We extend a path-integral approach to bosonization previously developed in the framework of equilibrium Quantum Field Theories, to the case in which time-dependent interactions are taken into account. In particular we consider a noncovariant version of the Thirring model in the presence of a dynamic barrier at zero temperature. By using the Closed Time Path (Schwinger–Keldysh) formalism, we compute the Green's function and the Total Energy Density of the system. Since our model contains the Tomonaga–Luttinger model as a particular case, we make contact with recent results on nonequilibrium electronic systems.
We extend a path-integral approach to bosonization previously developed in the framework of equilibrium Quantum Field Theories, to the case in which time-dependent interactions are taken into account. In particular we consider a noncovariant version of the Thirring model in the presence of a dynamic barrier at zero temperature. By using the Closed Time Path (Schwinger–Keldysh) formalism, we compute the Green's function and the Total Energy Density of the system. Since our model contains the Tomonaga–Luttinger model as a particular case, we make contact with recent results on nonequilibrium electronic systems.
We extend a path-integral approach to bosonization previously developed in the framework of equilibrium Quantum Field Theories, to the case in which time-dependent interactions are taken into account. In particular we consider a non covariant version of the Thirring model in the presence of a dynamic barrier at zero temperature. By using the Closed Time Path (Schwinger-Keldysh) formalism, we compute the Green's function and the Total Energy Density of the system. Since our model contains the Tomonaga Luttinger model as a particular case, we make contact with recent results on non-equilibrium electronic systems.
We present a path-integral bosonization approach for systems out of equilibrium based on a duality transformation of the original Dirac fermion theory combined with the Schwinger-Keldysh time closed contour technique, to handle the non-equilibrium situation. The duality approach to bosonization that we present is valid for D 2 space-time dimensions leading for D = 2 to exact results. In this last case we present the bosonization rules for fermion currents, calculate current-current correlation functions and establish the connection between the fermionic and bosonic distribution functions in a generic, nonequilibrium situation.
This chapter examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).
The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function (free energy).
The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function ("free energy").