Thermal and Nonequilibrium QFT

Thermal and nonequilibrium quantum field theory asks how quantum fields behave in states with temperature, density, preparation history, driving, or environmental exchange—and which reduced description remains trustworthy at the scale of the question. This volume connects equilibrium ensembles and KMS structure to real-time contours, kinetic and stochastic reductions, hydrodynamics, thermalization, open dynamics, hot gauge plasmas, and QCD collision inference. Its organizing rule is simple: every conclusion names the state, observable, scale, approximation, validation test, and evidence supporting it.

A broad real- and imaginary-time synthesis with compatible domain warnings is given by Landsman and van Weert 1987, §§ 2–4, pp. 149–218.

Helpful background. Partition Functions and Thermodynamic Response supplies equilibrium generating functions. Closed-Time-Path Grammar supplies normalized in–in evolution. Beta Functions, Running Masses, and Field Anomalous Dimensions supplies running-coupling control. Power Counting and Predictive Order supplies scale separation and truncation.

Choose a description from the question

No single formalism is “the thermal theory.” Begin with the observable and its time and length scales.

Reader’s question	Natural first description	Essential assumption	Early warning that another description is needed
equilibrium pressure or susceptibility	Gibbs/KMS state and Euclidean generating functional	equilibrium state and controlled thermodynamic limit	phase coexistence, sign problem, or infrared scale invalidates the expansion
static screening or phase structure	thermal EFT or dimensionally reduced theory	separation from nonzero Matsubara modes	critical or magnetostatic retained sector becomes strongly coupled
causal response after a perturbation	retarded correlator on a closed time path	prepared initial state and causal source protocol	memory, initial correlations, or nonanalytic kernels cannot be neglected
evolution of two-point functions	Kadanoff–Baym equations	specified self-energy closure and renormalized initial data	Ward identities, conservation, or memory convergence fail
dilute or quasiparticle transport	kinetic equation	narrow shells, gradient expansion, and controlled collision kernel	widths, coherence, or correlations are of leading order
longest-wavelength conserved dynamics	hydrodynamics	separation between conserved and nonhydrodynamic timescales	instability, acausality, loss of hyperbolicity, or extra slow modes
stochastic critical dynamics	Langevin, Fokker–Planck, or MSRJD theory	justified coarse graining and noise calculus	colored, multiplicative, or non-Markovian noise changes the universality class
subsystem or driven evolution	influence functional or quantum master equation	declared system–environment split and reduction	non-Markovianity, positivity failure, or conditional/full dynamics are conflated
hot non-Abelian plasma	scale-specific HTL, EQCD/MQCD, ultrasoft, or kinetic EFT	an actual hierarchy among $T$, $gT$, and $g^2T$	adjacent scales overlap or the magnetic sector dominates
properties inferred from nuclear collisions	multistage forward model and uncertainty-aware inference	complete likelihood, covariance, priors, discrepancy, and validation	parameter degeneracy or stage dependence prevents identification

The table gives entry points, not equivalences. Euclidean data can define exact analytic functions under strong hypotheses, but finite noisy Euclidean samples do not by themselves determine transport. A hydrodynamic fit can constrain a model while leaving microscopic thermalization underdetermined. A completely positive master equation can be a consistent ansatz without being a microscopic derivation.

Six connected routes through the volume

Equilibrium, density, and thermal calculation

Chapter	What it establishes
Equilibrium States, Ensembles, and Thermodynamic Limits	ensembles, field configurations, response, correlations, fluctuations, phases, and limit order
KMS States, Imaginary Time, and Thermal Spectra	Gibbs/KMS equivalence in its domain, thermal boundary conditions, Matsubara sums, spectral representations, and continuation limits
Finite Density and Conserved Charges	chemical potentials, background holonomy, charged KMS relations, susceptibilities, superfluid onset, and access boundaries
Thermal Perturbation Theory and Renormalization	thermal scale counting, sum-integrals, state-independent counterterms, pressure, masses, cuts, widths, and infrared failure
Thermal EFT, Screening, and Resummation	matched modes, screening, ring and variational reorganizations, static reduction, dissipative matching, and double-counting tests
Thermal Phases, Metastability, and Nucleation	physical phase criteria, metastability, bounce and determinant factors, wall dynamics, and completion conditions

Microscopic real time and controlled reductions

Chapter	What it establishes
Real-Time Contours, Keldysh Bases, and Response	normalized contours, $+/-$ and $r/a$ bases, unitarity identities, response, KMS, and nonlinear sources
Nonequilibrium Green Functions and Kadanoff–Baym Evolution	two-time equations, self-energy closures, initial correlations, conservation, Wigner transforms, and numerical checks
Critical and Stochastic Dynamics	Langevin fields, probability evolution, response functionals, detailed balance, dynamic universality, quenches, and aging
Kinetic Theory and Transport Equations	distribution functions, shell and gradient limits, collisions, entropy production, closures, coherence, and breakdown

Hydrodynamics, fluctuations, and transport

Chapter	What it establishes
Hydrodynamic Variables, Frames, and Ideal Modes	conserved variables, local equilibrium, constitutive tensors, hydrostatics, frames, sound, shear, and charge modes
Relativistic Dissipation, Transients, Stability, and Causality	viscous constitutive data, conventional first order, transient theories, BDNK, stability, causality, hyperbolicity, and attractors
Fluctuating, Generalized, and Schwinger–Keldysh Hydrodynamics	fluctuation kernels, SK actions, long-time tails, anomalies, superfluids, higher-form sectors, integrability, spin, and Hydro+
Response, Transport, and Inference	Kubo relations, contact and magnetization terms, order of limits, spectral peaks, sum rules, diffusion, viscosity, and inverse uncertainty

Thermalization and open dynamics

Chapter	What it establishes
Thermalization, Integrability Breaking, and Quantum Chaos	dephasing, ETH and exceptions, prethermalization, nonthermal fixed points, operator spreading, OTOCs, chaos bounds, and spectral evidence
Open QFT and Driven Dynamics	influence functionals, master equations, Lindblad fields, memory, driven steady states, positivity, causal consistency, and renormalization

Hot gauge theory and QCD evidence

Chapter	What it establishes
Hot Gauge Theory and Plasma EFTs	$T$, $gT$, and $g^2T$ sectors, dimensional reduction, HTLs, collective modes, ultrasoft color, kinetic theory, transport, and instabilities
QCD Matter and Multistage Collision Inference	versioned equation-of-state and phase evidence, pre-equilibrium, hydrodynamization, particlization, electromagnetic and hard probes, heavy flavor, quarkonium, and global inference

Readers can move linearly, but several shorter routes are coherent:

• for a graduate thermal core: Chapters 1 → 2 → 4 → 5 → 11 → 14;
• for finite density: Chapters 1 → 2 → 3 → 14 → 18;
• for microscopic real time: closed-time-path foundations → Chapters 7 → 8 → 10 → 11;
• for stochastic and open fields: Chapters 1 → 9 → 13 → 16;
• for phase transitions: Chapters 1 → 4 → 5 → 6;
• for plasma and collision theory: Chapters 2 → 3 → 4 → 5 → 10 → 12 → 14 → 17 → 18.

Shared conventions

The site uses natural units and the Lorentzian metric

$$\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1).$$

Its Fourier pair is

$$\widetilde f(p)=\int\mathrm d^d x\,e^{+ip\cdot x}f(x), \qquad f(x)=\int\frac{\mathrm d^dp}{(2\pi)^d} e^{-ip\cdot x}\widetilde f(p),$$

so $\partial_\mu\mapsto-ip_\mu$. With

$$G_R(t,\mathbf x)=-i\theta(t) \langle[A(t,\mathbf x),B(0)]\rangle,$$

and $\rho$ the Fourier transform of the commutator, the corresponding Hermitian two-point convention gives $\rho=-2\operatorname{Im}G_R$. Any page using another community convention states the translation and checks an invariant response or sum rule.

The equilibrium density operator is proportional to

$$e^{-\beta(H-\sum_a\mu_aQ_a)},$$

and a chemical potential is licensed only for a conserved charge within the declared state. Bosonic and fermionic thermal boundary conditions, contour branch order, $r/a$ normalization, Wigner coordinates, stochastic calculus, hydrodynamic frame, and transport limits are stated where first used because they vary across subfields. Conventions and normalizations is the site-wide reference.

How strong is the conclusion?

The same formula can support different claims depending on its derivation and input.

Basis	What may be concluded	What still requires more evidence
exact identity	equality under stated definitions and existence assumptions	numerical value or regime relevance
theorem	conclusion under every listed hypothesis	extension beyond those hypotheses
controlled expansion or EFT	result through a stated order with a power-counted remainder	extrapolation when scales collide
reproducible computation	result for frozen equations, inputs, algorithms, and tolerances	continuum, model, or physical identification outside the tested domain
experimental or observational dataset	measured estimator with calibration, covariance, and selection stated	unique microscopic explanation
multistage inference	posterior or likelihood statement conditional on model, priors, emulator, covariance, and discrepancy	model-independent “measurement” of an unidentifiable parameter

Mutable numerical values, method comparisons, and open disputes belong in the dated Thermal and Nonequilibrium Field Theory research field. An executable check should keep its equations, inputs, tolerances, expected failures, and preserved outputs together so the result can be inspected independently. Preparation gaps can be repaired through Learn quantum field theory. These resources extend the durable exposition; they do not silently raise the strength of a page’s conclusion.

Boundaries that matter throughout

Euclidean access is not automatically real-time access. Exact continuation and finite noisy reconstruction are separate problems.

Hydrodynamization is not isotropization or thermalization. Each is defined by different observables and can occur on a different timescale.

A conserving truncation is not automatically gauge consistent. Ward identities and vertex consistency require independent tests.

A bounce exponent is not a nucleation rate. Fluctuation determinants, zero and negative modes, statistical and dynamical prefactors, and completion dynamics remain visible.

Conventional relativistic Navier–Stokes is not every first-order theory. Stability, causality, and hyperbolicity are formulation- and frame-dependent questions.

An OTOC is not proof of chaos. Regulator, operator, time window, finite-size scaling, and counterexamples must be checked against independent diagnostics.

A Lindblad ansatz is not a microscopic derivation. Complete positivity does not establish the weak-coupling, Markov, secular, or locality approximations used to obtain it.

A fit is not identification. Agreement with collision data does not make a parameter unique or remove forward-model dependence.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: Which description answers a thermal or nonequilibrium question at the required scale and evidence level?

A declared state, observable, scale, and evidence requirement select among equilibrium, real-time, stochastic or kinetic, hydrodynamic or open, and plasma or collision-inference descriptions; each exit retains its own failure test. Solid connections show the primary relation; dashed outlines or arrows mark qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Check your preparation

You are prepared for the core route if you can

1. distinguish time-ordered, retarded, advanced, Wightman, and spectral correlators;
2. derive a response by differentiating a normalized generating functional;
3. explain why the thermodynamic and long-time limits may not commute;
4. identify a conserved current and its susceptibility;
5. perform power counting with more than one momentum scale; and
6. read a covariance matrix or uncertainty band without inferring model uniqueness.

If one item is unfamiliar, follow its linked prerequisite from the first chapter that uses it. The chapter overviews state both required and helpful background and finish with concrete mastery checks.

References

• Berges, Jürgen. “Introduction to Nonequilibrium Quantum Field Theory.” AIP Conference Proceedings 739 (2004): 3–62. doi:10.1063/1.1843591.
• Kamenev, Alex. Field Theory of Non-Equilibrium Systems. Cambridge: Cambridge University Press, 2011. doi:10.1017/CBO9781139003667.
• Kovtun, Pavel. “Lectures on Hydrodynamic Fluctuations in Relativistic Theories.” Journal of Physics A: Mathematical and Theoretical 45, no. 47 (2012): 473001. doi:10.1088/1751-8113/45/47/473001.
• Laine, Mikko, and Aleksi Vuorinen. Basics of Thermal Field Theory: A Tutorial on Perturbative Computations. Lecture Notes in Physics 925. Cham: Springer, 2016. doi:10.1007/978-3-319-31933-9.
• Landsman, N. P., and Ch. G. van Weert. “Real- and Imaginary-Time Field Theory at Finite Temperature and Density.” Physics Reports 145, nos. 3–4 (1987): 141–249. doi:10.1016/0370-1573(87)90121-9.