, COLLAPSE OF POSITRONIUM AND VACUUM INSTABILITY A.E. Shabada 6 0 P.N. Lebedev Physics Institute, Leninsky prospekt 53, Moscow 119991, Russia 0 V.V.Usovb 2 Center for Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel n Abstract.Ahypercriticalvalueforthemagneticfieldisdetermined,whichprovides a the full compensation of the positronium rest mass by the binding energy in the J maximum symmetry state and disappearance of the energy gap separating the electron-positron system from the vacuum. The compensation becomes possible 4 owing to the falling to the center phenomenon. The structure of the vacuum 2 is described in terms of strongly localized states of tightly mutually bound (or confined) pairs. Their delocalization for still higher magnetic field, capable of 1 screeningitsfurthergrowth,isdiscussed. v 2 It is accepted that magnetic fields are stable in pure quantum electrody- 4 5 namics(QED),andotherinteractions(weakorstrong)ormagneticmonopoles 1 have to be involvedto make the magnetized vacuum unstable [1]. In this Talk 0 (seeRef.[2]foradetailedversion)weestablish,withintheframeofQED,that 6 there exists a hypercritical value of the magnetic field 0 / h m2 π3/2 p Bh(1p)cr ≃ 4e exp(cid:26) √α +2CE(cid:27)≃1028B0 ≃1042G. (1) - o r Here α = e2/4π 1/137, B = m2/e = 4.4 1013G, m is the electron t ≃ 0 × s mass. The value (1) is less than the magnetic field (1047 1048) G expected a ∼ − to be present near superconductive cosmic strings [3] and the one ( 1047G) : v producedatthe beginning ofthe inflation[4]. The hypercriticalmagn∼eticfield i X leads to the shrinking of the energy gap between an electron and positron that takes place due to the falling to the center. The latter is caused by the r a ultravioletsingularityofthephotonpropagator(orCoulombpotential)[5]plus thedimensionalreductioninthemagneticfield[6]-[8]. Wediscussthevacuum structure around the hypercritical magnetic field and its possible decay under a further growth of the magnetic field - that may cause its screening. Werelyonthetheoryofthefallingtothecenterdevelopedin[9]thatimplies deviations from the standard quantum theory manifesting themselves when extremely large electric fields near the singularity become important. In that theory the singularity in the Schr¨odinger-like equation yields a singular mea- sure in the scalar product and hence the geometry of a black-hole-like object. (Stress, that the geometry is induced: no interaction of gravitational origin is included.) ae-mail: [email protected] be-mail: [email protected] We proceedfromthe (3+1)-dimensionalBethe-Salpeter (BS) equationin an approximation,whichis the ladderapproximationonce the photonpropagator (in the coordinate space) is taken in the Feynman gauge: D (x) = g D(x2), ij ij x2 x2 x2, g =(1, 1 1 1). In an asymptotically strong magnetic field ≡ 0− ii − − − this equation may be written [2] in the following (1+1)-form covariant under the Lorentz transformations along the axis 3: Pˆ Pˆ (i−∂→ˆ k m)Θ(t,z)( i←∂−ˆ k m) k− 2 − − k− 2 − P2 =i8πα D t2 z2 ⊥ g γ Θ(t,z)γ , (2) (cid:18) − − (eB)2(cid:19) ii i i iX=0,3 where Θ(t,z) is the 4 4 (Ritus transform of) BS amplitude, t = xe xp and z =xe xparethediff×erencesofthecoordinatesoftheelectron(e)an0d−pos0itron 3− 3 (p) along the time x andalong the magnetic field B=(0,0,B =B). P and 0 3 k P areprojectionsofthetotal(generalized)momentumofthepositroniumonto ⊥ the (0,3)- subspace and the (1,2)-subspace. Only two Dirac gamma-matrices, γ , are involved, ∂ˆ =∂ γ ∂ γ , Pˆ =P γ P γ . 0,3 k t 0 z 3 k 0 0 3 3 − − Equation(2)isvalidinthedomain,wheretheargumentofD isgreaterthan the electron Larmour radius squared (L )2 = (eB)−1. When B = , this B ∞ domain covers the whole exterior of the light cone z2 t2 0.The argument − ≥ of the original photon propagator (xe xp)2 has proved to be replaced in (2) by t2 z2 (xe xp)2 = t2 z2 −P2/(eB)2, where xe,p are the center of − − ⊥ − ⊥ − − ⊥ ⊥ orbit coordinates of the two particles in the transversalplane. Now that after the dimensionael reduection this subspace no longer existsethese substitute for the transversal particle coordinates themselves: xe,pare not coordinates, but ⊥ quantum numbers of the transverse momenta. The mechanism of replacement of a coordinate by a quantum number is the sameeas in [7]. Inderivingequation(2)the expansionoverthe completesetofRitus matrix eigenfunctions [10] was used in [2] that accumulate the dependence on the transversal spacial and spinorial degrees of freedom. This expansion yields an infinite setof equations,wheredifferent pairsofLandauquantumnumbers ne, np are entangled, Eq. (2) being the equationfor the (ne =np =0)-component that decouples from this set in the limit B = . ∞ Inthe ultra-relativisticlimitP =P =P =0equation(2)issolvedbythe 0 3 ⊥ most symmetric Ansatz Θ=IΦ, where I is the unit matrix. The ultraviolet singularity on the light cone (x2 = 0) of the free photon propagator, D(x2) = (i/4π2)(1/x2), after this expression is used in eq. (2), − leads to falling to the center in the Schr¨odinger-like differential equation d2Ψ(s) 1 4α 1 + m2 Ψ(s)= Ψ(s), (eB)−1/2 s , (3) − ds2 (cid:18) − 4s2(cid:19) π s2 ≪ ≤∞ towhichtheradialpartofequation(2)isreducedinthemostsymmetricalcase, when the wave function Φ(x) =s−1/2Ψ(s) does not depend on the hyperbolic angle φ in the space-like region of the two-dimensional Minkowsky space, t = ssinhφ, z =scoshφ, s=√z2 t2. − Thesolutionthatdecreasesats isgivenbytheMcDonaldfunctionwith →∞ imaginary index: Ψ(s) = √s K (ms), ν = i2 α/π 0.096i, oscillating ν ≃ when s 0. The falling to the center [11] holds fopr any positive α. → According to [9], the singular equation (3) should be considered as the gen- eralized eigenvalue problem with respect to α. The operator in the left-hand side is self-adjoint provided the standing wave condition is imposed, Ψ(s) =0, (4) |s=LB that treats the Larmour radius as the lower edge of the normalization box. The discrete eigenvalues α (L ) condense in the limit B = to become a n B ∞ continuum of states that form the (rigged) Hilbert space of vectors orthogonal with the singular measure s−2ds. The latter fact allows to normalize them to δ-functionsandinterpretasfreeparticlesemittedandabsorbedbythesingular center. AslongastheLarmourradiusismuchsmallerthantheonlycharacter- istic length in eq.(3), electronCompton length, L m−1 3.9 10−11 cm., B ≪ ≃ × thesmall-distanceasymptoticregimeisreached,andnothing”behindthehori- zon,” s<L , - where the two-dimensionalequations (2) and (3) are not valid B - may affect the problem. In this way the existence of the limit of vanishing regularizationlength, impossible in the standard theory, is achieved. For sufficiently large magnetic fields L becomes so small that the region B whereK (ms)oscillatesgetsinsidethedomainofvalidityofequation(3). The ν valueofthemagneticfieldwhenthishappensforthefirstnodeisjust(1). The corresponding Larmour radius is about fourteen orders of magnitude smaller than m−1 and makes 10−25 cm. Eq. (1) tells how large the magnetic ∼ field should be in order that the boundary problem (3), (4) might have a solution, in other words, that the point P = P = 0 might belong to the 0 spectrum. Therefore, if the magnetic field exceeds the first hypercritical value the positronium ground state existsc with its rest energy compensated for by the mass defect. The ultra-relativistic state P = 0 has the internal structure of what was µ called a ”confined state”, belonging to kinematical domain called ”sector III” in[9],i.e. theonewhosewavefunctionbehavesasastandingwavecombination of free particles near the lower edge of the normalization box and decreases as cA relationlike(1) ispresent in[8]. There, however, a different problem isstudied and, correspondingly, a different meaning is attributed to that relation: it expresses the mass gained dynamically - in the course of spontaneous breakdown of the chiral invariance in massless QED - by a massless Fermion in terms of the magnetic field applied to it. Later, in [12] the authors revised that relation in favor of a different approximation. Supposedly, therevisedrelationmaybeofuseintheproblemofultimatemagneticfielddealtwithhere. exp( ms) at large distances. The effective ”Bohr radius”, i.e. the value of − s that provides the maximum to the wave function makes s = 0.17m−1. max This is certainly much less than the standard Bohr radius (e2m)−1. Taken at the level of 1/2 of its maximum value, the wave function is concentrated within the limits 0.006 m−1 <s<1.1 m−1. But the effective region occupied by the confined state is still much closer to s = 0, since - in accord with the aforesaid - the probability density of the confined state is the wave function squared weighted with the measure s−2ds singular in the origin [9] and is hence concentrated near the edge of the normalization box s 10−25cm, and ≃ not in the vicinity of the maximum of the wave function. The electric fields at such distances are about 1043 Volt/cm. Certainly, there is no evidence that the standard quantum theory should be valid under such conditions. This fact encourages the use of a theory that admits deviations from the standard quantum theory that close to the singularity point. AtB =B(1) thetotalenergyandmomentumofapositroniumintheground hpcr state are zero. This state is not separated from the vacuum by an energy gap, and has maximum symmetry in the coordinate and spin space. Hence, it may be related to the vacuum and responsible for its structure. AtB >B(1) theeigenvaluesoftheBSequation(2)forthetotal2-momentum hpcr components P of the e+-e− system are expected to shift into the space-like 0,3 region (we keep P =0), whereas for B <B(1) the c.m. 2-momentum of the ⊥ hpcr real pair was, naturally, time-like. If P = 0, at least for far space-like P , 0,3 k 6 the situation can be modelled by the same equation as (3), but with the large negative quantity m2+P2/4 substituted for m2. Then the wave function k would contain two oscillating exponentials for large space-like intervals, P2 x +x exp±isvuu(cid:12)(cid:12)m2+ 4k (cid:12)(cid:12)exp(cid:26)iPk e 2 p(cid:27), s≫(−Pk2)−1/2, (5) t(cid:12) (cid:12) (cid:12) (cid:12)  (cid:12) (cid:12) and two oscillating exponentials exp( 2ilns α/π) for small ones, s L . B ± ∼ In the Lorentz frame, where P = 0,P = 0p, and with the time arguments 0 3 in the two-time BS amplitude equal to o6ne another: xe = xp, the solution 0 0 oscillatesalongthemagneticfieldwithrespecttotherelativecoordinatexe xp (mutually free particles) and with respect to the c.m. coordinate xe3+−x3p 3 3 (vacuum lattice). We are now in the kinematical domain called sector IV, or deconfinement sectorinRefs.[9]. Heretheconstituentsarefreeatlargeintervalsandnearthe singular point s = 0. The wave incoming from infinity is partially reflected, andpartiallypenetratestothesingularpoint,theprobabilityofcreationofthe delocalized(free)statesbeingdeterminedbythebarriertransmissioncoefficient [9]. Such states may existif one succeeds to satisfy e.g. periodic conditions, to be imposed on the lower and upper boundaries of the normalization volume, insteadofcondition(4),appropriateinsectorIII.Thepossibilitytoobeythem is provided again by the falling to the center. Now, the delocalized states in two-dimensional Minkowsky space correspond to electron and positron that circle alongLarmourorbits withvanishing radiiin the plane orthogonalto the magnetic field and simultaneously perform, when the interval between them is large, a free motion along the magnetic field. They have magnetic moments and seem to be capable of screening the magnetic field. This provides the mechanism that may prevent the classical magnetic field from being larger than a second hypercritical field, for which the delocalization first appears. No sooner than the delocalized states are found in our present problem one may definitely claim the instability of the vacuum with the second hypercritical magnetic field or - which is the same - the instability of such field under the pair creation that might provide the mechanism for its diminishing. For the present, we state that the first hypervalue (1) is such a value of the magnetic field, the exceeding of which would already cause restructuring of the vacuum. Acknowledgments ThisworkwassupportedbytheRussianFoundationforBasicResearch(project no05-02-17217)andthePresidentofRussiaProgramme(LSS-1578.2003.2),as well as by the IsraelScience Foundationof the IsraelAcademy ofSciences and Humanities. References [1] M. Banderand H.R. Rubinstein, Phys.Lett. B 280,121 (1992);289,385 (1992); R.C. Duncan, astro-ph/0002442; [2] A.E. Shabad and V.V. Usov, hep-th/0512236. [3] E.Witten, Nucl.Phys. B 249,557(1985);E.M.Chudnovsky,G.B.Field, and D.N. Spergel, Phys.Rev. D 34, 944 (1986). [4] S. Kawati and A. Kokado, Phys.Rev. D 39, 3612 (1989); D. Grasso and H.R. Rubinstein, Phys.Rep. 348, 163 (2001). [5] J.S. Goldstein, Phys.Rev. 91, 1516 (1953); P.I. Fomin, V.P. Gusynin, V.A. Miransky and Yu.A. Sitenko, Rivista Nuovo Cimento 6, numero 5 (1983); N. Setˆo, Progr.Theor.Phys.Suppl. 95, 25 (1988). 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