Classical Fields and Green-Hyperbolic Dynamics

A free field on curved spacetime begins as a covariant differential equation, but the equation alone is not yet a dynamical theory. One must identify admissible initial data, prove causal existence and uniqueness, construct advanced and retarded inverses, quotient sources or gauge representatives that describe the same solution, and equip the resulting solution space with the bilinear form that quantization will use. This chapter develops that chain without choosing particles or a vacuum.

The local PDE bridge needed here is compact. Test sources are smooth and compactly supported; a retarded or advanced Green operator returns a solution supported only in the causal future or past of that source. A spacelike-compact homogeneous solution meets each Cauchy surface in a compact set. Those support statements make Green identities and Cauchy-surface integrals meaningful.

Helpful background. Flat-Spacetime Quantum Field Theory provides canonical fields, mode normalization, and the flat-space comparison; Partial Differential Equations and Green Functions supplies distributional inverses, energy estimates, and support language; Causality and Global Structure establishes Cauchy hypersurfaces, domains of dependence, and global hyperbolicity.

From a field equation to a causal solution space

Fix a smooth, oriented, time-oriented globally hyperbolic spacetime $(M,g)$ with signature $(-,+,+,+)$. For a real scalar, the model operator is

$$P=\Box-m^2-\xi R.$$

Its principal symbol depends only on $g$, while mass and curvature coupling enter at lower differential order. Normal hyperbolicity and global hyperbolicity imply a unique Cauchy evolution with finite propagation speed. They also imply unique retarded and advanced Green operators $G_{\mathrm{ret}}$ and $G_{\mathrm{adv}}$. We use the chapter convention

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}.$$

Because $PG_{\mathrm{ret}}f=PG_{\mathrm{adv}}f=f$, the causal propagator sends a compact source $f$ to a homogeneous solution $Ef$. Sources differing by $Ph$ produce the same solution, and every spacelike-compact solution arises in this way under the stated hypotheses. Thus the analytic theorem becomes the exact quotient

$$\frac{C_0^\infty(M)}{P C_0^\infty(M)} \simeq \mathrm{Sol}_{\mathrm{sc}}(P).$$

The complete existence, support, and exact-sequence argument is given in Bär, Ginoux, and Pfäffle 2007, §§3.2–3.5, pp. 73–102. This quotient is the chapter’s central construction: local source data become global but causally controlled solutions, with equation-of-motion redundancy removed.

Route through the six topics

The six pages form one cumulative argument.

1. Scalar Fields on Curved Spacetimes begins with the action, derives the curvature-coupled Klein–Gordon equation, and checks its conserved current and stress tensor. It supplies a single example that reappears at every later step.

2. Normally Hyperbolic Operators separates the metric principal symbol from lower-order terms and states the global Cauchy theorem with its actual smoothness, support, and causal hypotheses.

3. Advanced, Retarded, and Causal Propagators turns that theorem into Green operators, proves the source-to-solution exact sequence, and fixes the sign of $E$ by an explicit Minkowski kernel.

4. Covariant Phase Space for Linear Fields shows that the conserved current defines a Cauchy-surface-independent symplectic form and that the same form can be evaluated by pairing sources through $E$.

5. Dirac and Maxwell Fields compares two essential extensions: a first-order spinor operator whose square is normally hyperbolic, and a gauge field whose hyperbolic representatives must still satisfy constraints and be quotiented by gauge freedom.

6. Boundaries, Non-Global Hyperbolicity, and Well-Posedness tests the hypotheses at their edge. Timelike boundaries require a choice of admissible boundary dynamics; compactly generated Cauchy horizons present a different causal and microlocal obstruction.

A first reading should follow this order. Readers already comfortable with hyperbolic PDE may begin at the propagator page, but the scalar conventions and the theorem hypotheses should remain available for sign and scope checks.

Why the hypotheses do real work

The construction rests on distinctions that should not be blurred.

Local hyperbolicity versus global prediction. A metric principal symbol fixes the null characteristics and confines propagation to the causal cone, including possible interior tails. Global hyperbolicity is what turns local solutions into a unique evolution over all of $M$. Dropping it can create missing boundary data or more severe causal pathologies. A Robin boundary can conserve total energy and still create a negative spatial eigenvalue, so boundary well-posedness and stability must also be distinguished.

Equation versus representative. The scalar quotient removes sources of the form $Ph$. Maxwell theory also removes gauge transformations, and its initial data obey Gauss’ constraint. A gauge-fixed wave equation without these reductions is larger than the physical theory.

Presymplectic versus symplectic. A conserved surface form can have a radical. Only after quotienting its degeneracies may one call it symplectic. For Dirac fields the corresponding object is a positive Hermitian form, not an antisymmetric scalar-field form.

Support versus decay. “Compactly supported source,” “spacelike-compact solution,” and “decays at infinity” are different conditions. The proof of Cauchy-surface independence uses support strongly enough to eliminate flux at spatial infinity; qualitative decay is not automatically sufficient.

These distinctions mirror the algebraic formulation of a free scalar field: the equation, causal propagator, symplectic space, and field algebra are connected explicitly in Hollands and Wald 2015, §§1–2.1. The limitation is equally important. None of these classical results chooses a state, defines particles, renormalizes composite fields, or permits metric backreaction.

One scalar example across the chapter

Take the conformally coupled massless scalar in four dimensions, $m=0$ and $\xi=1/6$. Its equation is

$$\left(\Box-\frac16R\right)\phi=0.$$

The curvature term alters amplitudes under conformal rescaling but not the metric principal symbol, so causal propagation still follows the null cones of $g$. Compact Cauchy data determine one spacelike-compact solution. A compact source $f$ produces $Ef$, and changing $f$ by $Ph$ leaves that solution unchanged. For two solutions, the current

$$j^\mu(\phi_1,\phi_2) =\phi_1\nabla^\mu\phi_2-\phi_2\nabla^\mu\phi_1$$

has zero divergence. Stokes’ theorem then makes its integral over a Cauchy surface independent of the chosen surface, provided the support condition excludes flux at spatial infinity. The example displays the entire chain with no particle interpretation: action, hyperbolic equation, causal propagation, quotient solution space, and symplectic structure.

It also supplies a counterexample to an overly strong inference. Conformal covariance of this massless equation does not make a generic curved spacetime stationary and therefore does not select a preferred positive-frequency splitting or vacuum. Those are quantum-state questions, developed only after the classical phase space has been fixed.

From classical solutions to quantization

At the end of the chapter the input to free-field quantization is precise: a real symplectic space for bosonic solutions, or a complex Hermitian space for Dirac solutions, together with causal localization. The next step may impose CCR or CAR relations and study representations and states. It must not revisit the sign of the causal propagator, silently enlarge the solution class, or forget a gauge quotient.

Generic flat-spacetime scattering, interacting gauge theory, and renormalization are outside this chapter. Metric response is also excluded: the stress tensor derived here is a classical observable on a fixed background, not yet the renormalized expectation value that sources a semiclassical Einstein equation. Boundaries are included only far enough to show exactly which global theorem has failed and what additional data or obstruction replaces it.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, 2007. Open PDF, arXiv:0806.1036.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. arXiv:1401.2026.

Integrative diagnostic

Let $P$ be formally self-adjoint and normally hyperbolic on a globally hyperbolic spacetime. Suppose $f,f',h\in C_0^\infty(M)$ satisfy $f'=f+Ph$. Show that they determine the same spacelike-compact solution under $E$, and explain why the causal-propagator pairing depends only on both source classes. Then state the hypothesis that would fail on a spacetime with an unprescribed timelike boundary.

Solution

The Green identities give $EP=0$, so $Ef'=Ef+EPh=Ef$; therefore the source-to-solution map factors through the quotient by $PC_0^\infty(M)$. For another source $g$, formal skew-adjointness of $E$ and $PE=EP=0$ show that replacing either entry by an equation-of-motion term leaves $\int f\,Eg\,d\mathrm{vol}_g$ unchanged. This checks that the symplectic pairing is defined on source classes rather than chosen representatives. With an unprescribed timelike boundary, local normal hyperbolicity may remain true, but global hyperbolicity—and hence uniqueness of the causal Green operators from Cauchy data alone—fails. A boundary condition or another admissible dynamical prescription is then additional physical input.