A Model of Electron-Positron Pair Formation

Transcription

1 Volume PROGRESS IN PHYSICS January, 8 A Moel of Electron-Positron Pair Formation Bo Lehnert Alfvén Laboratory, Royal Institute of Technology, S-44 Stockholm, Sween Bo.Lehnert@ee.kth.se The elementary electron-positron pair formation process is consiere in terms of a revise quantum electroynamic theory, with special attention to the conservation of energy, spin, an electric charge. The theory leas to a wave-packet photon moel of narrow line with an neele-raiation properties, not being available from conventional quantum electroynamics which is base on Maxwell s equations. The moel appears to be consistent with the observe pair prouction process, in which the create electron an positron form two rays that start within a very small region an have original irections along the path of the incoming photon. Conservation of angular momentum requires the photon to possess a spin, as given by the present theory but not by the conventional one. The nonzero electric fiel ivergence further gives rise to a local intrinsic electric charge ensity within the photon boy, whereas there is a vanishing total charge of the latter. This may explain the observe fact that the photon ecays on account of the impact from an external electric fiel. Such a behaviour shoul not become possible for a photon having zero local electric charge ensity. Introuction During the earliest phase of the expaning universe, the latter is imagine to be raiation-ominate, somewhat later also incluing particles such as neutrinos an electron-positron pairs. In the course of the expansion the free states of highly energetic electromagnetic raiation thus become partly conense into boun states of matter as etermine by Einstein s energy relation. The pair formation has for a long time both been stuie experimentally [] an been subject to theoretical analysis []. When a high-energy photon passes the fiel of an atomic nucleus or that of an electron, it becomes converte into an electron an a positron. The orbits of these create particles form two rays which start within a very small volume an have original irections along the path of the incoming photon. In this paper an attempt is mae to unerstan the elementary electron-positron pair formation process in terms of a revise quantum electroynamic theory an its application to a wave-packet moel of the iniviual photon [3, 4, 5, 6]. The basic properties of the latter will be escribe in Section, the intrinsic electric charge istribution of the moel in Section 3, the conservation laws of pair formation in Section 4, some questions on the vacuum state in Section 5, an the conclusions are finally presente in Section 6. A photon moel of revise quantum electroynamics The etaile euctions of the photon moel have been reporte elsewhere [3, 4, 5, 6] an will only be summarize here. The corresponing revise Lorentz an gauge invariant theory represents an extene version which aims beyon Maxwell s equations. Here the electric charge ensity an the relate electric fiel ivergence are nonzero in the vacuum state, as supporte by the quantum mechanical vacuum fluctuations an the relate zero-point energy. The resulting wave equation of the electric fiel E then has c r E + c r (iv E) = ; () which inclues the effect of a space-charge current ensity j = " (iv E)C that arises in aition to the isplacement current The velocity C has a moulus equal to the velocity c of light, as expresse by C = c. The inuction law still has the form curl with B staning for the magnetic fiel strength. The photon moel to be iscusse here is limite to axisymmetric normal moes in a cylinrical frame (r; '; z) =. A form of the velocity vector () C = c (; cos ; sin ) (3) is chosen uner the conition < j cos j, such as not to get into conflict with the Michelson-Morley experiments, i.e. by having phase an group velocities which only iffer by a very small amount from c. The fiel components can be expresse in terms of a generating function G G = E z + (cot )E ' ; G = R() e i(!t+kz) ; (4) where G is an amplitue factor, = r=r with r as a characteristic raial istance of the spatial profile, an! an k staning for the frequency an wave number of a normal 6 Bo Lehnert. A Moel of Electron-Positron Pair Formation

2 January, 8 PROGRESS IN PHYSICS Volume moe. Such moes are superimpose to form a wave-packet having the spectral amplitue k A k = exp z(k k ) ; (5) k where k an = k = c are the main wave number an wave length, an z represents the effective axial length of the packet. Accoring to experimental observations, the packet must have a narrow line with, as expresse by k z. The spectral averages of the fiel components in the case j cos j are then an Here E r = ie R 5 ; (6) E ' = E (k r )(sin )(cos )R 3 ; (7) E z = E (k r )(cos ) R 4 (8) B r = R 3 = D R ; R 4 = R R 8 = c sin E ' ; (9) B ' = c (sin ) E r ; () B z = c (cos ) R 8 R 5 E r : () + R 3 ; D = R 3 ; R 5 = (R R 3) ; () + (3) an E = e f ; e = g p kr ; z G = g (cos ) ; (4) f = cos(k z) + i sin(k z) exp z z ; (5) where z = z c(sin )t. Choosing the part of the normalize generating function G which is symmetric with respect to the axial centre z = of the moving wave packet, the components ( E ' ; E z ; B r ) become symmetric an the components ( E r ; B ' ; B z ) antisymmetric with respect to the same centre. Then the integrate electric charge an magnetic moment vanish. The equivalent total mass efine by the electromagnetic fiel energy an the energy relation by Einstein becomes on the other han m " + c r W m e f z ; (6) W m = R 5 ; where expression (5) has to be replace by the reuce function z f = sin(k z) exp z (7) ue to the symmetry conition on G with respect to z =. Finally the integrate angular momentum is obtaine from the Poynting vector, as given by + s " r E r B z rz = = " + c (cos )r3 W s e f z ; (8) W s = R 5 R 8 : Even if the integrate (total) electric charge of the photon boy as a whole vanishes, there is on account of the nonzero electric fiel ivergence a local nonzero electric charge ensity = e f " r (R 5) : (9) Due to the factor sin(k z) this ensity oscillates rapily in space as one procees along the axial irection. Thus the electric charge istribution consists of two equally large positive an negative oscillating contributions of total electric charge, being mixe up within the volume of the wave packet. To procee further the form of the raial function R() has now to be specifie. Since the experiments clearly reveal the pair formation to take place within a small region of space, the incoming photon shoul have a strongly limite extension in its raial (transverse) irection, thus having the character of neele raiation. Therefore the analysis is concentrate on the earlier treate case of a function R which is ivergent at =, having the form R() = e ; : () In the raial integrals of equations (6) an (8) the ominant terms then result in R 8 R 5 an W m = R5 = m 5 m ; () W s = R5 = s 5 s + ; () where m an s are small nonzero raii at the origin =. To compensate for the ivergence of W m an W s when m an s approach zero, we now introuce the shrinking parameters r = c r " ; g = c g " ; (3) where c r an c g are positive constants an the imensionless smallness parameter " is efine by < ". From relations Bo Lehnert. A Moel of Electron-Positron Pair Formation 7

3 Volume PROGRESS IN PHYSICS January, 8 (4) (8), () (3), the energy relation mc = h, an the quantum conition of the angular momentum, the result becomes m = " c 5 kz c " g J m = m h c ; (4) s = " c 5 kz c gc r (cos ) "+ s J m = h (5) with + J m = p f z z : (6) Here we are free to choose = which leas to s m = " : (7) The lower limits m, an s of the integrals () an () then ecrease linearly with " an with the raius r. This forms a similar set of geometrical configurations, having a common shape which is inepenent of m, s, an " in the range of small ". Taking ^r = r as an effective raius of the configuration (), combination of relations (3) (5) finally yiels a photon iameter ^r = " (8) j cos j being inepenent of. Thus the iniviual photon moel becomes strongly neele-shape when " j cos j. It shoul be observe that the photon spin of expression (5) isappears when iv E vanishes an the basic relations reuce to Maxwell s equations. This is also the case uner more general conitions, ue to the behaviour of the Poynting vector an to the requirement of a finite integrate fiel energy [3, 4, 5, 6]. 3 The intrinsic electric charge istribution We now turn to the intrinsic electric charge istribution within the photon wave-packet volume, representing an important but somewhat speculative part of the present analysis. It concerns the etaile process by which the photon configuration an its charge istribution are broken up to form a pair of particles of opposite electric polarity. Even if electric charges can arise an isappear in the vacuum state ue to the quantum mechanical fluctuations, it may be justifie as a first step to investigate whether the total intrinsic photon charge of one polarity can become sufficient as compare to the electric charges of the electron an positron. With the present strongly oscillating charge ensity in space, the total intrinsic charge of either polarity can be estimate with goo approximation from equations (7) an (9). This charge appears only within half of the axial extension of the packet, an its average value iffers by the factor from the local peak value of its sinusoial variation. From equation (9) this intrinsic charge is thus given by q = z r q f r = p z " 3 " kz c g ; (9) q where the last factor becomes equal to unity when = an the limit q = " for a similar set of geometrical configurations. Relations (9) an (4) then yiel an q = 8 3 " c z m = 8 3 " ch z z (3) q e 4. z = : (3) With a large an a small line with leaing to z, the total intrinsic charge thus substantially excees the charge of the create particle pair. However, the question remains how much of the intrinsic charge becomes available uring the isintegration process of the photon. A much smaller charge woul become available in a somewhat artificial situation where the ensity istribution of charge is perturbe by a 9 egrees phaseshift of the sinusoial factor in expression (7). This woul a a factor exp 4 (z = ) to the mile an right-han members of equation (3), an makes q e only for extremely large values of an for moerately narrow line withs. 4 Conservation laws of pair formation There are three conservation laws to be taken into account in the pair formation process. The first concerns the total energy. Here we limit ourselves to the marginal case where the kinetic energy of the create particles can be neglecte as compare to the equivalent energy of their rest masses. Conservation of the total energy is then expresse by mc = hc = m e c : (3) Combination with equation (8) yiels an effective photon iameter "h ^r = m e cj cos j : (33) With " j cos j we have ^r m being equal to the Compton wavelength an representing a clearly evelope form of neele raiation. The secon conservation law concerns the preservation of angular momentum. It is satisfie by the spin h of the photon in the capacity of a boson particle, as given by expression (5). This angular momentum becomes equal to the sum of the spin h of the create electron an positron being fermions. In principle, the angular momenta of the 4 two 8 Bo Lehnert. A Moel of Electron-Positron Pair Formation

4 January, 8 PROGRESS IN PHYSICS Volume create particles coul also become antiparallel an the spin of the photon zero, but such a situation woul contraict all other experience about the photon spin. The thir conservation law eals with the preservation of the electric charge. This conition is clearly satisfie by the vanishing integrate photon charge, an by the opposite polarities of the create particles. In a more etaile picture where the photon isintegrates into the charge particles, it coul also be conceive as a splitting process of the positive an negative parts of the intrinsic electric charge istributions of the photon. Magnetic moment conservation is satisfie by having parallel angular momenta an opposite charges of the electron an positron, an by a vanishing magnetic moment of the photon [5, 6]. 5 Associate questions of the vacuum state concept The main new feature of the revise quantum electroynamical theory of Section is the introuction of a nonzero electric fiel ivergence in the vacuum, as supporte by the existence of quantum mechanical fluctuations. In this theory the values of the ielectric constant an the magnetic permeability of the conventional empty-space vacuum have been aopte. This is because no electrically polarize an magnetize atoms or molecules are assume to be present, an that the vacuum fluctuations as well as superimpose regular phenomena such as waves take place in a backgroun of empty space. As in a review by Gross [7], the point coul further be mae that a vacuum polarization screens the point-chargelike electron in such a way that its effective electrostatic force vanishes at large istances. There is, however, experimental evience for such a screening not to become important at the scale of the electron an photon moels treate here. In the vacuum the electron is thus seen to be subject to scattering processes ue to its full electrostatic fiel, an an electrically charge macroscopic object is also associate with such a measurable fiel. This woul be consistent with a situation where the vacuum fluctuations either are small or essentially inepenent as compare to an external isturbance, an where their positive contributions to the local electric charge largely cancel their negative ones. To these arguments in favour of the empty-space values of the ielectric constant an the magnetic permeability two aitional points can also be ae. The first is ue to the Heisenberg uncertainty relation which implies that the vacuum fluctuations appear spontaneously uring short time intervals an inepenently of each other. They can therefore harly have a screening effect such as that ue to Debye in a quasi-neutral plasma. The secon point is base on the fact that static measurements of the ielectric constant an the magnetic permeability result in values the prouct of which becomes equal to the inverte square of the measure velocity of light. 6 Conclusions The basis of the conservation laws in Section 4 is rather obvious, but it nevertheless becomes nontrivial when a comparison is mae between conventional quantum electroynamics base on Maxwell s equations on one han an the present revise theory on the other. Thereby the following points shoul be observe: The neele-like raiation of the present photon moel is necessary for unerstaning the observe creation of an electron-positron pair which forms two rays that start within a small region, an which have original irections along the path of the incoming photon. Such neele raiation oes not come out of conventional theory [3, 4, 5, 6]; The present revise theory leas to a nonzero spin of the photon, not being available from conventional quantum electroynamics base on Maxwell s equations; [3, 4, 5, 6]. The present moel is thus consistent with a photon as a boson which ecays into two fermions; The nonzero ivergence of the electric fiel in the present theory allows for a local nonzero electric charge ensity, even if the photon has a vanishing net charge. This may inicate how the intrinsic electric photon charges can form two charge particles of opposite polarity when the photon structure becomes isintegrate. Such a process is supporte by the experimental fact that the photon ecays into two charge particles through the impact of the electric fiel from an atomic nucleus or from an electron. This coul harly occur if the photon boy woul become electrically neutral at any point within its volume. Apart from such a scenario, the electromagnetic fiel configuration of the photon may also be broken up by nonlinear interaction with a strong external electric fiel; The present approach to the pair formation process has some similarity with the breaking of the stability of vacuum by a strong external electric fiel, as being investigate by Frakin et al. [8]. References Submitte on November, 7 Accepte on November 9, 7. Richtmyer F. K. an Kennar E. H. Introuction to moern physics. McGraw-Hill Book Comp. Inc., New York an Lonon, 947, pages 6 an Heitler W. The quantum theory of raiation. Oxfor, Clarenon Press, 954, Ch. 5, paragraph Lehnert B. Photon physics of revise electromagnetics. Progress in Physics, 6, v., Bo Lehnert. A Moel of Electron-Positron Pair Formation 9

5 Volume PROGRESS IN PHYSICS January, 8 4. Lehnert B. Momentum of the pure raiation fiel. Progress in Physics, 7, v., Lehnert B. Revise quantum electroynamics with funamental applications. In: Proceeings of 7 ICTP Summer College on Plasma Physics (Eite by P. K. Shukla, L. Stenflo, an B. Eliasson), Worl Scientific Publishers, Singapore, Lehnert B. A revise electromagnetic theory with funamental applications. Sweish Physics Archive (Eite by D. Rabounski), The National Library of Sween, Stockholm, 7; an Bogoljubov Institute for Theoretical Physics (Eite by A. agorony), Kiev, Gross D. J. The iscovery of asymptotic freeom an the emergence of QCD. Les Prix Nobel, 4, Eite by T. Frängsmyr, Almquist och Wiksell International, Stockholm 4, Frakin E. S., Gavrilov S. P., Gitman D. M., an Shvartsman Sh. M. Quantum electroynamics with unstable vacuum. (Eite by V. L. Ginzburg), Proceeings of the Lebeev Physics Institute, Acaemy of Sciences of Russia, v., 995, Nova Science Publishers, Inc., Commack, New York. Bo Lehnert. A Moel of Electron-Positron Pair Formation