Nonequilibrium Green Functions and Kadanoff–Baym Evolution

Nonequilibrium Green-function theory evolves correlations at two independent times. It is therefore more general than a Boltzmann equation, but also more demanding: a result is meaningful only after the self-energy closure, initial correlations, renormalization, memory treatment, spectral normalization, and numerical convergence have all been exposed. This chapter develops that chain without assuming quasiparticles or time-translation invariance.

From exact two-time identities to controlled reductions

Start with two-time Green functions and self-energies to separate local mass terms from nonlocal collision kernels. 2PI effective actions then provide self-consistent closures and a precise, limited conservation theorem. The Kadanoff–Baym equations turn the contour Dyson equation into causal initial-value equations for the statistical and spectral correlators.

The remaining pages each test one reduction or input:

• initial correlations determine which boundary vertices and counterterms enter at the preparation time;
• memory kernels identify when a closed-system history integral may be shortened or localized;
• the Wigner and gradient expansion states the small parameter needed to pass toward transport theory;
• spectral and statistical evolution separates unequal-time coherence from any quasiparticle occupation number; and
• numerical validation supplies the convergence and benchmark tests required before interpreting apparent damping or thermalization.

Reading routes through two-time nonequilibrium evolution

Question	Start with	Necessary check	Stop condition
What is being evolved?	Two-time functions	Contour, sign, and equal-time conventions	Do not Fourier transform in relative time before approximate stationarity is demonstrated.
Is the closure conserving?	2PI truncations	Invariant functional and renormalized stationary equation	Conservation alone does not establish Ward identities, gauge independence, crossing, or accuracy.
Can memory be reduced?	Memory and Wigner pages	Correlation-time and gradient studies	Long tails, narrow widths, finite-time edges, or rapid phases invalidate a local kinetic reduction.
Is the computation trustworthy?	Numerical validation	Independent step, grid, cutoff, memory, and truncation studies	A smooth curve or small solver residual is not a continuum or closure test.

Conventions and scope

For a real scalar field this chapter uses

$$F(x,y)=\frac12\langle\{\phi(x),\phi(y)\}\rangle, \qquad \rho(x,y)=i\langle[\phi(x),\phi(y)]\rangle .$$

Thus $F(y,x)=F(x,y)$, $\rho(y,x)=-\rho(x,y)$, and canonical normalization gives $\rho(t,t)=0$ and $\partial_t\rho(t,t')|_{t=t'}=1$ mode by mode. Other sign conventions are common; importing an equation without transforming all propagators, self-energies, and boundary conditions is a frequent source of wrong signs. Spatial homogeneity is introduced only in checked examples, not in the formal derivations.

The chapter treats closed-system two-point evolution. Open-system influence kernels, stochastic unravellings, gauge-covariant kinetic theory, and downstream phenomenology require additional assumptions and are not consequences of a two-time solution alone.

References

• Baym, G. (1962). “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127, 1391–1401. DOI.
• Berges, J. (2004). “Introduction to Nonequilibrium Quantum Field Theory.” AIP Conference Proceedings 739, 3–62. arXiv:hep-ph/0409233; DOI.
• Kadanoff, L. P., and Baym, G. (1962). Quantum Statistical Mechanics. New York: W. A. Benjamin. Internet Archive record.