Interacting and Nonequilibrium Quantum Fields

Interacting quantum fields on curved spacetime are best built from local questions outward. Compactly supported interactions avoid assuming an eternal coupling; causal renormalization extends time-ordered distributions only where points coincide; local covariance identifies curvature counterterms; and a closed-time path computes expectation values from initial data. Late-time resummation enters only after a particular secular variable and approximation regime have been demonstrated.

Helpful background. Flat-Spacetime Quantum Field Theory supplies perturbative vertices and standard renormalization; Partial Differential Equations and Green Functions provides retarded evolution; Causality and Global Structure establishes globally hyperbolic causal domains; Quantum Fields in Curved Spacetime defines the Hadamard and local-algebra setting.

Follow the construction in order

The six pages form one calculation sequence:

1. Perturbative Algebraic Interactions constructs relative S-matrices and interacting local algebras as formal power series, without a global adiabatic limit.
2. Epstein–Glaser Renormalization on Curved Spacetime extends causally determined distributions across total diagonals and classifies the finite local ambiguity.
3. Renormalization Group and Local Couplings compares scaled local covariant prescriptions and distinguishes their cocycle from a constant-coupling beta function.
4. Schwinger–Keldysh and In-In Dynamics doubles sources and fields to obtain normalized, real, causal expectation equations from an initial density operator.
5. Kinetic, Secular, and Resummation Regimes asks when fixed order fails and which additional assumptions support kinetic, dynamical-RG, large-N, or stochastic approximations.
6. Interacting fields on curved backgrounds distinguishes methods that carry over from flat spacetime from effects of curvature and dynamical gravitational backreaction.

The order matters conceptually. One cannot discuss running couplings until renormalized local products exist, and one cannot interpret a secular expectation value until the state, contour, observable, and finite renormalization have been fixed.

The local algebraic spine

Assume a smooth globally hyperbolic background, a free Hadamard algebra, a compactly supported local interaction $V$, and renormalized time-ordered products. With the convention $S(V)=\mathcal T e^{iV}$, define

$$S_V(F)=S(V)^{-1}_{\star}\star S(V+F), \qquad R_V(F)=\left.\frac1i\frac{d}{dt}S_V(tF)\right|_{t=0}.$$

Causal factorization makes $R_V(F)$ insensitive to future changes of $V$ and identifies cutoff choices that agree near the causal region of $F$. This is how local interacting algebras exist without assuming a global S-matrix. Brunetti and Fredenhagen give the construction and its local-net consequence in Brunetti–Fredenhagen 2000, §§3–8.

The statement is coefficientwise. Every algebra, map, and interacting field is a formal power series in the coupling. The construction does not prove convergence or nonperturbative existence, and it does not choose a state or representation.

Renormalization and scaling stay local

Causal induction determines an $n$-point time-ordered distribution away from the total diagonal. Finite scaling degree permits extension to coincidence, with an ambiguity supported on that diagonal. On curved spacetime, normal-coordinate scaling separates tangent-space singular distributions from local curvature coefficients and a softer remainder. Local covariance then forbids arbitrary coordinate-dependent subtractions.

For a scalar theory, this logic admits operator mixing involving $R\phi^2$ and identity-valued curvature polynomials in addition to flat counterterms. Hollands and Wald construct the needed scaling expansion and extension under explicit microlocal, locality, covariance, scaling, unitarity, and field-independence hypotheses Hollands–Wald 2002, §§2–4, especially §4.1, Theorem 4.1.

Metric rescaling selects a one-parameter cocycle among finite local renormalizations. A familiar flat beta function is its projection in a flat region with constant coupling. This local ultraviolet coefficient does not require a global adiabatic limit; such a limit is an additional issue when constructing a global state or scattering matrix. Away from the constant-coupling region, powers of the source and its derivatives matter—for example, two quartic vertices produce a local term proportional to $\lambda^2f^2\phi^4$, not $\lambda^2f\phi^4$. A single running number cannot replace the full operator-mixing problem.

Expectation values require a return branch

Fixing the algebra is not yet fixing the physical prediction. A finite-time expectation value also needs an initial density operator and an in-in contour. The generating functional

$$Z[J_+,J_-]=\operatorname{Tr}(U_{J_+}\rho_0U_{J_-}^{\dagger})$$

obeys $Z[J,J]=1$ for normalized $\rho_0$. Its plus and minus variations produce the same mean field with opposite source signs, and their combinations give retarded response. Schwinger’s expectation-value method begins with precisely this forward-and-return structure Schwinger 1961, complete article, pp. 407–432.

The causal-propagator convention remains

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}, \qquad [\Phi(f),\Phi(g)]=iE(f,g)\mathbf1.$$

With $G^{ab}=-i\langle T_C\phi_a\phi_b\rangle$, the combination $G^{++}-G^{+-}$ equals the retarded Green function for the free field. If the action source is $+\int J\phi$, however, the classical equation is $P\phi=-J$ and the susceptibility is $-G_{\mathrm{ret}}$. Declaring the source sign prevents a false discrepancy.

An in-out effective action instead generates transition matrix elements and Feynman boundary conditions. It is not a substitute for an in-in initial-value equation merely because the same vertices occur.

Secular growth is a regime diagnosis

A term $\lambda L(t)$ with $L(t)$ growing invalidates fixed order when its magnitude becomes order one. It does not prove that the exact observable diverges or determine which infinite subset should be resummed. Burgess and collaborators demonstrate both an explicit de Sitter secular logarithm and a dynamical-RG leading-log resummation for selected scalar models Burgess et al. 2010, §§2.1–2.4, §§3.1–3.3, and §§4.1–4.3.

A kinetic equation additionally assumes a narrow spectral width, a gradient hierarchy, and sufficiently short memory. A stochastic reduction assumes a justified long/short split and controlled coarse graining. Large-N and two-particle-irreducible approximations have their own field-content and truncation conditions. Early-time matching, conservation or Ward identities, regulator independence, and comparison in an overlap regime are independent checks; no method receives universal priority from the word “secular.”

A useful check needs no collision calculation: a homogeneous collisionless distribution in FLRW must satisfy $\partial_t f-Hp\,\partial_p f=0$ when $p$ is physical momentum. Its solution $f(t,p)=F(ap)$ expresses redshift. A proposed curved-spacetime transport equation that omits the momentum derivative fails before interactions are considered. Likewise, $e^x$ and $(1-x)^{-1}$ have the same first secular correction but different higher terms, so a single growing logarithm cannot choose its own resummation.

Scope and failure tests

The shared hypotheses are a prescribed globally hyperbolic geometry, suitable local interactions, a Hadamard ultraviolet state class, and declared renormalized products. The pAQFT and renormalization results are formal perturbative constructions. The Schwinger–Keldysh equations require a specified initial state and matching at the return time. Kinetic or resummed conclusions require further scale hierarchies stated for a particular observable.

Several counterexamples mark the boundary. A coordinate-dependent subtraction violates local covariance even if it cancels a divergence in one chart. A non-Hadamard initial state can create initial-surface divergences not removed by bulk counterterms. A broad spectral function defeats a quasiparticle kinetic closure. A global in-out amplitude need not exist on a generic curved spacetime. Finally, computing $\langle T_{\mu\nu}\rangle$ on a fixed metric does not solve the metric-response equation; that question belongs to Semiclassical Gravity.

These limitations are not defects of the local framework. They identify which additional theorem, state condition, approximation, or gravitational equation is needed before the physical claim can be strengthened.

Solved integrative diagnostic

A researcher wants $\langle\phi^2(x)\rangle$ at a finite time after a compact $\lambda\phi^4$ interaction is activated in an expanding, prescribed spacetime. The first loop contains a growing logarithm. What sequence of choices and checks is required before calling it a late-time prediction?

Solution

First choose a Hadamard initial density operator and a compact switching function, then construct the interacting local observable through the relative S-matrix. Extend its coincident time-ordered distributions with a local covariant prescription, including allowed curvature mixing for $\phi^2$. Use a Schwinger–Keldysh contour, not an in-out amplitude, so the result is an expectation value with retarded support and verify equal-source normalization. Compare the logarithm with the tree term to locate fixed-order breakdown. Only then select a resummation whose state, geometry, observable, width, gradient, and memory assumptions hold; require early-time matching and the applicable conservation identities. Because the metric is prescribed, no conclusion about self-consistent expansion follows without a separate semiclassical-gravity equation.

References

• Brunetti, Romeo, and Klaus Fredenhagen. “Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds.” Communications in Mathematical Physics 208 (2000): 623–661. DOI.
• Burgess, C. P., R. Holman, L. Leblond, and S. Shandera. “Breakdown of Semiclassical Methods in de Sitter Space.” Journal of Cosmology and Astroparticle Physics 2010, no. 03 (2010): 033. DOI.
• Hollands, Stefan, and Robert M. Wald. “Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 231 (2002): 309–345. DOI.
• Schwinger, Julian. “Brownian Motion of a Quantum Oscillator.” Journal of Mathematical Physics 2 (1961): 407–432. DOI.