The two-body electromagnetic pulsar Miroslav Pardy Department of Theoretical Physics and Astrophysics, Masaryk University, Faculty of Science, Kotlsk 2, 611 37 Brno, Czech Republic, e-mail: [email protected] (February1, 2008) The power spectrum formula of the synchrotron radiation generated by the electron and positron moving at the opposite angular velocities in homogenous magnetic field is derived in the Schwinger version of quantumfield theory. The asymptotical form of this 0 formula is found. It is surprising that the spectrum depends periodically on radiation 0 frequencyωwhichmeansthatthesystemcomposedfromelectron,positronandmagnetic 0 field forms thetwo-body electromagnetic pulsar. 2 PACS numbers: 12.20., 41.20.J, 41.60.B, 11.10 n a J 7 I. INTRODUCTION 2 1 The production of photons by circular motion of charged particle in accelerator is one v 7 of the most interesting problems in the classical and quantum electrodynamics. 7 2 Inthispaperweareinterestedinthesynergicphotonproductioninitiatedbythecircular 1 motion of electron and positron in the homogenous magnetic field. It is supposed that 0 0 electron and positron are moving at the opposite angular velocities. This process is the 0 / generalization of the one-charge synergic synchrotron Cˇerenkov radiation which has been h p calculated in source theory two decades ago by Schwinger et al. [1]. We will follow also - p the article[2]as thestarting point. Althoughour final problem isthe radiationof thetwo- e h charge system in vacuum, we consider, first in general, the presence of dielectric medium, : v which is represented by the phenomenological index of refraction n and it is well known i X that this phenomenological constant depends on the external magnetic field. Introducing ar the phenomenological constant enables to consider also the Cˇerenkovian processes. We will investigate here how the original Schwinger et al. spectral formula of the synergic synchrotron Cˇerenkov radiation of the charged particle is modified if we consider the electron and positron moving at the opposite angular velocities. This problem is an analogue of the linear problem solved recently by author [3] also in source theory. We will show that the original spectral formula of the synergic synchrotron-Cˇerenkov radiation is modulated by function 4sin2(ωt) where ω is the frequency of the synergic radiation produced by the system and it does not depend on the orbital angular frequency of electron or positron. We will use here the fundamental ingredients of Schwinger source theory to determine the power spectral formula. Source theory [4–6] was initially constructed for a description of the particle physics situations occurring in high-energy physics experiments. It enables simplification of the calculationsintheelectrodynamics andgravitywheretheinteractions aremediatedbythe photon or graviton, respectively. It simplifies particularly the calculations with radiative corrections [6,7]. 1 II. FORMULATION OF A PROBLEM The basic formula of the Schwinger source theory is the so called vacuum to vacuum amplitude: 0+ 0− = eh¯iW, (1) h | i where in case of the electromagnetic field in the medium, the action W is given by the following formula: 1 W = (dx)(dx′)Jµ(x)D (x x′)Jν(x′), (2) 2c2 +µν − Z where µ Dµν = [gµν +(1 n−2)βµβν]D (x x′), (3) + c − + − where βµ (1,0), Jµ (c̺,J) is the conserved current, µ is the magnetic permeability ≡ ≡ of the medium, ǫ is the dielectric constant od the medium and n = √ǫµ is the index of refraction of the medium. Function D is defined as follows [1]: + i ∞ sin nω x x′ D (x x′) = dω c | − |e−iω|t−t′|. (4) + − 4π2c x x′ Z0 | − | The probability of the persistence of vacuum follows from the vacuum amplitude (1) in the following form: 2 0+ 0− 2 = e−h¯ImW, (5) |h | i| where Im W is the basis for the definition of the spectral function P(ω,t) as follows: 2 P(ω,t) d Im W = dtdω . (6) − h¯ − h¯ω Z Now, if we insert eq. (2) into eq. (6), we get after extracting P(ω,t) the following general expression for this spectral function: ω µ sin nω x x′ P(ω,t) = dxdx′dt′ c | − | −4π2n2 " x x′ # × Z | − | n2 cos[ω(t t′)][̺(x,t)̺(x′,t′) J(x,t) J(x′,t′)]. (7) − − c2 · Let us recall that the last formula can be derived also in the classical electrodynamical context as it is shown for instance in the Schwinger article [8]. The derivation of the power spectral formula from the vacuum amplitude is more simple. 2 III. THE POWER SPECTRAL FORMULA OF MOTION OF OPPOSITE CHARGES Now, we will apply the formula (7) to the two-body system with the opposite charges moving at the oposite angular velocities in order to get in general synergic synchrotron- ˇ Cerenkov radiation of electron and positron moving in a uniform magnetic field. While the synchrotron radiation is generated in a vacuum, the synergic synchrotron- Cˇerenkov radiationcanproducedonlyinamediumwithdielectricconstantn. Wesuppose the circular motion with velocity v in the plane perpendicular to the direction of the constant magnetic field H (chosen to be in the +z direction). The condition for the existence of the Cˇerenkov electromagnetic radiation is that the velocity of a charged particle in a medium is faster than the speed of light in this medium. ˇ This radiation was first observed experimentally by Cerenkov [9] and theoretically inter- preted by Tamm and Frank [10] in the framework of classical electrodynamics. A source theoretical description of this effect was given by Schwinger, Tsai and Erber [1] at the zero-temperature regime and the classical spectral formula was generalized to the finite temperature situation in electrodynamics and gravity in the framework of the source the- ory by Pardy [11, 12]. Here we derive the general formula of the radiation generated by the motion of two-body system in a uniform magnetic field. Later we consider only the process in vacuum. We can write the following formulas for the charge density ̺ and for the current density J of the two-body system with opposite charges and opposite angular velocities: ̺(x,t) = eδ(x x1(t)) eδ(x x2(t)) (8) − − − and J(x,t) = ev1(t)δ(x x1(t)) ev2(t)δ(x x2(t)) (9) − − − with x (t) = x(t) = R(icos(ω t)+jsin(ω t)), (10) 1 0 0 x (t) = R(icos( ω t)+jsin( ω t) = x( ω ,t) = x( t). (11) 2 0 0 0 − − − − The absolute values of velocities of both particles are the same, or v (t) = v (t) = v, 1 2 | | | | where (H = H ,E = energy of a particle) | | βE v(t) = dx/dt, ω = v/R, R = , β = v/c, v = v . (12) 0 eH | | After insertion of eqs. (8)–(9) into eq. (7), and after some mathematical operations we get ω µ ∞ 2 v (t) v (t′) P(ω,t) = e2 dt′cos(t t′) ( 1)i+j 1 i · j n2 −4π2n2 Z−∞ − i,j=1 − " − c2 # × X 3 sin nω x (t) x (t′) c | i − j | . (13) ( xi(t) xj(t′) ) | − | Using t′ = t+τ, we get for x (t) x (t′) =d A , (14) i j ij − ω τ +α A = [R2 +R2 2RRcos(ω τ +α )]1/2 = 2R sin 0 ij , (15) ij 0 ij | | − 2 (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) where α were evaluated as follows: (cid:12) (cid:12) ij (cid:12) (cid:12) α = 0, α = 2ω t, α = 2ω t, α = 0. (16) 11 12 0 21 0 22 Using v (t) v (t+τ) = ω2R2cos(ω τ +α ), (17) i · j 0 0 ij and relation (15) we get with v = ω R 0 ω µ ∞ 2 n2 P(ω,t) = e2 dτ cosωτ ( 1)i+j 1 v2cos(ω τ +α ) −4π2n2 Z−∞ i,j=1 − " − c2 0 ij # × X sin 2Rnω sin (ω0τ+αij) c 2 . (18)  h2Rsin (ω(cid:16)0τ+αij) (cid:17)i  2  (cid:16) (cid:17) Introducing new variable Tby relation  ω τ +α = ω T (19) 0 ij 0 for every integral in eq. (18), we get P(ω,t) in the following form ω e2 µ ∞ 2 P(ω,t) = dT ( 1)i+j −4π22Rn2 − × Z−∞ i,j=1 X ω c2 sin 2Rnω sin ω0T cos(ωT α ) 1 v2cos(ω T c 2 . (20) − ω0 ij " − n2 0 # h sin ω0T(cid:16) (cid:17)i  2  (cid:16) (cid:17) The last formula can be written in the more compact form,  ω µ e2 2 n2 P(ω,t) = ( 1)i+j P(ij) v2P(ij) , (21) −4π2n22R − ( 1 − c2 2 ) i,j=1 X where ω ω (ij) (ij) (ij) P = J cos α +J sin α (22) 1 1a ω ij 1b ω ij 0 0 and 4 ω ω (ij) (ij) (ij) P = J cos α +J sin α , (23) 2 2A ω ij 2B ω ij 0 0 where ∞ sin 2Rnω sin ω0T (ij) c 2 J = dT cosωT , (24) 1a Z−∞  h sin ω02T(cid:16) (cid:17)i (cid:16) (cid:17)   ∞ sin 2Rnω sin ω0T (ij) c 2 J = dT sinωT , (25) 1b Z−∞  h sin ω02T(cid:16) (cid:17)i (cid:16) (cid:17)   ∞ sin 2Rnω sin ω0T (ij) c 2 J = dT cosω T cosωT , (26) 2A Z−∞ 0  h sin ω02T(cid:16) (cid:17)i (cid:16) (cid:17)   ∞ sin 2Rnω sin ω0T (ij) c 2 J = dT cosω T sinωT , (27) 2B Z−∞ 0  h sin ω02T(cid:16) (cid:17)i (cid:16) (cid:17) Using   ω T = ϕ+2πl, ϕ ( π,π), l = 0, 1, 2, ..., (28) 0 ∈ − ± ± we can transform the T-integral into the sum of the telescopic integrals according to the scheme: ∞ 1 l=∞ π dT dϕ. (29) −→ ω Z−∞ 0 l=−∞Z−π X Using the fact that for the odd functions f(ϕ) and g(l), the relations are valid π l=∞ f(ϕ)dϕ = 0, g(l) = 0, (30) Z−π l=−∞ X we can write 1 π ω ω sin 2Rnω sin ϕ (ij) c 2 J = dϕ cos ϕcos2πl , (31) 1a ω0 Xl Z−π (cid:26) ω0 ω0(cid:27) h sin ϕ2 (cid:16) (cid:17)i (cid:16) (cid:17)   (ij) J = 0. (32) 1b For integrals with indices A, B we get: 1 π ω ω sin 2Rnω sin ϕ (ij) c 2 J = dϕcosϕ cos ϕcos2πl , (33) 2A ω0 Xl Z−π (cid:26) ω0 ω0(cid:27) h sin ϕ2 (cid:16) (cid:17)i (cid:16) (cid:17)   (ij) J = 0, (34) 2B 5 So, the power spectral formula (21) is of the form: ω µ e2 2 v P(ω,t) = ( 1)i+j P(ij) n2β2P(ij) ; β = , (35) −4π2n22R − 1 − 2 c iX,j=1 n o where ω (ij) (ij) P = J cos α (36) 1 1a ω ij 0 ω (ij) (ij) P = J cos α . (37) 2 2A ω ij 0 Using the Poisson theorem ∞ ω ∞ cos2π l = ω δ(ω ω l), (38) 0 0 ω − l=−∞ 0 k=−∞ X X (ij) (ij) we get for J and J (z = 2lnβ): 1a 2A π sin(zsin(ϕ/2)) (ij) J = dϕcosϕcoslϕ , (39) 1a l Z−π ( sin(ϕ/2) ) X π sin(zsin(ϕ/2)) (ij) J = dϕcosϕsinlϕ , (40) 2A l Z−π ( sin(ϕ/2) ) X Using the definition of the Bessel functions J and their corresponding derivation and 2l integral 1 π ϕ dϕcos zsin coslϕ = J (z), (41) 2l 2π 2 Z−π (cid:18) (cid:19) 1 π ϕ dϕsin zsin coslϕ = J′ (z), (42) 2π 2 − 2l Z−π (cid:18) (cid:19) 1 π sin zsin ϕ z 2 dϕ coslϕ = J (x)dx, (43) 2π si(cid:16)n(ϕ/2)(cid:17) 2l Z−π Z0 and using equations 2 ω ( 1)i+jcos α = 2(1 cos2ωt) = 4sin2ωt, (44) ij − ω − i,j=1 0 X and the definition of the partial power spectrum P l ∞ P(ω) = δ(ω lω )P , (45) 0 l − l=1 X we get the following final form of the partial power spectrum generated by motion of two-charge system moving in the cyclotron: 6 e2 ωµω 2lnβ P (ω,t) = [4(sinωt)2] 0 2n2β2J′ (2lnβ) (1 n2β2) dxJ (x) . (46) l πn2 v 2l − − Z0 2l ! So we see that the spectrum generated by the system of electron and positron is formed in such a way that the original synchrotron spectrum generated by electron is modulated by function 4sin2(ωt). This formula is analogical to the formula derived in [3] for the linear motion of the two-charge system emitting the Cˇerenkov radiation. The derived formula involves also the synergic process composed from the synchrotron radiation and the Cˇerenkov radiation for electron velocity v > c/n in a medium. Our goal is to apply the last formula in situation where there is a vacuum. In this case we can put µ = 1,n = 1 in the last formula and so we have e2ωω 2lβ P (ω,t) = 4sin2(ωt) 0 2β2J′ (2lβ) (1 β2) dxJ (x) . (47) l π v 2l − − Z0 2l ! So, we see, that final formula describing the opposite motion of eletron and positron in accelerator is of the form P (ω,t) = 4sin2(ωt)P (ω), (48) l l(electron) whereP isthespectrumofradiationonlyofelectron. Theresult issuprising because electron we naively expected that the total radiation of the opposite charges should be P (ω,t) = P (ω,t)+P (ω,t). (49) l l(electron) l(positron) So, we see that the resulting radiation can not be considerred as generated by the isolated particles but by a synergical production of a system of particles and magnetic field. At the same time we cannot interprete the result as a result of interference of two sources because the distance between sources radically changes and so, the condition of an interference is not fulfilled. Using the approximative formulae 1 1 3 2/3 J′ (2lβ) K (l/l ), l 1, (50) 2l ∼ √3π 2l 2/3 c ≫ (cid:18) c(cid:19) 2lβ 1 1 ∞ J (y)dy K (y)dy, l 1, (51) Z0 2l ∼ √3π Zl/lc 1/3 ≫ with [1] 3 l = (1 β2)−3/2, (52) c 2 − substituting eqs. (50) and (51) into eq. (47), respecting the high-energy situation for the high-energy particles where (1 β2) 0, and using the recurence relation − → 1 K′ = (K +K ), (53) 2/3 −2 1/3 5/3 7 and definition function κ(ξ) ∞ κ(ξ) = ξ K (y)dy, ξ = l/l , (54) 5/3 c Zξ or, π κ(ξ) ξ1/2e−ξ, ξ 1, (55) ≈ 2 ≫ r we get power spectrum formula of electron-positron pair as follows: ωe2 π 3 2/3 ω P (t) = 4sin2(ωt) ξ1/6e−ξ; l = . (56) l π2R 6 2l ω r (cid:18) (cid:19) 0 For l 1 the emitted spectrum is in some sense continual and it can be expressed by ≫ the following formula 1 P(ω,t) = P(l= ω )(t). (57) ω ω0 (cid:18) 0(cid:19) Alsothisformulainvolves thefactthatthetotalspectrumofradiationcannotbewritten as a sum of spectra of isolated sources but it is the result of synergical process of a system which consists of magnetic field and two particles moving in it. The classical electrodynamics is not broken by this formula but our naive image on the processes in the magnetic field is broken. From the last formula also follows that at time t = πk/ω there is no radiation of the frequency ω. At every frequency ω the spectrum oscillates with frequency ω. If the radiation were generated not synergically, then the spectral formula would be composed from two parts corresponding to two isolated sources. IV. DISCUSSION We have derived in this article the power spectrum formula of the synchrotron radia- tion generated by the electron and positron moving at the opposite angular velocities in homogenous magnetic field. We have used the Schwinger version of quantum field theory, for its simplicity. It is suprising that the spectrum depends periodically on radiation fre- quency ω which means that the system composed from electron, positron and magnetic field behaves as a pulsar. While such pulsar can be represented by a terrestrial exper- imental arrangement it is possioble to consider also the cosmological existence in some modified form. To our knowledge, our result is not involved in the classical monographies on the elec- tromagnetic theory andat the same time it was not stillstudied by the accelerator experts investigating the synchrotron radiation of bunches. This effect was not described in text- books on classical electromagnetic field and on the synchrotron radiation. We hope that sooner or later this effect will be verified by the accelerator physicists. The radiative corrections obviously influence the synergic spectrum of photons [2,7]. However, the goal of this article is restricted only to the simple processes. 8 The particle laboratory LEP in CERN uses instead of single electron and positron the bunches with 1010 electrons or positrons in one bunch of volume 300µm 40µm 0.01 × × m. So, in some approximation we can replace the charge of electron and positron by the charges Q and -Q of both bunches in order to get the realistic intensity of photons. Nevertheless thesynergic character oftheradiationoftwo bunches moving at theopposite direction in a magnetic field is conserved. The more exact description can be obtained in case we consider the internal structure of both bunches. But this is not the goal of our article. [1] J. Schwinger, W. Y. Tsai and T. Erber, Ann.Phys. (NY)96 (1976) 303. [2] M. Pardy,Phys. Lett. A 189 (1994) 227. [3] M. Pardy,Phys. Rev. A 55 No. 3 (1997) 1647. [4] J. Schwinger, Particles, Sources and Fields, Vol.I, (Addison-Wesley,Reading, Mass., 1970). [5] J. Schwinger, Particles, Sources and Fields, Vol.II, (Addison-Wesley,Reading, Mass., 1973). [6] W. Dittrich,Fortschritte derPhysik 26 (1978) 289. [7] M. Pardy,Phys. Lett. B 325 (1994) 517. [8] J. Schwinger, Phys. Rev. 75 (1949) 1912. [9] P.A. Cˇerenkov, C.R. Acad. Sci. (USSR)3 (1936) 413. [10] I.E. Tamm and I. M. Frank,Dokl. Akad.Nauk USSR14 (1937) 107. [11] M. Pardy,Phys. Lett. A 134 (1989) 357. [12] M. Pardy,Int.J. of Theor. Phys. 34, No. 6 (1995) 951. 9