QFT for gravity and cosmology

Use this pathway to produce one curved-spacetime result whose geometry, state, observable, renormalization, approximation, and validity boundary are explicit. It is not a sequence that turns a fixed-background calculation into quantum gravity merely by continuing to more advanced topics.

On a general curved spacetime there need not be a preferred vacuum, a unique particle notion, or asymptotic regions that support an ordinary S-matrix. Local correlators, detector responses, renormalized stress tensors, and late-time cosmological correlators can still be well-defined when their required data are supplied.

Fix the geometry, state, and observable

Take these three actions before opening a long chapter.

1. Choose the level of description. State what is prescribed, what is dynamical, and what is quantized.
2. Specify one result. Name the spacetime and causal domain, field or perturbation, state, observable, subtraction prescription, expansion parameters, and target accuracy.
3. Reconstruct one common calculation. Derive the field or mode equation and its normalization, then select the branch whose observable you actually need.

The first choice prevents several distinct theories from being called “quantum gravity.”

Description	Metric	Quantum variables	Typical result	What it does not establish
QFT on a prescribed curved background	Fixed classical geometry	Matter fields	Two-point function, detector response, or renormalized matter stress tensor	Backreaction or quantum metric fluctuations
Semiclassical gravity	Classical geometry solved self-consistently	Matter fields through renormalized expectation values	Mean geometry satisfying a semiclassical field equation	Full probability law for the metric
Cosmological perturbation theory	Chosen background plus constrained perturbations	Gauge-invariant matter and metric combinations at a stated order	In-in correlators or spectra on a chosen time slice	A background-independent theory or universal initial state
Gravity as EFT	Background and low-energy metric perturbations may both be dynamical	Matter and gravitons below a cutoff	Low-energy amplitudes, correlators, or long-distance quantum corrections	A microscopic ultraviolet completion
Quantum-gravity claim	Depends on the proposed definition	Geometry may be intrinsically quantum	Relational, boundary, or other theory-specific observables	Anything beyond the stated definition, regime, and evidence

These descriptions can overlap. Cosmological perturbations may be treated within semiclassical gravity or gravitational EFT, depending on which fluctuations and loops are retained. A matter loop on a fixed metric remains fixed-background QFT; a curvature-dependent answer is not automatically a graviton loop. The regime distinctions are developed in Hollands and Wald 2015, §§2–4 and the low-energy metric expansion in Donoghue 1994, §§II–IV.

Common preparation for every branch

Start from Core QFT at the first result below that you cannot reproduce. Do not repeat an earlier topic when its needed calculation is already secure.

Question to answer	Preparation
Can you vary a covariant action while retaining boundary terms and derive the field equation?	Classical fields, actions, and local dynamics
Can you distinguish the field algebra, state, correlator, particle basis, and observable?	Quantum fields, states, and observables and Functional integrals and correlators
Can you state the relevant covariance, conservation, gauge, or Ward identity together with boundary and anomaly terms?	Symmetry, currents, and Ward identities
Can you separate UV, IR, and threshold behavior and define a finite composite quantity with scale dependence controlled?	Loops and regularization and Renormalization and the renormalization group
Can you name the low scales, cutoff, operator expansion, first omitted term, and threshold matching when gravity is treated effectively?	Effective field theory and matching

Tensor calculus, Green operators, variational reasoning, causal structure, and quantum states are used immediately. If one of those operations is uncertain, use the matching readiness diagnostic and return to the calculation that exposed the gap.

Before choosing a state, complete Fields and Local Algebras on Curved Backgrounds. It fixes the spacetime, causal propagation, field operator, boundary data, and observable algebra. Then use States, Hadamard Structure, and Microlocal Control to separate state admissibility from state selection. Global hyperbolicity can supply advanced and retarded propagators; it does not select a vacuum.

Running example: a scalar mode on an expanding background

Consider a real scalar on a prescribed four-dimensional spatially flat FLRW geometry,

$$\mathrm ds^2 = a(\eta)^2 \left( \mathrm d\eta^2-\mathrm d\mathbf x^2 \right),$$

with action

$$S_\phi = \frac12 \int\mathrm d^4x\,\sqrt{-g} \left[ \nabla_\mu\phi\nabla^\mu\phi -\left(m^2+\xi R\right)\phi^2 \right].$$

The metric is fixed in this calculation. Varying $\phi$ while retaining the surface term gives the bulk equation

$$\left( \Box_g+m^2+\xi R \right)\phi=0.$$

The site convention has

$$R=-6\frac{a''}{a^3}.$$

Expand the field as

$$\begin{aligned} \phi(\eta,\mathbf x) ={}& \frac{1}{a(\eta)} \int\frac{\mathrm d^3k}{(2\pi)^{3/2}} \bigl[ a_{\mathbf k}v_k(\eta)e^{i\mathbf k\cdot\mathbf x} \\ &\qquad\qquad +a_{\mathbf k}^{\dagger}v_k^*(\eta) e^{-i\mathbf k\cdot\mathbf x} \bigr]. \end{aligned}$$

The rescaled modes obey

$$v_k'' + \omega_k^2(\eta)v_k =0, \qquad \omega_k^2 = k^2+a^2m^2-(1+6\xi)\frac{a''}{a}.$$

Canonical commutation fixes the Wronskian,

$$v_kv_k^{*\prime}-v_k'v_k^*=i.$$

This normalization follows from the symplectic form; it does not decide which normalized solution should be called positive frequency. The mode equation and Wronskian are derived without a particle interpretation in Parker 1969, §§II–III, pp. 1059–1065.

A second normalized mode basis has the form

$$\widetilde v_k = \alpha_kv_k+\beta_kv_k^*, \qquad |\alpha_k|^2-|\beta_k|^2=1.$$

If the operators are transformed inversely, this is merely a change of basis for the same field and state. If $\widetilde v_k$ instead defines new annihilation operators while the original basis defines the reference state, then $|\beta_k|^2$ is a relative occupation number. Its meaning requires the two state or asymptotic prescriptions being compared. Geometry alone supplies neither one.

From normalized modes to a finite observable

For the Gaussian state annihilated by the displayed $a_{\mathbf k}$, the two-point function is

$$\begin{aligned} W(x,x') ={}& \frac{1}{a(\eta)a(\eta')} \int\frac{\mathrm d^3k}{(2\pi)^3}\, v_k(\eta)v_k^*(\eta') \\ &\times e^{i\mathbf k\cdot(\mathbf x-\mathbf x')}. \end{aligned}$$

The coincidence limit diverges. A local field fluctuation can instead be defined by Hadamard point splitting,

$$\left\langle\phi^2(x)\right\rangle_{\mathrm{ren}} = \lim_{x'\to x} \left[ W(x,x')-H(x,x') \right] +c_1m^2+c_2R,$$

where $H$ is the locally constructed singular parametrix and $c_1,c_2$ encode finite renormalization conditions. A renormalized stress tensor requires additional derivatives, local curvature counterterms, and a conservation condition. Normal ordering relative to an arbitrary instantaneous mode basis is not a generally covariant replacement for this construction. Local covariant composite fields and their finite curvature freedom are described in Hollands and Wald 2015, §§3–4.

This example now exposes the full input set:

• geometry: the FLRW scale factor, causal patch, curvature convention, and whether the metric is prescribed or solved;
• state: a positive Hadamard two-point function plus any symmetry, asymptotic, adiabatic, or preparation criterion used to select it;
• observable: a correlator, detector response, spectrum, or renormalized local composite—not an unqualified particle count;
• approximation: free or interacting order, background and perturbation order, derivative expansion, duration, and any adiabatic assumption; and
• validity boundary: failed state regularity, large backreaction or stress fluctuations, loss of a scale hierarchy, large secular terms, or departure from the causal domain.

The Hadamard condition fixes the universal short-distance singular structure, not the smooth state-dependent remainder, so it does not choose one state. This distinction and the absence of a generic preferred vacuum are central to curved-spacetime QFT Birrell and Davies 1982, chs. 3 and 6.

Select the branch that matches the observable

Particles, detectors, and horizons

Choose this branch when the target is a transition probability, an asymptotic in/out comparison, a Hawking flux, or horizon thermality.

Start with Particles, Detectors, and Nonadiabatic Production. A switched two-level detector with gap $\Omega$ has a leading response of the form

$$\begin{aligned} P_{g\to e} ={}& g_d^2 |\langle e|\mu(0)|g\rangle|^2 \int\mathrm d\tau\,\mathrm d\tau'\, \chi(\tau)\chi(\tau') \\ &\times e^{-i\Omega(\tau-\tau')} W\!\left(x(\tau),x(\tau')\right) +O(g_d^4). \end{aligned}$$

The geometry enters through the worldline and two-point function; the state through $W$; and the apparatus through its gap, switching, smearing, and coupling. A detector click is therefore an operational response, not proof of a basis-independent particle density Unruh 1976, pp. 870–892.

Continue to Horizons and Hawking Radiation only after stating the horizon type, state, collapse or stationary assumptions, asymptotic region, greybody propagation, and backreaction range. An ordinary S-matrix is available only when suitable asymptotic states and regions exist. Otherwise the appropriate outputs are local response, correlators, or fluxes in a declared state.

Work product. Compute one detector response, Bogoliubov coefficient, or renormalized flux. Include the state, observer or trajectory, switching or asymptotic prescription, subtraction, energy balance, and the time or curvature range in which the fixed-background approximation remains valid.

Cosmological fields and correlators

Choose this branch when the target is a power spectrum, unequal-time correlator, particle-production estimate, or non-Gaussian correlation in an expanding universe.

Begin with Quantum Fields in Cosmology and then Inflationary Correlators and Cosmological Perturbation Theory. Carry the background solution, constrained variables, gauge choice or gauge-invariant combination, initial density matrix, interaction picture, in-in contour, renormalization prescription, and late-time evaluation surface.

For a curvature perturbation $\mathcal R$, state the spectrum convention rather than quoting a bare mode amplitude:

$$\left\langle \mathcal R_{\mathbf k}\mathcal R_{\mathbf k'} \right\rangle = (2\pi)^3 \delta^{(3)}(\mathbf k+\mathbf k') \frac{2\pi^2}{k^3} \mathcal P_{\mathcal R}(k).$$

The map from $\mathcal P_{\mathcal R}$ to a late-time observation adds transfer functions, matter content, and statistical inference. Gauge invariance at linear order does not remove initial-state, loop, reheating, or projection assumptions. The constraint reduction and gauge-invariant mode construction are reviewed in Mukhanov, Feldman, and Brandenberger 1992, §§II–V.

Work product. Derive one normalized mode equation and compute one two-point function or spectrum at a declared time. Show the subhorizon or adiabatic limit, one gauge or constraint check, the subtraction used for any composite quantity, and the approximation that controls interactions and backreaction.

Gravity EFT and validity

Choose this branch when metric fluctuations are quantized below a cutoff, a heavy threshold is integrated out, or a controlled low-energy gravitational correction is required.

Start with Gravity as Effective Field Theory and end with Validity, Breakdown, and Quantum-Gravity Handoffs. A schematic four-dimensional action is

$$\begin{aligned} S_{\mathrm{eff}} ={}& \int\mathrm d^4x\,\sqrt{-g}\, \Bigl[ -\frac{M_{\mathrm{Pl}}^2}{2}(R-2\Lambda) +c_1R^2 \\ &\qquad +c_2R_{\mu\nu}R^{\mu\nu} +\cdots \Bigr] +S_{\mathrm m}. \end{aligned}$$

The omitted terms are ordered by a declared derivative, curvature, loop, and threshold expansion. Local coefficients depend on matching and scheme; nonanalytic long-distance terms can be insensitive to short-distance details within the EFT assumptions. A higher-derivative pole near or above the cutoff is not automatically a new propagating state of the truncated theory.

Work product. Choose one amplitude, background response, or relational quantity. List the active fields, gauge treatment, operator basis, matching scale, renormalization scheme, first omitted terms, small parameters such as $E/\Lambda_{\mathrm{EFT}}$ and $R/\Lambda_{\mathrm{EFT}}^2$, and the observable-specific stopping condition. Donoghue’s construction shows how low-energy quantum-gravity predictions can be controlled without claiming a UV completion Donoghue 1994, §§II–IV.

Semiclassical gravity is a separate calculation

If the renormalized matter stress is allowed to change the geometry, the schematic mean equation is

$$G_{\mu\nu} +\Lambda g_{\mu\nu} +\sum_i c_iH_{\mu\nu}^{(i)} = 8\pi G \left\langle T_{\mu\nu}\right\rangle_{\mathrm{ren}}.$$

This requires renormalized gravitational couplings, a state, initial or boundary data, and a causal prescription. It predicts a mean geometry when the higher-curvature, response, and fluctuation corrections are controlled. It does not give the full quantum distribution of the metric. A small mean stress is also insufficient if stress fluctuations drive a large metric response.

A fixed-background solution can be used as the zeroth iterate of this equation, but that does not retroactively include backreaction. State the comparison being made and the parameter that makes the next iterate small.

Produce one curved-background result sheet

Leave the pathway with a short, inspectable calculation rather than a list of topics. Use this template:

framework:
  prescribed background / semiclassical gravity / cosmological perturbations /
  gravitational EFT

geometry and causal domain:
quantized variables and algebra:
state or density matrix:
observable and operational meaning:
regularization, subtraction, and finite conditions:
approximation order and small parameters:
derived equation or correlator:
two independent checks:
first omitted contribution:
stopping condition:
strongest conclusion supported:

A satisfactory result contains one derived equation, one finite observable, two checks that could have failed, and one sentence stating where the claim stops. For a quantum-gravity interpretation, also specify the proposed theory, observable map, regime, and evidence. The Quantum-Gravity Claims, Observables, and Evidence chapter helps classify such statements; it does not upgrade a semiclassical or EFT calculation into a nonperturbative definition.

Check your understanding

1. Verify the conformal scalar sign

In the site convention, set $m=0$ and $\xi=-1/6$. Show what happens to the FLRW mode equation, and state what the result does and does not imply.

Solution

The scale-factor term is

$$-(1+6\xi)\frac{a''}{a} = -(1-1)\frac{a''}{a} = 0.$$

Thus

$$v_k''+k^2v_k=0, \qquad v_k=\frac{e^{-ik\eta}}{\sqrt{2k}}$$

is a normalized positive-frequency solution for the conformally transported Minkowski state. There is no Bogoliubov mixing caused by $a(\eta)$ in this basis. This does not imply that every detector has zero transient response, that the renormalized stress vanishes, or that the trace anomaly is absent. Those are different observables with additional renormalization and apparatus data.

2. Check a Bogoliubov transformation

Let $\widetilde v=\alpha v+\beta v^*$ and suppose $v v^{*\prime}-v'v^*=i$. Derive the condition for $\widetilde v$ to have the same Wronskian. When may $|\beta|^2$ be called a particle number?

Solution

Direct substitution gives

$$\widetilde v\widetilde v^{*\prime} -\widetilde v'\widetilde v^* = \left( |\alpha|^2-|\beta|^2 \right) \left( v v^{*\prime}-v'v^* \right).$$

The normalized Wronskian is therefore preserved exactly when

$$|\alpha|^2-|\beta|^2=1.$$

If the two mode sets define distinct annihilation operators and suitable state or asymptotic prescriptions, then the original vacuum has expected occupation $|\beta|^2$ in the transformed mode. If modes and operators are both transformed inversely, the field and state are unchanged and no new particle population has been created. Without an asymptotic, adiabatic, or operational definition, $|\beta|^2$ is only a basis-relative number.

3. Classify the gravitational regime

A calculation evaluates a scalar one-loop effective action on a prescribed FLRW metric and retains local curvature terms through second order. Another calculation varies the renormalized action and solves for the mean metric. Which regimes are these, and has either quantized gravity?

Solution

The first is a matter-loop calculation in QFT on a prescribed curved background. Dependence on $R$ does not make it a graviton loop because the metric was not integrated over. The second is semiclassical gravity if the metric remains classical and is sourced self-consistently by renormalized matter expectation values and gravitational counterterms.

Neither calculation has quantized metric fluctuations. A gravitational-EFT calculation would include controlled metric perturbations, gauge fixing, ghosts, and graviton loops or matched metric operators below a cutoff. Even that would be a low-energy quantum theory, not automatically a UV completion.

References

• N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge University Press, 1982, doi:10.1017/CBO9780511622632.
• John F. Donoghue, “General Relativity as an Effective Field Theory: The Leading Quantum Corrections,” Physical Review D 50 (1994), 3874–3888, doi:10.1103/PhysRevD.50.3874.
• Stefan Hollands and Robert M. Wald, “Quantum Fields in Curved Spacetime,” Physics Reports 574 (2015), 1–35, doi:10.1016/j.physrep.2015.02.001.
• Viatcheslav F. Mukhanov, H. A. Feldman, and Robert H. Brandenberger, “Theory of Cosmological Perturbations,” Physics Reports 215 (1992), 203–333, doi:10.1016/0370-1573(92)90044-Z.
• Leonard Parker, “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183 (1969), 1057–1068, doi:10.1103/PhysRev.183.1057.
• William G. Unruh, “Notes on Black-Hole Evaporation,” Physical Review D 14 (1976), 870–892, doi:10.1103/PhysRevD.14.870.

Choose one branch now and open its first exact page: particles and detectors, quantum fields in cosmology, or gravity as effective field theory. Return to Learning pathways only if the intended observable belongs to a different physical domain.