Numerical Kadanoff–Baym Evolution and Validation

A trustworthy Kadanoff–Baym computation is a convergence study wrapped around a causal two-time solver. A concrete time-propagation construction is given by Stan, Dahlen, and van Leeuwen 2009, §§ II–III. Stability at one timestep is not enough: time discretization, spatial grid and UV cutoff, memory window, initial correlations, renormalization, finite volume, and self-energy truncation probe different errors and must be varied independently.

Required background. Use the Kadanoff–Baym equations for causal limits, initial correlations for preparation terms, and 2PI truncations for the closure and conserved functional.

Causal storage and marching

For a homogeneous system, store $F_{\mathbf p}(t_i,t_j)$ on the triangular domain $0\le j\le i\le N_t$ and reconstruct its transpose by symmetry. Store $\rho$ similarly using antisymmetry. At a new time $t_{n+1}$:

1. predict the new row from the differential equations;
2. evaluate local masses and self-energies consistently with that predictor;
3. compute memory quadratures only over their causal domains;
4. correct the new row until the nonlinear residual meets tolerance;
5. impose diagonal canonical data for $\rho$ through a scheme consistent with the integrator; and
6. update the conserved energy or charge from the same truncation.

Full history requires $O(N_t^2N_p)$ storage and typically at least quadratic time work. Symmetry halves storage but not the physical memory. Low-rank compression, adaptive history, or a finite memory window changes the approximation and needs its own convergence test.

Verification hierarchy

Algebraic tests. At each step verify $F_{ij}=F_{ji}$, $\rho_{ij}=-\rho_{ji}$, $\rho_{ii}=0$, and the near-diagonal derivative sum rule. Unit tests should evaluate the same memory convolution both by branch components and by the $F/\rho$ formula on a small artificial kernel.

Exactly solvable tests. With self-energies off, reproduce free oscillators and mass quenches. For a quadratic time-dependent Hamiltonian, compare against direct mode-function evolution and its Wronskian. These tests isolate causal limits and integrator errors from closure physics.

Conservation tests. Evaluate the explicit 2PI energy, including memory contributions. Plot its relative drift against $\Delta t$ on a log–log scale and recover the integrator’s expected order before reducing any other error. Small instantaneous residuals do not substitute for bounded long-time drift.

Continuum and volume tests. Change spatial spacing and physical volume separately. At fixed renormalized parameters, repeat with a higher UV cutoff and refit counterterms according to the declared scheme. Moving both cutoff and bare parameters without holding renormalized observables fixed is not a continuum test.

Closure tests. Compare successive controlled orders—loop, $1/N$, coupling, or nPI level—on the same initial state and regulator. Numerical convergence solves the chosen equations; it says nothing about the omitted skeletons.

The canonical record is the conservation and numerical validation matrix.

A reproducible convergence design

Choose a scalar target such as the damping time extracted from $F_{\mathbf p=0}(t,t)$ over a preregistered fit window, and a functional target such as the full spectral correlator over $0\le t-t'\le s_{\max}$. Build a reference grid by varying one control at a time:

Independent numerical controls for a two-time calculation

Control	Hold fixed	Compare	Typical hidden coupling
Timestep $\Delta t$	Grid, cutoff, memory duration, closure	Observed order, sum rule, energy drift	A memory window specified in number of steps changes physical duration.
Spatial grid or momentum cutoff	Renormalized parameters, volume, timestep	Local observables and spectral moments	Bare parameters must change with the regulator.
Physical volume	Spacing, renormalized parameters	Mode sums, recurrence time	Changing point count alone may change both volume and cutoff.
Memory duration	Timestep and total simulated time	Energy, unequal-time tails, fitted rates	Kernel amplitude can be small while its oscillatory integral remains important.
Initial cumulants	One- and two-point data, bulk closure	Initial slip and late observables	Different preparations may not represent the same physical state.
Truncation order	All numerical controls and renormalization conditions	Conserved and nonconserved observables	Each order may require a distinct counterterm solution.

Report the raw sequence, not only the extrapolated value. A useful error estimate is the envelope of the last resolved changes from numerical controls, kept separate from the difference between truncation orders.

Memory-cutoff diagnostic

Let $O_{\tau_m}(t)$ be an observable computed with history duration $\tau_m$. Define

$$\epsilon_m=\max_{t\in[t_a,t_b]} \frac{|O_{2\tau_m}(t)-O_{\tau_m}(t)|} {O_{\mathrm{scale}}}.$$

The scale must be specified and nonzero. Repeat for the conserved energy and $\rho$ sum rule. Convergence of one occupation estimator while energy continues to drift is insufficient. If the kernel has a power-law tail, no finite sequence of short windows should be extrapolated exponentially without evidence.

Benchmark boundaries

Free theory checks normalization but not collision quadrature. A large-$N$ or exactly diagonalizable finite system can test scattering and memory, but match volume, initial covariance, coupling normalization, and observables. A Boltzmann equation is a comparison for the kinetic regime, not ground truth for the two-time theory. Agreement obtained only after fitting a relaxation time is calibration, not prediction.

“Thermalization” should be reserved for a joint result: stationary correlators, KMS consistency, common temperature, conserved charges, volume stability, and truncation control. Visually smooth approach to a Bose–Einstein-shaped curve establishes none of these alone.

Exercise

A second-order solver gives energy drifts $8.0\times10^{-4}$, $2.1\times10^{-4}$, and $5.5\times10^{-5}$ as $\Delta t$ is halved twice. What does this test establish, and what does it not?

Solution

Each halving reduces the drift by factors about $3.8$, consistent with second-order temporal convergence. It supports the time integrator and energy evaluation in that range. It does not test the UV cutoff, volume, memory duration, initial-state consistency, renormalization, or accuracy of the self-energy truncation.

Continue

Only after this validation should the full two-time solution be reduced toward Wigner transport or interpreted through spectral and occupation diagnostics.

References

• Aarts, G., and Berges, J. (2001). “Nonequilibrium Time Evolution of the Spectral Function in Quantum Field Theory.” Physical Review D 64, 105010. arXiv:hep-ph/0103049; DOI.
• Stan, A., Dahlen, N. E., and van Leeuwen, R. (2009). “Time Propagation of the Kadanoff–Baym Equations for Inhomogeneous Systems.” Journal of Chemical Physics 130, 224101. DOI.