Nonequilibrium Green Functions and Kadanoff–Baym Evolution

Nonequilibrium Green functions track how excitations and their populations evolve together. Their two-time equations retain memory and initial correlations; controlled truncations make this dynamics computable without automatically guaranteeing accuracy.

Enter this chapter

Begin with Two-Time Green Functions and Self-Energies to identify the statistical and spectral functions needed for two-time evolution.

Helpful background. Two-Time Green Functions and Self-Energies and 2PI and nPI Effective Actions develop the definitions used by the chapter’s opening pages.

Topics

1. Two-Time Green Functions and Self-Energies — Defines the correlators and self-energies needed to close a nonequilibrium problem.
2. 2PI Effective Actions and Conserving Truncations — Derives self-consistent self-energies from a two-particle-irreducible functional.
3. Kadanoff–Baym Evolution Equations — Resolves contour evolution into causal equations with memory integrals.
4. Initial Correlations and Boundary Terms — Represents higher initial correlations as boundary interactions.
5. Memory Kernels in Closed-System Evolution — Explains how nonlocal self-energies retain scattering history.
6. Wigner Transformation and Gradient Expansion — Rewrites pair-time dependence in center and relative coordinates before approximating gradients.
7. Spectral and Statistical Evolution — Connects changing spectra to occupations and coherent correlations.
8. Numerical Kadanoff–Baym Evolution and Validation — Tests time stepping, memory truncation, conservation, and convergence.

How the ideas fit together

A self-energy determines both the available excitations and their evolution, so changing the statistical function while freezing an inconsistent spectrum can spoil the approximation. A symmetry-preserving two-particle-irreducible truncation gives self-consistent conservation laws, but conservation alone does not establish accuracy or complete gauge identities Berges 2004, §§ 2–3.

The Wigner transform is an exact change of variables. A kinetic equation requires additional gradient, memory, and spectral-shell approximations. Each removes information, which is why convergence of a two-time solver and validity of a kinetic reduction are different checks.

Where to go next

Continue to Critical and Stochastic Dynamics to relate a growing correlation length to a diverging relaxation time.

References

• Berges, J. (2004). “Introduction to Nonequilibrium Quantum Field Theory.” AIP Conference Proceedings 739, 3–62. arXiv:hep-ph/0409233; DOI.