Response, Transport, and Inference

Transport is not a number read directly from a correlator. It is a statement about a specified operator and source, after contact and magnetization terms have been fixed, a frequency–momentum limit has been chosen, conserved overlaps have been separated, and the surviving low-frequency information has been shown to be identifiable. This chapter builds that statement from linear response through uncertainty-aware inference.

Enter this chapter

The chapter uses the retarded convention

$$G^R_{AB}(t,\mathbf x)=-i\theta(t)\langle[A(t,\mathbf x),B(0)]\rangle, \qquad \rho_{AB}(\omega,\mathbf k)=-2\,\operatorname{Im}G^R_{AB}(\omega,\mathbf k).$$

For a Hamiltonian perturbation $H\mapsto H-\int f_B B$, the physical response kernel is $\mathcal R_{AB}=-G^R_{AB}$ before source-dependent contact terms are added. Frequencies and momenta follow the site’s forward $+ip\cdot x$ and inverse $-ip\cdot x$ Fourier convention; hydrodynamic modes therefore vary as $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$. These signs are fixed once here because an unnoticed convention change reverses spectral slopes and Kubo formulae.

Readers should already be comfortable with background sources and Ward identities. Closed-time-path differentiation is especially useful for keeping causal orderings and contacts straight; the opening page supplies the working derivation.

Routes through response and inference

If the question is…	Begin with	Continue with
What is the retarded object and which limit defines the coefficient?	Sources, Linear Response, and Kubo Formulae	Contact Terms, Magnetization Currents, and Order of Limits
How does a quadratic or higher response become a causal multipoint function?	Nonlinear Response and Higher-Order Kubo Relations	Return to the source and contact conventions before taking zero-frequency limits.
Is a narrow feature a delta function, a relaxation peak, a pole, or a continuum?	Spectral Functions and Transport Peaks	Transport Sum Rules and Ultraviolet Constraints
How are charge and heat diffusion related to measured conductivities?	Diffusion, Conductivity, and Susceptibility	Check momentum overlap and the open- or closed-circuit condition.
Which stress channel defines shear or bulk viscosity?	Shear and Bulk Viscosity	Check pressure contacts, sound mixing, and critical slow modes.
Which nearly conserved operators control a broad incoherent system?	Memory Functions and Slow-Mode Projection	Compare the chosen slow basis with kinetic or hydrodynamic modes.
Does the available dataset determine a value, an interval, or only an integral?	Transport Extraction, Inverse Problems, and Error Budgets	Report covariance, resolution, prior sensitivity, and model discrepancy with the claim.

The transport-extraction covariance reference gives a common record for comparing these routes. It is not an extraction algorithm: it states what must be disclosed for two transport claims to be scientifically comparable.

A transport claim has four layers

The definition layer specifies the operator, its source, normalization, tensor projector, thermodynamic variables held fixed, and the complete response including local terms. The dynamical layer specifies poles, cuts, exact conserved overlaps, and the order of $\omega\to0$, $k\to0$, volume, and regulator limits. The measurement layer specifies what the data actually constrain, including covariance and resolution. The interpretation layer states whether the result is a coefficient, a broad interval, a moment, a bound, or merely compatibility with an ansatz.

Confusing these layers creates familiar false conclusions: a hydrostatic circulation is called a transport current; a finite-volume line is called a Drude peak; a prior-dominated continuation is reported as a precise viscosity; or a scalar Einstein relation is applied to coupled charge–heat diffusion. Each page supplies an independent check that catches one of these errors.

The route combines the classic response construction Kubo 1957, §§2–3, hydrodynamic correlation and projection methods Forster 1995, chs. 3–6 Mori 1965, §§2–4, and the resolution-aware transport perspective reviewed by Meyer 2011, §§2–4, Open PDF.

Evidence ceiling

Exact Ward identities and sum rules constrain response without choosing a microscopic approximation, but they rarely reconstruct an entire spectral function. Hydrodynamics controls sufficiently small $\omega$ and $k$ after its slow variables are complete; kinetic theory and memory matrices require their own truncations. Euclidean continuation is an ill-conditioned inverse problem. Accordingly, a valid result must be no stronger than the weakest link joining definition, forward model, data resolution, and systematic-error control.

Review the chapter

1. Starting from a source coupling, can you state the sign relating $G^R$ to the physical response and identify every contact term?
2. Can you explain why $k\to0$ before $\omega\to0$ need not equal the reverse order?
3. Can you distinguish an exact delta function from a narrow resolved peak and from an unresolved family of spectra?
4. For coupled densities, can you construct the susceptibility and conductivity matrices and identify the basis-invariant diffusion rates?
5. Can you state which combination of a spectral function is actually identifiable after covariance, prior, resolution, and model discrepancy are propagated?

References

• Forster, Dieter. 1995. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions. Advanced Book Classics. Boca Raton, FL: CRC Press. DOI.
• Kubo, Ryogo. 1957. “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems.” Journal of the Physical Society of Japan 12 (6): 570–586. DOI.
• Meyer, Harvey B. 2011. “Transport Properties of the Quark–Gluon Plasma: A Lattice QCD Perspective.” European Physical Journal A 47: 86. DOI. Open PDF.
• Mori, Hazime. 1965. “Transport, Collective Motion, and Brownian Motion.” Progress of Theoretical Physics 33 (3): 423–455. DOI.