Green-Hyperbolic Operators and Causal Propagators

On a globally hyperbolic spacetime, every normally hyperbolic operator has unique retarded and advanced Green operators with causal support. Their difference maps compactly supported sources to spacelike-compact homogeneous solutions and produces an exact sequence; global hyperbolicity is the condition that turns local hyperbolic propagation into this global statement.

Required background. Globally hyperbolic spacetimes and the Loc categories supplies causal geometry; microlocal calculus for quantum fields supplies distributional kernels.

Helpful background. Time-slice axiom and relative Cauchy evolution uses the solution theory; isotony, additivity, duality, and primitive causality gives the algebraic consequence; Green operators, causal propagators, and state-dependent two-point functions supplies the physical application.

Green-hyperbolic operators

Let $(M,g)$ be oriented, time-oriented, and globally hyperbolic, and let $P:C^\infty(M,E_1)\to C^\infty(M,E_2)$ be a differential operator between vector bundles. Retarded and advanced Green operators $G_+$ and $G_-$ satisfy

$$PG_\pm f=f,\qquad G_\pm Pf=f, \qquad \operatorname{supp}G_\pm f\subset J^\pm(\operatorname{supp}f)$$

for compactly supported smooth $f$. An operator is Green hyperbolic when it and its formal dual possess such maps. Normally hyperbolic operators, whose principal symbol is $g^{ab}k_ak_b$ times the identity, are the basic examples; Dirac-type operators are included through a Green-hyperbolic square.

The support property gives uniqueness: the difference of two retarded solutions is homogeneous and supported to the future of a compact set; the global energy estimate across Cauchy surfaces forces it to vanish. Existence is obtained from the global Cauchy problem and finite propagation speed. The Green maps extend to past- or future-compact supports while retaining inverse and support identities; Bär 2015, Theorem 3.8, pp. 12–13.

Define the causal propagator $G=G_+-G_-$. Then

$$0\longrightarrow C_c^\infty(M,E_1) \xrightarrow{P}C_c^\infty(M,E_2) \xrightarrow{G}C_{sc}^\infty(M,E_1) \xrightarrow{P}C_{sc}^\infty(M,E_2)$$

is exact. In particular, $Gf$ solves $P(Gf)=0$; every spacelike-compact smooth solution is $Gf$ for some compactly supported $f$; and $Gf=0$ exactly when $f=Ph$ with compactly supported $h$. This is Bär 2015, Theorem 3.22, p. 17. Hence the classical solution space is naturally $C_c^\infty(M,E_2)/P C_c^\infty(M,E_1)$.

Ultrastatic Klein–Gordon construction

Let $M=\mathbb R\times\Sigma$ with $g=dt^2-h$ and compact Cauchy surface $\Sigma$. For $P=\partial_t^2+A$, where $A=-\Delta_h+m^2+\xi R$ is positive after the stated parameter choice, spectral calculus gives the solution of $Pu=f$ with vanishing past data:

$$(G_+f)(t)=\int_{-\infty}^t A^{-1/2} \sin((t-s)A^{1/2})f(s)\,ds.$$

The advanced map integrates from $t$ to $+\infty$ with the corresponding sign. Differentiating twice verifies $PG_\pm f=f$ and the jump of the first derivative fixes normalization. Finite propagation speed, not evident from spectral support alone, sharpens the time-order support to $J^\pm(\operatorname{supp}f)$. Compactness of $\Sigma$ makes every solution spacelike compact, so the exact sequence identifies the complete smooth solution space.

This worked result feeds directly into Green operators, causal propagators, and state-dependent two-point functions. The antisymmetric form $E(f,h)=\int f(Gh)d\mathrm{vol}$ supplies the canonical commutator, while a state-dependent symmetric bisolution is extra data. The causal propagator does not choose a state.

An independent check expands in an $A$-eigenfunction with eigenvalue $\omega^2$. The formula reduces to the one-dimensional retarded oscillator kernel $\theta(t-s)\sin\omega(t-s)/\omega$, whose derivative jump is one and whose retarded support is manifest.

The exact sequence also verifies the phase-space construction. For a formally self-adjoint $P$, define

$$\sigma([f],[h])=\int_M f\,Gh\,d\mathrm{vol}_g$$

on compactly supported test functions modulo $P C_c^\infty(M)$. Replacing $f$ by $f+Pk$ changes the integral by $\int k\,PGh=0$, so the form descends to the quotient. Formal adjointness and $G_\pm^*=G_\mp$ make it antisymmetric. Under the quotient-to-solution isomorphism, this is the usual Cauchy-surface symplectic form. Its value is independent of the chosen Cauchy surface because the associated current is conserved and the solution is spacelike compact. This check ties the analytic exact sequence to the canonical commutation relations without adding a complex structure or a vacuum choice.

The same support argument gives the classical time-slice mechanism. A cutoff between two Cauchy surfaces lets one replace any quotient class by a source supported in an arbitrarily small globally hyperbolic neighborhood of a Cauchy surface. Thus observables generated there recover the full free solution algebra. This conclusion uses both global hyperbolicity and the two-sided Green identities; finite propagation alone is not enough.

Adversarial loss of global hyperbolicity

Remove a timelike worldtube from Minkowski space and impose no boundary condition. Waves reaching the boundary can be reflected, absorbed, or extended in inequivalent ways, all compatible with the interior equation. Retarded solutions are not unique, so neither a canonical $G_+$ nor the same exact sequence exists. A chosen self-adjoint boundary condition may restore a well-posed propagator, but it defines a different theorem with boundary data.

The converse also fails: a formal kernel with $PG=0$ and causal-looking support need not have the two-sided inverse properties required of Green operators. Nor does Green hyperbolicity establish positivity, a Hadamard state, or an interacting QFT.

Exercises

1. Quotient independence. Show that $G(f+Ph)=Gf$ for compactly supported $h$.

Solution

$GPh=(G_+-G_-)Ph=h-h=0$. Therefore $G$ factors through the quotient by $P C_c^\infty$.

2. Homogeneous image. Verify that $Gf$ solves the homogeneous equation.

Solution

$PGf=PG_+f-PG_-f=f-f=0$. Its support lies in $J^+(\operatorname{supp}f)\cup J^-(\operatorname{supp}f)$, which is spacelike compact on a globally hyperbolic spacetime.

References

• Bär, Christian. “Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes.” Communications in Mathematical Physics 333 (2015): 1585–1615. DOI. Open PDF.
• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, 2007. DOI.