Field Theoretic Formulation of Kinetic Theory: Basic Development

We show how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. The need for a self-consistent approach is suggested by our interest in investigating ergodic-nonergodic transitions in dense fluids. The formal structure we develop has been implemented in detail for the simpler case of Smoluchowski dynamics. One aspect of the approach is the identification of a core problem spanned by the variables ?? the number density and B a response density. In this paper we set up the perturbation theory expansion with explicit development at zeroth and first order. We also determine all of the cumulants in the noninteracting limit among the core variables ?? and B.     

Resummation in hot field theories

There has been significant progress in our understanding of finite-temperature field theory over the past decade. In this paper, we review the progress in perturbative thermal field theory focusing on thermodynamic quantities. We first discuss the breakdown of naive perturbation theory at finite temperature and the need for an effective expansion that resums an infinite class of diagrams in the perturbative expansion. This effective expansion which is due to Braaten and Pisarski, can be used to systematically calculate various static and dynamical quantities as a weak-coupling expansion in powers of g. However, it turns out that the weak-coupling expansion for thermodynamic quantities are useless unless the coupling constant is very small. We critically discuss various ways of reorganizing the perturbative series for thermal field theories in order to improve its convergence. These include screened perturbation theory (SPT), hard-thermal-loop perturbation theory, the Φ-derivable approach, dimensionally reduced (DR) SPT, and the DR Φ-derivable approach.     

Instability of collapsing source under expansion-free condition in einstein Gauss–Bonnet gravity

In this work, we have investigated the dynamical instability of spherically symmetric gravitating object under expansion-free condition in Einstein Gauss–Bonnet gravity. In this context, the field equations and dynamical equations have been established in the Gauss–Bonnet gravity. The linear perturbation scheme has been used on the dynamical equations to construct the collapse equation. The Newtonian, post Newtonian and post Newtonian approximations have been applied to investigate the general dynamical (in)stability equations. It has been observed that the instability range of the collapsing source is independent of adiabatic index Γ (stiffness of the fluid does not play any role). The instability range can be determined by the pressure anisotropy, energy density profile, Gauss–Bonnet parameter α and some constraints at Newtonian, post Newtonian and post Newtonian order.     

Phase transition in the three-dimensional chiral field

The essentially nonlinear chiral field in three dimensional space time non-renormalizable in the usual perturbation expansion is studied. We consider systematic $\text{1}\text{N}$ expansion for this model. In the framework of the latter a phase transition takes place: above the critical point the theory is in the O(N) symmetric phase, below it the O(N) symmetry breaks. The $\text{1}\text{N}$ renormalized expansion for both phases is described and the connection between the non-renormalizability of the conventional perturbation theory and the non-analytic dependence on the coupling constant is established.     

Self-consistent perturbation theory for dynamics of valence fluctuations

A perturbation-theoretic scheme is developed for dynamics of valence fluctuations in rare-earth systems with unstable 4f shells. The theory is formulated in close analogy to the standard Green-function method for many-body systems but without use of the linked-cluster theorem. This formulation regards hybridization between 4f and conduction-band states as perturbation and naturally incorporates the strong on-site 4f-electron correlation. Some favorable features are: (i) the approximation scheme automatically satisfies conservation laws required for response functions; (ii) realistic 4f-shell structures with crystalline-electric-field effects can be taken into account; (iii) the theory does not have divergence difficulties over the whole temperature range. In the lowest-order self-consistent approximation, explicit formulae for dynamical susceptibilities and 4f-electron density of states are presented. At high temperatures, the theory reproduces previous results obtained by the Mori method.     

Das Konsistenzproblem der Greenfunktion eines A-Hyperons in Kernmaterie

An iteration procedure is derived for determining a self-consistent approximation for the one-particle Green's function in any order, starting from perturbation theory. This method is used to calculate the second Born approximation of the self energy of a Λ-particle in nuclear matter. After one iteration the approximation is nearly self-consistent. The value of the binding energy of the Λ-paiticle is only slightly decreased from the value of perturbation theory. Because of the vanishing particle density of the Λ-hyperon the quasiparticle energy depends on the square of the momentumk, the width of the spectral function is proportional tok ⁴ and the imaginary part of the self energy operator is proportional to (Ω-μ)^5/2 for Ω approaching the chemical potential μ.     

Perturbative calculation of critical exponents for the Bose–Hubbard model

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose–Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a ₆ into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose–Hubbard model, deviate by less than 1 % from the best known estimates computed so far for the three-dimensional XY universality class.     

Kinetic equations within the formalism of non-equilibrium thermo field dynamics

After reviewing the real-time formalism of dissipative quantum field theory, i.e. non-equilibrium thermo field dynamics (NETFD), a kinetic equation, a self-consistent equation for the dissipation coefficient and a “mass” or “chemical potential” renormalization equation for non-equilibrium transient situations are extracted out of the two-point Green's function of the Heisenberg field, in their most general forms upon the basic requirements of NETFD. The formulation is applied to the electron-phonon system, as an example, where the gradient expansion and the quasi-particle approximation are performed. The formalism of NETFD is reinvestigated in connection with the kinetic equations.     

The Vilkovisky-DeWitt effective action for quantum gravity

The Vilkovisky-DeWitt effective action for gauge theories is reviewed and then discussed in the context of N-dimensional quantum gravity and quantum Kaluza-Klein theory. The formalism gives an effective action which is gauge-independent and gauge and field-parametrization invariant. These features are illustrated for the vacuum energy of N-dimensional gravity. The Bunch-Parker local momentum space approach is used to calculate also the induced Ricci scalar term in the expansion of the effective action in powers of the curvature. The effective field equations are applied to the self-consistent dimensional reduction of five-dimensional Kaluza-Klein theory. A solution exists, but is found to be physically unacceptable.     

Finite width in out-of-equilibrium propagators and kinetic theory

We derive solutions to the Schwinger–Dyson equations on the Closed-Time-Path for a scalar field in the limit where backreaction is neglected. In Wigner space, the two-point Wightman functions have the curious property that the equilibrium component has a finite width, while the out-of equilibrium component has zero width. This feature is confirmed in a numerical simulation for scalar field theory with quartic interactions. When substituting these solutions into the collision term, we observe that an expansion including terms of all orders in gradients leads to an effective finite-width. Besides, we observe no breakdown of perturbation theory, that is sometimes associated with pinch singularities. The effective width is identical with the width of the equilibrium component. Therefore, reconciliation between the zero-width behaviour and the usual notion in kinetic theory, that the out-of-equilibrium contributions have a finite width as well, is achieved. This result may also be viewed as a generalisation of the fluctuation–dissipation relation to out-of-equilibrium systems with negligible backreaction.     

Instability ranges of collapsing star through adiabatic index in f(T, Θ) theory of gravity

The main purpose of this paper is to study the stability analysis of collapsing star in the scenario of Newtonian and post Newtonian approximations. We consider the cylindrical symmetry of collapsing object which is filled with anisotropic fluid. The f(T, Θ) theory of gravity, where T is the torsion scalar and Θ is the trace of energy-momentum tensor is taken into account. The dynamical field equations are constructed which help to derive collapse equation by applying the perturbation method. The stability behavior of collapsing star is defined in both Newtonian and post Newtonian approximations with the help of an adiabatic index.     

Nonperturbative Renormalization in Light-Front Dynamics and Applications

We present a general framework to calculate the properties of relativistic compound systems from the knowledge of an elementary Hamiltonian. Our framework provides a well-controlled nonperturbative calculational scheme which can be systematically improved. The state vector of a physical system is calculated in light-front dynamics. From the general properties of this form of dynamics, the state vector can be further decomposed in well-defined Fock components. In order to control the convergence of this expansion, we advocate the use of the covariant formulation of light-front dynamics. In this formulation, the state vector is projected on an arbitrary light-front plane ω·x =  0 defined by a light-like four-vector ω. This enables us to control any violation of rotational invariance due to the truncation of the Fock expansion. We then present a general nonperturbative renormalization scheme in order to avoid field-theoretical divergences which may remain uncancelled due to this truncation. This general framework has been applied to a large variety of models. As a starting point, we consider QED for the two-body Fock space truncation and calculate the anomalous magnetic moment of the electron. We show that it coincides, in this approximation, with the well-known Schwinger term. Then we investigate the properties of a purely scalar system in the three-body approximation, where we highlight the role of antiparticle degrees of freedom. As a non-trivial example of our framework, we calculate the structure of a physical fermion in the Yukawa model, for the three-body Fock space truncation (but still without antifermion contributions). We finally show why our approach is also well-suited to describe effective field theories like chiral perturbation theory in the baryonic sector.     

Zimmermann normal products and the Wilson expansion

The principal part and asymptotic form of the Wilson short distance expansion is derived in the framework of Bogoliubov-Parasiuk-Hepp-Zimmermann renormalized perturbation theory. Use is made of the Zimmermann normal product techniques with all work being performed in the scalar A⁴ model. The expansion of the product of four field operators is given as an explicit example.     

Spontaneous symmetry breaking in massless field theories,finite temperature and criticality

We study the behaviour at finite temperature of massless field theories exhibiting spontaneously broken solutions. We establish the occurence of a phase transition of the first kind at some critical point T_c which can be calculated to any finite order in perturbation theory. Similarly, perturbative methods can be used for thermodynamic functions in all regions, including the critical region. For the case of a gauge theory, we demonstrate the gauge independence of the critical point, the thermodynamic potentials and the order parameter to all orders of perturbation theory.     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

Remarks concerning the derivation and the expansion of the master equation

The present work consist of two parts: In the first part we apply the method of quasilinearization to the differential equation describing the time development of the quantum-mechanical probability density. In this way we derive the master equation without resorting to perturbation theory. In the second part of the paper, for a general form of the master equation which is an integro-differential equation, we test the accuracy of the Fokker-Planck approximation with the help of a solvable model. Then we study an alternative way of reducing the integro-differential equation to a partial differential equation. By expanding the transition probability W(q, q′), and the distribution function in terms of a complete set of functions, we show that for certain forms of W(q, q′), the master equation can be transformed exactly to partial differential equations of finite order.     

Wilson-Zimmermann expansions and the normal-product algorithm for the Ward identity of the axial-vector current in meson-nucleon dynamics

Zimmermann's normal-product algorithm is developed for the renormalizable γ₅ interaction of meson-nucleon dynamics. With the help of Zimmermann's algebraic identities Wilson's short-distance expansion and the Zimmermann identity are verified in perturbation theory for a fermion-fermion operator product. We demonstrate the power of this method in clarifying problems which seemed to appear in earlier, less rigorous, treatments of a generalized Ward-identity for the axial vector in such a field theory.     

Perturbation theory at large order in reggeon field theory

We estimate the behaviour of the perturbation expansion and the ? expansion at large orders in reggeon field theory. The perturbation expansion is divergent but Borel summable for α₀ < 1. At fixed total rapidity, the Borel summability extends into α₀ > 1.     

Loop quantum cosmology and spin foams

Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs). Specifically, it is shown that: (i) the physical inner product in the timeless framework equals the transition amplitude in the deparameterized theory; (ii) this quantity admits a vertex expansion a la SFMs in which the M-th term refers just to M volume transitions, without any reference to the time at which the transition takes place; (iii) the exact physical inner product is obtained by summing over just the discrete geometries; no ‘continuum limit’ is involved; and, (iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of group field theory. This sum over histories reformulation of LQC also addresses certain other issues which are briefly summarized.     

On the Landau diamagnetism

The grand-canonical partition function of an assembly of free spinless electrons in a magnetic field enclosed in a box (Dirichlet boundary conditions) is shown to be an entire function of the fugacityz and the magnetic fieldH, as a consequence of the trace-norm convergence of the perturbation series for the statistical semigroup. This allows to derive analyticity properties of the pressure as a function ofz andH, and to express the coefficients of its power series expansion aroundz=H=0 by means of the unperturbed semigroup. Hence, the magnetic susceptibility at zero field and fixed density is expressed in terms of Green functions of the heat equation. Its asymptotic expansion for Λ→∞ (Fisher) along parallelepipedic domains is obtained up to 0 (S(Λ)/V(Λ)). The volume term of this expansion is the Landau diamagnetism.