Using a quantum kinetic equation coupled to Maxwell's equation we study the possibility that focused beams at proposed X-ray free electron laser facilities can generate electric field strengths large enough to cause spontaneous electron-positron pair production from the QED vacuum. Our approach yields the time and momentum dependence of the single particle distribution function. Under conditions reckoned achievable at planned facilities, repeated cycles of particle creation and annihilation take place in tune with the laser frequency. However, the peak particle number density is insensitive to this frequency and one can anticipate the production of a few hundred particle pairs per laser period. Field-current feedback and quantum statistical effects are small and can be neglected in this application of non-equilibrium quantum mean field theory. The QED vacuum is unstable in the presence of a strong external field and decays by emitting electronpositron pairs. The pair production rate was first calculated for a static, homogeneous electric field in the early part of the last century [1] and since then many aspects have been studied in detail. Using the Schwinger formula [1] one finds that a sizeable rate requires a field: The possibility of spontaneous pair creation from the vacuum is of particular interest in ultra-relativistic heavy ion collisions [2,3]. Since the QCD string tension is large ( √ σ ∼ 400 MeV), flux tube models of the collision generate a background field that is easily strong enough to initiate the production process via a Schwinger-like mechanism. Feedback between the external field and the field created by the produced particles' motion drives plasma oscillations. This is the back-reaction phenomenon, which has been much discussed [4,5,6]. While this Vlasov-equation-based approach has met with some phenomenological success, a rigorous justification in QCD is wanting.Pair creation by laser beams in QED has also been discussed [7,8], and proposed X-ray free electron laser (XFEL) facilities at SLAC [9] and DESY [10], which could generate field strengths [11, 12] E ≈ 0.1E cr , promise to provide a means to explore this phenomenon.Vacuum decay is a far-from-equilibrium, time-dependent process and hence kinetic theory provides an appropriate descriptive framework. For spatially homogeneous fields, a rigorous connection between kinetic theory and a mean-field treatment of QED has been established [13,14]. The derivation makes plain that the true kinetic equation's source term is intrinsically nonMarkovian, and this is expressed in properties of the solution [13,14,15,16]. Herein we use this quantum Vlasov equation to obtain a description of the time evolution of the momentum distribution function for particles produced via vacuum decay at the planned XFEL facilities.A gauge and Lorentz invariant description of an electromagnetic (e.m.) field is obtained usingAn e.m. plane wave always fulfills F = G = 0 and such a light-like field cannot produce pairs [17]. Therefore, to produce pairs it is necessary t...

We solve the quantum Vlasov equation for fermions and bosons, incorporating spontaneous pair creation in the presence of back reactions and collisions. Pair creation is initiated by an external impulse field and the source term is non-Markovian. A simultaneous solution of Maxwell's equation in the presence of feedback yields an internal current and electric field that exhibit plasma oscillations with a period pl . Allowing for collisions, these oscillations are damped on a time scale r determined by the collision frequency. Plasma oscillations cannot affect the early stages of the formation of a quark-gluon plasma unless r ӷ pl and pl ϳ1/⌳ QCD ϳ1 fm/c. ͓S0556-2821͑99͒06123-8͔

A quantum kinetic equation coupled with Maxwell's equation is used to estimate the laser power required at an XFEL facility to expose intrinsically quantum effects in the process of QED vacuum decay via spontaneous pair production. A 9 TW-peak XFEL laser with photon energy 8.3 keV could be sufficient to initiate particle accumulation and the consequent formation of a plasma of spontaneously produced pairs. The evolution of the particle number in the plasma will exhibit nonMarkovian aspects of the strong-field pair production process and the plasma's internal currents will generate an electric field whose interference with that of the laser leads to plasma oscillations.PACS numbers: 42.55. Vc, 41.60.Cr, 11.15.Tk X-ray free electron laser (XFEL) facilities are planned at SLAC [1]: namely the Linac Coherent Light Source (LCLS), and as part of the e − e + linear collider project (TESLA) at DESY [2]. They propose to provide narrow bandwidth, high power, short-length laser X-ray pulses, with good spatial coherence and tunable energy. It is anticipated that the realisable values of these parameters will enable studies of completely new fields in X-ray science, with applications in atomic and molecular physics, plasma physics, and many other fields [2].A unique ability of these facilities is to provide very high peak power densities. For example, a P = 0.2 TWpeak laser at a wavelength of λ = 0.4 nm, values which are reckoned achievable with current technology [2], can conceivably produce a peak electric field strengthBoosting P to 1 TW and reducing λ to 0.1 nm, which is theoretically possible [3], would yield an order-ofmagnitude increase: E g = 1.1 × 10 17 V/m. Electric fields of this strength are sufficient for an experimental verification of the spontaneous decay of the QED vacuum [4,5,6,7].It is a long standing prediction that the QED vacuum is unstable in the presence of a strong, constant electric field, decaying via the production of e − e + pairs [8]. In such fields, appreciable particle production is certain if the strength exceeds E cr := m 2 e /e = 1.3 × 10 18 V/m. (We subsequently useh = c = 1.) The proposed XFEL facilities could generate E ≈ 0.1 E cr . (NB. Here "constant" means that the field must be uniform over timeand length-scales much greater than the electron's Compton wavelength: 1/m e ≈ 0.4 pm.)A single laser beam cannot produce pairs [9]. (For a light-like field F µν F µν = 0 and hence the vacuum survival probability is equal to one.) Nevertheless, if two or more coherent beams are crossed and form a standing wave at their intersection, one can hypothetically produce a region in which there is a strong electric field but no magnetic field. The radius of this spot volume is diffraction limited to be larger than the laser beams' wavelength: r σ > ∼ λ, and the interior electric field could be approximately constant on length-scales approaching this magnitude. The period of the electric field is also determined by λ. Hence at an XFEL facility one might satisfy the length-scale uniformity conditions no...

Aspects of the formation and equilibration of a quark-gluon plasma are explored using a quantum kinetic equation, which involves a non-Markovian, Abelian source term for quark and antiquark production and, for the collision term, a relaxation time approximation that defines a time-dependent quasi-equilibrium temperature and collective velocity. The strong Abelian field is determined via the simultaneous solution of Maxwell's equation. A particular feature of this approach is the appearance of plasma oscillations in all thermodynamic observables. Their presence can lead to a sharp increase in the time-integrated dilepton yield.

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