Flat-Spacetime Quantum Field Theory for Gravitation

Quantum field theory combines local relativistic fields with quantum states and observables. For gravitation, the essential flat-spacetime bridge is narrower than a complete particle-physics course: understand how classical modes become oscillators, what makes a particle interpretation possible, how two-point distributions separate fluctuations from causal response, why perturbation theory needs renormalization, and how gauge constraints remove unphysical variables.

The chapter follows a free real scalar field as its main example. Electromagnetism supplies the gauge example. Interacting, fermionic, non-Abelian, algebraic, thermal, and curved-spacetime theories appear only where they expose a boundary that later gravity pages must respect.

Helpful background. Calculus and Linear Algebra for Relativity supplies the linear-algebra language for mode spaces; Quantum States, Observables, and Measurement supplies Hilbert-space states and observables; Classical Fields and Their Actions supplies actions, canonical fields, and gauge redundancy; Events, Intervals, and Minkowski Geometry supplies causal geometry; The Wave Equation and Characteristics supplies hyperbolic propagation. The calculations use the relativistic mass-shell and Fourier-mode pages directly.

Check your preparation

Use these three questions to choose a starting point.

1. Can you vary a quadratic field action, identify its canonical momentum, and diagonalize a self-adjoint spatial operator? If not, review Classical Fields and Their Actions and Fourier Modes, Dispersion, and Group Velocity before starting with the modes page.
2. Can you move consistently between Schrödinger, Heisenberg, and interaction pictures? If not, review Unitary Dynamics and Pictures of Motion before the interactions page.
3. Can you distinguish a future-supported Green function from a time-ordered boundary value? If not, review Retarded and Advanced Green Functions before the propagator page.

Use any uncertainty to choose what to review: the first three pages rebuild the free-field chain explicitly, while the last three assume that chain and develop interactions, scale separation, and constrained state spaces.

Field conventions

We use

$$\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1), \qquad c=\hbar=1,$$

and positive-frequency time dependence $e^{-i\omega t}$. For a real scalar,

$$\mathcal L =-\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2, \qquad (\Box-m^2)\phi=0.$$

This differs in displayed signs from sources using $(+,-,-,-)$. We translate the complete action, mass shell, propagator, and Fourier convention together rather than changing isolated signs.

The six-page spine

1. Classical Field Modes and Canonical Variables derives canonical momentum, the Hamiltonian, mode normalization, and the oscillator decomposition.
2. Free Fields, Creation Operators, and Fock Space turns positive-frequency modes into ladder operators, derives the spacetime commutator, and fixes relativistic one-particle normalization.
3. Correlation Functions and Propagators distinguishes homogeneous Wightman and commutator distributions from retarded, advanced, and Feynman fundamental solutions, including their pole prescriptions.
4. Interactions, Time Ordering, and Perturbation Theory derives the bounded Dyson series, Wick combinatorics, vacuum projection, and the boundary between in–out and in–in questions.
5. Renormalization and Effective Field Theory separates regulator, subtraction, and matching scales, then turns higher-dimension operators into a power-counted low-energy expansion.
6. Gauge Symmetry, Constraints, and Physical States counts Maxwell degrees of freedom, imposes Gauss’s law on states, and checks gauge independence between conserved currents.

Two routes through the chapter

• Curved-spacetime QFT route. Take modes, Fock space, and propagators first. Focus on the dependence of vacuum and particles on a positive-frequency choice, because generic curved spacetimes need not provide a preferred global time translation.
• Quantum-gravity EFT route. Take classical canonical variables, interactions, renormalization/EFT, and gauge constraints. Focus on power counting, matching, nonanalytic low-energy effects, and the distinction between controlled low-energy quantum gravity and an unknown ultraviolet completion.

The first route continues through Complex Structures and One-Particle Spaces, Advanced, Retarded, and Causal Propagators, and Why Curved Spacetime Has No Preferred Vacuum. The second continues through General Relativity as a Quantum Effective Field Theory and Covariant Gauge Fixing and Ghost Fields.

Both routes require causal response. The commutator and retarded Green function are more robust than a particle basis and later organize propagation on curved backgrounds. When the target is a finite-time expectation value rather than an in–out amplitude, the interaction route branches to Schwinger–Keldysh and In-In Dynamics.

Review the chapter

Conceptual synthesis: when is a mode excitation a particle?

Why does writing a free field as independent oscillators not by itself define a background-independent particle concept?

Solution

The real classical solution space first needs a compatible positive-frequency split. That split turns the conserved but indefinite Klein–Gordon form into a positive one-particle inner product, defines annihilation operators, and selects a vacuum. Minkowski time translations provide a preferred positive-energy split preserved by the proper orthochronous Poincaré group.

On a generic time-dependent or curved background, different admissible splits can mix modes through Bogoliubov coefficients and need not define unitarily equivalent Fock representations. The oscillator form of the classical equations can survive even when a preferred global particle-number operator does not.

Calculational synthesis: recover a causal inverse

Let $K=\Box-m^2$ and let the free-field commutator distribution satisfy

$$C(0,\mathbf r)=0, \qquad \partial_\tau C(0,\mathbf r) =-i\delta^{(3)}(\mathbf r).$$

Show that $G_{\mathrm R}=-i\theta(\tau)C$ obeys $KG_{\mathrm R}=\delta^{(4)}$. Why does the Feynman function on this chapter instead obey $KG_{\mathrm F}=i\delta^{(4)}$?

Solution

Because $C(0,\mathbf r)=0$, the second time derivative of $\theta C$ has no surviving $\delta'$ term. Using $KC=0$ gives

$$K[\theta(\tau)C] =-\delta(\tau)\partial_\tau C(0,\mathbf r) =i\delta^{(4)}.$$

Multiplication by $-i$ yields $KG_{\mathrm R}=\delta^{(4)}$. The time-ordered function joins $G^+$ for $\tau\gt 0$ to $G^-$ for $\tau\lt 0$; their derivative jump is the same $-i\delta^{(3)}$, but there is no prefactor $-i$. Hence $KG_{\mathrm F}=i\delta^{(4)}$. The different normalization accompanies a different boundary value: future support for $G_{\mathrm R}$ versus opposite-side pole displacements for $G_{\mathrm F}$.

Transfer synthesis: power counting with gauge redundancy

Around a weakly curved background with characteristic inverse length $E$, compare a curvature-squared term $cR^2$ with the Einstein–Hilbert term $\overline M_{\mathrm{Pl}}^2R/2$. Estimate the relative correction, and explain why changing a covariant gauge parameter cannot change a conserved-source amplitude.

Solution

With $R\sim E^2$, the two terms scale as $cE^4$ and $\overline M_{\mathrm{Pl}}^2E^2/2$. Up to an order-one normalization,

$$\frac{\delta\mathcal A}{\mathcal A_{\mathrm{EH}}} \sim c\frac{E^2}{\overline M_{\mathrm{Pl}}^2}.$$

This is controlled only when momenta, masses, and curvature derivatives lie below the actual EFT cutoff and every contribution at the chosen order is retained. A change of covariant gauge modifies the propagator by longitudinal terms proportional to $k_\mu k_\nu$. Between conserved sources those terms give $(k\cdot J)(k\cdot J')=0$. The off-shell propagator may change, but the physical conserved-source amplitude does not.

Boundaries that matter

The vacuum is a state relative to an algebra and, in a mode treatment, a positive-frequency split; generic curved geometry need not select one. Wald 1995, printed pp. 6–10, PDF develops that representation boundary. A Feynman function is not a retarded response function, and an in–out amplitude is not a finite-time in–in expectation value. A regulator is not a physical cutoff unless the model says so, and the regulator, subtraction scale, and matching scale are distinct.

“Nonrenormalizable” does not mean “unpredictive”: it means that progressively higher accuracy requires more symmetry-allowed local coefficients. Neubert 2020, §4, printed pp. 31–39, PDF develops this operator expansion. In gravity, local short-distance effects and nonanalytic massless propagation have different predictive roles; Donoghue 1995, §§5–8, printed pp. 13–18, PDF gives the corresponding power counting and analytic–nonanalytic split. Gauge fixing changes intermediate variables and auxiliary states, not gauge-invariant observables.

This chapter does not establish interacting QFT nonperturbatively, prove the spin–statistics or CPT theorems, construct the Standard Model, or quantize gravity. It also does not carry the flat-space Fock construction unchanged into a generic curved spacetime. Those omissions are deliberate scientific boundaries.

The flat-spacetime derivations follow Tong 2006–2007, Chapters 1–3 and 6, printed pp. 7–80 and 124–149, PDF, translated as a complete set from Tong’s opposite metric convention and restricted to the structures needed for gravitation.

After this chapter, Complex Structures and One-Particle Spaces is the most direct continuation toward state selection and Bogoliubov transformations. For the low-energy gravity route, continue to General Relativity as a Quantum Effective Field Theory.

References

• John F. Donoghue, “Introduction to the Effective Field Theory Description of Gravity,” lectures presented at the Advanced School on Effective Theories, Almuñécar, Spain (1995), arXiv:gr-qc/9512024, §§5–8, printed pp. 13–18. Author-manuscript PDF.
• Matthias Neubert, “Renormalization Theory and Effective Field Theories,” Les Houches lecture notes, arXiv:1901.06573v2 [hep-ph] (2020), §4, printed pp. 31–39. Author-manuscript PDF.
• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes, Michaelmas Terms 2006 and 2007, Chapters 1–3 and 6, printed pp. 7–80 and 124–149. Official author PDF.
• Robert M. Wald, “Quantum Field Theory in Curved Spacetime,” arXiv:gr-qc/9509057 (1995), printed pp. 6–10. Author-manuscript PDF.