Electron-positron vacuum instability in strong electric fields. Relativistic semiclassical approach
uuniverse
Article
Electron-positron vacuum instability in strong electricfields. Relativistic semiclassical approach.
D. N. Voskresensky Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia National Research Nuclear University (MEPhI), Kashirskoe sh. 31, Moscow 115409, Russia * Correspondence: [email protected] Editor: nameReceived: date; Accepted: date; Published: date
Abstract:
Instability of electron-positron vacuum in strong electric fields is studied. First, falling tothe Coulomb center is discussed at Z > Z >
137 for electron. Then,focus is concentrated on description of deep electron levels and spontaneous positron productionin the field of a finite-size nucleus with the charge Z > Z cr (cid:39) (cid:101) = − m in the field of a supercharged nucleus with Z (cid:29) Z cr . Finally, attention is focused on many-particleproblems of polarization of the QED vacuum and electron condensation at ultra-short distances froma source of charge. We argue for a principal difference of cases, when the size of the source is largerthan the pole size r pole , at which the dielectric permittivity of the vacuum reaches zero, and smaller r pole . Some arguments are presented in favor of the logical consistency of QED. All problems areconsidered within the same relativistic semiclassical approach. Keywords: electron-positron production; supercritical atoms; electron condensation; polarization ofvacuum, zero-charge problem
Contents1 Introduction 3 A µ = ( A , (cid:126) ) A µ = ( A , (cid:126) ) Universe , xx a r X i v : . [ nu c l - t h ] F e b niverse , xx , x 2 of 56 (cid:101) < − m . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6.1 Energy spectrum for | (cid:101) | − m (cid:28) m . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6.2 Energy spectrum for | (cid:101) | (cid:28) − m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.7 Exponential estimate of probability of spontaneous production of positrons . . . . . . . 224.8 Critical charge of nucleus for muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 e (cid:28) Z (cid:28) e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.4 Strong screening, Ze (cid:29) r = r > r L . . . . . . . . . . . . . . 499.1.1 Electron condensation is not included . . . . . . . . . . . . . . . . . . . . . . . . . 499.1.2 Electron condensation on levels of upper continuum is included . . . . . . . . . 50 niverse , xx , x 3 of 56 r = r < r L . Polarization of vacuum and electron condensationon levels in lower continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3 Distribution of charge of electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
10 Conclusion 53References 541. Introduction
I dedicate this review to the blessed memory of Vladimir Stepanovich Popov who recently leavedus as the result of a many-year hard illness, which prevented him to work actively last years. Theproblem of the electron-positron pair production when the ground-state electron level dives belowenergy − mc ( m is the electron mass, c is speed of light) was of his interest starting from the end of1960-th. Especially he contributed to this problem during 1970-th. For his outstanding contributions tothe theory of ionization of atoms and ions in the field of intense laser radiation and to the theory ofthe creation of electron-positron pairs in the presence of superstrong external fields V. S. Popov wasawarded the I. Y. Pomeranchuk Prize in 2019.We worked together with Vladimir Stepanovich on problems of supercritical atoms with thecharge Z > Z cr = −
173 during 1976-1978 when we developed semiclassical treatment of thisproblem. These works, cf. [1–7], became a part of my PhD thesis [8] defended in 1977 under theguidance of Arkadi Benediktovich Migdal.As follows from the Dirac equation in the Coulomb field of a point-like nucleus with Z > e (inunits ¯ h = c =
1, which will be used in this paper, e (cid:39) Z and reaches the energy − m . Then two positrons with the energies > m go offto infinity and electrons with (cid:101) < − m screen the field of the nucleus by the charge − e . Typicaldistance characterizing electrons of the vacuum K shell is ∼ ( m ) (cid:29) R nucl , cf. [7]. Reference [9]essentially overestimated value of the critical charge. The problem got a new push in the end of1960-th. Independently W. Pieper and W. Greiner [10] (in numerical analysis) and V. S. Popov [11–15](in analytical and numerical studies) evaluated value of the critical charge to be Z cr (cid:39) − Z / A . Thenthere appeared idea to observe positron production in heavy-ion collisions, where the supercriticalatom is formed for a short time [16,17]. As the reviews of these problems, I can recommend [18–20].In 1976 with the inauguration of the UNI-LAC accelerator in GSI, Darmstadt, it became possibleto accelerate heavy ions up to uranium below and above the Coulomb barrier. Instead of a positronline associated with spontaneous decay of the electron-positron vacuum, mysterious line structureswere observed, which in spite of many attempts did not get a reasonable theoretical interpretation. Theexperimental results on the mentioned positron lines proved to be erroneous. New experiments weredone during 1993 – 1995, cf. [21–23]. Presence of the line structures was not observed. Events, whichcould be interpreted as the effect of the decay of the QED vacuum with the spontaneous production ofthe electron-positron pair, were not selected. In spite of the effect of the spontaneous production ofpositrons in the electric field of the supercharged nucleus has been predicted many decades ago, it wasnot yet observed experimentally in heavy-ion collisions.The spontaneous pair creation could be observed in heavy-ion collisions provided existence of anuclear sticking [24]. However the ideas on the sticking of nuclei did not find a support. Besides aspontaneous production of positrons a more intensive induced production of pairs occurs due to anexcitation of nuclear levels. Therefore, the key question is how to distinguish spontaneous productionof positrons originated in the decay of the electron-positron vacuum from the induced production andother competing phone processes. New studies of low-energy heavy-ion collisions at the supercritical niverse , xx , x 4 of 56 Figure 1.
Typical effective Schrödinger potential of a charged particle in an electric central-symmetricpotential well, r ± are turning points, r corresponds to maximum of U . Dashed line describesquasi-stationary level with (cid:101) < − m . regime are anticipated at the upcoming accelerator facilities in Germany, Russia, and China [25–27].This possibility renewed theoretical interest to the problem [28]. As one can see from numericalresults reported in [28], these results support those obtained in earlier works, although comparisonwith analytical results derived in [1–7] was not performed. Also let me notice that recently thereappeared statements that spontaneous production of positrons should not occur in the problem underconsideration. I see no serious grounds for these revisions and thereby will not review these works. States with | (cid:101) | < m correspond to the energy E = ( (cid:101) − m ) /2 m <
0, see Fig. 1. In terms of theSchródinger equation these are ordinary bound states. Let the ground state level is empty and weare able to adiabatically increase the charge of the nucleus Z . The latter means that either the time τ Z characterizing increase of Z is much larger compared to 1/ | (cid:101) − (cid:101) njm | , where (cid:101) njm are energies ofother bound states in the potential well, or τ Z > m for the case of transitions from the ground-statelevel, (cid:101) , to the continues spectrum. Empty level with (cid:101) < − m becomes quasi-stationary, see Fig. 1.Penetrating the barrier between continua, see Fig. 2 below, two electrons (with opposite spins) areproduced, which occupy this level, whereas two positrons of opposite energy go off through the barrierto infinity. In standard interpretation, cf. [15], the electron states, ψ ∝ e − i (cid:101) t , with (cid:101) = (cid:101) + i Γ ( (cid:101) ) /2for (cid:101) < − m , Γ >
0, cf. Eqs. (3.5), (3.6) in [29], are occupied due to a redistribution of the charge ofthe vacuum. The vacuum gets the charge 2 e < (cid:101) e + = − (cid:101) − i Γ ( (cid:101) ) /2 go off to infinity after passage of a time ∼ τ e Γ t , τ ∼ R ,where R is the size of the potential well for R > ∼ m , as it occurs for any decaying quasi-stationarystate, producing a diverging spherical wave ψ ∝ e ikr , k = (cid:113) (cid:101) e + − m for the positron. For far-distantpotentials situation is similar to that for the charged bosons, cf. [30]. For the case V = − Ze / r for r > R nucl , one obtains that Γ ( − m ) = Z < Z cr electrons of the lower continuum (with (cid:101) < − m ) fill all energy levels according to theDirac picture of the electron-positron vacuum. Spatially they are distributed at large distances. For Z > Z cr the process of the tunneling of the electron of the lower continuum to the empty (a localized)state prepared in the upper continuum with (cid:101) < − m can be treated as the tunneling of the virtualpositron (electron hole) with (cid:101) e + = − (cid:101) − i Γ /2 from the region of the well to infinity, where it alreadycan be observed. If one scatters an external real positron with a resonance energy (cid:101) e + (cid:39) − (cid:101) > m onsuch a potential, this positron for a short time forms a resonance quasi-stationary state in the effectivepotential, which after passage of a time ∼ Γ is decayed. As the result, positron goes back to infinity.After that, during a time of the same order of magnitude two positrons, being produced in a fluctuationtogether with two electrons, go off to infinity and those two electrons fill the stationary negative-energystate, as it was explained. niverse , xx , x 5 of 56 If the ground state level was initially occupied by two electrons of opposite spins, then at adiabaticchange of the potential (in the sense clarified above) they remain on this level (cid:101) = (cid:101) . Productionof pairs does not occur. During a time ∼ Γ their charge 2 e < (cid:101) − Γ ( (cid:101) ) /2 < ∼ (cid:101) < ∼ (cid:101) + Γ ( (cid:101) ) /2. This charge is localized at distances ( ∼ ( m ) typicalfor the ground state in the Coulomb field [7]). If the experimenter scatters an external positron with (cid:101) e + (cid:39) − (cid:101) > m on such a potential, the positron annihilates with one of the two electrons haveoccupied the ground-state level. After passage of time ∼ Γ there occurs spontaneous production ofthe one new pair, the electron fills empty state (after that again two electrons occupy the ground-statelevel) and the positron goes to infinity. Semiclassical approximation is one of the most important approximate methods of quantummechanics [31]. Classical and semiclassical ideas are widely used in quantum field theory in problemsdealing with spontaneous vacuum symmetry breaking of bosons, cf. [20,30,32], in condensed matterphysics, cf. [33–35], and in physics of nuclear matter [36,37].As a consequence of the instability of the boson vacuum in a strong external field, there appearsreconstruction of the ground state and there arises condensate of the classical boson field [38,39].Many-particle repulsion of particles in the condensate provides stability of the ground state. Afterthat excitations prove to be stable, cf. [36,37]. They are also successfully described using semiclassicalmethods, e.g. such as the loop expansion [30,40].For fermions there exist two possibilities. In the first situation, fermions heaving attractiveinteraction, being rather close to each other, may form Cooper pairs, cf. [34]. In the second situation,which we focus on here, electron-positron pairs, being produced in a strong static electric field, arewell separated from each other by the potential barrier. As the result, the electric potential attractsparticles of one sign of the charge and repels antiparticles. Due to the Pauli principle each unstablesingle-particle state is occupied by only one fermion. Therefore it is natural to prolong a single-particledescription in a overcritical region (till there appeared still not too many dangerous states). Classicalapproximation does not work for fermions but semiclassical methods prove to be working. As known,semiclassical approach is applicable for calculation of the energy of levels with large quantum numbersand for spatially smooth potentials, when d ˜ λ / dx (cid:28)
1, where ˜ λ = p is the reduced electron DeBroglie length, p is the momentum, x is coordinate. For the Coulomb field for the ground-state levela rough estimate yields d ˜ λ / dr ∼ ( Ze ) for r →
0. However even for d ˜ λ / dx ∼ ∼ π , cf. [31]. Instability of the vacuum near a nucleus heaving a supercritical charge.
It proves to be thatthe semiclassical approximation is applicable with an appropriate accuracy for the description of theelectron energy levels in supercritical field of a nucleus with the supercritical charge Z > ( − ) .Semiclassical approximation allows to find rather simple expressions for the critical value of the charge,cf. Refs. [8,41,42], for energies of deep levels as a function of Z and for probabilities of traversing ofthe barrier between continua, cf. [3–7]. Spontaneous positron production in low-energy heavy-ion collisions.
Comparison of theoryand experiment should check application of QED in region of strong fields outside the applicability ofthe perturbation theory. Description of spontaneous production of positrons in heavy-ion collisionsneeds solution of two-center problem for the Dirac equation. Since variables are not separated inthis case, the problem does not allow for the analytical treatment and numerical calculations arecumbersome. However usage of semiclassical approximation results in simple analytical expressionsfor the energies of the electron levels, cf. [6,7], valid with error less than few %. Thereby, this is onemore example of the efficiency of the semiclassical approach.
Electron condensation in a field of a supercharged nucleus.
In supercritical fields many energylevels cross the boundary of the lower continuum and the problem of the finding of the vacuum niverse , xx , x 6 of 56 charge density becomes of purely many-particle origin. It can be considered within the relativisticThomas-Fermi method, cf. [2]. All initially empty states, which crossed the boundary (cid:101) = − m , arefilled after a while. In this sense one may speak about “electron condensate”. Vacuum polarization and electron condensation at super-short distances from Coulomb center.
In spite of the successes in explanation of all purely electrodynamical phenomena, QED is principallyunsatisfactory theory, since relations between bare mass and charge and observable ones containdiverging integrals [43]. As the result, as one thinks, there is no not contradictive manner to pass fromsuper-short to long distances. In spite of this, as is well known, it is possible to remove divergenciesfrom all observable quantities with the help of the renormalization procedure.One of central problems related to renormalization of the charge is the problem of so-called“zero charge”. Considering square of the charge of electron e ( r ) as a function of the radius r andassuming finite value of the bare charge e ( r ) = e > r →
0, one derives e ( r → ∞ ) → e ( r → ∞ ) → e = n ext = Z δ ( r − r ) for r →
0, cf. [3]. The problem of a distribution of the charge near an external source of the charge withradius R (cid:28) m , as well as the problem of the distribution of the charge of the electron at distances r (cid:28) m are the key principal problems of QED. Semiclassical approach proves to be very promisingin calculation of the vacuum dielectric permittivity in strong inhomogeneous electric fields [44]. Thedensity of the polarized charge is supplemented by the density from electron condensation [3,36]. Theproblem proves to be specific and depends on weather the radius of the external source of the charge islarger than a distance r pole , where dielectric permittivity decreases to zero, or smaller r pole , cf. [45,46].References [45,46] argued for the condensation of electron states in the upper continuum at distanceslarger than r pole for r > r pole and for the condensation of electron states originated in the lowercontinuum at distances smaller than r pole (for r < r pole ), at which the dielectric permittivity proves tobe negative and e <
0. Semiclassical consideration of this problem allows to present arguments infavor of a logical consistency of QED.
Similar effects in semimetals and in stack of graphene layers.
Existence of the Weyl semimetals,i.e. materials with the points in Brillouin zone, where the completely filled valence and completelyempty conduction bands meet with linear dispersion law, (cid:101) = v F p , where the Fermi velocity is v F ∼ − , has been predicted in [47]. Systems with relativistic dispersion law are likely to be realized insome doped silver chalcogenides, pyrochlore iridates and in topological insulator multilayer structures.Weyl semimetals are 3-dimensional analogs of graphene [48], where the energy of excitations is alsoapproximately presented by the linear function of the momentum but the electron subsystem is twodimensional one, whereas the photon subsystem remains three dimensional. Even though the massof excitations m = m (cid:54) =
0, can be induced in many ways [49] resulting in the gapped dispersion relation (cid:101) = p v + m v ,In difference with a small value of the fine structure constant in QED, e = α ef = e / v F ε , where ε is the dielectric permittivity of thesubstance. Coupling constant α ef can be as (cid:28) > ∼
1, depending on the substance, and both weakand strong coupling regimes are experimentally accessible. Thus Weyl semimetals and infinite stack ofgraphene layers make it possible to experimentally study various effects have been considered in 3+1quantum electrodynamics (QED) for weak and effectively strong couplings, cf. [50].Also, electron-positron pair production from the vacuum can be triggered by the laserelectromagnetic fields. However it seems unlikely to realize such a possibility at least in the nearestfuture, cf. [51] and references therein.Below, attention is focused on a semiclassical description. I describe instabilities of the bosonand fermion vacua in static potentials, in particular in the Coulomb field. Then focus is concentratedon the description of the spontaneous positron production in low-energy heavy-ion collisions. Next,many-particle semiclassical description of electron condensation is considered. Finally, modification of niverse , xx , x 7 of 56 the Coulomb field at super-short distances due to vacuum polarization and electron condensation arestudied.The paper is organized as follows. Sect. 2 starts with a brief discussion of instability for thecharged bosons in static electric fields, in particular in the Coulomb field of a point-like nucleus withthe charge Z > Z cr = ( e ) . Behavior of deeply bound electrons obeying the Dirac equation inthe strong static electric fields is considered in Sect. 3. First I consider the case of a one-dimensionalfield and then of a spherically symmetric field. The Dirac equation is transformed to equivalentSchrödinger form in an effective potential and interpretation of the solutions is discussed. Thenin Subsect. 3.5 I demonstrate exact solution of the problem of bound states in the strong Coulombfield of a point-like center. The focus is made on the problem of the falling of the electron to thecenter for a nucleus with the charge Z ≥ e . Subsect. 3.6 describes how the problem is resolvedtaking into account that nuclei have a finite size. In Sect. 4 I introduce a semiclassical approachto the Dirac equation, being transformed to the second-order differential equation. Electron levelscrossed the boundary of the lower continuum are considered. The mean radius of the K-electronshell and the critical charge of the nucleus are found for (cid:101) = − m , as well as the number of levelscrossed the boundary of the lower continuum and their energies. The critical charge of the nucleusfor the muon is also found. Comparison of semiclassical expressions with much more cumbersomeexact expressions permits to understand merits of the semiclassical approach. In Sect. 5 semiclassicalapproximation is developed for the system of linear Dirac equations. Semiclassical wave functionsin classically allowed and forbidden regions are introduced and the Bohr-Sommerfeld quantizationrule is formulated. Next, probability of the positron production is calculated. Then, semiclassicalapproximation is applied to non-central potentials. In Sect. 6 focus is concentrated on problems ofspontaneous positron production in low-energy collisions of heavy ions. Energies of deep levels as afunction of the distance between colliding nuclei and angular distribution of the positron productionare found employing semiclassical approach. Then I consider screening of the charge at collisionsof not fully striped nuclei. Semiclassical approximation (imaginary time method) is adequate todescribe dynamics of the tunneling of electrons from lower continuum to the upper one. In such away a correction on non-adiabaticity to the probability of production of positrons is found. Electroncondensation in the field of a supercharged nucleus is considered in Sect. 7. Effects associated withpolarization of the electron-positron vacuum in weak and strong fields are studied in Sect. 8. Thenin Sect. 9 I focus on description of the charge distribution at super-short distances from the chargesource. Effects of polarization of vacuum and electron condensation in upper and lower continua willbe considered. Sect. 10 contains a conclusion.
2. Relativistic spinless charged particle in static field A µ = ( A , (cid:126) ) Consider a spinless negatively charged boson placed in a stationary attractive potential well V .The Klein-Gordon-Fock equation renders ∆ φ + [( (cid:101) − V ) − m ] φ = E = (cid:101) − m m , U ef = − V − (cid:101) V m , (2)we may rewrite Eq. (1) in the form of the Schrödinger equation, ∆ φ + m ( E − U ef ) φ = niverse , xx , x 8 of 56 Figure 2.
Schematic picture of deformation of the upper and lower continua in a strong external electricfield (the boundaries of the continua are shaded). Electrons belonging to the vacuum shell in uppercontinuum fill the cross-hatched region. The states below the curve (cid:101) − / m = V ( r ) / m − W shows an artificial cutoff energy. As we see from Eq. (2), for relativistic particles there appears an attractive term in the effective potential − V / ( mc ) , even for a purely repulsive potential V . In the limit case E (cid:28) m and | V | (cid:28) m we have (cid:101) (cid:39) m + E and U ef (cid:39) V , and we recover Schrödinger equation for a nonrelativistic particle. For | (cid:101) | < m the “nonrelativistic” energy is E < − m < (cid:101) < m . For a sufficiently deep potential well the energy of the ground state level maycross the boundary (cid:101) = − m . In a deeper potential other levels cross this boundary. For (cid:101) < − m , hereRe E >
0, the levels become quasi-stationary, see Fig. 1.A comment is in order (D. N. Voskresensky 1974, see comment in [52]). For a spinless particleunder consideration, the ground-state single-particle level crosses the boundary (cid:101) = − m only forfar-distant potentials, when − V ( r → ∞ ) > C cr / r , for a constant C cr >
0. For potentials obeyingcondition − V ( r → ∞ ) < C cr / r , there appears a bound state for antiparticle. In both cases for abroad potential well of a typical radius R (cid:29) m the vacuum instability occurs at | V | (cid:39) | V | cr (cid:39) m ( ± O ( ( m R )) either at (cid:101) cr = − m or at (cid:101) cr (cid:39) − m ( − O ( ( m R )) . In case of a broad potentialwell solutions of many-particle problems in both cases are almost the same, cf. [30]. For − V > − V cr there appears production of pairs. Positively charged antiparticles go off to infinity and negativelycharged particles form a condensate, see [30,36].Let us illustrate how deformation of boundaries of upper and lower continua occurs in a staticelectric field forming a broad potential well for a negatively charged particle. To be specific considerspherically symmetric field. Boundaries of continua, (cid:101) ± , are determined by (cid:126) p ( r ) = ( (cid:101) ± − V ) − m = p ( r ) >
0, these are classically allowedregions. In the gap between continua p ( r ) <
0. This is classically forbidden region. For V < V cr = − m − O ( ( m R )) there arises a region of the overlapping of the continua that means that negativelycharged particle may penetrate from the lower continuum (from exterior of the potential well) to theupper one (to interior of the well). niverse , xx , x 9 of 56 With an exponential accuracy the probability of a passage of the one-dimensional barrier isdetermined by W ∼ e − Im S ∼ e − (cid:82) x x | p | dx , (5)where x and x are the turning points at which p ( x ) =
0. This expression is applicable for W (cid:28) eE = −∇ V = const , | eE | (cid:28) m . Then we have p (cid:39) (cid:112) ( (cid:101) + eEx ) − m . From Eq. (5) we immediately obtain W ∼ e − π E / E , E = m . (6)This expression coincides with the first term of the infinite series solution [53].A question arises is it possible to observe a process of production of pairs already in a weakattractive electric field with the strength | E | (cid:28) m at − δ V > m ? Critical difference − δ V (cid:39) − m can be easily reached in the field of the condenser, where ∇ A = const , at increase of the distance d between plates. Employing |∇ A | = | (cid:126) E | ∼ V/cm, the value, which is easily produced inelectrical engineering, we estimate | δ V | > m π already for d > ∼ cm. Here m π (cid:39)
140 MeV is themass of the lightest charged boson, the pion. However the probability of the production of the pairs W ∼ e − Im S , Im S = (cid:82) x x | p | dx , is negligibly small at these conditions. Indeed for V = − eEx we getIm S = (cid:82) x x | p | dx = π E E . for pions E (cid:39) · V/cm. For electrons E (cid:39) · V/cm.
In case of the Coulomb field of a point-like nucleus, V = − Ze / r , with the help of the replacement φ ( (cid:126) r ) = R ( r ) Y lm we obtain equation for the radial wave function R ( r ) in the form ∆ R r + m (cid:20) E + ( Ze ) mr − l ( l + ) mr + (cid:101) Ze mr (cid:21) R = ∆ r = r ∂ ( rR ) ∂ r , (7)where E = (cid:101) − m m is effective nonrelativistic Schrödinger energy of the particle, U ef ( r ) = − ( Ze ) mr + l ( l + ) mr − (cid:101) Ze mr (8)is the effective potential, now depending on l . Eq. (7) and the ordinary Schrödinger equation for theradial function in the effective potential coincide after doing replacements l ( l + ) − ( Ze ) = λ ( λ + ) , (cid:101) Ze / m → Z (cid:48) e (9)in the former one. Thus, instead of the energy of the Schrödinger particle in the Coulomb field wederive E n r , l = − ( Z (cid:48) e ) m ( n r + λ + ) . (10)Here n r + λ + = n + λ − l , n r =
0, 1, ... is the radial quantum number. Solving Eq. (9) and retainingsolution with positive-sign square root, λ = − + (cid:113) ( l + ) − ( Ze ) , because for Z = l = λ =
0, we find the Sommerfeld formula for a spinless particle, (cid:101) n r , l = m + Z e ( n − l − + √ ( l + ) − ( Ze ) ) . (11)There are two square-root solutions of this equation. Solution, which yields (cid:101) → m for Ze (cid:28) n = l =
0, describes negatively charged particle in the attractive Coulomb field ( Z > ) . Solution, niverse , xx , x 10 of 56 which yields (cid:101) → − m for Ze (cid:28) n = l = Z >
0, after change of (cid:101) → − (cid:101) describes positivelycharged particle of the same mass in the field Z <
0, since Eq. (1) does not change under simultaneousreplacement (cid:101) → − (cid:101) and Z → − Z .In the limit Ze (cid:28) n = l = (cid:101) → m − ( Ze ) m n in accordance with the result for the Schrödinger particle.For Z > Z cr = ( e ) the particle, being in the ground state ( n = Ze = + δ for 0 < δ (cid:28)
1. Then choosing positive-sign square root of solution (11) we have for Ze = + δ , (cid:101) (cid:39) m ( + i δ ) √ and the wave function φ ∝ e − i (cid:101) t ∝ e m δ t / √ → ∞ for t → ∞ ,being not normalized, reflecting the fact of the falling of the negatively charged particle to the Coulombcenter with Z > Z < (cid:101) (cid:39) − m alreadyfor small Z >
0. However note that the negative-root solution of Eq. (11) − m ( + i δ ) √ for the negativelycharged particle near the Coulomb center for Z > Z cr = ( e ) yields φ ∝ e − m δ t / √ , i.e., decreasing at t → ∞ . This implies a possibility of a multi-particle interpretation of the (cid:101) < Z >
0. We return to this question in Subsection 9.2.The value Z cr = Z cr =
68, if nuclei were point-like. As we have mentioned, the lightest spinless meson is the pion.The radius of the real nucleus with atomic number A is found from the condition 4 πρ R /3 = A ,where ρ (cid:39) − (cid:39) m π . For a symmetric nucleus A (cid:39) Z we estimate R > a π = ( m π Ze ) (radius of the ground-state orbit for pion) already for Z >
40. Then the lowest pion orbit enters insidethe nucleus and approximation of point-like nucleus becomes invalid.Note that for Z = Z cr , (cid:101) part + (cid:101) a.part = m √ > R (cid:54) = R (cid:28) m π , such that V = − Ze / r for r > R and V = − Ze / R (the model I) or for V = − Ze R ( − r R ) , the model II at r < R , the ground state particle level continues to decrease withincreasing Z and decreasing R and for Z = Z cr ( R ) > Z cr , it reaches (cid:101) = − m . At Z = Z cr ( R ) the sum (cid:101) part + (cid:101) a.part is zero that corresponds to the spontaneous production of the pairs for Z ≥ Z cr ( R ) , at R < R cr . Sommerfeld formula for electron.
Electron has spin 1/2. In absence of the magnetic field spinand orbit spaces are orthogonal. Thus one may expect that expression (11) continues to hold also forelectron after replacement l + → | (cid:126) J | + = | κ | , where κ = −
1, 0, 1... is integer number, sinceaxial vectors of angular momentum and spin are summed up, (cid:126) L → (cid:126) J = (cid:126) L + (cid:126) s . Then we have (cid:101) n r , κ = m + Z e ( n r + √ κ − ( Ze ) ) , (12)where n r = n − | κ | =
0, 1, ... is a radial quantum number. Now falling to the center appears when theground state level reaches the value (cid:101) =
0. It occurs for Z = Z cr = e = R (cid:54) =
0, e.g. for the case V = − Ze / r for r > R and V = − Ze / R for r < R , the ground state levelcontinues to decrease with increasing Z and for Z = Z cr ( R ) > Z cr it reaches (cid:101) = − m . After that thesum (cid:101) part + (cid:101) a.part reaches zero that corresponds to spontaneous production of the electron-positronpairs. Two electrons may occupy the ground-state level and two positrons with − (cid:101) > m go off toinfinity,Note that the same expression (12) is derived from the exact solution of the Dirac equation in theCoulomb field, as we will see in Subsect. 3.5. niverse , xx , x 11 of 56
3. Dirac equation for particle in static electric field, A µ = ( A , (cid:126) ) Now we are at the position to focus on the problem of our main interest in this paper, i.e., todescribe behavior of electrons in a strong static electric field.Interaction with 4-vector field A µ = ( A , (cid:126) A ) is constructed with the help of minimal coupling ( ˆ p + e ˆ A / c − mc ) Ψ = A = γ µ A µ , γ µ are ordinary Dirac matrices. In case of a static one-dimensional electric field ( (cid:126) A =
0) using replacement Ψ = e − i (cid:101) t ˜ ψ ( x ) werewrite Eq. (13) as ( (cid:101) − V + i γ (cid:126) γ ddx − γ m + V ( x )) ˜ ψ ( x ) = ψ ( x ) = G ( x ) χ ± − iF σ x χ ± , χ + = , χ − = , (15)we may rewrite Eq. (14) as ψ (cid:48) = ¯ h − ˆ D ψ , ˆ D = m + (cid:101) − Vm − (cid:101) + V , ψ = GF . (16)For the further convenience we retained here dependence on ¯ h . Introducing ψ jlm = r G ( r ) Ω jlM ( (cid:126) n ) iF ( r ) Ω jl (cid:48) M ( (cid:126) n (cid:48) ) , Ω jl (cid:48) M = − (cid:126) σ (cid:126) n Ω jlM ( (cid:126) n ) , (17)where Ω jlM is the spherical spinor, j , M are full angular momentum and its projection, j = l ± l isorbital angular momentum, l + l (cid:48) = j , (cid:126) n = (cid:126) r / r .After separation of angular and spin variables the Dirac system becomes ψ (cid:48) = ¯ h − ˆ D ψ , ˆ D = − ˜ κ / r m + (cid:101) − Vm − (cid:101) + V ˜ κ / r , ψ = GF , (18)˜ κ = ¯ h κ , | κ | = j + κ = −
1. One-dimensional result, see (16), followsfrom (18) provided one puts κ = d / dr → d / dx . With the help of replacement φ = ( m + (cid:101) − V ) − G , (19)Eq. (18) is reduced to equation of the second-order in r -derivative, similar to the Schrödinger equation, φ (cid:48)(cid:48) + p ( r ) φ = p = m ( E − U ef ( r )) , (20) niverse , xx , x 12 of 56 where E = (cid:101) − m m , U ef ( r ) = (cid:101) Vm − V m + κ ( + κ ) r m + U s , (21) U s = m (cid:34) V (cid:48)(cid:48) m + (cid:101) − V + (cid:18) V (cid:48) m + (cid:101) − V (cid:19) − κ V (cid:48) r ( m + (cid:101) − V ) (cid:35) (22)is the term appeared due to the spin. If U s were zero, after the replacement κ → l we would recoverthe Klein-Gordon-Fock equation for a spinless particle.At r →
0, for V = − Ze / r we have U s → − + κ mr . For 1s level κ = − U s → mr . In the lattercase U ef ( r ) → − ( Ze ) mr + mr (23)for r →
0. The falling to the center in such a Schrödinger potential occurs when U ef ( r ) < − ( mr ) ,cf. [54], that corresponds to Ze > Dirac equation describes electron and positron simultaneously. Therefore at appearance of thebound state in a potential well there arises a question does it relate to the electron or to positron. Asexample consider case of a weak external static central-symmetric electric field produced by a staticsource of a positive charge distributed in a range r . Then V = − ζ v ( r ) < ζ > ζ theDirac equation, as the Klein-Gordon-Fock equation, can be transformed to the Schrödinger equationfor a nonrelativistic particle. The bound state for the electron appears first at a certain value of ζ . Atdecreasing ζ , this state is diluted in the continues spectrum with (cid:101) ≥ m .System of Dirac equations (18) is symmetric in respect to replacements (cid:101) → − (cid:101) , V → − V , κ → − κ , G → F . Equation describing energy levels does not depend on G and F . Thereby it is symmetricrespectively replacements (cid:101) → − (cid:101) , V → − V , κ → − κ . In case of the source of a positive chargethe electron undergoes attraction. In the field of the opposite-sign charge ( V → − V ) the electronundergoes repulsion. Since in the attractive field there appears the electron energy level going from theupper continuum, in the repulsive field there appears the electron energy level originating from thelower continuum. However, since the Dirac equation describes simultaneously electron and positron,if the electron moves in a repulsive field, then positron moves in an attractive one. Thereby the electronlevel going in a repulsive field from the lower continuum can be interpreted as the positron level( (cid:101) → − (cid:101) , κ → − κ ) going from the upper continuum (now in the field of attraction to the positron). Itis natural to think that in a weak repulsive field for the electron for small ζ a deeply bound level with (cid:101) (cid:39) − m should not exist. Since such a state nevertheless exists in the full set of solutions of the Diracequation, it after the replacement (cid:101) → − (cid:101) , κ → − κ should be interpreted as the positron state. Thisinterpretation is confirmed experimentally. In the field of a proton there are electron bound states lyingnear the boundary of the upper continuum but there are no positron states with (cid:101) (cid:39) − m . Vise versa,in the field of an antiproton there exist positron levels with (cid:101) (cid:39) m but there are no electron levels with (cid:101) (cid:39) − m . This picture is established also by minimization of the energy in mentioned cases. Namely, inthe field of a positive charge presence of the bound electron is more energetically favorable comparedto presence of the positron.Statements done above seem obvious except the case, which I shall consider below in Subsect. 9.2,when polarization of the vacuum may result in a negative dielectric permittivity and attraction isreplaced by repulsion. niverse , xx , x 13 of 56 Consider discrete spectrum (cid:101) < m of the Dirac equation in the potential V = − Ze / r . We search G and F in Eq. (18) as G = √ m + (cid:101) e − ˜ r /2 ˜ r g ( Q + Q ) , F = −√ m − (cid:101) e − ˜ r /2 ˜ r g ( Q − Q ) , (24)where ˜ r = r (cid:112) m − (cid:101) , g = (cid:113) κ − ( Ze ) . (25)This form of the solution follows from asymptotic behavior of G , F ∼ r ± g at r → G , F ∼ e − ˜ r /2 at r → ∞ . Solutions G , F ∼ C r − g are dropped (i.e. we put C = ) due to divergence of theircontribution to the probability ( (cid:82) | ψ | dr → ∞ ).Setting (24) in Eq. (18) we obtain system of equations˜ rQ (cid:48) + (cid:16) g − Ze (cid:101) √ m − (cid:101) (cid:17) Q + (cid:16) κ − Ze m √ m − (cid:101) (cid:17) Q = rQ (cid:48) + (cid:16) g + Ze (cid:101) √ m − (cid:101) − ˜ r (cid:17) Q + (cid:16) κ + Ze m √ m − (cid:101) (cid:17) Q = rQ (cid:48)(cid:48) + ( g + − ˜ r ) Q (cid:48) − (cid:16) g − Ze (cid:101) √ m − (cid:101) (cid:17) Q = rQ (cid:48)(cid:48) + ( g + − ˜ r ) Q (cid:48) − (cid:16) g + − Ze (cid:101) √ m − (cid:101) (cid:17) Q = (cid:101) → − (cid:101) and Ze → − Ze .Finite solution for ˜ r → Q = AF (cid:18) g − Ze (cid:101) √ m − (cid:101) , 2 g +
1, ˜ r (cid:19) , Q = BF (cid:18) g + − Ze (cid:101) √ m − (cid:101) , 2 g +
1, ˜ r (cid:19) , (28)where F ( α , β , z ) is the degenerate hypergeometric function. Setting ˜ r = B = − g − Ze (cid:101) √ m − (cid:101) κ − Ze m √ m − (cid:101) A . (29)Both hypergeometrical functions in (28) are reduced to polinomials, otherwise they would grow as e ˜ r for ˜ r → ∞ , that results in divergence of the probability. From this requirement follows that α in F ( α , β , z ) equals to a non-positive integer number, i.e., g − Ze (cid:101) √ m − (cid:101) = − n r , n r =
1, 2, ... (30)For n r = g = Ze (cid:101) √ m − (cid:101) and Ze m √ m − (cid:101) = | κ | .If κ <
0, then B = Q = κ >
0, then B = − A and Q is a divergent function at n r =
0. Thereby permitted are n r =
0, 1, ... for κ < n r =
1, 2, ...for κ >
0. From (30) also follows solution for negatively charged particle with (cid:101) < Z <
0. Ina single particle problem under consideration one should drop such a solution, since it describesa strongly bound particle already in a weak field. However such a solution can be appropriatelytreated within a many-particle picture with taking into account vacuum polarization and electroncondensation originated in lower continuum, as we argue below in Subsect. 9.2. niverse , xx , x 14 of 56 From (30) we obtain Sommerfeld expression (cid:101) = ± m (cid:34) + ( Ze ) ( (cid:112) κ − ( Ze ) + n r ) (cid:35) − , (31)cf. Eq. (12). Note that for Z > Z < e , only solution (cid:101) > n r + g > + ” sign solutions (31) correspond to particles (electrons) in the field of positively chargedCoulomb center (or to antiparticles – positrons in the field of negatively charged Coulomb center). The“ − ” sign solutions (31), after replacements (cid:101) → − (cid:101) , κ → − κ (after that “ − ” sign branch coincides with“ + ” sign branch) describe antiparticles with (cid:101) > Z < Z > κ = − n r =
0. Its energy is (cid:101) = mg , g = (cid:113) − ( Ze ) . (32)At Ze ≥ r → G = a r g + a r − g , F = b r g + b r − g . (33)For Ze = + δ > g = i √ δ becomes imaginary and solutions oscillate as C cos ( | g | ln r ) + C sin ( | g | ln r ) , (34)that corresponds to not normalized probability (cid:82) ∞ | ψ | dr . At Ze = + δ , 0 < δ (cid:28)
1, solution ofEq. (32) yields (cid:101) = + im √ δ and the electron wave function grows as Ψ ∝ e + m √ δ t indicating fallingof the electron to the center. Solution of opposite sign (see Eq. (31)) rises from negative continuumat V →
0. In single-particle problem a negative-energy solution should be dropped. Note that at Ze = + δ , it yields (cid:101) = − im √ δ and Ψ → t → ∞ that may suggest an interpretation. Howeveran appropriate interpretation proves to be possible only beyond the single-particle problem, as will beshown in Subsect. 9.2.Solutions (31) and (32) hold formally for the positron in the Coulomb potential of the nucleus withthe charge Z <
0. Within the single-particle problem under consideration, appropriate interpretationagain exists for the solution, which energy is originated from the upper continuum decreasing withincreasing − Z , rather than negative-energy solution, similarly to that happened for the electron at Z >
0. Only two (due to Pauli principle) electrons, if they have occupied the ground state, undergofalling to the Coulomb center for Ze =
1. For levels with quantum number n r > (cid:101) n r , κ > Z = e . Now assume that the ground-state level was empty and we adiabatically increase Z .There is no appropriate solution of the single-particle problem for the point-like nucleus with Z > e in this case. Avoiding problem of falling to the center.
A reasonable interpretation may appear, only ifone assumes that the nucleus has a size R (cid:54) = R . Assume first that R (cid:28) r Λ = m . In the limit Λ = ln ( r Λ / R ) (cid:29) (cid:101) ( ζ < ) = mg /th ( Λ g ) , for ζ = Ze < ζ < ( Λ g ) (cid:39) + e − Λ g rapidly tends to unity and Eq. (35) coincides with (32). For R (cid:54) = ζ = (cid:101) ( ζ ) . Eq. (35) is analyticallycontinued in the region ζ >
1. For ζ close to unity we have (cid:101) ( ζ > ) = m ˜ g /tg ( Λ ˜ g ) , for ζ = Ze > niverse , xx , x 15 of 56 where ˜ g = (cid:112) ζ −
1. One gets d (cid:101) ( ζ ) / d ζ → ∞ for R →
0. At ˜ g = π / Λ , Eq. (35) has a spurious pole.At fixed R the curve (cid:101) ( ζ > ) continues to decrease with increasing ζ and reaches the boundary ofthe lower continuum. It occurs at ζ cr = + π / ( Λ ) + O ( Λ − ) .A comment is in order. Single-particle solution for R → R assmall as R ∼ r L (cid:39) r Λ e − π / ( e ) , the multi-particle effects of the polarization of the vacuum should beincluded and the problem goes beyond the single-particle one, see below consideration in Sect. 8. For the Coulomb field with the charge Z < e , the electron in the ground state is typicallysituated at distances ∼ a = ( Z obs e m ) > m and distribution of the charge Z ( r ) at distances r ∼ R nucl (cid:28) a almost does not affect the electron motion. In the realistic problem the nucleus has afinite size, R nucl (cid:39) r N A (cid:28) a , where A is atomic number, r N (cid:39) r < R nucl . The falling to the centrum does not occur, as it has been mentioned. Even for Z (cid:29) e the electron density remains to be distributed at finite distances.With taking into account of distribution of the charge inside the nucleus we have V ( r ) = − ζ f ( r / R nucl ) / R nucl for 0 < r < R nucl , V = − ζ / r , for r > R nucl . (37)Two models have been employed in the literature: model I, when f ( x < ) = f ( x < ) = ( − x ) /2 that describesdistribution of protons with the constant volume density.The energy shift of the electron level can be found with the help of the perturbation theory appliedto the Dirac system (18). Following [15], β = ∂(cid:101)∂ζ = (cid:90) V ( r )( G + F ) dr / ζ < (cid:101) ( ζ ) decreases monotonically with increasing ζ and crosses the boundary of the lowercontinuum with a finite value β . After that (cid:101) ( ζ ) acquires an exponentially small imaginary part.Since exact solution of the Coulomb problem for r > R looks rather cumbersome and for r < R is impossible for a realistic cut of the potential, it is natural to use approximate methods. Mosteconomical is semiclassical approach. Here we should notice that the replacement (19) becomessingular for (cid:101) < − m in the point V ( r ) = m + (cid:101) <
0. Due to this the effective potential U ef ( r , (cid:101) ) = m ( r − r ) − + ... → ∞ , for r → r , (39)and semiclassical expressions loose their sense due to divergency of the integral (cid:82) r m ( E − U ef ( r , (cid:101) )) dr . However this is only a formal problem since initial Dirac system (18) has no singularityat r → r . To avoid the problem one should bypass the singular point in the complex plane, as oneusually does bypassing turning points, or one may apply semiclassical consideration straight to thelinear Dirac equations. Note that in the one-dimensional case corresponding to κ =
0, see Eq. (16), thementioned singularity occurs in the turning points and one may use standard semiclassical methods.Probability of the spontaneous production of positrons is determined by the width of thecorresponding electron level, Im (cid:101) , for Re (cid:101) < − m . Thus the width is found from the solution ofthe Dirac equation. The value Γ , which determines probability of the positron production, W ∼ e Γ t ,can be expressed directly through components of the Dirac bispinor ( G and F ). It yields the flux ofparticles going to infinity (at normalization on one particle): Γ = (cid:90) ψ † γ (cid:126) γψ d (cid:126) f = ( FG ∗ ) | r → ∞ . (40) niverse , xx , x 16 of 56
4. Semiclassical approach to Dirac equation transformed to second-order differential equation
Substituting ψ = Ae iS /¯ h , where A and S are real quantities, in equation¯ h ∆ ψ + p ( r ) ψ = h ∆ A + p A = A ( ∇ S ) , i ¯ h ( ∇ A ∇ S + A ∆ S ) = h is here recovered. The Hamilton-Jacobi equation for the action ( ∇ S ) = p is obtained provided¯ h A (cid:48)(cid:48) p A ∼ ¯ h ( pl ) ∼ (cid:18) d ˜ λ dr (cid:19) (cid:28) λ = ¯ hp , (43)where l is typical size of the potential V . For the Coulomb potential at typical distances r ∼ ( m ) characterising ground-state electron with (cid:101) (cid:39) − m we have p ∼ ˜ g / r . From estimate (43) we see thatsemiclassical approximation for the wave function for such distances is accurate up to terms 1/ ˜ g ,˜ g = (cid:112) ζ − κ for ζ > | κ | .Using the Bohr-Sommerfeld quantization rule we have¯ h ( pl ) ∼ ¯ h ( (cid:82) r − r dr ) ∼ π ( n r + γ ) , (44)where the phase γ ∼ n r =
0, 1, ..., r and r − are the turning points separating the classically allowedregion. Thus even in calculation of the energy of the levels with small quantum numbers one mayconsider on the error not larger that 10% .Finally let us notice that the transition from the Dirac equation in the external field to thecorresponding more simple Hamilton-Jacobi equation has been used in many investigations, cf.[55–57]. The case of deep levels with binding energy > ∼ m was studied in [3–7]. In the field V = − ζ / r for ζ < | κ | semiclassical method results in exact expression for the energyspectrum. Let us show this. For that we do replacements G = (cid:114) m + (cid:101) r ( χ + χ ) , F = (cid:114) m − (cid:101) r ( χ − χ ) . (45)Then the system of two Dirac equations (18) reduces to equations χ (cid:48)(cid:48) i + p i ( r ) χ i = i =
1, 2 , (46)with p i ( r ) = (cid:34) (cid:101) − m − (cid:101)ζ ± √ m − (cid:101) r + ζ − κ + r (cid:35) . (47) niverse , xx , x 17 of 56 Adding the Langer correction to the effective potential that results in replacements p i → p ∗ i we find p ∗ i ( r ) = (cid:113) − a + b / r − g / r , a = m − (cid:101) , b = (cid:101)ζ ± (cid:112) m − (cid:101) , g = κ − ζ . (48)Then applying the Bohr-Sommerfeld quantization rule we have (cid:90) r − r p ∗ i dr = ( a − b − g ) π = ( n r + ) π . (49)From here we recover the exact result (31). To get (31) from exact solution of the Dirac equationswe have performed a cumbersome analysis of hypergeometric functions, whereas the semiclassicalapproach needs taking only one simple integral.After replacements b → (cid:101)ζ , g → ( l + ) − ζ , Eq. (48) is valid also for spinless bosons.Performing integration leads us to the exact expression (11). Certainly it is possible to apply semiclassical approach also to Eq. (20) with effective potential inthe form (21), (22). In the range, where the parameter of applicability of semiclassical approximationis ∼
1, usage of Dirac equations presented in different forms leads to slightly different results. Forinstance applying (20) to the Coulomb field does not yield the exact result for the energy of the levelsalthough accuracy of the approximation proves to be appropriate. For (cid:101) < − m replacement (19) leadsto the singularity in the point r where V ( r ) = m + (cid:101) < (cid:82) [ m ( E − U ef )] dr . However, asit was mentioned, this circumstance is not reflected on the calculation of the energy levels, since r issituated under the barrier, where wave functions prove to be exponentially small.Electron energy levels can be found with the help of the Bohr-Sommerfeld quantization rule [3]applied to the Dirac equation presented in the form (20) with effective potential in the form (21), (22).We have (cid:90) r − r p ∗ dr = ( n r + γ (cid:48) ) π . (50)Value p ∗ is obtained from expression (20) after taking into account the Langer correction, i.e. afterdoing the replacement κ ( + κ ) / r → ( κ + ) / r in the expression for the effective potential. Thevalue of the phase γ (cid:48) depends on whether the turning point is inside the nucleus or outside it. In thelatter case the potential is V = − ζ / r and γ (cid:48) = κ = − γ (cid:48) = κ (cid:54) = − r − < r < r + : W = (cid:90) r + r − ( G + F ) dr . (51)To be specific let us put (cid:101) = − m , and consider ζ (cid:29) | κ | . The wave function in the classically allowedregion is [31]: χ = ( c / (cid:112) p ∗ ) sin ( (cid:90) rr p ∗ dr + π /4 ) dr . (52)Constant c is found from the normalization condition [2],2 (cid:90) r − r ( (cid:101) − V ) χ dr / m (cid:39) niverse , xx , x 18 of 56 Then we expand the effective potential (21) near the turning point. For V = − ζ / r we obtain U ef = ζ / r − ˜ g / ( r m ) = U ( r − ) + m ( r − r − ) / ζ + ..., ˜ g = (cid:113) ζ − κ , (53)for r − r − (cid:28) r − ∼ ζ . Solution of Eq. (20) in potential (53) is expressed through the Airy function χ ( r ) = ( − V ) − G = c Ai ( ζ − ( r − r − )) . (54)The probability to find the particle in the sub-barrier region is W = − (cid:90) ∞ r − ( V / m + ) χ dr (cid:39) c (cid:90) ∞ Ai ( x ζ − ) dx = c ζ − , (55)where c = Γ ( ) /16 π (cid:39) ζ ∼ ζ . This justifies that we neglected the contribution ofthe region r > r − at normalization of the wave functions (taking r < r < r − ). Note that quantizationrule remains applicable with a larger accuracy, 1/ ζ , since at its derivation it was not used how wavefunctions are normalized. Strictly speaking, in case of quasistationary levels the quantization rule isslightly modified, due to Im (cid:101) (cid:54) =
0, cf. [58]. However changes of the energy levels are exponentiallysmall due to exponential smallness of the penetrability of the barrier.With the semiclassical χ function we obtain expression for the averages r λ , For (cid:101) = − m and ζ (cid:29) | κ | one has [3], r λ = ζ λ π ( λ + ) Γ ( λ + ) m λ λ + Γ ( λ + ) (cid:20) − ( λ + ) κ ( λ + ) ζ + ... (cid:21) . (56) Γ ( x ) is the Euler Γ -function. For ζ ∼ ζ .The quantity r characterizes mean radius of the bound state at (cid:101) = − m , values r λ at λ = Z cr due to a screening of the charge by otherelectrons of the ion (if they are), see below in Subsect. 6.4. Comparison of the semiclassical expressionswith the exact solutions found numerically shows an appropriate accuracy of the semiclassical resultseven for ζ ∼ | κ | ∼
1. For ζ (cid:29) κ (cid:29) Let us calculate the critical charge of the nucleus (when the electron level with quantum numbers n , κ reaches (cid:101) = − m ). Using the Bohr-Sommerfeld quantization rule in the form (50) one obtains, cf.[42], mR nucl = ˜ g / ( ζ ch y ) , (57)where y is positive root of the equation y − th y = ( n r + γ ) π − ˜ γ g , ˜ γ = arcctg ( Ξ / ˜ g ) , (58) n r =
0, 1, ... radial quantum number, γ = ns levels and γ = κ (cid:54) = − Ξ was found from matching of the exact solution inside the nucleus andsemiclassical one outside the nucleus. As was shown in [8], usage of the semiclassical solutions bothinside and outside the nucleus does not spoil the accuracy of the result. Therefore we further followconsideration of [8]. niverse , xx , x 19 of 56 For the model I, the semiclassical solution inside the nucleus coincides with the exact one and wefind Ξ = β ctg β β = (cid:113) ζ ( ζ − R nucl m ) . (59)Note here that a first estimate of R cr in this model was performed in [41], where it was taken ˜ γ = ζ ,that differs from that follows from (58), (59).For the model II, analytical expression can be found expanding p ( r < R ) in the parameter ζ ,˜ γ = arcctg [(( p ∗ ( R nucl ) / ( ˜ gm )) ctg (cid:90) R nucl p ∗ dr ] , (60) (cid:82) R nucl p ∗ dr = (cid:82) dx [ ζ f ( x ) − ζ R nucl m f ( x ) − ( f ( x ))] = ζ (cid:20) − c ζ − R nucl m ζ + O (cid:18) ζ , R m ζ (cid:19)(cid:21) , c = (cid:16) + √ arth √ (cid:17) , κ = − f ( x ) follows Eq. (37), here for the model II. Although parameter of applicability of semiclassicalexpressions to the Coulomb field is ˜ g (cid:29)
1, difference of the above obtained expression with the resultof the exact calculation is less than few percents even at ζ = ζ cr (cid:39) ζ ∼ y and dropping numerically small term e − y , from Eq. (57) wefinally find R nucl (cid:39) g ζ cr m (cid:20) exp (cid:18) π ( n r + γ ) − ˜ γ ˜ g cr + (cid:19) + (cid:21) − . (62)from where we find Z cr ( R nucl ) . Now let us find the number of levels n κ with fixed quantum number κ and the total number oflevels N , which have crossed the boundary (cid:101) = − m . For this aim [5] we need to use Bohr-Sommerfeldquantization rule at (cid:101) = − m . For ˜ g (cid:29)
1, we have d ˜ λ / dr (cid:28)
1. For ζ (cid:29) ζ − | κ | (cid:29) ζ − ,i.e., semiclassical approximation can be violated only for states with the momenta at which ζ − | κ | < ∼ ζ − . Accuracy of the semiclassical expressions for the wave function is ∼ ζ , cf. [2]. With takinginto account these approximations employing the Bohr-Sommerfeld quantization rule we obtain n κ = π (cid:90) ( V + Vm − κ / r ) dr . (63)For the potential given by Eq. (37), for R nucl m (cid:28) n κ = g π [ ( Arth (cid:112) − η − (cid:112) − η ) + h ( ρ ) , (64)where ρ = | κ | / ζ , η = R nucl / r − = R nucl m / ( ζ ( − ρ )) , r − is the turning point in the effectivepotential, h ( ρ ) takes into account integral over the interior region of the nucleus 0 < r < R nucl , h ( ρ ) = ( − ρ ) − (cid:90) x [ f ( x ) − ρ x − ] dx , (65)where x = x ( ρ ) is the root of equation x f ( x ) = ρ .For Ze (cid:28) (cid:101) < − m of the supercritical atom over the momenta j = | κ | − j corresponds to r − = R nucl , η = κ max = ζ − R nucl m + O ( R m / ζ ) . (66) niverse , xx , x 20 of 56 Figure 3.
Number of levels with (cid:101) < − m for the potential of the model II. The stepwise broken linerepresents a numerical solution of the Dirac equation, while the curve Q was computed according tothe semiclassical Eq. (68). Total number of levels with (cid:101) < − m , N = ∑ κ n κ , (67)can be found by replacing the summation by the integration. We should take into account that in theDirac equation | κ | ≥
1. Thereby we sill should subtract spurious term κ =
0. Thus N = (cid:82) dr [ ( V + Vm ) r − π ( V + Vm ) ]= A ζ ln ζ R nucl m + A ζ + A ζ ln ζ R nucl m + A ζ + A + ..., (68)where A = A = (cid:82) f ( x ) xdx − ln 2 − A = − π , A = − π ( (cid:82) f ( x ) dx + ln 2 − ) .For the model II, result of this calculation is shown in Fig. 3. We again observe excellent accuracyof the semiclassical result, even for ζ ∼ (cid:101) < − m | (cid:101) | − m (cid:28) m Expand effective potential in m + (cid:101) , cf. [3]: U ef ( r , (cid:101) ) = ∞ ∑ n = ( m + (cid:101) ) n u n ( r ) , (69)where U ef ( r , (cid:101) ) can be taken following Eq. (21). Here u ( r ) = U ef ( r , (cid:101) = − m ) . For n ≥ u n = Vm n δ n + mV n (cid:34) − V (cid:48)(cid:48) V + ( n + ) (cid:18) V (cid:48) V (cid:19) + κ V (cid:48) rV (cid:35) , (70)where δ n is the Kronecker symbol. niverse , xx , x 21 of 56 Energy of the levels is found from the Bohr-Sommerfeld quantization condition (cid:90) r − (cid:112) − mu dr + ( m + (cid:101) ) (cid:90) r − (cid:112) − u dr + O (( + (cid:101) / m ) ) = ( n r + γ (cid:48) ) π . (71)As before, γ = κ = − γ = κ (cid:54) = −
1. With the help of (71) we find (cid:101) = − m + β ( ζ cr − ζ ) + ... , (72) β = f / f , f = (cid:90) r − (cid:112) − m u dr , f = (cid:90) r − ζ f / ( mR nucl ) − f √− u / m R nucl dr .Comparison of numerical calculation done following these expressions with that for the exact Diracequation again shows a good agreement. Note that the value β determines the threshold behavior ofthe probability of the production of positrons.4.6.2. Energy spectrum for | (cid:101) | (cid:28) − m This spectrum has been found in [5]. For ζ (cid:29) ζ cr many levels have energies | (cid:101) | (cid:28) − m . In thiscase, as follows from Eq. (21), (22), the terms ∝ κ in centrifugal potential and in spin term cancel eachother. Approximately we have p ∗ ( r ) (cid:39) [( (cid:101) − V ) − κ / r ] . (73)For k = √ (cid:101) − m < ζ / R nucl the turning point r − lies outside the nucleus, r − > R nucl . Employingthe Bohr-Sommerfeld quantization condition we get k n (cid:39) | (cid:101) n | = c ζ R − e − n π / ζ = ζ R − e − ( n − n ∗ ) π / ζ , n > n ∗ , (74) c = exp ( (cid:82) f ( x ) dx − ) , n ∗ = ζπ − ( (cid:82) f ( x ) dx − ) .For deeper levels, k > ζ R − , classically permitted region r < r < r − is completely inside the nucleus.Thereby the spectrum is entirely determined by the expression for f ( x ) : k n = ζ R − f ( Ξ n ) , 1 (cid:28) n (cid:28) n ∗ , (75)where Ξ n is the root of equation (cid:90) Ξ f ( x ) dx − Ξ f ( Ξ ) = n π / ζ . (76)For example, for the model II at 1 (cid:28) n (cid:28) n ∗ we have k n = ζ R nucl (cid:104) − ( n / n ∗ ) (cid:105) , n ∗ = ζ π . (77)From these expressions it is easy to find expression for the level density dn / d (cid:101) . For model II wefind dn / d (cid:101) = Cy − , for 0 < y < dn / d (cid:101) = C ( − y ) , for , 1 < y < y = kR nucl / ζ , C = const . From here we see the crowding of levels toward the boundary (cid:101) = − m ( k → niverse , xx , x 22 of 56 For levels with arbitrary angular momenta the “Coulomb” part of the spectrum gets the form (cid:101) n κ = − ζ R − c ( ρ ) exp ( − n π / ˜ g ) , (80)where ρ = | κ | / ζ , 0 < ρ <
1. Pre-exponential factor c ( ρ ) = exp (cid:104) ln ( ( e ρ ) − ( − ρ )) − ( − ρ ) − Arth ( − ρ ) + h ( ρ ) (cid:105) , (81)where h ( ρ ) given by Eq. (65) depends on the f ( x ) , e = c ( ρ ) monotonically decreases with increase of ρ from 1 for model I and from (cid:39) ρ = ρ = r − lies inside the nucleus. Condition ofapplicability of Eq. (80) is ˜ g / π (cid:28) n < n κ . Since n κ (cid:39) ( ˜ g / π ) ln ( ζ / R nucl ) , then due to large values ofthe logarithm this equation describes most of levels crossed the boundary (cid:101) < − m .The exponential dependence of (cid:101) n on n and crowding of levels near (cid:101) = − m , as follows fromEqs. (74), (80), are related to the fact that U ef (cid:39) − ˜ g / r for r →
0. If R was zero, the electrons wouldcollapse to the center. The spectrum of the Schrödinger equation in such a potential behaves as [59], E n = E e − π n / ˜ g , (82)where E is the energy of the lowest level. In our case E (cid:39) (cid:101) /2 m and thereby we recover Eq. (80) for c ( ρ ) = Since following Dirac the process of production of e − e + pairs can be treated as the penetrationof electrons of the lower continuum into the upper continuum through classically forbidden region( p < G and F for r → ∞ . This single-particle picture is distortedwith deepening of the level and with increase of the number of levels crossed the boundary (cid:101) = − m .We may use Eqs. (20), (21), (22) with taking into account the Langer correction, which improvesapplication of semiclassical expressions.In the threshold region of positron energies setting (cid:101) (cid:39) − m in the expression for the spin term U s ,we obtain p ∗ ( r ) (cid:39) ( (cid:101) − V ) − m − κ / r , (83)cf. with Eq. (73) we have used for description of the very deep levels. In case of the Coulomb field V = − ζ / r , replacing (83) in (5) we obtain W ∼ exp (cid:34) − πζ (cid:32) ( m + k ) k − ( − ρ ) (cid:33)(cid:35) , ρ = κ / ζ , k = (cid:112) (cid:101) − m (cid:28) m , (84)that coincides with asymptotic of exact solution of the Coulomb problem. For the electron one has R nucl (cid:28) m , since 1/ m (cid:39)
386 fm and R nucl (cid:39) r A (cid:39) A / m π , m π (cid:39) m . For muon R nucl (cid:29) m µ , m µ (cid:39) m e .In order to find critical charge for the muon, ζ µ cr , when µ − level reaches (cid:101) = − m µ , we continue toapply semiclassical approximation. For the model I, the turning point lies outside the nucleus. Let us niverse , xx , x 23 of 56 expand U ef ( r , (cid:101) ) near the turning point. Using Eq. (54), after the replacement r − → r , and matchingsolutions G (cid:48) / G at r = R nucl we find [3]:Ai (cid:48) ( ) Ai ( ) ζ (cid:39) β ctg β m µ R nucl , β = (cid:113) ζ ( ζ − R nucl m µ ) (85)for the ns level. From here follows ζ µ cr (cid:39) Rm µ + ( n π ) R nucl m µ ( + a ( R nucl m µ ) − + ... ) , a = − − π Γ ( ) , (86)that coincides with expression, which follows from direct solution of the Dirac equation at (cid:101) = − m µ .In the model II we obtain ζ µ cr (cid:39) Z µ cr (cid:39) Z µ cr (cid:39)
5. Semiclassical approximation to system of linear Dirac equations
Let us apply semiclassical expansion to Eq. (18), cf. [7]. Parameter of expansion ˜ λ / l is ∝ ¯ h , where l is the typical length for the change of the potential. We present ψ = φ e (cid:82) ydr , (87) y ( r ) = h y − ( r ) + y ( r ) + ... , φ = ∞ ∑ n = ¯ h n φ ( n ) , (88)and arrive at the chain of equations for y n and φ ( n ) : ( ˆ D − y − ) φ ( ) = ( ˆ D − y − ) φ ( ) = φ ( ) (cid:48) r + y φ ( ) , ... (89)Usually one restricts expansion by consideration of first two terms. Since semiclassical series isasymptotic one, retaining of too many terms may worsen convergence of the series to the exactsolution.In order the system of homogeneous equations (89) to have nontrivial solution, y − ( r ) should bean eigenvalue and φ ( ) ≡ φ i , i =
1, 2, the eigenfunction of one of two-component eigenvectors of thematrix ˆ D ( r ) . From the condition det ˆ D = y − ≡ λ i = ± i (cid:113) ( (cid:101) − V ) − m − ˜ κ / r ≡ ± q . (90)Replacing y − back to Eq. (89) we obtain φ i = A m + (cid:101) − V λ i + κ / r = A λ i − κ / rm − (cid:101) + V (91)where A and A are normalization constants.Since the matrix ˆ D is not symmetrical, besides the right-hand eigenvectors φ i we should introducethe left-hand eigenvectors ˜ φ i : ( ˆ D − λ i ) φ i = ˜ φ i ( ˆ D − λ i ) = φ i = A ( m − (cid:101) + V , λ i + κ / r ) = A ( λ i − κ / r , m − (cid:101) − V ) . niverse , xx , x 24 of 56 Note that left eigenvectors do not coincide with transposed right eigenvectors ( ˜ φ i (cid:54) = φ Ti ) and left-handand right-hand vectors are mutually orthogonal, ( ˜ φ i , φ j ) = ∑ α = ( ˜ φ i ) α ( φ j ) α ∼ δ ij . (93)To determine y let in Eq. (89) put φ ( ) = φ i and multiply both sides of equation from the left by ˜ φ i . Asfollows from the first equation (92), the term with φ ( ) vanishes, and we obtain y = − ( ˜ φ i , φ (cid:48) i ) / ( ˜ φ i , φ i ) . (94)Further calculations entail no difficulty, cf. [7,60]. The resulting wave functions of thequasistationary state with energy (cid:101) < − m in the region of classically allowed motion r < r < r − havethe form: G = C (cid:104) (cid:101) + m − Vp (cid:105) sin θ , F = sgn κ · C (cid:104) (cid:101) − m − Vp (cid:105) sin θ , (95) p = − iq = (cid:113) ( (cid:101) − V ) − m − κ r , θ = (cid:82) r r ( p + κ wpr ) dr + π /4 , θ = (cid:82) r r ( p + κ ˜ wpr ) dr + π /4 , w = (cid:16) V (cid:48) m + (cid:101) − V − r (cid:17) , ˜ w = (cid:16) V (cid:48) m − (cid:101) + V + r (cid:17) .Here C is normalization constant. As it was discussed, semiclassical wave functions can be normalizedneglecting penetration of the particle into the classically forbidden regions r < r and r > r − , i.e. (cid:82) r − r ( G + F ) dr =
1. Thus we find C = (cid:20) (cid:90) r − r (cid:101) − Vp dr (cid:21) − = (cid:18) T (cid:19) , (96)where T is the period of the particle motion in classically allowed region.In sub-barrier region r − < r < r + , where p < p = iq and q , y − and y are real, wave functionsattenuate exponentially with increasing r . Resulting expressions have different forms in dependenceon the sign of κ . For κ <
0, i.e. for κ = −
1, we have ψ = GF = C − ( Qq ) − exp (cid:20) − (cid:90) rr − (cid:18) q − V (cid:48) m Qq (cid:19) dr (cid:21) m + (cid:101) − V − Q (97)with Q = q − κ / r .For κ >
0, we have ψ = C + ( Qq ) − exp (cid:20) − (cid:90) rr − (cid:18) q + V (cid:48) m Qq (cid:19) dr (cid:21) − Qm − (cid:101) + V (98)with Q = q + κ / r , C ± are normalization constants.In the region r > r + quasistationary state describes outgoing positron and represents a divergingwave. For κ < ψ = iC − ( Pp ) − exp (cid:20) − (cid:90) rr + (cid:18) ip − V (cid:48) m Pp (cid:19) dr (cid:21) m + (cid:101) − ViP (99)with P = p − i κ / r . The flux of particles going off to infinity is then given by Γ = lim Im ( F ∗ G ) at r → ∞ . niverse , xx , x 25 of 56 For κ > ψ = iC + ( Pp ) − exp (cid:20) − (cid:90) rr + (cid:18) ip + V (cid:48) m Pp (cid:19) dr (cid:21) iPm − (cid:101) + V (100)with P = p + i κ / r . C ± are normalization constants.The obtained formulas are valid for all r except regions δ r ∝ ζ near the turning points. Theusual procedure is employed to match semiclassical solutions. The solution is either expressed interms of an Airy function or one may use the Zwaan’s method. As the result, we have C ± = − iC ± = − sgn κ C (cid:34) | κ | mr − + ( κ + r − m ) (cid:35) − sgn κ /2 exp (cid:20) − (cid:90) r + r − (cid:18) q + sgn κ V (cid:48) m Qq (cid:19) dr (cid:21) .(101)Note that the effective potential, which we have used in (20), can be presented employing function w appeared in (95): U ef = − V m + (cid:101) Vm + κ ( κ + ) r m − κ rm w + m ( w (cid:48) + w + wr ) . (102)The terms in Eq. (102), which contain the function w , are due to the electron spin. For | V | (cid:29) m they aresmall compared to the first three terms. Then the expression for the effective potential takes the sameform as for a scalar particle. At the turning points r − and r + the effective potential is not singular.The action becomes S = (cid:90) r dr (cid:113) m ( E − U ef ) = (cid:90) r dr (cid:20) p + κ wpr − m ( w (cid:48) + w + wr ) (cid:21) . (103)Expanding S in 1/ ζ (cid:28) S = (cid:90) r dr (cid:20) p + κ wpr + O ( ζ ) (cid:21) , (104)that coincides with Eqs. (95) – (100), which we have derived in this section.Using Eqs. (95)–(100) for (cid:101) = − m we obtain r = ( ζ − κ + )( ζ + κ /3 − κ /3 + ) ζ ( ζ + ( κ − κ /2 + ) /2 ) , (105)This expression yields r = m for the ground-state level, whereas exact result gives 0.303/ m . For ζ (cid:29) | κ | (cid:29) To be specific consider case κ < (cid:101) = (cid:101) − m and the variable ˜ q = ( q + κ / r ) (cid:39) q + κ qr let us transform the factor in exponent(97) as (cid:90) rr − dr (cid:18) q − V (cid:48) m Qq (cid:19) = (cid:90) rr − dr (cid:20) ˜ q − q ( V (cid:48) / Q + κ / r ) (cid:21) = (cid:90) rr − dr (cid:20) ˜ q − ( ln Q ) (cid:48) (cid:21) , (106)where Q = q − κ / r . The latter term in the integral cancels with the pre-exponential factor Q − . Nowlet us take into account that κ ( + κ ) = l ( + l ) . a Then we have˜ q ( r ) = (cid:104) m ( − ˜ (cid:101) + V ( r ) − l ( l + ) / ( mr ) (cid:105) , G ( r ) = C ˜ q ( r ) exp ( − (cid:90) rr − dr ˜ q ( r ) , (107) niverse , xx , x 26 of 56 where C = const that reproduces the Schr ¨dinger wave function in this region. Note that ˜ q ( r ) entersnot κ = ∓ ( j + ) but orbital moment l . We formally considered case κ < κ > κ (cid:54) = − q ( r ) the Langer correction. From (97), (98) we derive [7], (cid:90) r − r dr ( p + κ wpr ) = ( n + γ (cid:48) ) . (108)As we have mentioned, value γ (cid:48) depends on the fact does r lie inside the nucleus or outside it. In thelatter case γ (cid:48) = κ (cid:54) = − γ (cid:48) = κ = − (cid:101) n κ . It differs from ordinary Bohr-Sommerfeld ruleused in nonrelativistic quantum mechanics by expression for relattivistic momentum p ( r ) and bythe term ∝ w appeared due to the spin-orbital interaction. Account of the term ∝ w is legitimatewithin semiclassical scheme. Let us show it on example of the Coulomb field V = − ζ / r . Then w ( r ) = − m + (cid:101) ( ζ +( m + (cid:101) ) r ) and p ( r ) is determined by Eq. (95). For r < r < r − the momentum p ( r ) ∼ ˜ g / r and the ratio | κ wp r | ∼ | κ ˜ g − rw | ∼ | κ | / ζ for deep levels. Since semiclassical approximation for wavefunctions is valid up to 1/ ζ , in case of deep levels | (cid:101) | (cid:29) m the second term in the integral (108) shouldbe retained for | κ | (cid:29) | κ | ∼
1. For (cid:101) = − m , we have w = Let us calculate probability of spontaneous production of positrons, Γ = − (cid:101) . Replacing (99),(100) in (40) we find Γ = Γ e − (cid:82) r + r − q ( r ) dr , Γ = T − e κ Pr (cid:82) r + r − wdr / ( qr ) . (109)The last integral is understood in the sense of the principal value, denoted as Pr, due to singularity atthe point where V ( r ) = m + (cid:101) .In the nonrelativistic limit the value Γ = T has meaning of number of impacts per unit timeof the particle (localized inside the region r < r < r − ) against the potential barrier at r = r − , andthe exponential is the probability of the penetration of the barrier in each impact. Allowance for therelativistic effects and the spin changes the expression for the period of the oscillations and adds to(109) a factor depending on the sign of κ .With taking into account that in the region of the barrier V is purely Coulomb field, for w = Γ = Γ exp (cid:104) − πζ (cid:16) ( m + k ) / k − ( − ρ ) (cid:17)(cid:105) , (110)1/ Γ = ζ k (cid:20) ( − ρ ) ( m + k ) − m k Arth (cid:18) k (cid:16) − ρ m + k (cid:17) (cid:19)(cid:21) .For the positron momentum k = √ (cid:101) − m → Γ = c = [ ζ ( − ρ ) ( + ρ )] , for k → ∞ , Γ = c k = k / [ ζ ( − ρ ) ] . For k (cid:28) ζ m the width Γ is exponentially small for any κ . For | κ | (cid:29) ( ζ / π ) expression simplifies as Γ (cid:39) k [ ζ ( − ρ )] − exp [ − πζ ( − ( − ρ ) )] . (111)For | κ | < ∼ κ = ( ζ / π ) , the exponential factor in Γ becomes of the order of unity, and thequasiclassical approximation becomes invalid. Note that κ / κ max = √ πζ . Therefore number of niverse , xx , x 27 of 56 levels diffused in the continuum, for which Γ is not exponentially small, is tiny for ζ (cid:29) We described the spectrum of the quasistationary levels in the lower continuum for a sphericalnucleus with charge Z > Z cr . The results can be generalized to the case, when the potential does notobey spherical symmetry [7]. Let us present the Dirac equation as − i ( ˆ (cid:126) α ∇ ) ψ = ¯ h − ˆ D ψ , ˆ D = (cid:101) − m ˆ β − V ( (cid:126) r ) , (112)where ˆ (cid:126) α = γ (cid:126) γ , ˆ β = γ are Dirac matrices and we recovered dependence on ¯ h . Let us present bispinor ψ as ψ = φ e i σ and expand real quantities φ and σ in the parameter proportional to ¯ h : σ = ¯ h − σ − + σ + ..., φ = φ ( ) + ¯ h φ ( ) + ... (113)Replacing these series to Eq. (112) we obtain the chain of equations [ ˆ D − ( ˆ (cid:126) α ∇ σ − )] φ ( ) = [ ˆ D − ( ˆ (cid:126) α ∇ σ − )] φ ( ) = ˆ (cid:126) α ∇ ) φ ( ) + ( ˆ (cid:126) α ∇ ) σ ) φ ( ) , ...Condition of existence of nontrivial solution φ ( ) ,det [ ˆ D − ( ˆ (cid:126) α ∇ σ − )] = ( ∇ σ − ) = ( (cid:101) − V ) − m . (116)In difference with spherically-symmetric case, matrixˆ D − ˆ (cid:126) α ∇ σ − = (cid:101) − V ( (cid:126) r ) − m ˆ β − ˆ (cid:126) α ∇ S (117)is Hermitian, therefore its left-hand, ˜ φ i , and right-hand, φ i , eigenvectors are Hermitian conjugates,˜ φ i = φ † i , and ( ˆ D − ˆ (cid:126) α ∇ S ) φ i = φ † i ( ˆ D − ˆ (cid:126) α ∇ S ) = i =
1, 2, 3, 4 . (118)With the help of this equation, from (114) we find system of equations for σ , φ † i ( ˆ (cid:126) α ∇ σ ) φ j = − φ † i ˆ (cid:126) α ∇ φ j . (119)Bispinors φ i are found by diagonalizing the matrix ˆ D − ˆ (cid:126) α ∇ S , so that the right-hand side of (119)contains known quantities. Determining from this equation σ , we obtain the quasiclassical solution ofthe Dirac equation ψ = φ i exp ( ¯ h − σ − + σ ) . (120)In practice calculation of the functions σ − and σ for noncentral potentials is a complicatedmathematical problem requiring the solution of first-order differential equations in partial derivatives.In contrast to the case when V is spherically symmetric, in general case the result is not expressed inquadratures. If a parameter of “non-sphericity” is small one may develop a perturbation theory. niverse , xx , x 28 of 56
6. Spontaneous production of positrons in heavy-ion collisions
Minimal distance between colliding nuclei with charges Z and Z is as follows [17,61], R min = ( Z + Z ) e / ( E c.m. ) + (cid:113) ( Z + Z ) e / ( E c.m. ) + b ,where E c.m. is the kinetic energy of colliding nuclei in c.m. reference frame, b is the impact parameter.In order the energy of the electron, (cid:101) , in the quasi-molecule would become < − m the colliding heavynuclei should reach distances | (cid:126) r − (cid:126) r | = R < R cr , where R cr (cid:39)
33 fm for central U + U collisions, seebelow. Thus R cr is approximately twice larger than 2 R nucl , where R nucl (cid:39) A fm is the radius ofthe single nucleus (cid:39) R cr (cid:28) r K (cid:39) ζ cr , where r is estimated using Eq. (105).For U + U collisions R cr / r K ∼ v A ∼ ( − ) , cf. [17], whereasthe electron of the K-shell has typical velocity v e (cid:39)
1. Thereby one may use adiabatic approximation,i.e., we may use (cid:101) ( R ( t )) . Since R cr / r K ∼ (cid:28)
1, the anisotropy of the potential is not as large and wemay present V ( r ) = − (cid:18) Z e r + Z e r (cid:19) = − Ze r (cid:18) + R ( r ) P ( cos θ ) + ... (cid:19) , (121)where Z = Z + Z , (cid:126) r = | (cid:126) r ± (cid:126) R /2 | , P is the second Legendre polinomical, R ( t ) is the distancebetween centers of nuclei. In the second equation and further we for simplicity consider the case Z = Z . Otherwise there appear odd-power terms in the expansion. In inclusive experiments thisanisotropy disappears due to the averaging. However for event-by-event collisions such terms maylead to forward-backward anisotropy reflecting in some observable effects. In the first approximationin ( R /2 r K ) , the problem is reduced to that we have considered above for the spherical nucleus withthe charge Z = Z + Z . The effective nucleus radius now is 2 R nucl .The process of the spontaneous production of positrons also can be described in adiabaticapproximation since as we have argued we may use that (cid:101) ( R ( t )) and since 1/ Γ ( (cid:101) ( R ( t ))) (cid:29) τ col > ∼ R cr / v A . The most serious experimental problem is to separate spontaneous production of positronsfrom the phone processes. For example, the parameter 2 R nucl / R cr ∼ ( − ) is not as small.Therefore a serious competing time-dependent process is associated with an induced production ofpositrons occurring due to excitation of the nuclear levels, cf. [19] and references therein. Howeverdifference between characteristics of the induced and spontaneous production of positrons is significant.The induced positron production exists both in subcritical and the supercritical regimes. When theelectron level crosses the boundary (cid:101) = − m there appears a narrow energy-line in the positronspectrum owing to the switching on of the spontaneous positron production occurring in the tunnelingprocess. Thus there is principal difference between the subcritical and the supercritical regimes thatmay help to the experimental identification of the spontaneous positron production.Another effect is associated with presence of a magnetic component of the field. First indicationon presence of strong magnetic fields in heavy ion collisions was performed in [62]. For peripheralcollisions of heavy ions at collision energies < ∼ GeV · A it yields h ∼ H π ( Ze ) for R (cid:39) A / m π , v nucl ∼ H π = m π / e . More generally, replacing 1 → v γ , γ = √ − v we have eh ∼ Ze v γ / R . (122)For collisions with low energies E ∼ ( − ) MeV · A of our interest here it follows that h ∼ G, for R ∼ R cr (cid:39) ( − ) fm, and v nucl ∼ − , cf. also [63]. niverse , xx , x 29 of 56 In presence of a “weak” homogeneous magnetic field, the reduction of Z cr in case of thesupercritical atom has been found by using the perturbation theory [64], ζ cr ( h ) = ζ cr ( ) − π µ ( R ) hH , (123) H = m / e (cid:39) · G, µ (cid:39) ζ = ζ cr .For strong fields, numerical evaluations [64], see also [18], yielded Z cr =
165 for h = H , and Z cr =
96 at h = H . For h = G one gets Z cr =
41. This effect appears because of the exactcompensation of the diamagnetic and paramagnetic contributions to the ground state for the electron.Although these estimates are performed for the case of purely uniform static magnetic field, they showthat a magnetic effect also should be carefully studied for the case of realistic time-space configurationof the field.Below I focus only on the description of the spontaneous production of positrons and simplifyingconsideration I also ignore mentioned magnetic effects.
Usage of the Bohr-Sommerfeld quantization rule allows to consider the problem analytically [8],cf. [7]. From (19), (20), (21) with taking into account the Langer correction resulting in the replacement p → p ∗ , we have p ∗ ( r ) = F ( r , (cid:101) ) / r , F ( r , (cid:101) ) = (cid:20) ( (cid:101) − m ) r + (cid:101)ζ r + κ + a −
34 ˜ a − ( κ + ) + ζ (cid:21) , (124)where ˜ a = + r ( m + (cid:101) ) / ζ , ζ = Ze , Z = Z + Z . Applying the quantization rule (50) first for (cid:101) (cid:54) = − m and then for (cid:101) = − m and subtracting one result from the other we obtain (cid:90) r (cid:101) R /2 drF ( r , (cid:101) ) / r = (cid:90) r − m R cr /2 drF ( r , (cid:101) = − m ) / r . (125)Here r (cid:101) is the turning point for the given (cid:101) and r − m is the turning point for (cid:101) = − m . I used that inintegration over the regions r < R /2, r < R cr /2 dependence on (cid:101) can be dropped since at | (cid:101) | ∼ m ofour interest we have | V | (cid:29) | (cid:101) | . Thereby, the specifics of the behavior V ( r ) in the region r < R cr /2almost does not affect the result. To be specific, we may use V = const for r < R cr /2. Integrals undergologarithmic diverge at the lower limit. After their regularization the dependence on R and R cr isseparated in the explicit form: (cid:90) r (cid:101) dr [ F ( r , (cid:101) ) − F ( r , (cid:101) = − m )] / r + (cid:90) r (cid:101) r − m drF ( r , (cid:101) = − m ) / r = ˜ g ln RR cr . (126)Integrals in (126) are calculated numerically. Comparison with exact solution of two-center Diracproblem shows that the error of the semiclassical result does not exceed 0.1%. We can proceed furtherusing that r | m + (cid:101) | / ζ < r (cid:101) | m + (cid:101) | / ζ (cid:28) | (cid:101) | ∼ m of our interest. Thereby we expand ˜ a inEq. (124) in the series of r . As the result we find F ( r , (cid:101) ) = ( ˜ g + br + cr ) , (127) b = (cid:101)ζ − ( κ − )( m + (cid:101) ) / ζ , c = (cid:101) − m + ( κ − )( (cid:101) + m ) / ζ .From (126) and (127) we obtain RR cr = − ζ b (cid:18) + ˜ g c b + O ( c ) (cid:19) . (128) niverse , xx , x 30 of 56 Figure 4.
Solution (cid:101) ( R / R cr ) of Eq. (130) for various values of the parameter ζ . For | (cid:101) + m | (cid:28) m we find (cid:101) = − m − β m ( R − R cr ) / R cr , β = (cid:18) − κ − ζ − ˜ g ζ (cid:19) − . (129)For U + U collisions for the ground-state level we find ζ (cid:39) β (cid:39) β determines the probability of the production of positrons for | (cid:101) + m | (cid:28) m . The semiclassicalapproximation reproduces the Z dependence of β correctly, the difference with exact calculation donewithin solution of the two-center problem for the Dirac equation [65] is about (3-4)%.Setting c = ( ) we obtain very simple and on the other hand accurate result [6–8]: (cid:101) ( R ) = (cid:101) ( R / R cr ) = − m R cr / R − ( κ − ) / ( ζ ) − ( κ − ) / ( ζ ) . (130)Difference of this simple expression with exact solution of the two-center Dirac equation [65] is lessthan (1-2)% already for ζ → c , which has still higher accuracy. It may becurious to notice that, when in 1976 I showed the result (130) to Vladimir Stepanovich Popov, hedid not believe in it, saying that one of his collaborators during a year is trying to solve the Diracequation for the two-center problem numerically on ITEP big computer and yet obtained only theresult, for ζ =
1. He took slide rule (that time there were no PCs) and confirmed that the wholecurve (130) fully coincides with the result of the exact numerical calculation. Then, the result (130)was reflected in our publications [6,7]. Result (130) is shown in Fig. 4. For ζ = κ = − − (cid:101) ( R / R cr ) = ( R cr / R ) + R cr , can be found from Eq. (62) for a sphericalnucleus after replacement of the nucleus radius R nucl by R /2, where now R is the distance betweennuclei and Z → Z + Z . As the result, we find R cr = g ζ m (cid:20) exp (cid:18) π ( n + γ ) − ˜ γ ˜ g + (cid:19)(cid:21) − . (131)For the case of U + U collisions, in the model I we obtain R cr (cid:39)
33 fm, whereas exact solution of theDirac equation [65] yields R cr (cid:39) The potential of the system of two nuclei (121) contains at r (cid:29) R a quadrupole correction. In thesub-barrier region the correction is < ∼ ( R cr / ( r − )) < ∼ − . Therefore the problem is reduced to the niverse , xx , x 31 of 56 calculation of the penetrability of a three-dimensional barrier that differs only little from a sphericallysymmetrical one. Thus we may use expansion V = V + m R V , S = S + m R S . (132)We substitute these expressions to the Hamilton-Jacobi equation and obtain ( ∇ S ) = m ( E − U ) , ∇ S ∇ S = − U , (133) U ( r ) = − (cid:16) ζ r m + ζ(cid:101) rm (cid:17) , U ( r ) = − ζ r m (cid:16) (cid:101) + ζ r (cid:17) P ( cos θ ) .The first equation is easily integrated resulting in S ( r , θ ) = (cid:90) r pdr + κθ . (134)Taking into account of the first term leads to exponential term in (110). Second term in (134) is due toanisotropy of the potential.Equation for S in the under-barrier region r − < r < r + gets the form iq ∂ S ∂ r + κ r ∂ S ∂θ = − U ( r , θ ) , p = iq , (135)and it is solved by the method of separation of the variables. Supposing r U ( r , θ ) = u ( r )(
34 cos ( θ ) + ) (136)and taking into account the boundary condition Im S ( r − , θ ) =
0, for r = r + we obtainIm S ( r + , θ ) = aP ( cos θ ) + a , (137) a = (cid:82) r + r − dr u ( r ) m q ( r ) ch (cid:16) κ (cid:82) r + r dr (cid:48) q ( r (cid:48) ) r (cid:48) (cid:17) , a = − (cid:82) r + r − dr u ( r ) m q ( r ) sh (cid:16) κ (cid:82) r + r dr (cid:48) q ( r (cid:48) ) r (cid:48) (cid:17) .For the angular asymmetry of the positron production, the constant a is immaterial.A remarkable fact is that the expression for a acquires a hyperbolic cosine that enhances theangular anisotropy of the emitted particles compared with the anisotropy of the potential. Thecause of this effect is that the sub-barrier trajectory of a tunneling particle with nonzero angularmomentum is not a straight line due to κ (cid:54) =
0. This constitutes the substantial difference between thethree-dimensional and the one-dimensional problems.For a Coulomb field integrals (137) can be calculated exactly. However result looks cumbersome.An estimate shows that W ( θ ) (cid:39) exp ( − S ) = C exp ( α P ( cos θ )) , where C is a constant, α ∼ m R η − sh η (cid:29) m R , η = πκ / g . For U + U collisions α ∼ | κ | (cid:29)
1. However,as it always occurs, even for | κ | ∼ If the colliding nuclei are not fully stripped, the quasi-molecule is surrounded by an electron cloud.Screening weakens the attraction of the K -electron to the nuclei in the quasi-molecule. As a result, thecritical distance R cr , at which the K-electron level crosses the boundary (cid:101) = − m , is decreased. This niverse , xx , x 32 of 56 effect can be calculated using nonrelativistic many-particle semiclassical approximation (Thomas-Fermimethod), cf. [7,8]. Let us use that R cr (cid:28) r K (cid:28) a TF = ( π /128 ) ( Ze ) − / m (cid:39) ζ − / m , (138)where a TF is the mean radius of the Thomas-Fermi atom. The shift of the ground-state electron energylevel can be found with the help of the perturbation theory. We have ∆ (cid:101) (cid:39) V ( (cid:126) r ) − V ( (cid:126) r ) , (139)where V ( (cid:126) r ) is the potential of the two striped nuclei (121) and V ( (cid:126) r ) is the potential of the two not fullystriped ions. The typical size for the change of δ V is a TF . Therefore with accuracy ∼ ( R cr / a TF ) ∼ − the perturbation can be considered as spherically symmetric. Thus V ( r ) = V ( r i ) − Ze φ ( r ) r , V ( r i ) = − Z e r i , (140) r i = x a TF is the radius of the ion, φ ( r ) is the solution of the Thomas-Fermi equation [54], φ (cid:48)(cid:48) x = x − φ (141)with boundary conditions φ ( ) = φ ( x ) = x = r / a TF , and Z = − Zx φ (cid:48) x ( x ) is the observedcharge of the two partially screened nuclei.Expansion φ ( x → ) yields [54]: φ ( x ) = + φ (cid:48) x ( ) x + x + ... (142)For the case of neutral atoms φ (cid:48) x ( ) = − ∆ (cid:101) = V ( r i ) + φ (cid:48) x ( ) Ze a TF = Ze a TF [ φ (cid:48) x ( ) − φ (cid:48) x ( x )] + ζ a r + ... (143)Values φ (cid:48) x ( ) and − φ (cid:48) x ( x ) are tabulated. We estimate | ∆ R cr / R cr | ∼ | ∆ (cid:101) / (cid:101) | (cid:39)
10% for ionizationparameter q = ( Z + Z − N ) / ( Z + Z ) (cid:39) (cid:39)
12% for q =
0, where N is the total number ofelectrons in the quasi-molecule. V ( x , t ) .The Lagrangian is as follows L = − m (cid:112) − ˙ x − V ( x , t ) + V . (144)The constant is added to recover Lorentz invariance of the action S = (cid:90) t t Ldt , (145)since t is not a scalar. At initial time-moment particle was in the point x ( t ) and at final moment, in x ( t ) . niverse , xx , x 33 of 56 In the semiclassical approximation the wave function is ψ ( x ) ∼ e iS ( x , x ) = e i Re S ( x , x ) − Im S ( x , x ) . (146)The action is found from the Hamilton-Jacobi equation.In the imaginary-time method the sub-barrier motion is formally considered at imaginary valuesof the time variable. Performing variable replacement τ = it we arrive at the Euclidian action S E = (cid:90) τ τ [ m (cid:113) + ( dx / d τ ) + V ( x , τ ) − V ] d τ . (147)The trajectory x ( τ ) in under-barrier motion, where S E is real, is determined by condition δ S =
0. Fromhere one finds equation of motion, which has a meaning of the Newton equation d ˜ pd τ = dd τ mdx / d τ (cid:112) + ( dx / d τ ) = − ∂ V E ( x , τ ) ∂ x , V E = − V . (148)With exponential accuracy the probability to find the particle in the turning point of the exit fromthe barrier, if it initially were in the point of the entrance in the barrier, is given by W ( x , x ) = e − Im S ( x , x ) = e − S E ( x ( τ ) , x ( τ )) . (149)This expression can be generalized to take into account pre-exponential coefficient. However we willrestrict ourself by consideration of the exponential term.It is essential that the sub-barrier trajectory satisfies the classical equation of motion but now inthe Euclidian time. To find it and to calculate S and W we may use formally the known equations ofthe classical physics.6.5.2. Tunneling in slowly time-dependent potentialCase of space-dependent and slowly time-dependent fields was considered in [7]. For simplicityconsider a scalar particle in a one-dimensional field. Let the probability of the tunneling in the staticlimit is known, W = e − (cid:82) x x | p | dx , (150)where x and x are entrance and exit turning points, i.e. p ( x ) = p ( x ) =
0. Variation of the actiondue to a weak dependence of the potential on time V ( x , t ) yields δ S = δ (cid:82) t t [ − m ( − ˙ x ) − V ( x , t )] dt (151) = (cid:82) t t [ p δ ˙ x − ( ∂ V / ∂ x ) δ x − δ V ( t )] dt = − (cid:82) t t δ V ( x ( t )) dt .We used equation of motion and integration by parts. The last integral can be calculated usingimaginary-time method. Thus we obtain δ S E = (cid:90) τ τ δ V E ( x ( τ )) d τ . (152)Dependence x ( τ ) is determined from (148) as τ ( x , x ) = (cid:90) x x dx (cid:112) m − ˜ p ˜ p = (cid:90) x x dx V − (cid:101) (cid:112) m − ( (cid:101) − V ) , (153)where we used relation ˜ p = m − ( (cid:101) − V ) and that (cid:101) may only adiabatically change with time, i.e. itmay depend on τ via dependence of one of the parameters. niverse , xx , x 34 of 56 ( r + − r − ) √ m + k / k ) the potential V and (cid:101) did not have a time to change. Here please do not mixtypical time, for which the particle passes the barrier, cf. [66], and time 1/ Γ , being inversed probabilityto observe the positron. As we see from this simple estimate, adiabatic approximation does not hold atleast for k →
0, i.e. in the vicinity of the boundary of the continua, | (cid:101) | (cid:39) m .Let us find correction to the penetrability of the Coulomb barrier due to finite speed of thecolliding nuclei [7]. Following (121) the R ( t ) dependent correction to the static Coulomb potential is asfollows δ V = − ζ r P ( cos θ ) R ( t ) . (154)Further consider the case when positrons are emitted along the axis that joins the nuclei, P ( ) = P ( π ) =
1. Then, probability of their production is maximal. Expanding R ( t ) near the closest approachpoint we obtain R ( t ) = R + v t / ( R ) . (155)From (154) and (155) we have δ V = − ζ v r t . (156)The imaginary time τ = it is found from Eq. (153). Thus we obtain τ = ζ k [ m φ + ( m + k ) ( m + ρ k ) sin φ ] , (157)where we introduced variable φ = [( r + − r ) / ( r − r − )] , 0 ≤ φ ≤ π , r = r + cos ( φ /2 ) + r − sin ( φ /2 ) , values τ = φ = τ t = πζ m / k , i.e., τ t → ∞ for the electron energy (cid:101) → − m ,whereas for deep electron levels τ t strongly diminishes.Replacement of (157) in (152) yields δ S E = δ Im S = − ZAm N ζ R v p I ( (cid:101) p , η ) , (158)where (cid:101) p = − (cid:101) , v p = ( − m / (cid:101) p ) is the speed of the positron, I ( (cid:101) p , η ) = − (cid:82) π d φ (cid:20) sin φ +( − v p ) η cos φ + η (cid:21) φ +( − v p ) η cos φ + η , (159) η = [ − ( − ρ ) v p ] .The ratio δ = Im δ S Im S , (160)where Im S = πζ [ v − p − ( − ρ ) ] , for collisions U + U ( ζ = (cid:101) p . It is seen that δ < (cid:101) p > m . The adiabatic approximation in theproblem of spontaneous production of positrons becomes invalid near (cid:101) p = m , where the positronproduction cross section is any case tiny.Numerical calculations [17,42] have shown that R cr increases rapidly with increasing charge Z = Z + Z of colliding nuclei. The cross section of the spontaneous production of positrons increasesin this case ∝ R , while the correction for the non-adiabaticity of the tunneling decreases as 1/ R cr ata fixed (cid:101) p . Therefore it would be more convenient to perform experiments with heavier nuclei, for niverse , xx , x 35 of 56 Figure 5.
Correction on non-adiabaticity of the motion of nuclei, δ , for collisions U + U as a function ofthe positron energy (cid:101) p . which R cr is larger.
7. Many-particle semiclassical approximation. Electron condensation in upper continuum
In a many-particle problem, most of electrons in spherically symmetric potential well, V < l (cid:29) p max = (cid:113) ( (cid:101) bound − V ) − m (161)with (cid:101) bound ≥ − m . If there is sufficient amount of external electrons, the resulting system ischarge-neutral. In this case we should put (cid:101) bound = m . Then p max = √− mV + V , and takinginto account that each cell of the phase space can be occupied only by two electrons of opposite spinwe have n e = p max π = ( − mV + V ) π . (162)Thus the relativistic Thomas-Fermi equation renders ∆ V = π e (cid:34) n nucl − ( − mV + V ) π (cid:35) , (163) n nucl is the charged density of the nucleus. It is curious to note that such equation for neutral atomhas been introduced long ago [67] but relativistic term was then treated as a small correction innonrelativistic limit | V | (cid:28) m .7.1.1. Filling of the vacuum shell by electronsNote that even in absence of external electrons, which may fill the empty states, in case when thepotential well V < − mc electrons and positrons can be created already from the vacuum in absenceof any external electrons. Positrons go off to infinity, whereas electrons screen the initial positive charge niverse , xx , x 36 of 56 of the source. In this case we should put (cid:101) bound = − m . Then the relativistic Thomas-Fermi equationrenders, cf. [1,2,68], ∆ V = π e (cid:34) n nucl − ( mV + V ) π θ ( mV + V ) (cid:35) , (164)where θ ( x ) is the step-function, with the boundary conditions on the boarder of the ion V ( r i ) = − m = − Z i e / r i , V (cid:48) ( r i ) = Z i e / r i , (165)and with V ( r ) = − Z i e / r for r > r i . Reference [68] presented numerical solutions. The thoroughanalytical and numerical study of the problem of the filling of the vacuum shell by many electronswas performed in an independent study [1,2]. This phenomenon was called “electron condensation”demonstrating that all vacuum levels are filled by electrons of the lower continuum, cf. [36]. Electron density can be found by direct summation of squared of the wave functions [2]: n e = − ∑ n κ m | ψ n κ m | , (166)where ψ n κ m are semiclassical wave functions presented in Eqs. (95) – (100). Actually, we need wavefunctions in classically allowed region given by (95).Differentiating quantization rule (108) over n we obtain ∂(cid:101)∂ n (cid:90) r − r (cid:101) − Vp dr (cid:39) π , (167)where we dropped the term ∂∂ n κ wpr , which leads only to a small correction | w | / V ∼ ζ , cf. [5].From (167) and (96) we obtain C = (cid:18) π ∂(cid:101)∂ n (cid:19) . (168)Using that ∑ jm = − j | Y lm | = ( j + ) / ( π ) , where Y lm is the spherical function, from Eq. (166) we have n e ( r ) = − ∑ n κ j + π ∂(cid:101)∂ n (cid:101) − Vpr . (169)Here we replaced sin θ and sin θ by 1/2 due to multiple oscillations. Replacing summation in n byintegration we find n e = − π ∑ κ N j = − π ∑ κ ( j + ) r (cid:18)(cid:113) ( (cid:101) − V ) − m − ( j + ) / r (cid:19) (cid:101) bound (cid:101) m , (170) (cid:101) m corresponds to the zero of the under-square-root expression. Doing further integration in j with (cid:101) bound = − m we recover (164).Now let us estimate number of electrons in the vacuum shell, for which single-particleapproximation fails, i.e. number of levels, for which the width has no exponential smallness.Integrating (170) over the volume we find number of levels with momenta j ≤ κ − δ ( κ ) = N e κ − ∑ j = N j = c κ , (171) niverse , xx , x 37 of 56 where c = I / ( I ) , I = (cid:90) ( V + mV ) dr , I = (cid:90) ( V + mV ) r dr , (172) V ≥ − mV . In particular for V = − ζ / r with logarithmic accuracy we obtain I = ζ ln ( ζ / R nucl ) , I = ζ ln ( ζ / R nucl ) , c = ( ζ ) . (173)For ζ (cid:29) κ = ( ζ / π ) it follows that δ ( κ ) = ( πζ ) (cid:28) | κ | < κ , whereas not all of them have exponentiallysuppressed Γ . Taking into account of a correction (174) leads to appearance of a numerical factorln ( κ / R nucl ) / ln ( ζ / R nucl ) (cid:39) ζ (cid:29) R ∝ ζ . We estimate δ (cid:39) ζ , i.e., δ ∼
1% for Z ∼ e . Smallness of δ characterizes accuracy of Eq (164).Taking into account of the exchange and correlation corrections in the relativistic Thomas-Fermiequation is conveniently done by means of a variational method analogously to that is performed forthe nonrelativistic Thomas-Fermi equation [69]. We arrive at n e (cid:39) − π [( V + mV ) − ν ( V + m )] θ ( V + mV ) , (175) ν (cid:39) e / π . For Ze (cid:28) Ze (cid:29) e → e ( + e / π ) , cf. [2].Also, a correction appears due to that the dielectric permittivity of the vacuum, ε ( eE ) , differsfrom unity, e (cid:126) E = −∇ V . Thus one should replace ∆ V → ∇ ( ε ( E ) ∇ V ) in Eq. (164). However thiscorrection, as the correlation correction, is tiny, since ε ( eE ) = − ( e / ( π )) ln ( eE / m ) , and at distances r > ∼ ( a Z ) of our interest ε ( eE ) (cid:39) + O ( e / ( π )) , cf. [43] and Eq. (243) below. e (cid:28) Z (cid:28) e Consider screening of the positively charged nucleus of initial proton number Z and radius R (typically R nucl (cid:39) A / m π , A ∼ Z ). Assume that inside the nucleus the proton charge density is n p = const . Introducing ψ = − V / m − V < − m ( ψ ≥ Z , from Eq. (164) we obtain ∆ ( m ψ ) = e m π ( ψ − ) θ ( ψ − ) − π n p θ ( R nucl − r ) , (176) θ ( x ) is the step-function. For r > R nucl with the help of the replacement x = r / r i we obtain ψ (cid:48)(cid:48) x + x − ψ (cid:48) x = µ ( ψ − ) , ψ ( ) = ψ (cid:48) = − µ = e m r i π = ( Z obs e ) π . (177)Here Z obs is the charge seen at infinity. Since µ (cid:28)
1, we may use expansion ψ ( x , µ ) = ψ ( x ) + µψ ( x ) + ... (178)Then we have equations ∆ x ψ = ψ ( ) = ψ (cid:48) ( ) = ∆ x ψ = ( ψ − ) ψ ψ , ψ ( ) = ψ (cid:48) ( ) = ψ ( x , µ ) = x − [ + µ ( − ln x + C ) + O ( x , µ )] , C = − niverse , xx , x 38 of 56 Inside the nucleus at condition Ze (cid:28) ψ = ζ y ( Ξ ) / ( R nucl m ) , Ξ = r / R we obtain y (cid:48)(cid:48) Ξ + y (cid:48) Ξ Ξ = e R πζ (cid:18) ζ y R − m (cid:19) − π e ζ n p , r < R nucl . (182)Using that inside the nucleus | V | ∼ ζ / R nucl ∼ Z m (cid:29) m and π R n p = Z we get y (cid:48)(cid:48) Ξ + y (cid:48) Ξ Ξ = − + ν y , ν = ( Ze ) π . (183)Since ν (cid:28) y = y ( Ξ ) + ν y ( Ξ ) + ... (184)and get y ( Ξ ) = ( − Ξ ) , y ( Ξ ) = C + Ξ − (cid:90) Ξ y ( x ) x ( Ξ − x ) dx . (185)Matching of V and V (cid:48) at the edge of the nucleus yields C = − − (cid:90) y ( x ) xdx , (186)and Z obs = Z (cid:20) − π ( Ze ) ( ln ζ R + C ) + ... (cid:21) , (187) C = ln 2 − + (cid:90) y ( x ) x dx (cid:39) (cid:29) Z ∼ A /2 and R nucl (cid:39) Z / m π . Since R grows with Z , onemay expect that for sufficiently large Z most of electrons enter the nucleus and the interior becomescharge-neutral, as infinite matter. For the bare nucleus, the energy associated with the electric field, E el = (cid:90) ( ∇ V ) π e d x ∼ Z e / R nucl ∼ Z e m π , (189)increases with Z more rapidly compared to the binding energy ∼ A ∼ Z , thereby the volume-chargedsystems do not exist. The charge, if exists, is repelled to the surface.To approximately solve Eq. (164) we now introduce variables x = ( r − R nucl ) / l and V = − V χ ( x ) .Constant V is found from the condition of the charge neutrality at x → − ∞ , i.e., V / ( π ) = n p for V (cid:29) m . Thus in new variables Eq. (164) renders χ (cid:48)(cid:48) x l − + χ (cid:48) x l − / ( x + R nucl / l ) = π e n p V [ χ − θ ( − x )] , (190)with boundary conditions χ ( − ∞ ) = χ ( ∞ ) =
0. The latter condition just means that typical decreaseof the potential occurs already at x ∼ l near the nucleus boundary, whereas the transition to theCoulomb law occurs at x (cid:29) l . Solution at such large distances can be found only numerically.Since in dimensionless equation with dimensionless boundary conditions typical | x | ∼
1, for R nucl (cid:29) l , that we assume, we can neglect the second term in l.h.s. of Eq. (190). In this case geometrybecomes one-dimensional and Eq. (190) reduces to χ (cid:48)(cid:48) = χ − θ ( − x ) , (191) niverse , xx , x 39 of 56 where we determined the length l as l − = π e n p / V = e ( π /3 ) ( n p ) . (192)With taking into account of the boundary conditions the first integral of Eq. (191) is as follows2 χ (cid:48) = χ + ( − χ + ) θ ( − x ) , (193)and final solution is χ ( x ) = − [ + − sh ( a − x / √ )] − x < a = √ χ ( x ) = ( x + b ) − , x > b = √ x <
0. To get it we write χ = + ψ , ψ (cid:28) χ ( x ) (cid:39) − C (cid:48) e x √ . (196)Using the boundary conditions at x = C (cid:48) (cid:39) E max = π √ (cid:18) π (cid:19) ( n p ) (cid:39) · V/cm ,that (cid:39) E QED = m c / ( e ¯ h ) (cid:39) · V/cm. Note that toget this conclusion we essentially used the relation R nucl ∼ Z / m π .The energy of the system can be recovered by integration of Eq. (164). For | V | (cid:29) m we have E = (cid:90) (cid:20) − ( ∇ V ) π e − V π − n p θ ( R nucl − r ) V (cid:21) d x . (197)Expression (189) is obtained, after one puts zero the term V π related to the electron condensation andemploys the partial integration and the Poisson equation.In our case ∇ V = R nucl (cid:29) l and V = ( π n p ) . With these values Eq.(197) yields E = V π · π R . (198)So, the energy is reduced to the kinetic energy of the degenerate relativistic electron gas filling vacuumenergy levels with (cid:101) < − m . One should add to it the energy associated with the strong interaction ofnucleons resulting in the binding of the ordinary atomic nuclei. In such a way we get limit transitionto the description of infinite matter. We see that not taking into account a pion condensate or someother complex processes we have E > A ∼ Z we would assume validity of the β equilibrium conditions, n ↔ p + e + ¯ ν , we would get A (cid:29) Z and taking into account of the gravity and the filling of allelectron levels up to (cid:101) = m , we would recover the description of the ordinary neutron-star matter, cf.[37]. niverse , xx , x 40 of 56 Figure 6.
Pre-exponential factor D ( µ ) in Eq. (201). For V = − Ze / r , the number of electrons filling vacuum shell is N e (cid:39) (cid:90) r | V | π d x ∼ ln 1/ ( rm ) → ∞ (199)for r → r < r i with boundary conditions (165)corresponding to that for r > r i we deal with the Coulomb law with the charge equal to the observablecharge Z obs . As we shall see, such a problem has a unique solution independently on the charge Z putin the center, i.e. at r →
0. It proves to be that the exact solution of Eq. (164) has the pole singularityalready at a finite value r = r pole ( µ ) . In a weak screening limit from Eq. (176) for x → x , x = r / r i , weget [3], ψ ( x , µ ) = Ax − x [ + a ( x − x ) / x + a ( x − x ) / x + ... ] , A = ( µ /2 ) − , (200) a = − a = + µ x /6, ... Substitution of (200) in Eq. (176) allows to find coefficients a n butdoes not allow to recover dependence x ( µ ) . To get full solution of the problem we need to solve Eq.(176) with the boundary conditions (165) in the whole interval 1 > x > x ( µ ) . Numerical solutionyields x ( µ ) = r pole ( µ ) / r i = D ( µ ) e − ( µ ) , µ → D ( µ ) is shown in Fig. 6. For Z obs (cid:29) ( e ) with increasing Z obs the polemoves towards the value 1/ m .We conclude that in the many-particle problem including electron condensation but not includingpolarization of the vacuum, the falling to the center manifests itself in the presence of the pole ata distance r pole ( µ ) . Thus we have found relation between Z obs and Z ( r ( µ )) for r > r pole ( µ ) . Thesolution is cut for r < r , e.g., using either model I or model II.At this instance we should recall about existence of the Landau pole for r = r L (cid:39) e − π / ( e ) / m ,which appears within the multi-particle problem of the polarization of the electron-positron vacuumnear the Coulomb center, cf. [43]. Comparison of exponential factors shows that for Z obs < ( e ) we have r L > r pole ( µ ) and for Z obs > ( e ) we have r L < r pole ( µ ) . Thus, in the case Z obs < ( e ) ,with decreasing r first polarization of the vacuum becomes effective and only at r in a narrow vicinityof r L , where Z ( r ) > e , the electron condensation comes into play. For Z obs > e , first electron niverse , xx , x 41 of 56 condensation becomes effective and only at r in a narrow vicinity of r pole ( µ ) > r L polarization of thevacuum begins to contribute, see below a detailed discussion in Sect. 9.Note that the value Z obs e plays a role of an effective coupling in description of semimetals anddiscussed effects might be relevant in this case, cf. [50].It is curious to note that inclusion of gravitational field of the source into consideration,modifies the QED problem of the distribution of the charge with taking into account of the electroncondensation, cf. [70]. Solution (200) is modified at x very near x . After a growth, solution continuesup to x → V → − Z e / r with Z ∼ Z / ( eGm ) , where G is the gravitational constant. Also,the pole solution (200) disappears in case of electron condensation in a strong uniform magnetic field,cf. [71].
8. Polarization of vacuum
In absence of external electromagnetic fields electrons of the lower continuum have infinite energy E = ∑ (cid:126) p σ (cid:101) − (cid:126) p σ , (202)where (cid:101) − (cid:126) p σ = − (cid:112) m + (cid:126) p are negative-sign solutions of the dispersion relation of the free Diracequation. In pure QED, i.e. at ignorance of gravitational effects, infinite constant (202) has no sense,being subtracted within renormalization procedure. In presence of the electric and magnetic fieldsenergy levels of the lower continuum, (cid:101) − (cid:126) p σ are changed. The difference E − E = ∑ (cid:126) p σ (cid:101) − (cid:126) p σ − ∑ (cid:126) p σ (cid:101) − (cid:126) p σ (203)has the physical meaning.Heisenberg and Euler considered polarization of the electron-positron vacuum in the staticuniform stationary electric and magnetic fields [72], cf. [43,73]. For the case of uniform purelymagnetic field calculation is more transparent. Eigenvalues of the Dirac equation are (cid:101) ± (cid:126) p σ = ± (cid:113) m + p z + | e | H ( n + ) − eH σ n =
0, 1, ..., σ = ± − ” sign solution. To calculate sum (203) one uses that number ofstates in interval dp z in uniform magnetic field is given by | eH | ( π ) dp z V , (205)cf. [33]. Taking into account double degeneracy of levels with n , σ = n + σ = − n = σ = −
1, with (cid:101) − (cid:126) p σ solution one obtains E = − (cid:90) ∞ − ∞ ∞ ∑ n = (cid:113) m + p z + | e | Hn dp z V + | e | H ( π ) (cid:90) ∞ − ∞ (cid:113) m + p z dp z V . (206)Divergence of integrals is removed by subtraction of E . Straightforward calculations result inexpressions for the H dependent energy and the Lagrangian L = −E . In uniform stationary fields (cid:126) E and (cid:126) H the Lagrangian density L can be function only of Lorentz-invariants (cid:126) E − (cid:126) H and (cid:126) E (cid:126) H . Note niverse , xx , x 42 of 56 that in presence of the sources of the current the Lagrangian density additionally depends on j µ A µ . Inthe case under consideration employing arguments of dimensionality and parity in (cid:126) H one can write L ( H ) = L ( H ) + L (cid:48) ( H ) = − H π + m f ( H / m ) . (207)First term is the ordinary Lagrangian density in magnetic field, whereas second term is contribution ofthe polarization of the vacuum in magnetic field. In case of uniform static magnetic and electric fieldsfunction f ( H ) in (207) should be replaced by f ( H , E ) = f ( H − E , ( (cid:126) E (cid:126) H ) ) . (208)At H =
0, thereby f ( E ) = f ( − E , 0 ) . At E = f ( H , 0 ) = f ( H , 0 ) . From here we see that f ( E ) = f ( H = iE , 0 ) , i.e. expression (207) for the case H (cid:54) = E =
0, remains valid after replacement H → iE . Note that f ( − E , 0 ) has a small imaginary part associated with a possibility of the tunnelingof electrons, which occupy levels of the lower continuum, to the upper continuum.In case of strong electric and magnetic fields | eE | / m (cid:29) | eH | / m (cid:29)
1, with a logarithmicaccuracy [72] found expressions for the dielectric and diamagnetic permittivities: ε ( E ) = − e π ln ( | eE | / m ) + O ( e ) , µ ( H ) = − e π ln ( | eH | / m ) + O ( e ) . (209)The corresponding contributions to the energy of the lower continuum are E E = (cid:90) d x ε ( ∇ V )( ∇ V ) π e , E H = (cid:90) d x µ ( H ) H π e . (210)Note that expressions (209) are derived with logarithmic accuracy, i.e. at assumptionln ( | eE | / m ) | (cid:29) ( | eH | / m ) | (cid:29)
1. They are invalid only in a narrow region of fields, where | e π ln ( | eE | / m ) | ∼ O ( e ) and | e π ln ( | eH | / m ) | ∼ O ( e ) . Thereby they are formally applicable alsofor negative values of ε and µ . The Green function of the free photon is given by iD µν ( x − x (cid:48) ) = < | ˆ T ˆ A int µ ( x ) ˆ A int ν ( x (cid:48) ) | > , (211)ˆ T is the ordinary time ordering, operators are in interaction picture, cf. [43]. Most general form is asfollows, D µν ( x − x (cid:48) ) = g µν D (( x − x (cid:48) ) ) − ∂ µ ∂ ν D ( l ) (( x − x (cid:48) ) ) , (212) g µν is the metric tensor. One usually uses the Feynmann gauge condition D )( l ) = D xx = − D , we have D ( k ) = π c ( k + i ) . (213)In the Feynmann gauge D µν ( k ) = − g µν π c ( k + i ) . (214)The free propagator of spin 1/2 electron is G ik = − i < | ˆ T ˆ Ψ i ( x ) ˆ Ψ k ( x (cid:48) | > , (215) niverse , xx , x 43 of 56 where Ψ i ( x ) satisfies Dirac equation ( ˆ p − m ) Ψ i ( x ) = G ( p ) = γ µ p µ − m , G ( p ) = γ µ p µ + mp − m . (216)We may turn the contour in p plane against clock arrow not touching poles and then we performreplacements ip = p , ix = x , px = − ˜ p ˜ x = − ( p x + (cid:126) p (cid:126) x ) , ˜ p = ( (cid:126) p , p ) , ˜ x = ( (cid:126) x , x ) , (cid:82) dp / i → (cid:82) dp .Let us present 1˜ p + m = (cid:90) ∞ e − α ( ˜ p + m ) d α , (217) G ( x ) = (cid:82) d p ( π ) e − ipx p − (cid:126) p − m + i δ = − i (cid:82) ∞ − ∞ d ˜ p ( π ) e i ˜ p ˜ x ˜ p + m = − i ∏ i = (cid:82) ∞ − ∞ d ˜ p π e i ˜ p i ˜ x i (cid:82) ∞ d α e − α ( ˜ p i + m ) = − i π (cid:82) ∞ due − m / u − ˜ x u /4 , (218) u = α . For ˜ xm (cid:28) m = G ( x ) = i π x , ˜ xm (cid:28) xm (cid:29) − m u − ˜ x u (cid:39) − m ˜ x − ˜ x ( u − u m ) m and we find G ( x ) = − i (cid:114) m π ˜ x e − m ˜ x , ˜ xm (cid:29) G ( x ) = (cid:90) d p ( π ) e − ipx γ µ p µ + mp − m = ( m + i γ µ ∂ µ ) G ( x ) . (221)Thus for ˜ xm (cid:28) G ( x ) = γ µ x µ π x , the electron Green function is odd function of itscoordinate argument. Taking into account the vacuum polarization diagrams in the first order perturbation theory in e the Dyson equation gets the form iD µν ( X − X ) = iD µν ( X − X ) (222) + (cid:82) d X d X iD µλ ( X − X ) Tr [( − ie ) γ λ iG ( X − X )( − ie ) γ ρ iG ( X − X )] iD ρν ( X − X ) .In the momentum representation we get iD µν ( k ) = iD µν ( k ) + iD µλ ( k ) (cid:90) d p ( π ) Tr [( − ie ) γ λ iG ( p + k )( − ie ) γ ρ iG ( p )] iD ρν ( k )( − ) . (223)The last factor ( − ) comes from the closed fermion loop. Next terms are constructed analogously.Closed fermion loops give zero. niverse , xx , x 44 of 56 Sum of all irreducible diagrams (which cannot be separated by a single photon line) is calledthe photon polarization operator, − i Π µν . Thereby, in the lowest order − i Π λρ = Tr [( − ie ) γ λ iG ( p + k )( − ie ) γ ρ iG ( p )] . In brief notations Eq. (223) renders D = D + D Π D . (224) e rather than physical one, e = r →
0. At the same time, r → r , with performingthe limit r → − i Π µν = Tr ( − ie γ µ ) iG ( x )( − ie γ ν ) iG ( − x ) . (225)At r > m , in case of weak external fields, the effects of polarization of vacuum should besuppressed, since the electron Green function and thereby the photon polarization operator decreaseexponentially in Euclidean variables, cf. Eq. (220). Therefore consider opposite limit case ˜ x (cid:28) m when effects of polarization of the vacuum can be significant. We recognize that at short distancesthere is no scale of length except the Compton wave length. Thus, G and Π µν should be power-lawfunctions of ˜ x . We have − i Π µν ( x ) = − e Tr [ γ µ ˆ x γ ν ˆ x π x ] = − e x µ x ν − x δ µν π x , (226) − i Π ( t , (cid:126) R ) = − e t + (cid:126) R π ( t − (cid:126) R ) . (227)In mixed ω , (cid:126) R representation: Π ( ω = (cid:126) R ) = (cid:90) d τ Π ( R ) = e π ( (cid:126) R ) . (228)Using Eq. (229) and (212) with D ( l ) =
0, we have A ( x ) = (cid:90) d x (cid:48) D ( x , x (cid:48) ) j ( x (cid:48) ) , (229)Multiplying Eq. (224) by e n ext ( (cid:126) r ) we arrive at the Poisson equation for the static field V ( (cid:126) r ) = e A n.ren0 = eA ren0 , expressed in terms of non-renormalized quantities, ∆ V ( (cid:126) r ) = π e ( − n ext ( (cid:126) r ) + π (cid:90) K ( ω = (cid:126) R ) d RV ( (cid:126) r + (cid:126) R )) , (230)where K ( ω = (cid:126) R ) = Π ( ω = (cid:126) R ) / e , K ( ω = (cid:126) R ) does not depend on e . As will be shownbelow, K ( ω = (cid:126) R ) diverges for r → ∆ V = − π e n ext ( (cid:126) r ) . niverse , xx , x 45 of 56 To perform this procedure of renormalization of the charge let us consider polarization of the vacuumin a weak field, i.e assuming n ext to be small. Then we may use expansion V ( (cid:126) r + (cid:126) R ) (cid:39) V ( (cid:126) r ) + ∇ V ( (cid:126) r ) (cid:126) R + ∂ V ∂ R i ∂ k R k R i R k + ... (231)We may drop convergent terms in the expansion (231) irrelevant for the renormalization procedure. Theterm (cid:82) K ( ω = (cid:126) R ) d RV ( (cid:126) r ) should be put zero since constant potential cannot produce polarizationcharges due to gauge invariance. The term (cid:82) K ( ω = (cid:126) R ) (cid:126) Rd R ∇ V ( (cid:126) r ) = ∆ V = − e π n ext , e = e = e + π e (cid:82) K ( ω = (cid:126) R ) (cid:126) R d (cid:126) R . (232)Finally we derived a formal relation between the bare coupling constant e and the physical one e = e . Thus, in the lowest approximation over e using Eq. (228) and relation between Π and K we obtain e = e + e π ln m r , e = e − e π ln m r , r → e > e → e . This is known as“the problem of the zero charge”, (or “Moscow zero”), cf. [43]. Strictly speaking, such a considerationsuffers of inconsistency since inverse relation given by the second equation has so called Landau polefor r = r L = m e − π / ( e ) . (234)From the second equation (233) follows the solution e → − π ln (( ( m r )) (cid:32) + π e ln ( ( m r )) (cid:33) , (235)corresponding to e < e . Similar procedure could be performed in 4-invariant formfor the 4-potential e A µ instead of e A .8.4.2. Case of a strong static electric fieldIn presence of a strong static electric field the electron polarization operator, even being consideredwith the only one-loop diagram, should be calculated with full electron Green functions, G , instead offree ones [36,44]. In this approximation expression (225) is repaced by − i Π µν = Tr ( − ie γ µ ) iG ( x )( − ie γ ν ) iG ( − x ) . (236)At this level the Ward-Takahashi identity is satisfied only approximately. It can be fulfilled exactlyafter taking into account of the higher order diagrams.Multiplying Eq. (224) by e n ext ( (cid:126) r ) we derive the Poisson equation for the static field V ( (cid:126) r ) = e A n.ren0 = eA ren0 , expressed in terms of non-renormalized quantities, ∆ V ( (cid:126) r ) = − π e ( n ext ( (cid:126) r ) − π (cid:90) K ( ω = (cid:126) r , (cid:126) R ) d RV ( (cid:126) r + (cid:126) R )) , (237) niverse , xx , x 46 of 56 where K ( ω = (cid:126) r , (cid:126) R , e ) = Π ( ω = (cid:126) r , (cid:126) R , e ) / e . Expressed in non-renormalized terms, both thesequantities depend on e . For G → G they transform to K ( ω = (cid:126) r , (cid:126) R ) = Π ( ω = (cid:126) r , (cid:126) R , e ) / e .We again use expansion (231). The term (cid:82) K ( ω = (cid:126) r , (cid:126) R ) d RV ( (cid:126) r ) should be put zero, sinceconstant potential cannot produce polarization charges due to gauge invariance. The term (cid:82) K ( ω = (cid:126) r , (cid:126) R ) (cid:126) Rd R = (cid:126) r ↔ (cid:126) r (cid:48) . So, we obtain ∆ V = − π e ( n ext + n ) , n = (cid:90) d RK ( ω = (cid:126) r , (cid:126) R ) R i R k ∂ i ∂ k V ( (cid:126) r ) + δ n , (238)where we retained the residual convergent term δ n .Let the field E ( (cid:126) r ) is locally directed in z direction. Then we rewrite (cid:82) d RK ( ω = (cid:126) r , (cid:126) R ) R i R k ∂ i ∂ k V ( (cid:126) r ) = (cid:82) K ( ω = (cid:126) r , (cid:126) R ) ρ d R ∆ V (239) − (cid:82) K ( ω = (cid:126) r , (cid:126) R ) ρ d R ∂ z V + (cid:82) K ( ω = (cid:126) r , (cid:126) R ) z d R ∂ z V ,where ρ = x + y . Renormalization of the charge is performed by addition and subtraction to n theterm 14 (cid:90) K ( ω = (cid:126) R ) ρ d R = (cid:90) K ( ω = (cid:126) R ) (cid:126) R d R ,where we used isotropy of the quantity K ( ω = (cid:126) R ) . Thus, we obtain ∆ V = − e π ( n ext + n ren1 ) , (240) n ren1 = (cid:82) ( K ( ω = (cid:126) r , (cid:126) R ) − K ( ω = (cid:126) R )) ρ d R ∆ V + (cid:82) K ( ω = (cid:126) r , (cid:126) R )( z − ρ ) d R ∂ z V + δ n ,where e = e .Now let us evaluate the electron Green function in a strong static electric field. For this it issufficient to use semiclassical expression for the Green function in mixed space G ( ω , (cid:126) r , (cid:126) r (cid:48) ) ∝ e iS ( (cid:126) r ) − iS ( (cid:126) r (cid:48) ) ,with S ( (cid:126) r ) − S ( (cid:126) r (cid:48) ) (cid:39) (cid:90) (cid:126) r (cid:48) (cid:126) r p ( l ) dl (cid:39) (cid:90) (cid:126) r (cid:48) (cid:126) r ( ω − V ( l )) dl ,where V = V + ∇ V (cid:126) R + ..., V is const. The quantity p ( l ) can be estimated from the Klein-Gordon-Fockequation ∆ ψ + (( ω − V ) − m ) ψ =
0, since in a strong field spin effects can be neglected with a certainaccuracy. Thus we estimate G ( ω , (cid:126) r , (cid:126) r (cid:48) ) ∝ e i ω (cid:48) | (cid:126) R |− ieE (cid:126) R C , where ω (cid:48) = ω − V , C ∼ | (cid:126) R | (cid:28) (cid:112) | eE | , eE = −∇ V , the Green function G , is reduced to G , and with alogarithmic accuracy Π ( ω = ) (cid:39) Π ( ω = ) . For | (cid:126) R | (cid:29) (cid:112) | eE | , the Green function G rapidlyoscillates and with a logarithmic accuracy Π ( ω = ) can be put zero. Thereby from (240) withlogarithmic accuracy we obtain n ren1 (cid:39) − ∆ V (cid:90) (cid:126) R > | eE | K ρ d R + δ n (cid:39) − ∆ V π ln | eE | + δ n . (241)Now we should take into account that (cid:82) n ( r ) d r = n = div (cid:126) P , where (cid:126) P is a polarization vector. Thus with our logarithmic accuracy weshould replace − ln | eE | π ∆ V → −∇ (cid:18) ln | eE | π ∇ V (cid:19) .So, finally we arrive at the Poisson equation ∇ ( ε ( E ) ∇ V ) = − π e n ext , (242) niverse , xx , x 47 of 56 with ε ( E ) = − e π ln | eE | = − e π ln ( Q ( r ) / r ) . (243)For Z = ε ( E ) (cid:39) − e π ln ( r ) + O ( e ln ( r )) that reproduces known Uehling law[43]. Eq. (243) This expression was calculated with one-loop diagram (although with full Greenfunctions). We used approximation e π ln ( r ) (cid:28)
1. Otherwise higher-loop order diagrams andvertex correction diagrams should be included. Reference [43] demonstrated that expression isactually valid with a higher accuracy, ε ( E ) (cid:39) − e π ln ( r ) + O ( e , e ln ( r )) , since for Z (cid:28) e it is recovered in the so called main logarithmic approximation ln ( r ) (cid:29)
1, which shows that ε ( E ) (cid:39) − e π ln ( r ) + O ( e ln ( r )) . It is therefore also formally valid for e ln ( r ) (cid:29) e ln ( r ) (cid:28)
1, i.e. in a region, where (cid:101) ( E ) <
0. We may also use another argument in favor of aformal validity of this expression at ε ( E ) <
0. For this let us consider theory with N (cid:29) ∼ m and let the coupling is e / N , cf. [74]. Then instead of Eq. (243) weimmediately arrive at expression ε ( E ) = − N e π N ln | eE | + O ( N e N ln | eE | ) = − e π ln | eE | + O ( N ) , (244)being valid both in the region, where ε ( E ) >
0, as well as for ε ( E ) < | H / H (cid:48) | (cid:29) R H = (cid:113) | eH | , | E / E (cid:48) | (cid:29) R E = (cid:113) | eE | , (245)where R H = (cid:112) | eH | is the typical radius of the curvature of the charged particle trajectory inmagnetic field (Larmor radius) and R E = (cid:112) | eE | is the typical radius of the curvature of the chargedparticle trajectory in electric field. Thus for the electric field of the form E = Q ( r ) / r , criterionof applicability of approximation of a uniform field coincides with inequality Q ( r ) (cid:29) rQ (cid:48) (cid:28)
1. So, expression for the dielectric permittivity of the vacuum (209) derived for the case of theuniform field coincides with (243) with the logarithmic accuracy and with the same accuracy we maywrite interpolation expression ε ( E ) = − e π ln (( Q ( r ) + ) / ( r m )) + O ( e ) . (246)Once more notice that we may use Eqs. (209), (243), (246) both for ε ( E ) > ε ( E ) <
0. There existcorrections to Eq. (246) in the region, where | ε ( E ) | ∼ e , however there are no physical reasons toexpect presence of any singularities in this region. Therefore it seems reasonable to use the sameexpression (246) at all distances. In presence of charge sources the Lagrangian density is already not only function of (cid:126) E , as itwas in case of purely uniform field, but it contains the term n ext V . The charge sources always existin a realistic problem. Indeed, uniform electric field can be constructed only in a limited region ofspace, namely inside the condenser with length of plates l (cid:29) d , where d is the distance between theplates. Outside the condenser the field decreases to zero. The electron-positron pairs produced in thetunneling process inside the condenser go to the plates. Electrons are localized near positively chargedplate and positrons, near negatively charged one. niverse , xx , x 48 of 56 Recall that the energy of the electron in a smooth field V in classical approximation is given by (cid:101) = V ± (cid:113) (cid:126) p + m , (247)cf. Fig. 2 demonstrating boundaries of the upper and lower continua in the field V <
0. The upper signsolution corresponds to states that originate in upper continuum, which can be occupied in attractivefield for electrons, V < − m in case of a broad potential well, after the tunneling of electrons fromthe lower continuum. In the standard interpretation, see discussion in Subsect. 3.4, the lower signsolution corresponds to positrons after replacement (cid:101) → − (cid:101) . Let us study another interpretation whenthe lower sign solution corresponds to electron states that originate in the lower continuum, beingoccupied by the electrons. As we show below, this interpretation might be relevant in a specific case,when ε < V > (cid:101) → (cid:101) − V . Let us expand potential V ( (cid:126) r (cid:48) ) near a point (cid:126) r : V ( (cid:126) r (cid:48) ) = V ( (cid:126) r ) − e (cid:126) E ( (cid:126) r ) (cid:126) R + ... , (cid:126) R = (cid:126) r (cid:48) − (cid:126) r . (248)Assuming V ( (cid:126) r ) to be very smooth function of coordinates we may retain only these two terms in theexpansion. It is easy to ascertain consequences of the replacement − e (cid:126) E (cid:126) r → V ( (cid:126) r ) − e (cid:126) E (cid:126) r . The term − ∑ (cid:82) ψ ∗ e (cid:126) E (cid:126) R ψ d R was already taken into account in the problem solved by Heisenberg and Eulerin case of purely uniform electric field. Expressions for the Lagrangian and the energy of the lowercontinuum in uniform fields are more easily calculated for the case of purely magnetic field as we havementioned. We found Eq. (206), where typical momenta p z contributing to the sum are p z ∼ (cid:112) | eH | .In case of purely electric field the typical momenta contributing to the sum are p ∼ (cid:112) | eE | . Performingsummation in Eq. (206) Refs.[43,72] derived expression (207) and with the help of invariants recoveredEq. (208). After doing replacement H → iE , | eH | → | eE | one arrived at expressions (209).Now, see Eq. (247), in the expression for the energy there appears additional potential term δ E V = ∑ (cid:90) ψ ∗ V ( (cid:126) r ) ψ d r = ∓ | V | π , (249)since ∑ njm | ψ njm | = | V | π >
0. The upper sign is for V < − m and the lower sign is for V > | V | (cid:29) m .There is still a kinetic term in the energy, see Eq. (247), which we should add consideringcondensation of electrons, corresponding to the region of momenta p ∼ | V | (cid:29) m rather than to p ∼ (cid:112) | eE | , the latter term we have included. At least in limit cases V (cid:29) | eE | and V (cid:28) | eE | ,mentioned contributions are not overlapped. As the result, the kinetic term is δ E kin ( V ) = ± (cid:90) | V | p · π p dp ( π ) d r (cid:39) ± (cid:90) V π d r . (250)The upper sign corresponds to electron condensation on levels of the upper continuum occupiedduring the tunneling of electrons of the lower continuum in the field V <
0. This case we have studiedin Sect. 7. The lower sign solution corresponds to the electron condensation on levels of the lowercontinuum, may be possible for V >
0, cf. with the first term in Eq. (206), where it was summed up incase of the magnetic field.Finally in case of weakly inhomogeneous electric field we obtain E = E E + δ E V + δ E kin = − (cid:90) d x (cid:101) ( ∇ V )( ∇ V ) π e − (cid:90) n ext d x ∓ (cid:90) V π d x . (251) niverse , xx , x 49 of 56 From the semiclassical derivation one may see the difference between condensation of electrons onlevels of upper continuum crossed the boundary (cid:101) = − m , cf. Eq. (197) and condensation on levelsin the lower continuum in a repulsive field. In the former case vacant states with (cid:101) < − m are filledonly in the process of the tunneling of electrons from the lower continuum. In the upper continuumthe kinetic energy of electrons is positive E kin = + ∑ (cid:82) ψ ∗ p ψ d x , p >
0, whereas the kinetic energy ofelectrons occupying levels of the lower continuum is negative, E kin = − ∑ (cid:82) ψ ∗ p ψ d x , p >
0, cf. thefirst term in Eq. (206) has been used in case of the uniform magnetic field.Variation of the energy yields the Poisson equation, ∇ ( ε ∇ V ) = π e ( n ext − θ ( V + mV )( V + mV ) / ( π )) , (252)cf. Eq. (164), which described electron condensation in attractive potential of a supercharged nucleusat ε (cid:39)
1. Although we are interested in the case | V | (cid:29)
1, we recovered dependence on m in Eq. (252).Now, for ε > V < − m we deal with electron condensation on levels of the upper continuumcrossed the boundary (cid:101) = − m with increasing | V | , as it follows from the standard interpretation ofthe levels, being appeared from the upper continuum during an adiabatic increase of | V | . Below wewill argue for a possibility of the condensation of electrons originated in the lower continuum in theproblem of the screening of the positive-charge source at ultrashort distances from it (at r < r pole ),where ε ren ( r ) <
9. Distribution of charge at super-short distances from the Coulomb center = r > r L n ext = − Z δ ( (cid:126) r − (cid:126) r ) , Z >
0, that corresponds to the surface distributionof protons following model I. Neglect first a possibility of electron condensation and include only apolarization of the vacuum. We seek solution of Eq. (242) in the form V = − Q ( r ) / r < eE = −∇ V = Q ( r ) / r > Q ( r ) = C / ε ( r , Q ( r )) , ε ( r , Q ( r )) = − e π ln ( Q ( r ) / r ) , C = const . (254)For r > ∼ m we can set ε ( r , Q ( r )) (cid:39) C = Z obs e .The potential V is easily recovered in case of a smooth variation of the charge Q ( r ) , when Q ( r ) (cid:39) Q ( r ) . (255)This condition is fulfilled for | Q (cid:48) | (cid:28) | Q | / r that yields | ε ( r ) | (cid:29) e / ( π ) .Solution of Eq. (254) has two branches, one corresponds to ε ( r , Q ( r )) >
0, other relates to ε ( r , Q ( r )) <
0. We assume Z = Z obs for r > ∼ m and find Q ( r ) for decreasing r . Then we obtain Q ( r ) = Z obs e / ε ( r , Q ( r )) (256)on the positive branch of ε ( r , Q ( r )) . Expression (256) has a kink at r = ˜ r m , ε ( ˜ r m ) ∼ e / ( π ) and Q ( ˜ r m ) ∼ π Z obs (cid:29)
1. Therefore Eq. (256) has a meaning only for r > ˜ r m . Only then one can finda relation between Z obs and Z . Note however that actually Eq. (256) becomes invalid already at niverse , xx , x 50 of 56 a slightly larger r than ˜ r m , when (cid:101) ( r , Q ( r )) reaches values ∼ e . At these distances Eq. (254) for ε becomes invalid and approximation (255) we have used also fails.A comment is in order. Consider what would be, if we used Eq. (242) and (254) for r < ˜ r m . Thenwe would get Q ( r ) = − Z e / ε ( Q ) > ε ( Q ) <
0. This solution becomes invalid in the vicinity of˜ r m , where − ε ∼ e , now for r < ˜ r m , and it cannot be smoothly matched with the solution we havederived for r > ˜ r m .9.1.2. Electron condensation on levels of upper continuum is includedIn the region, where Q ( r ) >
1, besides the vacuum polarization, cf. Eq. (242), we should includethe electron condensation on levels of the upper continuum crossed the boundary (cid:101) < − m , cf. Eq.(252). Thus we have ∇ ( ε ( E ) ∇ V ) = π e V / ( π ) , at r > r , (257) − V (cid:29) m . Solution of this equation can be easily obtained in approximation (255). We have [36], Q ( r ) = C ε ( r , Q ( r )) − C . (258)To be specific consider the case Q obs (cid:28)
1. Constant C is determined from the condition Q ( r > ∼ m ) (cid:39) Q obs = Z obs e , since ε ( r > ∼ m ) (cid:39)
1. Thus we obtain Q ( r ) = Q ε ( r , Q ( r ))( Q + ) − Q (cid:39) Q ε ( r , Q ( r )) − Q . (259)This solution shows an apparent pole at r = r appole . Near this pole, in the region where ε ( r , Q ( r )) −√ Q obs < ∼ e / ( π ) the condition (255) is no longer fulfilled and solution (259) loses its meaning. Todetermine Q ( r ) in immediate vicinity of the pole and at still smaller r we, as before, assume that ε ( r , Q ( r )) is a smooth function of coordinates but now Q ( r ) > ∼ Q ( r ) . We have found the pole solutionof the relativistic Thomas-Fermi equation for ε ( r , Q ( r )) =
1, cf. Eq. (200) obtained above and [3]. Nowwith ε ( r , Q ( r )) (cid:39) const < Q ( r ) (cid:29) Q ( r ) we similarly get [45], V = − (cid:18) πε ( r , Q ( r )) e (cid:19) r ( r − r pole ) , 0 < r − r pole (cid:28) r pole . (260)The value ˜ r m is now irrelevant, since solution (256) is modified due to inclusion of the electroncondensation. At very short distances from the pole, at which ε ( r , Q ( r )) becomes < ∼ e / ( π ) , thissolution also fails, first since expression for ε ( r , Q ( r )) becomes invalid in this region and, second,because condition that ε ( r , Q ( r )) varies smoothly with r , which we used, is also violated.Finally, we stress that solutions (259), (260) correspond to the charge distribution near the barecharge Z for r > r pole . They loose their meaning for r < r pole . At fixed Z obs for r > ∼ m , the charge Z ( r ) related to this Z obs is increased with decreasing r . Even for Z obs (cid:28) e , at tiny distances, r ∼ r pole ∼ r L , the charge Q ( r ) becomes very large, Q ( r ) (cid:29)
1, and at these distances the electroncondensation on levels of the upper continuum crossed the boundary (cid:101) < − m comes into play. Asthe result, the charge distribution gets the pole, as we have found. Our solution fails for r < r pole .The value of r pole essentially depends on the value of Z obs . For Q obs > ∼ r pole increasesconsiderably, see Eq. (201) derived for ε (cid:39) niverse , xx , x 51 of 56 = r < r L . Polarization of vacuum and electron condensation on levels in lowercontinuum Since QED is the theory with a local interaction, the charge sources can be of arbitrary sizes,including r →
0. To attack the zero-charge problem let us reconsider interpretation of the electroncondensation in the field of the charged source of a very small size.Since Dirac equation in spherically symmetric field does not change under simultaneousreplacements (cid:101) → − (cid:101) , e → − e , i.e. V → − V , and κ → − κ , in the Coulomb field of a negativecharge Z < Z > | Z | the energy of such level, (cid:101) e goes up and at a value | Z | > −
170 (depending on r ) the level intersects the boundary of theupper continuum (cid:101) e = m . According to the traditional interpretation, which we have used considering r > r pole , the electron states with (cid:101) e > − m , which appeared from the lower continuum alreadyin a weak field of repulsion to the electron, should be regarded as unphysical, and they should bereinterpreted as positron states with energies (cid:101) e + = − (cid:101) e . As a consequence of such reinterpretation, fora nucleus with − Z > e , upon decreasing r the lowest positron level reaches the energy (cid:101) e + = − m .Then two positrons, after tunnelling from the lower continuum, fill this empty level and two electronsgo off to infinity. Similarly, positron states with (cid:101) e + > − m appeared from the lower continuum alreadyin a weak field of attraction to electron (for Z >
0) are regarded as unphysical, being interpreted aselectron states with energies (cid:101) e = − (cid:101) e + . As we have demonstrated, such interpretation allows to solvethe problem of the charge distribution only for r > r > r pole , even with taking into account of suchmultiparticle effects, as the polarization of the vacuum and (for Z >
0) the electron condensation onlevels of the upper continuum crossed the boundary (cid:101) e = − m .However beyond the framework of a single-particle problem there appears a possibility of anotherinterpretation [45,46]. Following this possibility, we may interpret the electron levels originated in thelower continuum in the weak repulsive field (for Z < ε ( r ) can be negative at smalldistances. Then, no preliminary tunneling occurs from one continuum to another. Near the positivelycharged center of radius r < r pole the desired repulsive potential for the electrons appears, sincedielectric permittivity of the vacuum expressed in terms of the physical charge e > r < r pole . In terms of a unrenormalized charge ε unren ( r → r → ) → e < V ( r ) >
0, cf. Eq. (235). Passage of the pole with decreasing r becomes possible because ofthe phenomenon of electron condensation on levels originated in the lower continuum even in a weakfield.Above, dealing with electron condensation on levels of the upper continuum, due to presence ofthe pole we could not get a continues solution for all r . Now, dealing with ε < r → r → Q ( r > r → ) and Q obs = Q ( r → ∞ ) .For ε < Z > V proves to be repulsive. Thus, for a positivelycharged center, due to change of the sign of ε there are electron levels coming from the lower continuum.Since the quantity | Z / ε ( r ) | increases with increasing r , in a certain range of r , where − Q ( r ) >
1, inthe bare potential there are many such levels. To count them one can use the relativistic Thomas-Fermiapproach, now employing the electron density − V / ( π ) for V >
0. We have ∇ ( ε ( r ) ∇ V ) = − θ ( V ) π e V / ( π ) + π Z e δ ( (cid:126) r − (cid:126) r ) , (261)cf. [45], and Eq. (252) derived above. Introducing tortoise coordinate, Ξ = ln ( r m ) , we obtain d ( Q (cid:101) ) d Ξ = − e π Q − π r Q δ ( (cid:126) r − (cid:126) r ) , Q = Q + dQ d Ξ , (262) niverse , xx , x 52 of 56 where Q = Z e >
0. With condition Q (cid:39) Q (justified by the resulting distribution), we have duu = − e π d Ξ ε , u = ε Q , r > r . (263)Integrating further we find Q ( r ) = C ε + C = Q ε + Q . (264)Choosing appropriate sign of the solution corresponding to the repulsive potential for the electron dueto ε < r < r pole , we arrive at Q ( r ) = − Q (cid:113) ε + Q . (265)For r → r → Q > Q ( r ) (cid:39) − Q / | ε | →
0. Thus a test particledoes not interact with the nucleus at ultrashort distances. Recall asymptotic freedom property in QCDfor r →
0. For r ∼ m we have ε (cid:39) Q ( r ) = Z obs e . Thus we obtain relation between the bareand observed charges Z obs = − Z / ( + ( Z e ) ) . (266)For Z (cid:28) e we get Z obs (cid:39) − Z . The maximum possible value of | Z ( r ) | is | Z max | (cid:39) ( √ e ) , (cid:101) ( r max ) = r (cid:29) r pole the potential looks like an ordinary Coulomb potential.Individual charges situated at these distances, each with Z obs (cid:28) e , can be summed up to the totalcharge Z > Z cr ∼ e . At these distances ε > r (cid:29) r pole , and there may appearelectron condensation on the levels in the upper continuum crossed the boundary (cid:101) = − m . Theselevels become filled by electrons, after tunneling from the lower continuum, as we have demonstratedin Sect. 7. Thus reconstruction of the interaction at r < r pole does not affect any phenomena that can beobserved experimentally occurring at much larger distances.Note that solution (265) is similar to the solution obtained within QCD in the model [71], whichtook into account a possibility of the quark condensation near the external color-charge source. Theessential difference is in the dependencies of ε ( r ) in QCD and in QED. In QCD within a logarithmicapproximation ε QCD ( r ) (cid:39) b ln ( r Λ / r ) where b and r Λ are some constants, i.e., ε QCD ( r → ) → ∞ and ε QCD ( r → ∞ ) → − ∞ , whereas within QED we employed that ε QED ( r → ) → − ∞ and ε QED ( r → m ) →
1. In QCD there appears condensation of quarks on levels that originate in theupper continuum and in case under consideration in QED we included electron condensation on levelsthat originate in the lower continuum.Note that in terms of unrenormalized dielectric permittivity Eq. (261) renders ∇ ( ε n.ren ( r ) ∇ V ) = − θ ( V ) π e V / ( π ) + π e Z δ ( (cid:126) r − (cid:126) r ) , (267)with ε n.ren ( r ) = − e π ln ( r / r ) . Using relation e = f ( e ) we obtain ε n.ren ( r → r ) → ε n.ren < r > r pole . Thus at small distances r < r pole the non-renormalized dielectric permittivity is positive.Value Z e < Z >
0, that corresponds to V >
0, and − e V / ( π ) is positive. So, near a negativeexternal charge there appear positive charges and vise versa near a positive external charge there arisenegative charges, as expected in QED.Recall also that Hamiltonian, where one replaced p µ → p µ − e A unren µ should be Hermitianoperator, as well as the same Hamiltonian expressed in terms of renormalized charge, where one usesreplacement p µ → p µ − eA ren µ . Within ordinary second quantization scheme one expands ˆ A µ in seriesof plane waves, where appear the creation and annihilation operators, considering A µ as the realquantity. Since e is imaginary, A unren µ should be considered as purely imaginary quantity. Now, weshould perform expansion for e A n.ren µ , being real quantity. The energy is reduced to the energy of niverse , xx , x 53 of 56 stable oscillators only after performing renormalization, i.e., being expressed in terms of eA ren µ . Up to now we considered the charge distribution near the external charge source, which wasassumed to be infinitely massive. For description of the electron mass distribution, m ( r ) , one needsto study Dyson equation for the electron Green function, cf. [75]. As we argue below, at distancesof our interest | V | (cid:29) m ( r ) and dependence of m ( r ) does not influence the charge distribution in thelogarithmic approximation we have used. Equation for the mass is given by [76], dm ( Ξ ) d Ξ = − e π d t ( Ξ ) m ( Ξ ) , (268)where d t is the so called d -function of the photon, Ξ = ln ( ( r m )) is the tortoise coordinateintroduced above.A clarification is in order. As is known, presence of a zero in the expression for the dielectricpermittivity ˜ ε ( Ξ ) defined via the photon d -function, e d t = e ( Ξ ) , ˜ ε ( Ξ ) = e / e ( Ξ ) , (269)according to the Källen-Lehmann expansion, would correspond either to violation of the causality orto the instability of the vacuum [77]. However note that in our case the quantity ˜ (cid:101) ( Ξ ) does not havezero, ˜ ε ( Ξ ) = e e ( Ξ ) = ( ε ( Ξ ) + e ) , (270)as follows from (264) for Q = Z e , Z =
1. Thus the quantity ˜ (cid:101) ( Ξ ) does not coincide with ε ( Ξ ) . Thelatter quantity may vanish and even it can be negative, whereas “true” value ˜ ε ( Ξ ) > m ( Ξ ) = m (cid:32) ε ( Ξ ) + ( ε ( Ξ ) + e ) + ( + e ) (cid:33) , (271)where m is the observed electron mass. Thus m ( Ξ → ∞ ) → m ( Ξ → ∞ ) → m , i.e. in this case theentire electron mass is of purely electromagnetic origin.Concluding, we presented some arguments for the logical consistency of QED.
10. Conclusion
Most actively problem of a spontaneous production of positrons from the QED vacuum in strongfields has been attacked in theoretical works in Moscow (in the group of V. S. Popov in 1970-th)and in Frankfurt (in the group of W. Greiner in 70-th and 80-th of previous century). Experimentsperformed at GSI Darmstadt in 1980-th had turned out puzzling line structures in the energy spectra.These results were not confirmed by the subsequent experiments performed in 1990-th. Questionabout experimental confirmation of existence of the spontaneous positron production in low-energyheavy-ion collisions remained open. Now an interest to this problem is renewed [28] in connectionwith a possibility to perform new experiments at the upcoming accelerator facilities in Germany,Russia, and China [25–27]. Study of many-particle effects in description of the QED vacuum in strongfields is of a principal interest. The problem of the zero-charge remains to be one of the most importantfundamental problems of QED already about 70 years. In the given paper these problems were studiedwithin a common relativistic semiclassical approach developed in the reviewed papers. niverse , xx , x 54 of 56 In the given paper, first, the problems of the falling to the Coulomb center for the charged spinlessboson and for the fermion were first considered within the single-particle picture. Then, focus wasconcentrated on a case of spontaneous positron production in the field of a finite supercritical nucleuswith the charge Z > Z cr (cid:39) ( − ) . Behavior of deep electron levels crossed the boundary of thelower continuum and the probability of spontaneous positron production were studied. Then, similareffects were considered in application to the low-energy collisions of heavy ions, when for a short timethe electron level of the quasi-molecule crosses the boundary of the lower continuum (cid:101) = − m . Next,was studied the phenomenon of the electron condensation on levels of the upper continuum crossedthe boundary of the lower continuum in the field of a supercharged nucleus with Z (cid:29) Z cr . Then, focuswas concentrated on many-particle problems of the polarization of the QED vacuum and electroncondensation at ultra-short distances from the source of the charge. Arguments were presented forthe important difference of the cases, when the size of the source is larger than the pole size r pole , atwhich the dielectric permittivity of the vacuum reaches zero, and smaller r pole . Then, distributionsof the charge and mass of the electron were considered and arguments were given in favor of thelogical consistency of QED. Also I believe that at least some results reviewed in this paper can findapplications in description of semi-metals and stack of layers of graphene. References
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