Standard derivations of the functional integral in non-equilibrium quantum field theory are based on the discrete time representation. In this work we derive the non-equilibrium functional integral for non-interacting bosons and fermions using a continuum time approach by accounting for the statistical distribution through the boundary conditions and using them to evaluate the Green's function.
Standard derivations of the functional integral in non-equilibrium quantum field theory are based on the discrete time representation. In this work we derive the non-equilibrium functional integral for non-interacting bosons and fermions using a continuum time approach by accounting for the statistical distribution through the boundary conditions and using them to evaluate the Green's function.
Abstract We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values γm,mˉc γ m , m ¯ c of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound δC δ C and decay constant αC α C of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.
Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
The Statistical Mechanics of the N-particle system is derived from Quantum Random Field Theory. An asymptotically measure preserving, flow operator has been obtained from the QFT evolution operator. A basic assumption is that the field Lagrangian is a generalized, infinitely divisible random field, and the interaction Hamiltonian varies in time in a step-wise manner at the transition lattice space-time points. Macroscopic boundary conditions are imposed on the wave functions. The density matrix, the partition function, the Bose-Einstein, the Fermi-Dirac distributions and the thermodynamic temperature follow. The equilibrium or non-equilibrium temperature have been obtained as functionals of the interaction Hamiltonian. The chemical potential has also been derived from the theory.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
Preface 1. Quantum fields 2. Operators on the multi-particle state space 3. Quantum dynamics and Green's functions 4. Non-equilibrium theory 5. Real-time formalism 6. Linear response theory 7. Quantum kinetic equations 8. Non-equilibrium superconductivity 9. Diagrammatics and generating functionals 10. Effective action 11. Disordered conductors 12. Classical statistical dynamics Appendices: A. Path integrals B. Retarded and advanced propagators C. Analytic properties of Green's functions Bibliography Index.