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P. K. Panda, A. Mishra, J. M. Eisenberg, and W. Greiner,

We study here hot nuclear matter in the quark meson coupling (QMC) model
which incorporates explicitly quark degrees of freedom, with quarks coupled
to scalar and vector mesons. The equation of state of nuclear matter including
the composite nature of the nucleons is calculated at finite temperatures.
The calculations are done taking into account the medium-dependent bag
constant. Nucleon properties at finite temperatures as calculated here
are found to be appreciably different from the value at *T* = 0.

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G. Kälbermann, J. M. Eisenberg, and B. Svetitsky,

We study hot nuclear matter in a model based on nucleon interactions deriving from the exchange of scalar and vector mesons. The main new feature of our work is the treatment of the scale breaking of quantum chromodynamics through the introduction of a dilaton field. Although the dilaton effects are quite small quantitatively, they affect the high-temperature phase transition appreciably. We find that inclusion of the dilaton leads to a metastable high-density state at zero pressure, similar to that found by Glendenning who considered instead the admixture of higher baryon resonances.

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Y. Kluger, E. Mottola, and J. M. Eisenberg,

The adiabatic particle number in mean field theory obeys a quantum Vlasov equation which is nonlocal in time. For weak, slowly varying electric fields this particle number can be identified with the single particle distribution function in phase space, and its time rate of change is the appropriate effective source term for the Boltzmann–Vlasov equation. By analyzing the evolution of the particle number we exhibit the time structure of the particle creation process in a constant electric field, and derive the local form of the source term due to pair creation. In order to capture the secular Schwinger creation rate, the source term requires an asymptotic expansion which is uniform in time, and whose longitudinal momentum dependence can be approximated by a delta function only on long time scales. The local Vlasov source term amounts to a kind of Markov limit of field theory, where information about quantum phase correlations in the created pairs is ignored and a reversible Hamiltonian evolution is replaced by an irreversible kinetic one. This replacement has a precise counterpart in the density matrix description, where it corresponds to disregarding the rapidly varying off-diagonal terms in the adiabatic number basis and treating the more slowly varying diagonal elements as the probabilities of creating pairs in a stochastic process. A numerical comparison between the quantum and local kinetic approaches to the dynamical backreaction problem shows remarkably good agreement, even in quite strong electric fields, over a large range of times.

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J. M. Eisenberg, *Back reaction in the presence of thermalizing collisions*,
in L. C. Biedenharn Memorial Volume: Found. Phys. **27**, 1213 (1997)
[hep-ph/9609205].

Preequilibrium parton production following an ultrarelativistic nucleus–nucleus collision is studied in terms of the decay of a strong chromoelectric field which generates pairs through the Schwinger mechanism. Back-reaction of the partons with the field is included and a model transport equation containing a collision term is solved for the central rapidity region based on an approximation in which the partons relax to a thermal distribution.

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J. M. Eisenberg,

A derivation is given of the Boltzmann–Vlasov equation beginning from multiple scattering considerations. The motivation for the discussion, which is purely pedagogical in nature, is the current interest in understanding the origins of transport equations in terms of rigorous field-theory descriptions, or, as in this case, exact nonrelativistic formulations.

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J. M. Eisenberg,

A system is studied in which initially a strong classical electric field exists within an infinitely-long cylinder and no charges are present. Subsequently, within the cylinder, pairs of charged particles tunnel out from the vacuum and the current produced through their acceleration by the field acts back on the field, setting up plasma oscillations. This yields a rough model of phenomena that may occur in the pre-equilibrium formation phase of a quark–gluon plasma. In an infinite volume, this back-reaction has been studied in a field-theory description, and it has been found that the results of a full calculation of this sort are well represented in a much simpler transport formalism. It is the purpose here to explore that comparison for a situation involving a cylindrical volume of given radius.

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F. Cooper, J. M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky,

We study pair production from a strong electric field in boost-invariant coordinates as a simple model for the central rapidity region of a heavy-ion collision. We derive and solve the renormalized equations for the time evolution of the mean electric field and current of the produced particles, when the field is taken to be a function only of the fluid proper time. We find that a relativistic transport theory with a Schwinger source term modified to take Pauli blocking (or Bose enhancement) into account gives a good description of the numerical solution to the field equations. We also compute the renormalized energy–momentum tensor of the produced particles and compare the effective pressure, energy and entropy density to that expected from hydrodynamic models of energy and momentum flow of the plasma.

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Y. Kluger, J. M. Eisenberg, and B. Svetitsky,

We review recent achievements in the solution of the initial-value problem for quantum back-reaction in scalar and spinor QED. The problem is formulated and solved in the semiclassical mean-field approximation for a homogeneous, time-dependent electric field. Our primary motivation in examining back-reaction has to do with applications to theoretical models of production of the quark–gluon plasma, though we here address practicable solutions for back-reaction in general. We review the application of the method of adiabatic regularization to the Klein–Gordon and Dirac fields in order to renormalize the expectation value of the current and derive a finite coupled set of ordinary differential equations for the time evolution of the system. Three time scales are involved in the problem and therefore caution is needed to achieve numerical stability for this system. Several physical features, like plasma oscillations and plateaus in the current, appear in the solution. From the plateau of the electric current one can estimate the number of pairs before the onset of plasma oscillations, while the plasma oscillations themselves yield the number of particles from the plasma frequency.

We compare the field-theory solution to a simple model based on a relativistic Boltzmann–Vlasov equation, with a particle production source term inferred from the Schwinger particle creation rate and a Pauli-blocking (or Bose-enhancement) factor. This model reproduces very well the time behavior of the electric field and the creation rate of charged pairs of the semiclassical calculation. It therefore provides a simple intuitive understanding of the nature of the solution since nearly all the physical features can be expressed in terms of the classical distribution function.

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Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,

The initial-value problem for the quantum back-reaction in spinor QED is formulated and solved in the semiclassical mean field approximation, for a homogeneous but time-dependent electric field

E(t). We apply the method of adiabatic regularization to the Dirac equation in order to renormalize the expectation value of the current and derive a finite coupled set of ordinary differential equations for the time evolution of the system. We solve this system in (1+1) dimensions numerically and compare the solution to a simple model based on a relativistic Boltzmann–Vlasov equation, with a particle production source term inferred from the Schwinger particle creation rate and a Pauli-blocking factor. This model reproduces very well the time behavior of the electric field and the creation rate of electron–positron pairs of the semiclassical calculation.

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J. M. Eisenberg, Y. Kluger, and B. Svetitsky,

*Dedicated to Wieslaw Czyz on the occasion of his sixty-fifth birthday*

We present a summary of the present status of efforts to solve the problem in which pairs are produced in a strong electric field, are accelerated by it, and then react back on it through the counter-field produced by their current. This picture has been used by Bialas and Czyz and others as a model for effects that may possibly arise in the study of the quark–gluon plasma. We here give a didactic review of recent developments in this back-reaction problem. We first present a simple version of the theory of pair tunneling from a fixed electric field, and then sketch how this has been applied to the quark–gluon plasma. Then we turn to a field formulation of the problem for charged bosons, which leads to the need to carry out a renormalization program, outlined again in simple terms. Numerical results for this program are presented for one spatial dimension, the corresponding physical behavior of the system is discussed, and the implications for three spatial dimensions are considered. We exhibit a phenomenological transport equation embodying physics that is essentially identical to that of the field formulation, thus helping to tie the model of Bialas and Czyz for the quark–gluon plasma to a field-theory formulation. Last, we note the status of extensions to (i) the problem with three space dimensions; (ii) the fermion case; (iii) the formulation in terms of boost-invariant variables (as desirable for the quark–gluon plasma); and (iv) transport equations derived in a fundamental and consistent fashion.

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Y. Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,

We investigate the mechanism of pair creation in scalar QED from spatially homogeneous strong electric fields in 1+1 dimensions. Solution of the semiclassical field equations shows particle creation followed by plasma oscillations. We compare our results with a model based on a relativistic Boltzmann–Vlasov equation with a pair-creation source term related to the Schwinger mechanism. The time evolution of the electric field and the current obtained from the Boltzmann–Vlasov model is surprisingly similar to that found in the semiclassical calculation.

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J. M. Eisenberg and G. Kälbermann,

Background material on solitons and, especially, skyrmions is provided
and the applications of the latter to the derivation of the nucleon–nucleon
force is reviewed with attention to the use of the product ansatz, additional
terms in the lagrangian, baryon resonance admixtures, dilatons, and exact
two- or three-dimensional solutions for the *B* = 2 system in order to find
the sources of attraction in the central and spin–orbit potentials. We
discuss extensions to two-baryon systems with nonzero strangeness and address
applications to the behavior of the nucleon in nuclei achieved from skyrmions.

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J. M. Eisenberg and G. Kälbermann,

Background material on solitons and, especially, skyrmions is provided
and the applications of the latter to the derivation of the nucleon–nucleon
force is reviewed with attention to the use of the product ansatz, additional
terms in the lagrangian, baryon resonance admixtures, dilatons, and exact
two- or three-dimensional solutions for the *B* = 2 system in order
to find the sources of attraction in the central and spin-orbit potentials.
We discuss extensions to two-baryon systems with nonzero strangeness and
address applications to the behavior of the nucleon in nuclei achieved
from skyrmions.

J. M. Eisenberg and G. Kälbermann,

The applications of skyrmions to the derivation of the nucleon–nucleon force is reviewed with attention to the use of the product ansatz, additional terms in the lagrangian, baryon resonance admixtures, the instanton ansatz, dilatons, and exact two- or three-dimensional solutions for the B = 2 system in order to find the sources of attraction in the central and spin–orbit potentials. We also discuss extensions to two-baryon systems with nonzero strangeness and address possible insights into the behavior of the nucleon in nuclei achieved from the skyrmion approach.

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J. M. Eisenberg and G. Kälbermann,

The applications of skyrmions to the derivation of the nucleon–nucleon
force is reviewed with attention to the use of the product ansatz, additional
terms in the lagrangian, baryon resonance admixtures, the instanton ansatz,
dilatons, and exact two- or three-dimensional solutions for the *B* = 2
system in order to find the sources of attraction in the central and spin–orbit
potentials.

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G. Kälbermann and J. M. Eisenberg,

Within the skyrmion approach for the nucleon–nucleon force, difficulties have been experienced in obtaining an isoscalar attractive spin–orbit potential, in parallel to the problems of finding attraction in the isoscalar central potential. We here study the spin–orbit force using a skyrmion with four- and six-derivative stabilizing terms in the lagrangian as well as with the crucial addition of a dilaton. With these features present the spin–orbit force proves to be attractive as does the central potential. In the absence of the dilaton, attraction can also be found for the spin–orbit potential but only at the expense of a greatly over-emphasized term with six derivatives and a continuing absence of attraction in the central potential.

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G. Kälbermann and J. M. Eisenberg,

Earlier work reported on the existence of a term within a generalized skyrmion approach that yields appreciable spin content for the proton. Unfortunately there is no accessible experiment that can fix the coefficient of this term directly; plausible but highly uncertain values for it gave a result for the spin content loosely consistent with the currently measured ΔΣ = 0.273±0.13. We here attempt to narrow the range of values for this coefficient by performing global fits to all the parameters of the generalized Skyrme lagrangian while requiring reasonable results for the baryon octet and decuplet masses and octet magnetic moments. This requirement fixes the coefficient loosely, and we find that parameter sets that fit the baryon masses and magnetic moments yield proton spin content near ΔΣ ~ 0.15.

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G. Kälbermann, J. M. Eisenberg, and A. Schäfer,

It is well known that in lowest order the skyrmion model of the nucleon gives vanishing spin content. With new data indicating a proton spin content ΔΣ = 0.22±0.14, it is an increasing challenge to find ways in which the skyrmion can move away from the null result. We show here that a particular term in the skyrmion lagrangian in SU(3) involving six derivatives of the field can, with plausible parameters, yield a spin content consistent with present experiment.

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