Baym–Kadanoff Conservation Laws and Approximation Validity

Baym–Kadanoff consistency is a precise conservation statement, not a general certificate of accuracy. A self-energy, propagator, and response vertex derived from one invariant functional obey selected continuity equations; other properties require separate tests. This page assembles those tests into a validity assessment for approximate many-body Green functions. The scope of the conservation theorem is Baym 1962, pp. 1393–1397, while misleading skeleton-series branches are exhibited by Kozik, Ferrero, and Georges 2015.

Required background. Conserving Approximations and Φ-Derivable Functionals states the functional construction, and Current Vertices and Ward-Consistent Response gives the Ward identity.

Helpful background. Functional-Method Validation and Error Control supplies the broader validation framework used to separate formal consistency from numerical accuracy.

Conservation theorem and hypotheses

Choose a skeleton functional $\Phi[G]$ invariant under the continuous transformation associated with a conserved quantity, define

$$\Sigma=\frac{\delta\Phi}{\delta G},$$

and solve Dyson’s equation self-consistently. The resulting collision terms conserve particle number and, when the microscopic action and regulator have the relevant symmetries, momentum and energy. Differentiating the same self-energy produces the response kernel

$$I=\frac{\delta\Sigma}{\delta G}, \qquad \Gamma=\gamma+I\,GG\,\Gamma,$$

which satisfies the matching Ward identity. The theorem does not apply unchanged if the iteration is stopped away from self-consistency, counterterms violate the symmetry, or response is evaluated with an unrelated vertex.

Independent validity dimensions

Evaluate an approximation along distinct axes:

Property	Question	Representative check
Conservation	Do symmetry charges satisfy continuity equations?	$q_\mu K^{\mu\nu}=0$; conserved collision integral
One-particle causality	Is $G^R$ analytic with nonnegative $A$?	pole location and $A(\omega)\ge0$
Exact moments	Is total and weighted spectral strength correct?	$\int A/(2\pi)=1$; nested-commutator moments
Thermodynamics	Do derivative and response routes agree?	$V^{-1}\partial^2\Omega/\partial\mu^2$ versus static $\chi_{nn}$
Crossing	Are equivalent leg permutations consistent?	exchange and channel-crossing identities
Numerical control	Is the solution stable?	regulator, grid, basis, tolerance, and branch variation
Physical accuracy	Does it match controlled limits or data?	weak-coupling coefficients, benchmarks, experiment

Passing one row does not imply the others. In particular, conservation neither enforces spectral positivity nor supplies a bound on omitted vertex diagrams.

Example: self-consistent second Born

For a two-body interaction, the second-Born functional retains Hartree–Fock and the second-order skeletons built from full $G$. Differentiation produces the corresponding self-energy; iterating it is conserving. A density response must use the derivative kernel, which includes variations of every internal full propagator in the second-order self-energy.

Using the converged $G$ in a single bare bubble is cheaper but is not the Baym–Kadanoff response. Conversely, solving the consistent vertex equation can preserve the $f$-sum while the spectral lines remain quantitatively inaccurate at strong coupling.

Branches, convergence, and formal solutions

Nonlinear Dyson equations can have multiple stationary solutions. Convergence of an iteration proves only that a fixed point of that algorithm was reached. Compare grand potentials where the functional is trustworthy, continue from controlled parameter regions, test stability to perturbations, and disclose branch dependence.

At strong coupling, diagrammatic functionals can have nontrivial analytic structure and misleading branches. A formal skeleton series should therefore be checked against exactly solvable limits, sign-problem-free numerics where available, and observable-level constraints rather than accepted on diagram order alone.

A practical stopping rule

Do not attach a quantitative uncertainty to a conserving approximation without a variation that probes its dominant omission. Depending on the problem, compare vertex levels, self-consistency schemes, diagram orders, regulators, or an external benchmark. If those variations are not stable in the target regime, restrict the claim to the conservation property and qualitative trends actually demonstrated.

Common pitfalls

Calling a dressed bubble conserving. Conservation requires the response vertex generated by the self-energy, not dressed propagators alone.

Confusing fixed-point convergence with physical uniqueness. Different initial conditions can reach different branches of the same nonlinear equations.

Using a Ward identity as a complete validation. Ward consistency tests a symmetry relation; it does not establish crossing, positivity, or quantitative accuracy.

Exercises

Classify two approximations

Approximation A solves a Φ-derived Dyson equation self-consistently but uses a bare response vertex. Approximation B uses $G_0$ once in the same self-energy. Which theorem hypotheses fail?

Solution

A satisfies the stationary one-particle construction but its response omits $\delta\Sigma/\delta G$, so the response Ward identity is not guaranteed. B is not evaluated at the self-consistent stationary propagator, so the Baym conservation proof does not apply even before response is considered.

Separate conservation and accuracy

Give a result that can be exact in a conserving approximation while the spectrum is poor.

Solution

The integrated particle-number continuity equation or a corresponding sum rule can be exact because it follows from symmetry of $\Phi$. Peak positions, widths, and satellite weights can still have large errors because conservation does not fix their detailed frequency distribution.

Continue

Many-Body Green Functions and Response summarizes the correlator-to-response workflow. Universal Relations and Tan Contact gives nonperturbative short-distance checks. From Few-Body Inputs to Many-Body Predictions applies independent error sources to resonant gases.

References

• Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127 (1962): 1391–1401. DOI.
• Kozik, Evgeny, Michel Ferrero, and Antoine Georges. “Nonexistence of the Luttinger–Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models.” Physical Review Letters 114 (2015): 156402. DOI.

Further reading

• Baym, Gordon, and Leo P. Kadanoff. “Conservation Laws and Correlation Functions.” Physical Review 124 (1961): 287–299. DOI.
• Luttinger, J. M., and J. C. Ward. “Ground-State Energy of a Many-Fermion System. II.” Physical Review 118 (1960): 1417–1427. DOI.