A new theory of Aharonov-Bohm effect with a variant
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1. S Et 2mE rcos 0 0 0 533 65 6 7 8 9 10 534 G Lochak Fig 2 Aharonov Bohm scheme The motion of the electron is given by the Jacobi theorem sn Const z Const 11 The trajectories are the rays of the wave os 2mE xsin ycos6 U 12 00 The motion along the rays is given by Sat op 088 ysin8 1 13 1 that is to say with E xm A new theory of the Aharonov Bohm effect with a variant 535 xcos ysin v t t 14 We see that the rays electron trajectories defined by 12 are orthogonal to the moving planes 13 but they are not orthogonal to the equal phase surfaces 8 10 except far away from the magnetic string x gt co where the potential term becomes negligible Therefore despite the presence of the potential the electronic trajectories remain rectilinear and are not deviated because there is no magnetic field The velocity v Const is the one of the incident electrons because of the conservation of energy But the diffraction of the waves through the holes A and A creates for the electron trajectories an interval of possible angles 0 among which are the angles of the interference fringes modified by the magnetic potential So there is no deviation of the electrons but only a deviation of the angles of phase synchronization between the waves issued from At and A because a fieldless potential can only cha2. ae ae g Arctg an Arctg 2 x x x xX tly xL ly e ax G20 y c 06 x x 0 y c 9d yte Arct Arct a ai oy J x oy p x And in analogy with 24 y c yte e Arctg Arctg 27 x x Introducing 27 in 24 we get the equation 6 again with the complete integral 7 and finally a complete integral of 24 analogous to 8 S Et 2mE xcos ysin te 28 X se Aretg ae Arctg xX We shall not repeat the whole preceding theory The most important thing is to note that the electron trajectories are the same straight lines as before for the same reason the absence of magnetic field We find equations 12 13 14 again for the wave rays The Lagrange momenta de Broglie wave vectors up to a factor h are now os y c yte 72mEcos ox a tty 29 os x x 72mEsin d i oy x y cy x y c A new theory of the Aharonov Bohm effect with a variant 541 Fig 4 New experiment scheme The equations of the orthogonal lines of phase would be useless for the prediction of the physical effect it was interesting to perform the integration only once on the example 16 in order to show the difference between rays and phase lines The shifting of interference fringes Let us look once more at a plane wave coming from x 0 to the plane x b and diffracting through the holes A and A The angle 0 iS s
3. Waves with a Preface of Chen Ning Yang World Scientific Singapore 1998 4 Born M Wolf E Principles of Optics Pergamon Oxford 1964 5 Broglie L de Th se de 1924 Ann Fond Louis de Broglie 17 1992 p 1 108 6 The Feynman Lectures in Physics Vol 2 Electrodynamics Addison Wesley 1964 7 Lochak G Ann Fond Louis de Broglie 25 2000 p 107 127 8 Tamm I E Osnovy Teorii Electrichestva OGIZ Moscou 1946 9 Jackson J D Classical Electrodynamics 2 ed Wiley N Y 1975 Manuscrit re u le 11 f vrier 2002 r vis le 20 septembre 2002 The book of Akira Tonomura written in non technical terms is in principle a popular book but it is clear and profound and I highly appreciate it even if I can disagree with him on some points concerning gauge invariance
4. with 1 gt 0 because we have chosen a c gt gt in order for the string to be outside the trajectories Nevertheless the first shift dominates because the second trajectory is farther from the string than the first one so that the effect does exist And since we have two strings the effect is doubled hence the factor two before in 34 Let us take as an example c a Then we have n Arctg C Arctg so max n 0 52 35 V3 The maximum value of 7 is obtained for b ar Comparing the maximum value of the Aharonov Bohm shift in 20 T E A 1 57 with the maximum shift in 34 n 0 52 we see that the effect predicted here is three times smaller But this is not important because the aim was not to give another proof of the interference shift due to a fieldless potential the Aharonov Bohm proof is excellent but to prove that an effect of the same type can be obtained with an experiment which cannot be explained in terms of a line integral which here obviously vanishes 4 THE QUESTION OF GAUGE INVARIANCE There is only one problem in an interference phenomenon where are the fringes And the answer is given by the phase difference between two waves coming from two coherent sources Curiously the calculation of this phase difference is at the basis of all the interference phenomena except the Aharonov Bohm effect The interference is taken for granted and the only question is to find t
5. Annales de la Fondation Louis de Broglie Volume 27 no 3 2002 529 A new theory of the Aharonov Bohm effect with a variant in which the source of the potential is outside the electronic trajectories GEORGES LOCHAK Fondation Louis de Broglie 23 rue Marsoulan F 75012 Paris ABSTRACT A new theory of the Aharonov Bohm experiment based on the calculation of the phase difference between the electronic trajectories shows that the shifting of the interference fringes depends both on the gauge of the potential and of the location of its source with respect to the interference device A new experiment is then suggested in which the source of the potential is outside the electronic trajectories The line integral of the potential along the trajectories equals zero but the shifting ofthe fringes does not vanish RESUME Une nouvelle th orie de l effet Aharonov Bohm bas e sur le calcul de la diff rence de phase entre les trajectoires lectroniques montre que l effet d pend la fois de la jauge du potentiel et de la position de la source par rapport au dispositif interf rentiel On propose ensuite une nouvelle exp rience dans laquelle la source du potentiel est ext rieure aux trajectoires L int grale du potentiel le long des trajectoires est nulle mais le d placement des franges subsiste 1 INTRODUCTION The Aharonov Bohm experiment 1 2 3 was conceived in order to prove the effect of a fieldless magnetic potential on el
6. T We shall now suggest an experiment inspired by that of Aharonov Bohm but which is such that the circular integral along the electron trajectories equals zero an thus cannot have any relation with the fringe shift This experiment was already suggested in 6 but as an intuitive argument Here we give the exact calculation The idea is to substitute the magnetic string included between the electronic trajectories by two strings on both sides Fig 3 and 4 In A new theory of the Aharonov Bohm effect with a variant 539 principle one string would be enough but we shall see that the effect is smaller than Aharonov Bohm s so that it is useful to double it Owing to the new position of strings the magnetic flux through the closed line of trajectories will be equal to zero because the potential is still a gradient and its source is outside The effect remains but the problem of gauge invariance is clearly irrelevant The Hamilton Jacobi equation becomes here mS Ee e 5 VE ot dx x y cP x 4 y cy 24 dE dy x y x 4 y e We see that according to Fig 3 and 4 the magnetic strings are parallel to Oz and cut the plane xOz in two points at a distance c from Oz We suppose c gt 2 25 2 in order to put the strings outside the trajectories Fresnel M llenstedt biprism solenoids hk p mvteA Fig 3 New experiment 540 G Lochak Paralleling the relations 4 we now have
7. ectronic interferences The idea was to introduce between the electronic trajectories coming from two virtual coherent sources a magnetic string or a thin solenoid orthogonal to the trajectories and long enough so that the magnetic field emanating from the extremities cannot modify the electron trajectories Fig 1 530 G Lochak Fresnel M llenstedt biprism screen Fig 1 Aharonov Bohm experiment Theoretically in order for a magnetic flux to be trapped inside a string or a solenoid it must be infinitely long this is what is assumed in the calculations But in practice a few millimeters are sufficient because the transverse dimensions of the device are on the order of microns As this point was contested Tonomura 2 3 succeeded in substituting for the rectilinear string a microscopic toroidal magnet 10 um one electron beam passing through the hole of the torus and the other passing outside so that the magnetic lines may be regarded as beeing entirely enclosed in the magnet Nevertheless in what follows we shall restrict ourselves to an infinite magnetic string which is sufficient for our present object because the subtleties of Tonomura tori were invented in order to answer other arguments than those we are aiming at refuting in the present paper Let us give at first an intuitive interpretation of the Aharonov Bohm experiment Recall that the wave vector of an electron in a magnetic potential even f
8. entury that electromagnetic potentials are only mathematical intermediate entities And even more shocking is the fact that formula 1 imposes an electromagnetic gauge that can be measured experimentally The almost unanimous opinion that gauge invariance is an absolute law is so firmly fixed in prevailing thought that even distinguished physicists 4 are led to present a wrong formula for the wavelength writing A instead of formula 1 in the presence of a potential For my the same reason Feynman managed to relate the Aharonov Bohm effect not with the wavelength formula 1 but with the magnetic flux trapped in the string or in the solenoid saving in this way the gauge invariance 6 7 The aim of the present work is to prove that the shifting of the fringes depends on the distance from the solenoid to the experimental device and to suggest a new experiment in which the solenoid with its magnetic flux is outside the quadrilateral formed by the electronic trajectories which causes the integral of A to vanish and makes the argument of the magnetic flux enclosed by the trajectories ineffective Actually the quadrilateral itself will be removed from the calculations rejecting to infinity the electron source and the interference fringes which introduces negligible errors this approximation is usual in optics 532 G Lochak 2 A NEW THEORY OF THE AHARONOV BOHM EFFECT The commonly admitted theories of this effect are of
9. he shift without damaging the gauge invariance This is why the circular integral of A plays the central role But circular integral cannot give the interfringe 544 G Lochak Therefore the phenomenon is calculated in two parts a The free interference without potential b The shift due to the potential considered separately and which is absent from the calculation of the phase differences thus forgetting the geometry of the experiment It is for this reason that the location of the solenoid and the form under which the potential enters the expression of the phase are forgotten If there is something new in the present paper it is precisely an attempt to come back to the old problems and methods of interference phenomena owing to a simple calculation of phases Now we shall go back to the phases given by formulae 30 which include the case of 19 adding in an arbitrary gauge term f x y We find S Et sm x o y 2 re Arere areg s p x 0 x X G6 S Br In 5 y 2 te Areg Es Areg f x y 0 x x Close to the slits we have generalizing 31 S en o s c AS e i a aw Migs fa saline Hir and the phase difference 34 becomes A new theory of the Aharonov Bohm effect with a variant 545 28 2 py 4 28 r e ee Clearly except if f x y is even in y the phase difference is modified and that the phenomenon is not gauge invariant 38 5 APPENDIX The
10. ieldless is given by the de Broglie formula 5 n hik p my bed 1 p is the Lagrange momentum This formula is a direct consequence of the identification of the principles of Fermat and of least action it is one of the A new theory of the Aharonov Bohm effect with a variant 531 most reliable results of quantum mechanics Therefore it is a priori obvious that interference and diffraction phenomena will be influenced by the presence of a magnetic potential independently of the presence or not of a magnetic field This phenomenon follows from a simple change of wavelength and thus a change of phase as may be done in optics by introducing a plate of glass into a Michelson interferometer Besides the phenomenon is manifestly gauge dependent if we add something to A whether it be a gradient or not A is modified Of course it is true even when A 0 i e for the formula A in the vacuum which is thus gauge dependent mv too This fact was emphasized by de Broglie many years ago electron interferences are not gauge invariant In the case of the Aharonov Bohm experiment there are additive phases on both interfering waves and moreover they are in opposite directions which doubles the shift of interference fringes We furthermore give a new proof of all this This remarkable effect which proves the influence of a fieldless magnetic potential on electron waves is shocking for those who have been convinced for a c
11. magnetic potential of an infinetely thin and long solenoid or an infinite magnetic string We start from a classical formula in electromagnetism 8 9 expressing the vector potential created by a magnetic dipole at a distance l xl AoH 39 l Ina point P the potential is equal to pe xl_ dZ x MP A _7 40 k oj MP is the magnetic flux trapped in the string or in the solenoid and MP P x y z Z z 2 ai dZ x MP y4Z xdZ 0 Now we get from 40 546 G Lochak ofr y FP 42 Z slp and given that 2 Ip y ae ay a we find A 2 A 20 4 0 44 Fay x y Acknowledgements I would like to dedicate this attempt to understand a little better these difficult questions to my old master Louis de Broglie and to David Bohm with whom I worked in Paris at the Institute Henri Poincar and who was a friend of mine Let them be postumously thanked for their teaching And I would like to thank warmly my son Pierre Lochak whose valuable advice was for me of a great help Of course if all that is true the merit is shared if it is not the failure is mine Bibliography 1 Aharonov Y Bohm D Physical Review 115 485 1959 2 Olariu S Iovitsu Popescu I Reviews of Modern Physics 57 n 2 1985 p 339 436 A new theory of the Aharonov Bohm effect with a variant 547 3 Tonomura A The Quantum World Unveiled by Electron
12. mall again and we have owing to 28 and in analogy with 19 two waves 542 G Lochak S Et sm x b yF 2 ye yte e Arete areg t2 x x For we have up to a common constant additive term S e n C S e n 6 G1 with the definitions gat eee n E rT 2 G A 2 32 Disregarding as in 19 the small terms corresponding to the potential near the interference field great values of x we find the analogue of 22 for the phase differences for the waves coming from A and A St St Br V3n x 5 y 2 p e01 0 33 SoS Er VIE x 5 y 2 p e n h Introducing the wavelength A amE we can deduce the phase m difference between the two waves just as in 23 _ AS a h A 2E Ag a 7n 34 We again find a first term corresponding to the Young interferences and a second one analogous to the Aharonov Bohm effect This term is smaller for the obvious reason that each magnetic string produces a shift on the nearest trajectory but unfortunately it also produces a shift on the other A new theory of the Aharonov Bohm effect with a variant 543 one and this second shift is in the same direction as the first one because both trajectories are on the same side of the string whereas they were on opposite sides in the case of Aharonov Bohm so that the phase shifts on the trajectories were opposite too This is why we find now instead of a factor E the difference n
13. nge the phases not the trajectories This is the Aharonov Bohm effect that we now have to calculate Let us first look at the orthogonal lines to the equal phase surfaces S they are enveloped by the Lagrange momenta i e by the de Broglie wave vectors in accordance with the formula 1 while the rays 12 are the impulse lines mv The momenta are os Py 2mE cos0 ox x y 15 os x a a 2mE sinb E 5 gt oy x y Hence the equation d d 7 16 2mE cos0 gt 2mE sin e x y x y 536 G Lochak The integration is obvious thanks to the integral combinations xdx ydy and xdy ydx In polar coordinates we find r E in 0 0 A log c Const A S 17 r sin p N c Const P 17 and in Cartesian coordinates Jx ty ycos0 x Se OE 18 Comparing with 12 one can see that the orthogonal lines to the phase planes become parallel to the rays far from the magnetic string It is worth noting that phase orthogonal lines 17 or 18 and the phase velocity Vv V which we cannot calculate here because the frequency is correct only in relativity depends on the potential through the momentum p but this is not the case for the electron trajectories 12 and for the electron velocity in 14 In other words the electrons i e energy do not follow the phase propagation neither in velocity nor in trajectory The same happens in crystal optic
14. s the phase propagation depends on the inductions that is on the polarization of the medium while the propagation of energy is given by the Poynting vector which is only defined by fields and does not depend on the polarization 4 The shifting of interference fringes Let us consider a plane wave propagating along Ox 6 0 The holes A and A will emit in the half space x gt 0 two waves S and S According to 8 we have A new theory of the Aharonov Bohm effect with a variant 537 S Et 42mE b gt 2 Arctg gt as S Et V2mE b gt 2 Arctg gt X where we have taken into account the smallness of 0 cos 1 sin 0 Let us now suppose that f 0 when x b and let us write a Arctg 20 4 prs 20 The initial waves S and S in A and A x b y are Sj E S E 21 Now let us note that in all the known experiments the magnetic string or the solenoid was very close to A and A The authors say in the shadow of the electrostatic fiber of the M llenstedt biprism 2 as it is shown on the Fig 1 Therefore in A and A the distance b is T T T very small and amp 5 so that S E gt S gt Therefore we see that in Aand A at the beginning of the trajectories the phases defined by 21 depend on the potential exclusively through the value of On the contrary at the other end of the
15. ten complicated 2 but for the physical bases one can read the brillant book of Tonomura 3 Actually the geometrical optics approximation is sufficient to answer the true question Where are the fringes This is why we shall make use of it assuming that we are in the case of Young slits the other cases are topologically equivalent We shall define the phase of the de Broglie wave as 2 9 h 2 where S is the principal Hamilton function which obeys the Hamilton Jacobi equation 2 2 os S y os x 2m E E 3 ot z x y dy x y where Si and Pore are the components of the potential x Ty x Ty EA created by an infinite string along the Oz axis 2 is twice the magnetic flux trapped in the string or in the solenoid see Appendix and Fig 2 The electronic wave propagates from x oco to x The a Young slits are on a parallel to Oy at from the point C located at x b The potential appearing in 3 is a gradient because 9 Arctg 9 Arctg 4 x y x x y i x so that the equation 3 is easily integrated defining A new theory of the Aharonov Bohm effect with a variant S eArctg x which gives 2m 2 az or x oy We choose the complete integral X Et 2mE xcos ysin6 Hence we get a complete integral of 3 y S Et 2mE xcos ysin Arctg x Or in polar coordinates x rcos y rsin
16. trajectories on the interference fringes far from A and A7 the distance is of the order of 15cm while a b 10cm which justifies the approximation of parallel trajectories for the waves S and S in the vicinity of the fringes Close to the fringes the term 0 in 8 and 10 has practically the same value for S and S7 is very small and 00 would be of the third order so it disappears from 19 In other words on the fringes 538 G Lochak contrary to the origin the potential has no more influence Finally according to 19 and 20 the phase difference respectively undergone by the two waves propagating from A and A to the interference field will be defined by the quantities S Si Br In x 5 y 2 p e 22 SSS Er Vink x b y Sp e h Introducing the wavelengh A the phase difference V2mE between the two waves will be _AS _ a0 2e 23 h h Ga A The first term gives the standard Young fringes the second one is the Aharonov Bohm effect The formula 23 is not exactly in accordance with the classical theory because of the angle E which is absent from the classical one E is half the angle under which the Young slits are seen from the solenoid The presence of E entails a dependence of the effect on the position of the string which is in principle experimentally testable according to 20 the effect must decrease when the distance b increases 3 ANEW EXPERIMEN
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