Equations to Calculate Collapse Strength for High Collapse Casing

a r X i v :a s t r o -p h /0310718v 1 24 O c t 2003ESI 1392hep-th/0310718Axion bremsstrahlung from collisions of global stringsD.V.GAL’TSOV ∗,E.Yu.MELKUMOVA ∗and R.KERNER †∗Department of Physics,Moscow State University,119899,Moscow,Russia†Universit`e Pierre et Marie Curie,4,place Jussieu,Paris,75252,France(Dated:February 2,2008)Abstract We calculate axion radiation emitted in the collision of two straight global strings.The strings are supposed to be in the unexcited ground state,to be inclined with respect to each other,and to move in parallel planes.Radiation arises when the point of minimal separation between the strings moves faster than light.This effect exhibits a typical Cerenkov nature.Surprisingly,it allows an alternative interpretation as bremsstrahlung under a collision of point charges in 2+1electrodynamics.This can be demonstrated by suitable world-sheet reparameterizations and di-mensional reduction.Cosmological estimates show that our mechanism generates axion production comparable with that from the oscillating string loops and may lead to further restrictions on the axion window.PACS numbers:11.27.+d,98.80.Cq,98.80.-k,95.30.SfI.INTRODUCTIONFor more than twenty years the Peccei-Quinn axion[1,2,3]remains one of the most serious dark matter candidates.Its properties depend essentially on one unknown parameter, the vacuum expectation value f marking the energy scale of the U(1)symmetry breaking. The axion acquires a small massm a≃6µeV1012GeVformed under some of these collisions(note that in our case passing of strings through each other is not assumed).Meanwhile,as far as we are aware,this mechanism was not explored so far in the axion physics context.It works as follows.Neglecting the boundary effects, let us assume that two infinite straight strings move in parallel planes,the two strings be-ing generically inclined with respect to each other.Then the point of minimal separation between the strings,which marks the localization of the effective source of axion radiation, is allowed to move with any velocity,superluminal in particular.In this latter case one can expect Cerenkov axion emission to emerge.Similar mechanism was earlier suggested for gravitational radiation of local strings[34],but in that case the explicit calculations have led to zero result.Vanishing of the gravitational radiation,however,has a specific origin related to the absence of gravitons in the2+1gravity theory.Indeed,as it was explained in[34],two crossed superluminal strings can be“parallelized”by suitable coordinate trans-formations and world sheet reparameterizations,so the problem is essentially equivalent to the point particles’collision in2+1gravity theory.For otherfields,(e.g.electromagnetic) this mechanism does work;see[35]for calculation of the Cerenkov electromagnetic radiation generated in collisions of superconducting strings.So one can expect to have a non-vanishing axion bremsstrahlung from collisions of global strings as well.Our calculation follows the approach of[34]and it is essentially perturbative in terms of the string-axionfield interaction constant(equal to f)involving two subsequent iterations in the strings equations of motion and the axionfield equations.This is similar to calculation of the bremsstrahlung from charged ultrarelativistic particles.In this latter case the iteration sequence converges if the scattering angle is small.We assume the same condition to hold for the strings,though we do not enter here into a detailed investigation of the convergence problem.We would like to mention also a similar perturbative approach in terms of geodesic deviations[36].II.GENERAL SETTINGConsider a pair of relativistic stringsxµ=xµn(σa),µ=0,1,2,3,σa=(τ,σ),a=0,1,where n=1,2is the index labelling the two strings.The4-dimensional space-time is assumed to beflat and the signature is chosen as+,−−−and(+,−)for the string world-sheets.Strings interact via the axionfield Bµνas described by the action[13]S=S st+S B,(2)where the string term isS st=− n=1,2 µ0n−γγab∂a xµn∂b xνnηµν+2πf n Bµνǫab∂a xµn∂b xνn d2σn,(3)and thefield action is1S B=−γγab∂b xµn)=2πf n HµαβVαβn,(6) whereVαβn=ǫab∂a xαn∂b xβn,(7)and Hµis the totalfield strength due to both strings(no externalfield is assumed).The αβcorresponding potential two-form is the sumBµν=Bµν1+Bµν2,(8) of contributions Bµνn due to each string:Bµνn=4πJµνn.(9) Here the Lorentz gauge∂λBνλ=0is assumed,the4-dimensional D’Alembert operator is introduced as =−ηµν∂µ∂ν,and the source term readsf nJµνn=The self-action terms in the equations of motion(6)diverge both near the strings and at large distances,so two regularization parametersδand∆have to be introduced[37].These can be absorbed by classical renormalization of the string tension as followsµ=µ0+2πf2log(∆/δ).(11)The ultraviolet cutofflengthδis of the order of string thicknessδ∼1/f,while the infrared cutoffdistance∆can be taken as the correlation length in the string network.Assuming that such a renormalization is performed,we are left with the same equations of motion(6)with the physical tension parametersµn which contain only the mutual interaction terms on the right hand side.The constraint equations for each string read:∂a xµ∂b xνηµν−1σa n(σa n)(13)and under Weyl rescaling of the metricγab,one canfix theflat gaugeγab=ηab,(14)so that the constraint equations simplify to˙x2+x′2=0,˙xµx′νηµν=0(15)for each string,where dots and primes denote the derivatives overτandσas usual.In the gauge(14)the renormalized equations of motion read:µn ∂2τ−∂2σ xµn=2πf n Hµνλn′V n′νλ,n=n′,(16)corresponds to the contribution of the n′−th string(no sum over n′).It where thefield Hµνλn′is worth noting that imposingflat world-sheet metric we still do notfix the gauge completely, since one is free to perform two-dimensional linear transformations preserving the Minkowski metric(14)on the world sheets.The retarded solution to the wave equation can be presented in its standard formBµνn=4π G ret(x−x′)Jµνn(x′)d4x′,(17)where the Green’s function isG ret(x−x′)=1q2+2iǫq0d4q,(18)withǫ→+0.The advanced Green’s function is obtained by changing the sign ofǫ.The energy momentum tensor for the axionfield readsTµν= HµαβHναβ−1∂xν=−4πHµαβJαβ,(20) where Hµαβshould be taken as the half-difference of the retarded and advanced solution ofthe wave equation,Hµαβrad =(Hµαβadv−Hµαβadv)/2.The corresponding potential has then thefollowing Fourier transform:B radµν(k)=−4iπ2k0π kµk0√where eθand eϕare two unit vectors orthogonal to k and to each other:eϕ·eθ=0,k·e i=0.(27) Using antisymmetry and transversality of the current(23)and the completeness conditioneθi eθj+eϕi eϕj=δij−k i k j/ω2,(28)onefinds∆Pµ=1|k0||J ij(k)e ij|2δ(k2)d4k.(29)Integrating over k0,wefinally obtain∆Pµ=1|k||J(k)|2d3k,(30)whereJ(k)=J ij(k)e ij(31) with k0=|k|.In what follows we shall use the following explicit parameterization of three orthogonal vectors by spherical angles:k=ω[sinθcosϕ,sinθsinϕ,cosθ],eϕ=[−sinϕ,cosϕ,0],(32)eθ=[cosθcosϕ,cosθsinϕ,−sinθ].Our strategy consists in solving the system of equations(16,17)iteratively,by expanding all functions in powers of the coupling constant f:Bµν=∞l=1l Bµν,(33)Jµν=∞l=0l Jµν,(34)xµ(σ)=∞l=0l xµ(σ).(35)Such expansions have to be performed separately for each string.In our notation,the zeroth-order approximations for the string embedding functions0xµn(σ)give zeroth-order source terms0Jµνn,which generate thefirst orderfields1Bµνn originating on each string. These quantities,when substituted to the string equations of motion(16),will generate theFIG.1:In the chosen frame,thefirst string is at rest,the second moves in the parallel plane separated from thefirst by an impact distance d=d2−d1being inclined at the angleαwith respect to the string at rest.Its velocity v is perpendicular to the string itself.first order terms1xµn,and so forth.Convergence of this expansion(with a formal parameter f which has a dimension of an inverse length)is difficult to explore in general.This is likely to depend substantially on the choice of the zeroth-order solution of the string equations of motion.In this paper,the strings are assumed to be in the unexcited state in the zeroth order,that is they are straight and freely moving.In this case it can be expected that the power series solutions will give sensible results(at least in the lowest appropriate order when radiation appears)if the deviation angle of colliding strings due to their lowest order interaction is small.We shall come back to this condition later on.III.COLLISION KINEMATICSThe kinematics of collision is shown on Fig. 1.The unperturbed world-sheets of un-excited,freely moving straight(infinite)strings are described by linear functions ofτ,σ:xµn=dµn+uµnτn+Σµnσn,n=1,2,(36)with dµn andΣµn being space-like and uµn time-like constant vectors.Then the constraint equations(15)imply the orthogonality and normalization conditions(choosing the unit norm time-like4-velocity uµ)uµΣµ=0,uµuµ=1=−ΣµΣµ(37)for each string.We choose the reference frame such that thefirst string is at rest and is stretched along the x3axis:uµ1=[1,0,0,0],Σµ1=[0,0,0,1].(38)The second string is assumed to move in the plane x2,x3with the velocity v orthogonal to the string itself:uµ2=γ[1,0,−v cosα,v sinα],Σµ2=[0,0,sinα,cosα],(39)whereγ=(1−v2)−1/2.We also choose both impact parameters dµn to be orthogonal to uµandΣµand aligned with the axis x1,the distance between the planes being d=d2−d1.The angle of inclinationαof the second string with respect to thefirst one can be presented in the Lorentz-invariant form asα=arccos(−Σ1Σ2),(40) and the relative velocity of the strings asv=(1−(u1u2)−2)1=(u1u2)−1 (u1u2)2−12(42)sinαalong the x3-axis.This motion is not associated with propagation of signal of any kind,so the velocity v p may be arbitrary,in particular,superluminal.The case of parallel strings corresponds to v p=∞.Let us explore the residual gauge invariance consisting of the SO(1,1)transformations of the world-sheetsτn→τn coshχn+σn sinhχn,σn→τn sinhχn+σcoshχn,(43) preserving the constraints(15).It is easy to show that the relative orientation of two strings is not a gauge independent property[34].One can build the following matrix describing the relative orientation of the world-sheetsκab=∂a xµ1∂b xν2ηµν,(44)whose determinantκ=detκab=γcosα(45) is invariant under the transformation(43).But the relative velocity(41)and the inclination angle(40)separately are not invariant.In particular,for superluminal strings one can always find the world-sheet boosts which make the strings parallel.This is achieved by the following choice of parameters in(43)for two stringstanhχ1=sinαγv,(46)which exists if v p>1.Note that strings are superluminal in the above sense provided thatκ=γcosα≥1.(47) If this condition is not fulfilled,then the“parallelizing”transformation does not exist. IV.FIRST ORDER INTERACTIONAccording to the conventions chosen above,thefield equations(9)for thefirst orderfield terms1Bµνn read:1Bµνn=4π0Jµνn,(48) with the source terms(48)0Jµνn=f nThe retarded solution 1B µνof the Eq.(48)generated bythezero-order source is1B µνn (q )=−8π3f n V µνn e iqd nδ(qu n )δ(q Σn )µ1X µ1δ(qu 2)δ(q Σ2)e iq (d 2−d 1−u 1τ−Σ1σ)µ2X µ2δ(qu 1)δ(q Σ1)e iq (d 1−d 2−u 2τ−Σ2σ)0˙x 22.(59)11To calculate the quantity in the numerator one has to integrate in(56)over d4q.Integration over q0and q3is performed using delta-functions.Then,in the limitτ→∞,an asymptotic relationsin[(qu2)τ]limτ→∞=iπ,(61)q1the resulting estimate for the scattering angle being2π2f2δϕ∼.(63)log(∆/δ)γv2The smallness of this quantity may serve as a rough estimate of the validity of the per-turbation expansions(33-35).Therefore our iteration procedure works in the relativistic case v∼1,especially for ultrarelativistic collisions,γ≫1.But for global cosmological strings the quantity log(∆/δ)is large enough.This improves the convergence of our iter-ation scheme even forγ∼1.Therefore our approach seems to be reasonably justified in realistic cosmological situations.V.RADIATIONUsing thefirst order corrections to the world-sheets of the string due to their lowest order interaction terms one can evaluate higher order terms of the expansion.Radiation emerges in the second approximation for the axionfield2Bµνgenerated by the source termJµνwhich will be obtained from thefirst order contributions to the world-sheet embedding 1functions as follows:Jµν= n=1,2f n d2σ 0˙x[µn1x′ν]n+1˙x[µn0x′ν]n −11where brackets denote anti-symmetrization over indices with the factor1/2.This follows from the fact that the Fourier-transform of this current is non-zero on the axion mass-shellk2=0.The Fourier transform of thefirst-orderfield source can be presented in the following form:1Jµν(k)=−8π2f1f2 δ(qu2)δ(qΣ2)δ[(k−q)u1]δ[(k−q)Σ1]e i(d1k)+q(d2−d1)) f1Qµν1µ2 ,(65) where the two termsQµν1=(qΣ1)u[µ1Xν]1−(qu1)Σ[µ1Xν]1−1q2[(qΣ1)2−(qu1)2],(66)Qµν2=[(k−q)Σ2]u[µ2Xν]2−[(k−q)u2]Σ[µ2Xν]2−1(k−q)2{[(k−q)Σ2]2−[(k−q)u2]2}(67)could be associated with contributions of thefirst and the second string,respectively.It has to be noted,however,that in the second order the axionfield is generated collectively by a source term containing symmetrically the contributions of thefirst order coming from both strings.It can be checked that the antisymmetric tensors Qµνn satisfy the transversality conditionskµQµν1=kµQµν2=0(68) ensuring in turn the transversality of the current1Jµν(k).Integration over q-space is performed as follows.First,usingδ[(k−q)u1],we integrate over q0,whichfixes the valueq0=k0=ω.(69)Then,using the delta-functionsδ(qu2)andδ(qΣ2),one integrates over q2and q3obtainingq2=−ωcosαv.(70)The remaining delta-functionδ[(k−q)Σ1]does not furtherfix the vector qµ,but imposes an extra relation on the parameters:k3=ωsinαeffective source localized in the region surrounding the point of minimal separation between the strings where their deformation described by thefirst order interaction is maximal.This condition can be rewritten as the usual Cerenkov’s cone condition on the angle of emissioncosθ=1/v p,(72)which makes sense if v p≥1.So the axion radiation is emitted inside the Cerenkov cone oriented along the x3axis(Fig.2).In the limiting case v p=1,radiation is emitted strictly along the x3direction;this occurs when the string inclination angleαis related to the string Lorentz factor ascosα=γ−1,withγ=(1−(u1u2)−2)1(q1)2+κ21=πf1(−iκ1)e−κ1dγv.(75)Similarly,the second term in(65)gives the integralse−iq1d dq1f2(q1)κ2,(76)whereκ2=ωξωv.(77)Therefore,the whole integral over d4q of thefirst term in(65)is proportional to the value of the integrand in the complex point qµ=¯qµwhere¯qµ=ωv[0,i cosα,1/γ,0].(79)14For the second term Qµν2onefinds instead the following quantitieskµ−¯qµ=ωξv[v,−i cosα,−cosα,sinα].(81) Finally,projecting the current on the polarization tensor(26),we obtain the following lowest order approximation for the quantity(31):1J(k)=4√ω2γv sinθδ(cosθ−1/v p)·(82)f1e−ωd/vγ+ikd1µ2γ2ξ2 cosϕ−i(vξγ2sinθ+sinϕ),whereξ=cosα+v sinθsinϕ.Two terms here may be attributed to the contributions of thefirst(remaining at rest)and the second(moving)strings;in what follows we set f1=f2=f,µ1=µ2=µ.The total amplitude is therefore a superposition of two terms exhibiting different frequency cutoff.Thefirst term’s exponential factor leads to the isotropic conditionωvγdξ=v2γ2cosα 1+β2γ2cosα2 ],(86) whereβ=π/2+ϕ≪1.Due to the factorξ−2in the second term in the amplitude(31), radiation is peaked in the directionϕ=−π/2within the narrow regionβ γ−1.(87)15FIG.2:Radiation peaking on Cerenkov cone in the direction of motion of the relativistic sting.The exponential factor exhibits similar peaking,providedγcos α≫1,(88)so the frequency range extends up toωγ22π3f 32πδ(cosθ−1),(93)16one obtains after integration over angles in d3k=ω2dωd cosθdϕthe total radiated energy per unit length of the string at restP0µ2ωexp(−2ωdl =(2π)5f6vγ∆ .(95)For d≪vγ∆,P0µ2ln vγ∆vγ∆ ≈2d vγ∆ ,(97)in this case radiation is exponentially small.A particular value,Ei(1,1)=0.219,can be used to estimate the energy loss for intermediate impact parameters.For d=vγ∆one hasP0µ2.(98) For generic values of parametersα,v it is convenient to rewrite the amplitude in terms of the quantityκ=γcosαwhich is invariant under world-sheets reparameterizations char-acteristic of the relative motion of the strings.Setting without lack of generality d1=0we obtain:1J(k)=4√µω2√vγ (cosϕ−iκsinϕ)−−exp−ωdκ2−1 cosϕ−iκ κsinϕ+√κ+√FIG.3:The spectral function F(z).the main frequency range we obtain the spectral-angular distribution in the vicinity of this direction as follows1dωdβ=64π4f6κ2(1+κ2β2)4exp −ωd(1+κ2β2)d.(101)Choosing as an infrared cutoffan inverse length parameter∆and integrating and over frequencies we obtain the angular distribution of the total radiation1dβ=64π4f6κ2(1+κ2β2)4Ei 1,d(1+β2γκ)γκ.(103)One can also obtain the spectral distribution of radiation by integrating(100)over the variableβ,which can be extended to the full axis in view of the exponential decay of the integrand.The result reads:1dω=16π5f6dγκ,(104)whereF=(8z2+8z+6)e−zπz+(8z2+12z+6−3/z)(erf(√y(y)φ46810120.20.40.60.81FIG.4:Dependence of radiation energy on the impact parameter,y =d/(γκ∆).The spectrum is shown on the Fig. 3.For small frequencies ω≪γκ/d the function F (z )has a logarithmic divergenceF (z )∼3√lP 0=16π5f 6κγκ∆,(108)whereφ(y )=12π2F 212;32;−y−3ln 4y e C +7VI.BREMSSTRAHLUNG IN2+1ELECTRODYNAMICSClassical system of parallel strings interacting via the antisymmetric two-form potential Bµνcan be equivalently presented as the system of point charges interacting via the Maxwell field Aµin2+1dimensions(in what follows in this section the Greek indices run over µ=0,1,2).Indeed,substituting the constraint into the action(2)and assuming that all variables are x3-independent,one can rewrite the action asS=− n=1,2 (m n dτn+e n Aµdxµn)−1√2lB3µ,m n=µn l,e n=28π2 kµk08π2 kµPerturbation theory with respect to the formal charge parameter e (whose dimension in the 2+1electrodynamics is l −1/2in the units =c =1)is constructed in the same way as in the Sec.II,introducing the input world-lines0x µn =d µn +u µn τn ,(118)with space-like impact vectors d µn and unit time-like 3-velocities u µn :u µ1=[1,0,0],u µ2=γ[1,0,−v ].(119)After the first iteration one obtains1x µ1=ie 1e 2q 2(qu 1)2δ(qu 2)e iq (d −u 1τ1)d 3q,(120)where d =d 2−d 1,and similarly for the second charge.The first order current,which gives the lowest order contribution to radiation,reads1J µ= n =1,2e n dτn (1˙x µn −u µn 1x νn ∂ν)δ(x −0x n (τn )).(121)Its Fourier-transform is1J µ(k )=e 1e 2m 1Q µ1+e 2q 2(ku1)2,Q µ2=(ku 2)2u µ1−(ku 1)(ku 2)u µ2+(u 1u 2)[(kq )u µ2−(ku 2)(k −q )µ]2ωγv e 1e −ωd/vγ+ikd 1m 2γ2ξ2 cos ϕ−i (vξγ2+sin ϕ),(124)where ξ=1+v sin ϕ.As we argued in the Sec.3,our iteration scheme converges if the relative velocity is relativistic,and for cosmological applications one is interested by moderately relativisticvelocities whenγis not large.However,for generalγit is difficult to perform computation of the total energy loss analytically,so we continue calculations in the ultrarelativistic case γ≫1when additional simplifications can be made.Hopefully this should give a reasonable approximation forγof the order of several units.One can show that then the second term in (124)is dominant(the asymmetry of contributions from thefirst and the second charges is due to our choice of the reference frame:thefirst charge is at rest before the collision,while the second is moving with the relativistic velocity).The spectral distribution of radiation due to the second term extends tillω∼2γ2/d in the narrow angular regionδϕ∼1/γaround the directionϕ=−π/2,while thefirst term has a lower(∼γ/d)frequency cutoff.The total contribution of the second term to the radiation loss is greater than that of thefirst one by a factor ofγ,so in what follows we neglect thefirst term.Setting e1=e2=e,m1=m2=m wefind from the Eq.(116)the spectral-angular distribution of radiation as followsdP032π2e6ωγ6ξ4.(125)Forγ≫1,radiation is peaked in the direction of motion of the relativistic chargeϕ=−π/2, and in the vicinity of this direction(cf.(86))ξ≈1dωdβ=e6γ2(1+γ2β2)4exp −ωd(1+γ2β2)d.(128)Integrating over frequencies from the infrared cutoff∆−1we obtaindP08π2m2(1−γ2β2)2γ2∆ .(129)Taking into account relations(111)between parameters of3+1and2+1problems,one can see that this expression coincides with(102)forγ=κ,i.e.forα=0:1dβ=64π4f6γ2(1+γ2β2)4Ei 1,d(1+β2γ2)VII.COSMOLOGICAL ESTIMATESIn the standard axion cosmology scenario,which assumes the Peccei-Quinn phase transi-tion after inflation,the global string network is formed at the temperature T0∼f.Stringsinitially move with substantial friction[17,18,19]due to scattering of particles present in the hot cosmic plasma.At some temperature T∗<f the string network enters the scale-invariant regime[20,21,22]when strings move almost freely with relativistic veloci-ties[23,24]and are able to form closed loops via interconnections.Finally,at much lower temperature T1of the QCD phase transition,axions become massive and strings disappear. Our mechanism presumably should work in the temperature interval(T∗,T1),though an additional contribution from the damped epoch T0<T<T∗can also be expected.The scaling density of strings is determined from numerical experiments,it can be presented asζµρs=1012GeV −4,(132)t1(sec)∼2·10−7 fw(w)ξ0123450.20.40.60.81FIG.5:Dependence of the axion radiation rate on the Lorentz factor of the collision ξ(w ),w =L/(γ2∆).the normalization volume.Therefore,for the axion energy density generated per unit time we obtain:dεaL 2NV =ρsw2,−12,32,−12,12−3ln 4w e C w,(136)and using for µthe expression(11)with only the second (leading)term,we obtaindεat 3ln 2(tf ),(137)whereK =8γ2∆.(138)Since in the cosmological context L ∼∆∼t ,one can take w =γ−2for an estimate.The realistic value of γis of the order of unity,while our formulas were obtained in the γ≫1approximation,since we were keeping only the second term in the exact amplitude(99).But an independent calculation shows that forγ∼1the contribution from thefirst term is of the same order,so we can hope to get reasonable order of magnitude estimate using the above formulas forγnot large.The fullγ-dependence is given by the function γ3ξ(γ−2),whereξ(w)is plotted on Fig.5.Forγ=√ln2x dx=−x2ln x +2Li(x2)≈x22t2∗ln2(t∗/t0),(143)where the ratio t∗/t0can be estimated ast∗f 2.(144) Finally,dividing by the critical energy density at t=t1εcr=3m2Pl13 2fVIII.CONCLUSIONSIn this paper,we have suggested a new mechanism for the axion emission by the global string network:the bremsstrahlung under string collisions.As far as we are aware,this effect was not discussed in the context of the cosmic string scenario so far.Though it is of the second order in the axion coupling constant,rough cosmological estimates show that it is not small and gives a contribution of the same order of magnitude as radiation due to oscillating loops.We have found the radiation amplitude using the perturbation theory whose validity is restricted by relativistic(though not necessarily ultrarelativistic)velocities of colliding strings.The frequency spectrum has an infrared divergence which is not an artifact of our approximation,but rather is the effect of the space-time dimensionality:as we have shown, an equivalent description of the axion bremsstrahlung from strings in3+1dimensions is provided by the electrodynamics of point charges in2+1dimensions,where its origin lies in the logarithmic dependence on distance of the Coulomb potential.Unfortunately the integration over the spectrum and the angular distribution can be performed analytically only in the ultrarelativistic case,and thefinal formulas obtained relate to this situation. But,in view of a smooth dependence of the radiation efficiency on the Lorentz factor of the collision,we hope that our results give a reasonable estimate of the effect for values of the Lorentz factor of the order of several units as well.From a purely theoretical viewpoint two aspects are worth to be discussed.Thefirst is the Cerenkov nature of radiation which is associated with the possibility of the superluminal motion of an effective source located around the point of minimal separation between the strings.In perturbative setting,the strings,which are straight in zero order approximation, get deformed under axion interaction,with the deformation being maximal near this point. This region therefore acts as an effective source of radiation arising in the next order ap-proximation.This explains why radiation is concentrated on the Cerenkov cone directed along the trajectory of the effective source.Another interesting feature is that the effect has an alternative2+1interpretation as the bremsstrahlung of point charges.This is due to the fact that two strings inclined with respect to each other and moving in the superluminal regime can be made parallel by suitable reparameterizations of their world-sheets and the choice of the space-time Lorentz frame.This transformation is only feasible once the relativestring motion is superluminal in the sense described above.Classical dynamics of parallel strings interacting via the axionfield in3+1dimensions is equivalent to dynamics of point charges in2+1electrodynamics,therefore the string axion bremsstrahlung is reduced to the point particle bremsstrahlung in lower dimension.AcknowledgmentsThefinal version of this work was completed while one of the authors(DVG)was visiting Erwin Shr¨o dinger Institute(Vienna)under the programm”Gravity in two dimensions”.He is grateful to Organizers for the invitation and support and to participants of the Workshop for stimulating discussions.The work of DVG was also supported in parts by the Ministry of Education of Russia.[1]R.Peccei and H.Quinn,CP conservation in the presence of pseudoparticles,Phys.Rev.Lett.38,1440(1977),and Constraints imposed by CP conservation in the presence of pseudopar-ticles,Phys.Rev.D16,1791(1977).[2]S.Weinberg,A new light boson,Phys.Rev.Lett.40,223(1978).[3] F.Wilczek,Problem of strong P and T invariance in the presence of instantons,Phys.Rev.Lett.40,279(1978).[4]L.Abbot and P.Sikivie,A cosmological bound on invisible axion,Phys.Lett.B120,133(1983);J.Preskill,M.Wise,and F.Wilczek,Cosmology of the invisible axion,Phys.Let.B120127(1983);M.Dine and W.Fischler,The not so harmless axion,Phys.Lett.B120137 (1983).[5]M.Srednicki,Axions:past,present and future,in Proceedings of the International Conferenceon Continuous advances in QCD,Minneapolis2002,edited by K.A.Olive,M.A.Shifman, M.B.Voloshin(Singapore,World Scientific,2002.),p.509;hep-th/0210172.[6]P.Sikivie,Axions and their distribution in galactic halos,Invited talk at4th InternationalWorkshop on the Identification of Dark Matter(IDM2002),York,England,2002,10pp.;hep-ph/0211254.[7]J.E.Kim,Light pseudoscalars,particle physics and cosmology,Phys.Rep.150,1(1987);。

(24)
where the Lagrangian density satisfies the Euler-Lagrange equations of motions
(25)
such that the Euler-Lagrange equations of motion just give the Klein-Gordon equation (12) and its complex conjugate.
as the basic field equation of the scalar field.
The plane waves (10) are basic solutions and the field (9) is constructed by
a general superposition of the basic states.
Quantization
The challenge is to find operator solutions of the Klein-Gordon equation (12) which satisfy eq. (28). In analogy to the Lagrange density (24) , the hamiltonian is
Lecture 8
The Klein-Gordon equation
WS2010/11: ‚Introduction to Nuclear and Particle Physics‘
The bosons in field theory
Bosons with spin 0
scalar (or pseudo-scalar) meson fields
(23)

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薛定谔方程英语English: The Schrödinger equation, also known as the Schrödinger wave equation, is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is named after the Austrian physicist Erwin Schrödinger, who first formulated it in 1926. The equation is a partial differential equation that describes how the wave function of a physical system evolves over time, and it is central to the understanding of quantum mechanics. The Schrödinger eq uation is used to calculate the probability distribution of a particle in a given region of space, and it has been immensely successful in explaining the behavior of particles at the atomic and subatomic levels.中文翻译: 薛定谔方程，也被称为薛定谔波动方程，是量子力学中的基本方程，描述了一个物理系统的量子状态随时间的变化。

它以奥地利物理学家埃尔温·薛定谔的名字命名，他在1926年首次提出了这个方程。