1 β β. (3) In onsequene, in general w 1 u 1. The equality ase orresponds obviously to the ase u =, leading to w 1 = u 1 =. (4) Thus, the maximum allowed rate of separation between two bodies with regard a third one is, whih is also a relativisti invariant. 3. Measurements of pairs of orrelated photons We want to onsider now the problem of the pair of orrelated photons in a usual EPR experiment. Let us assume that green photons are sent to the left and red to the right. Calling v and h the vertial and horizontal polarizations, the entangled wavefuntion an be written as ψ = v(x, τ)> h( x, τ)>+ h(x, τ)> v( x, τ)>. (5) Equation (5) desribes an extended wavefuntion, whih ontains the idea of non-loality in a transparent way. We shall use in what follows the von Neumann - Penrose [9, 10] onepts onerning proedures U (unitary evolution) and R (measurement or redution of the wavepaket). The measurements of polarizations at the two symmetri left-hand A and right-hand B polarizers + detetors are two spae-like events. If the polarizer B is removed,a measurement at A, say v, means that the wavefuntion of the unmeasured photon at B jumps simultaneously to h. Let us name K, K the rest and moving systems respetively (K moving to the left, that is, with veloity V ). An essential onflit appears [10] between Quantum Mehanis (if we onsider the notion of instantaneous redution of the wavepaket) and the spirit of Speial Relativity (ovariane, relative simultaneity) when the proess of measurement at the left and wavefuntion jump at right is observed from the moving system K. The moving observer would find that the left photon polarization at A is measured at a time t 1 before the right photon arrives at B, and the simultaneous event at the right (jump of the wavefuntion) is expeted to our before the arrival of the signal at B; thus, a puzzle appears. An opposite onlusion would be obtained for another observer moving to the right at veloity V, leading to ontraditory statements to both observers.

3 15 J.KARLES,H.PEREZROJAS We want to give an operational way of haraterizing the two times of arrival of the photons at the detetors. In K, the times of arrival of the photons to the detetors are determined by the quotients of the lengths of the apparatus arms by the veloity of light. Let us name by L the ommon length of the arms of the polarizer and then t = L/ is the ommon value of the time the photon takes to travel along both arms in K. If a measurement is made, the R proedure on (5) leads to either ψ 1 = v(l, t) > h( L, t) > or ψ = h(l, t) > v( L, t) >. We want to arefully analize the onditions of both observers onerning the two photon detetion trying to find a partial understanding of the soure of the puzzle from the point of view of the observer in K. For observers in K the times of arrival of the photons to A and B an be measured as said before either a) by the quotient of the length of the (equal) arms by the veloity of light relative to the polarizer, whih is the veloity of light or b) by dividing the oordinates of the detetors by. One gets the ommon value t = L/ by following any of the proedures a) and b), whih means the simultaneity of both events at K. However, for observers in K proedures a) and b) do not lead to the same results. Thus, to have equivalent kinematial onditions for the measurements in K, K, we must have a set of polarizers at rest in K (whih we will onsider later) to be allowed to make a omplete omparison of the R proedure in both systems. Let us write the time of arrival of the two photons at the detetors A and B, as seen by an observer in K. The time of arrival of the photon to A an be written in K as t 1 = L( V ) 1 β = L 1 β. (6) + V From this equation it results t 1 = L 1 β 1 + β < L = t. If we all L = 1 β L the length of the arm as measured from the observer K, one an write and for the arrival time at B, we have, t 1 = L + V, (7) t = L V. (8) Thus, the observer O in K onludes that the photon arrives at A earlier than is indiated by the loks in K, and earlier than the photon at B, sine t = L V > t > t 1 = L + V. The observer O in K argues that this is due to the fat that the photon at the left moved along the length of the arm L = L 1 β (defined by the spaetime points (x + L, t ); (x, t )) with regard to the detetor A at the superluminal (tahyoni) veloity + V, faster than the photon in K and faster than the photon moving (with regard to K ) to B, whih traveled at the relative (sub-luminal or tardyoni) veloity V. Thus, the observer in K explains the inequality in the times t 1, t of ations of the R operation at A and B as some sort of tahyoni effet in A and tardyoni effet in B, onerning rates of separation between the light and the detetors. The wavefuntion at the right jumps later than that at the left beause it approahes B at a relative speed slower than sine the onditions under whih the observations are made in the moving frame K are not the same than the ones in K. (Note that t > t 1 is due to the usual lag of the moving lok C. It is assumed that the rest and moving loks were synhronized at the spae-time oordinates (0, 0), (0, 0 ), respetively, the moving lok being ompared with two stationary ones at K. The fat that t < t does not ontradits the lag of moving loks, and it results from the fat that t is measured in B with another lok C in K, previously synhronized with C.) If the removed polarizer were A, the fat that the wavepaket jumped at t 1 < t means nothing but an effet to be expeted when superluminal-subluminal rates of separation are involved. For an observer moving to the right at veloity V, these effets are exhanged. (From the point of view of relativisti quantum field theory, the polarizations of photons arriving at A, B, as being spaelike separated, onsidered as loal observables in quantum eletrodynamis, are not dependent upon the order in whih the measurements have been performed). 4. The shift of frequenies Let us write the oordinates x 1, x orresponding to the loation of the detetors at t 1, t. We have x 1 = L 1 β 1 + V = t 1, (9) x = L 1 β 1 V = t. (10) Thus x 1 x. These quantities are respetively the distanes traveled by the left and right photons along the x axis, aording to the observers in K. If we name R l,r = L/λ l,r, the quotient of the length of the arm by the wavelengths λ l,r of the left and right photons, i.e., the number of wavelengths ontained in L at left and right, then by naming λ l,r the wavelengths registered by the detetors A, B as observed from the K system, we expet that x 1 = R l λ l, and similarly, x = R r λ r. We get thus λ l,r = λ l,r 1 β 1 ± V. (11) Here we get the paradoxial result that the orresponding frequenies of the green (left) and red (right) photons are shifted to the violet and to the infrared, respetively. This is

4 SUPERLUMINALRATESOFSEPARATIONANDEPRPHOTONS 153 due to the fat that although the soure is moving to the right in the K system, the detetors are also moving to the right, the rates of separation between the light and the detetors being observed respetively as + V and V. We note also that in K it is needed either a moving observer, or a set of observers with synhronized loks in K to follow the motion of the detetors in K and to register the results of the measurements. To onlude, let us now turn our attention to investigate the onditions under whih the pair of photons an be observed in K in an equivalent way as in K. If the polarizations of the pair of photons emitted by the soure in K are measured with polarizers + detetors at rest in K, the entangled wavefuntion is now ψ = v(x, τ ) > h( x, τ ) > + h(x, τ ) > v( x, τ ) >. (1) Let us find the right and left photon wavelengths as measured by fixed detetors in K. As emitted by a soure moving to the right, their wavelengths are Doppler shifted to the values λ l,r = λ l,r 1 β 1 V. (13) Thus, the situation beomes the opposite than before: if measured by detetors at rest, the left photon is shifted to the red and the right photon to the violet. Now let us onsider the event of arrival of the left photon to A in the frame K and the simultaneous event of arrival of the right photon, as observed in the frame K. (Obviously, as A is moving to the right with regard to K, we onsider the measurement of the arrival of the left photon to A, observed from K, as equivalent to the one made by a polarizer + detetor A at rest in K, whih would oine with A at the spaetime point (x 1, t 1), being x 1/λ l = ( V )R l /( + V )). The right photon arrival must be measured with a detetor B fixed in K, at time t 1, and at a distane x = x 1 from the soure, the length of the wavetrain in units λ r observed in K being x = R r, (14) λ r whih is the same length of the wavetrain, in units λ r, observed by K at the detetor B. We onlude that if the left photon is measured at is arrival to A by both observers, the simultaneous redution of the right photon in the frames K, K would be deteted in a different and independent way respetively by detetors B, B. The deteted photons would have obviously different quantum numbers (respetively, momenta p = /λ r and p = /λ r ). But we have assumed that the proess of measurement does not hange the photon, whih is not true, sine the right photon in B is either absorbed or its momentum hanged by the measurement, and the detetor at B would measure either no photon or a photon with momentum p p. 5. Conlusions We have seen how the relative superluminal veloity onept is useful in understanding the onflit between simultaneous quantum measurements and relative simultaneity, arising when orrelated EPR pairs are observed in frames at rest K, and moving K : the relative rates of separation photondetetors are different for both observers. We have seen also that the simultaneous event to a measurement of the left photon in A, for the photon moving to the right (i.e., simultaneous redution of the wavepaket at the right), tested by detetors at rest with regard to different frames, ours independently for eah frame K,K and the R ation on the right photon in one frame disturbs in general the R ation on the same photon in any other frame (see Aharonov and Albert [11]). From all our disussion, we have an illustrative example of the statement made in [11] onerning the fat that in relativisti quantum field theories, the quantum states themselves make sense only within a given frame, (e.g., the entangled wavefuntions (5) and (1) are not onneted by a Lorentz transformation, and annot be, sine they represent non-loal, and in onsequene, non-ovariant objets) and that ovariane resides in the experimental probabilities (obtained from the results of the R ation). In this way we may understand the present oexistene of speial relativity and quantum mehanial onepts. Aknowledgement The authors thank H.H. Garía-Compeán, B. Mielnik, and R. Tarrah for disussion, ritiism and omments. One of the authors (H.P.R) thank Prof. M. Chaihian and the University of Helsinki, as well as Universidad Naional de Colombia, Medellin, for hospitality. The partial support from the Aademy of Finland under projet No is also greately aknowledged by H. P. R. 1. J.S. Bell, Speakable and unspeakable in quantum mehanis (Cambridge: University Press, 1987).. R. Chiao, Phys. Rev. A 48 (1993) R J.I. Latorre, P. Pasual, R. Tarrah, Nu. Phys. B 437 (1995) A. Einstein, Ann. der Physik 17 (1905) R.P. Feynman, R.S. Leighton and M. Sands M, The Feynman s Letures on Physis I, (Massahusetts: Addison Wesley 1963) 15.

5 154 J.KARLES,H.PEREZROJAS 6. V.A. Fok, The Theory of Spae Time and Gravitation (London: Pergamon Press, 1964) A.D. Linde, Inflation and Quantum Cosmology (Boston: Aademi Press, 1991). 8. J. Karles, Rev. Mex. Fis. 38 (199) J. von Neumann, Mathematial Foundations of Quantum Mehanis ( Prineton: Prineton University Press, 1955). 10. R. Penrose, The Emperor s New Mind, (New York:Penguin Books, 1991) 11. Y. Aharonov, D.Z. Albert Phys. Rev. D 4 (1981) 359.