Time-Slice Axiom and Relative Cauchy Evolution

Relative Cauchy evolution compares a theory on two metrics that agree outside a compact region. Its definition is purely algebraic once the time-slice axiom supplies four inverses, but its domain is geometric: both perturbed and unperturbed metrics must remain admissible globally hyperbolic backgrounds with compatible orientations.

Required background. Isotony, Additivity, Duality, and Primitive Causality distinguishes the time-slice property from other net axioms. Globally Hyperbolic Spacetimes and the Loc Categories defines Cauchy morphisms. Locally Covariant QFT as a Functor supplies the functor and its time-slice axiom.

Helpful background. Relative Cauchy Evolution and Background Response develops the physical interpretation, while Local Field Algebras, Causality, and the Time-Slice Property reviews causal propagation.

Four Cauchy maps around a metric perturbation

Let $h$ be a smooth compactly supported symmetric tensor such that $g+h$ is Lorentzian, time-orientable compatibly with $g$, and globally hyperbolic. Write $K=\operatorname{supp}h$ and denote the perturbed spacetime by $M[h]$. Choose causally convex globally hyperbolic regions $M^\pm$ lying respectively to the future and past of $K$, large enough to contain Cauchy surfaces of both $M$ and $M[h]$. Their inclusions give Cauchy morphisms

$$i^\pm:M^\pm\longrightarrow M, \qquad j^\pm:M^\pm\longrightarrow M[h].$$

The metrics agree on $M^\pm$, so the same source object may be used in each pair. Time-slice makes all four algebra maps isomorphisms. Define the retarded and advanced identifications

$$\tau_M^\pm[h]=\mathcal A(j^\pm)\,\mathcal A(i^\pm)^{-1}: \mathcal A(M)\longrightarrow\mathcal A(M[h]),$$

and the relative Cauchy evolution

$$\operatorname{rce}_M[h] =\bigl(\tau_M^-[h]\bigr)^{-1}\tau_M^+[h] \in\operatorname{Aut}\mathcal A(M).$$

This convention first identifies an observable through the future of the perturbation and returns through the past. Reversing both naming conventions yields the inverse automorphism, so signs in later response formulas must be tied to this definition. The construction and independence of the chosen Cauchy neighborhoods are proved in Brunetti, Fredenhagen, and Verch 2003, § 4.1, pp. 21–23.

Why the construction is well defined

If $M^+$ is replaced by another admissible future Cauchy neighborhood, a third neighborhood can be placed farther to the future and functoriality supplies commuting triangles. The time-slice inverses cancel, giving the same $\tau^+$. The past argument is identical. Thus $\operatorname{rce}_M[h]$ depends on the background perturbation, not on auxiliary slices.

For the free Klein–Gordon theory, classical retarded and advanced Møller maps compare solutions of $P_M\phi=0$ and $P_{M[h]}\phi_h=0$. Their scattering composition is a symplectic automorphism $R_h$ of the unperturbed solution space, and quantization gives

$$\operatorname{rce}_M[h]\Phi_M(f)=\Phi_M(R_h[f]),$$

where $[f]$ denotes the test-function class modulo $P_M C_c^\infty(M)$. This is the concrete four-map computation used in Relative Cauchy Evolution and Background Response: compose the four Cauchy isomorphisms and evaluate the result on a smeared field.

Checks, locality, and the failure boundary

Two checks are immediate. At $h=0$, one may take $i^\pm=j^\pm$, so $\operatorname{rce}_M[0]=\mathrm{id}$. If an observable is localized in a causally convex region spacelike to $K$, causal propagation permits the comparison maps to be chosen so that its image never meets the perturbation; relative Cauchy evolution fixes it. This localization property is the seed of the dynamical net.

The construction stops if $g+h$ is not an object of the source category. In particular, one may not perturb through a nonglobally hyperbolic metric and continue to write $\mathcal A(j^\pm)^{-1}$: the arrows $j^\pm$ or their time-slice inverses need not exist. Nor does the finite automorphism automatically have a functional derivative; differentiability is an additional analytic hypothesis.

Covariance and support of relative evolution

Relative Cauchy evolution is itself covariant under embeddings that contain the perturbation with its causal influence controlled. If $\psi:M\to N$ transports $h$ to $\psi_*h$ and the comparison regions fit inside the image, functoriality gives the intertwining relation

$$\mathcal A(\psi)\operatorname{rce}_M[h] =\operatorname{rce}_N[\psi_*h]\mathcal A(\psi).$$

This is proved by applying $\mathcal A$ to the same four geometric squares. It is not a statement about extending an arbitrary compact tensor by zero across every embedding: smoothness and admissibility of the target metric must still be checked.

The support statement also has a useful converse warning. If $\operatorname{rce}[h]$ fixes every observable in a region spacelike to $K$, one has causal insensitivity. One has not shown that every observable fixed by exterior perturbations was generated in $K$. Global zero modes and cohomological fluxes later defeat exactly that inference. Relative Cauchy evolution is therefore a detector of localization only after a model-specific dynamical-locality theorem.

For two separated perturbations $h_1,h_2$, causal factorization relations may follow by choosing compatible comparison regions. They depend on the order and admissibility of the combined metrics; the symbol $\operatorname{rce}[h_1+h_2]$ cannot be split as a product without verifying those geometric conditions.

Exercise

Show directly from the four-map definition that $\operatorname{rce}_M[0]=\mathrm{id}$ and that changing $M^+$ without changing its future placement does not change $\tau^+$.

Solution

For $h=0$, the two target spacetimes coincide and the paired inclusions may be chosen equal, hence $\tau^+=\tau^-=\mathrm{id}$. For two future regions, insert a third Cauchy region contained in their common future. Functoriality expresses each original inclusion through the third; after applying $\mathcal A$, the common Cauchy isomorphism and its inverse cancel. Both definitions of $\tau^+$ are therefore equal.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality and Covariance: What Makes a Physical Theory the Same in All Spacetimes?” Annales Henri Poincaré 13 (2012): 1613–1674. DOI; Open PDF.