Zero mode in the time-dependent symmetry breaking of λϕ4\lambda\phi^4 theory

We apply the quartic exponential variational approximation to the symmetry breaking phenomena of scalar field in three and four dimensions. We calculate effective potential and effective action for the time-dependent system by separating the zero mode from other non-zero modes of the scalar field and treating the zero mode quantum mechanically. It is shown that the quantum mechanical properties of the zero mode play a non-trivial role in the symmetry breaking of the scalar $\lambda \phi^4$ theory.Comment: 10 pages, 3 figure

An O(N) symmetric extension of the Sine-Gordon Equation

We discuss an O(N) exension of the Sine-Gordon (S-G)equation which allows us to perform an expansion around the leading order in large-N result using Path-Integral methods. In leading order we show our methods agree with the results of a variational calculation at large-N. We discuss the striking differences for a non-polynomial interaction between the form for the effective potential in the Gaussian approximation that one obtains at large-N when compared to the N=1 case. This is in contrast to the case when the classical potential is a polynomial in the field and no such drastic differences occur. We find for our large-N extension of the Sine-Gordon model that the unbroken ground state is unstable as one increases the coupling constant (as it is for the original S-G equation) and we determine the stability criteria.Comment: 21 pages, Latex (Revtex4) v3:minor grammatical changes and addition

Mean field theory for collective motion of quantum meson fields

Mean field theory for the time evolution of quantum meson fields is studied in terms of the functional Schroedinger picture with a time-dependent Gaussian variational wave functional. We first show that the equations of motion for the variational wavefunctional can be rewritten in a compact form similar to the Hartree-Bogoliubov equations in quantum many-body theory and this result is used to recover the covariance of the theory. We then apply this method to the O(N) model and present analytic solutions of the mean field evolution equations for an N-component scalar field. These solutions correspond to quantum rotations in isospin space and represent generalizations of the classical solutions obtained earlier by Anselm and Ryskin. As compared to classical solutions new effects arise because of the coupling between the average value of the field and its quantum fluctuations. We show how to generalize these solutions to the case of mean field dynamics at finite temperature. The relevance of these solutions for the observation of a coherent collective state or a disoriented chiral condensate in ultra-relativistic nuclear collisions is discussed.Comment: 31 pages, 2 Postscript figures, uses ptptex.st

Semiclassical Symmetries

Essential properties of semiclassical approximation for quantum mechanics are viewed as axioms of an abstract semiclassical mechanics. Its symmetry properties are discussed. Semiclassical systems being invariant under Lie groups are considered. An infinitesimal analog of group relation is written. Sufficient conditions for reconstructing semiclassical group transformations (integrability of representation of Lie algebra) are discussed. The obtained results may be used for mathematical proof of Poincare invariance of semiclasical Hamiltonian field theory and for investigation of quantum anomalies.Comment: LaTeX, 33 page

Quantum Decoherence, Entropy and Thermalization in Strong Interactions at High Energy

Entropy is generated in high-multiplying events by a dynamical separation of strongly interacting systems into partons and unobservable environment modes (almost constant field configurations) due to confinement.Comment: 45 pages, 2 figure

Effective action and the quantum equation of motion

We carefully analyse the use of the effective action in dynamical problems, in particular the conditions under which the equation \frac{\delta \Ga} {\delta \phi}=0 can be used as a quantum equation of motion, and the relation between the asymptotic states involved in the definition of \Ga and the initial state of the system. By considering the quantum mechanical example of a double-well potential, where we can get exact results for the time evolution of the system, we show that an approximation to the effective potential in the quantum equation of motion that correctly describes the dynamical evolution of the system is obtained with the help of the wilsonian RG equation (already at the lowest order of the derivative expansion), while the commonly used one-loop effective potential fails to reproduce the exact results.Comment: 28 pages, 13 figures. Revised version to appear in The European Physical Journal

Resumming the large-N approximation for time evolving quantum systems

In this paper we discuss two methods of resumming the leading and next to leading order in 1/N diagrams for the quartic O(N) model. These two approaches have the property that they preserve both boundedness and positivity for expectation values of operators in our numerical simulations. These approximations can be understood either in terms of a truncation to the infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a particular two-particle irreducible vacuum energy graph in the effective action of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the case of quantum mechanics where the Lagrangian is $L(x,\dot{x}) = (1/2) \sum_{i=1}^{N} \dot{x}_i^2 - (g/8N) [ \sum_{i=1}^{N} x_i^2 - r_0^2 ]^{2}$. The key to these approximations is to treat both the $x$ propagator and the $x^2$ propagator on similar footing which leads to a theory whose graphs have the same topology as QED with the $x^2$ propagator playing the role of the photon. The bare vertex approximation is obtained by replacing the exact vertex function by the bare one in the exact Schwinger-Dyson equations for the one and two point functions. The second approximation, which we call the dynamic Debye screening approximation, makes the further approximation of replacing the exact $x^2$ propagator by its value at leading order in the 1/N expansion. These two approximations are compared with exact numerical simulations for the quantum roll problem. The bare vertex approximation captures the physics at large and modest $N$ better than the dynamic Debye screening approximation.Comment: 30 pages, 12 figures. The color version of a few figures are separately liste

Schwinger-Dyson approach to non-equilibrium classical field theory

In this paper we discuss a Schwinger-Dyson [SD] approach for determining the time evolution of the unequal time correlation functions of a non-equilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0$. We focus on $\lambda \phi^4$ field theory in 1+1 space time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation [BVA], is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for 1/N expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\phi$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.Comment: 36 pages, 5 figure

Out-of-equilibrium evolution of quantum fields in the hybrid model with quantum back reaction

The hybrid model with a scalar "inflaton" field coupled to a "Higgs" field with a broken symmetry potential is one of the promising models for inflation and (p)reheating after inflation. We consider the nonequilibrium evolution of the quantum fields of this model with quantum back reaction in the Hartree approximation, in particular the transition of the Higgs field from the metastable "false vacuum" to the broken symmetry phase. We have performed the renormalization of the equations of motion, of the gap equations and of the energy density, using dimensional regularization. We study the influence of the back reaction on the evolution of the classical fields and of the quantum fluctuations. We observe that back reaction plays an important role over a wide range of parameters. Some implications of our investigation for the preheating stage after cosmic inflation are presented.Comment: 35 pages, 16 eps figures, revtex4; v2: typos corrected and references added, accepted for publication in Physical Review

Renormalization of Poincare Transformations in Hamiltonian Semiclassical Field Theory

Semiclassical Hamiltonian field theory is investigated from the axiomatic point of view. A notion of a semiclassical state is introduced. An "elementary" semiclassical state is specified by a set of classical field configuration and quantum state in this external field. "Composed" semiclassical states viewed as formal superpositions of "elementary" states are nontrivial only if the Maslov isotropic condition is satisfied; the inner product of "composed" semiclassical states is degenerate. The mathematical proof of Poincare invariance of semiclassical field theory is obtained for "elementary" and "composed" semiclassical states. The notion of semiclassical field is introduced; its Poincare invariance is also mathematically proved.Comment: LaTeX, 40 pages; short version of hep-th/010307