Relative Cauchy Evolution

Relative Cauchy evolution compares one quantum field theory on two metrics that differ only in a compact spacetime region. The time-slice axiom carries observables around the perturbation through an unchanged past and an unchanged future; composing those identifications produces an automorphism of the original algebra. Its functional derivative is generated by the renormalized stress–energy tensor. This is a response calculation on prescribed backgrounds, not an equation that makes the metric dynamical.

Required background. Flat-Spacetime Quantum Field Theory supplies stress–energy as metric response and the commutator algebra used in the free-field check.

Helpful background. Partial Differential Equations and Green Functions provides causal propagators and support control across the perturbed region. Causality and Global Structure defines the past and future Cauchy neighborhoods used in the comparison. Quantum Fields in Curved Spacetime provides the route from this algebraic response to renormalized local observables and semiclassical limits.

A compact metric perturbation

Let $M=(\mathcal M,g)$ be globally hyperbolic, and let $h_{ab}$ be a smooth, compactly supported symmetric tensor such that

$$M[h]=(\mathcal M,g+h)$$

is also globally hyperbolic with the same orientation and time orientation. Write $K=\operatorname{supp}h$. Choose globally hyperbolic subspacetimes $M^-$ and $M^+$ lying respectively to the past and future of $K$, each containing a Cauchy surface of both $M$ and $M[h]$. There are four Cauchy morphisms,

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

The metrics agree on $M^\pm$, so these maps identify the same local geometry. The time-slice property makes all four induced algebra maps isomorphisms. No coordinate identification inside $K$ is needed. The construction and its independence from the chosen Cauchy neighborhoods are given in Fewster and Verch 2015, §4, pp. 10–12.

This setup requires more than a small componentwise change in $g$. The perturbed tensor must remain Lorentzian and globally hyperbolic. Compact support supplies unchanged comparison regions; a perturbation extending to infinity would require separate asymptotic data and is not covered by this definition.

Composing the time-slice maps

With the convention that the observable first travels from $M$ through the future comparison region to $M[h]$ and returns through the past comparison region, define

$$\operatorname{rce}_M[h] =\mathcal A(\iota^-) \circ\mathcal A(j^-)^{-1} \circ\mathcal A(j^+) \circ\mathcal A(\iota^+)^{-1}.$$

Reading from right to left makes the domains clear:

$$\mathcal A(M) \longrightarrow\mathcal A(M^+) \longrightarrow\mathcal A(M[h]) \longrightarrow\mathcal A(M^-) \longrightarrow\mathcal A(M).$$

Thus $\operatorname{rce}_M[h]$ is an automorphism of the unperturbed algebra. Some authors use the inverse automorphism; all derivative signs must be translated with that choice. The formula above agrees with Fewster and Verch 2015, §4, equation (15), p. 11.

If $A$ is localized in a region causally disjoint from $K$, then

$$\operatorname{rce}_M[h](A)=A.$$

One can choose the comparison maps so that the observable never crosses the perturbed region. This locality of response is stronger information than the bare statement that the two global algebras are abstractly isomorphic.

Free scalar response

For the Klein–Gordon field, let $P_h=\Box_{g+h}-m^2-\xi R_{g+h}$ and $E_h=G_{h,\mathrm{ret}}-G_{h,\mathrm{adv}}$. First choose a representative $f$ supported in the unchanged future region $M^+$. The time-slice axiom allows this for every observable generator. With our operator and propagator conventions, the finite response is

$$\operatorname{rce}_M[h]\Phi_M(f) =\Phi_M(f)+\Phi_M\bigl((P_h-P)E_hf\bigr), \qquad \operatorname{supp}f\subset M^+.$$

The added test function is compactly supported in $K$, since the coefficients of $P_h-P$ vanish outside $K$. Its causal solution need not be supported in $K$. Compact support alone does not make a test function solve the field equation. The quotient and time-slice construction are what make this an automorphism of the on-shell algebra.

Here is the sign check. Put $w=G_{h,\mathrm{adv}}f$ and choose $\chi=1$ in the distant past, with $\chi=0$ before $K$ and thereafter. On the transition region $E_hf=-w$, and the past representative of $f$ is $-P_h(\chi E_hf)=P_h(\chi w)$. Since $P_hw=f$, this equals $f-(P_h-P)(1-\chi)w-P(1-\chi)w$. The last term vanishes in the unperturbed field algebra, and $(P_h-P)(1-\chi)w=-(P_h-P)E_hf$, giving the plus sign. The section $(1-\chi)w$ is compactly supported, which justifies this use of the field equation.

For a general $f$, first replace it by $f_+\sim f$ supported in $M^+$ and apply the finite formula to $f_+$. The source derivation explicitly makes this restriction Fewster and Verch 2015, §4, Eq. (17), p. 12, PDF. That source uses signature $(+,-,-,-)$, $P_{\mathrm{source}}=-P$, and advanced-minus-retarded $E_{\mathrm{source}}=E$ after translation; copying its minus sign without translating $P$ would give the wrong response here.

For a one-parameter family $g_s=g+s h+O(s^2)$, define the first variation

$$\dot P[h]\phi =\left.\frac{d}{ds}P_{s h}\phi\right|_{s=0}.$$

Differentiating the preceding formula at $s=0$ yields

$$\left.\frac{d}{ds} \operatorname{rce}_M[s h]\Phi_M(f)\right|_{s=0} =\Phi_M\bigl(\dot P[h]E_Mf\bigr).$$

Unlike the finite formula, this first-order expression applies directly to every compact $f$: if $f-f_+=Pk$, then $Ef=Ef_+$. For readers wishing to compute it, write $h=g^{ab}h_{ab}$ and raise indices with the unperturbed metric. Metric variation gives

$$\begin{aligned} \dot P[h]\phi ={}&-h^{ab}\nabla_a\nabla_b\phi -\left(\nabla_a h^{ab}-\frac12\nabla^b h\right)\nabla_b\phi -\xi\dot R[h]\phi,\\ \dot R[h] ={}&\nabla_a\nabla_b h^{ab}-\Box h-R_{ab}h^{ab}. \end{aligned}$$

Thus nonminimal coupling contributes a curvature-variation term; the minimal-coupling formula alone would be incomplete for $\xi\ne0$. The derivative on products satisfies the Leibniz rule. Its existence is understood in a suitable distributional or represented algebra topology, rather than in the operator norm of the abstract Weyl algebra.

Stress–energy as the generator

Use the metric-variation convention

$$\delta S_{\mathrm m} =\frac12\int_M T^{ab}\,\delta g_{ab}\,d\operatorname{vol}_g, \qquad T(h)=\int_M T^{ab}h_{ab}\,d\operatorname{vol}_g.$$

For a locally covariantly renormalized stress tensor and the rce orientation declared above, the free scalar calculation gives

$$\left.\frac{d}{ds} \operatorname{rce}_M[s h](A)\right|_{s=0} =\frac{1}{2i}[T_{\mathrm{ren}}(h),A] =-\frac{i}{2}[T_{\mathrm{ren}}(h),A].$$

The formula means equality of derivations; $T_{\mathrm{ren}}(h)$ need not belong to the original minimal polynomial algebra. For $A=\Phi_M(f)$, its commutator is $2i\Phi_M(\dot P[h]E_Mf)$, giving the derivative above. One can derive this from $S=\tfrac12\int\phi P\phi\,dV_g$: varying the operator and volume measure gives the symmetric quadratic expression for $T(h)$, and commuting one field through it gives two identical terms. Terms proportional to $PEf$ vanish. Finite geometric counterterms multiply the identity and therefore do not change the commutator. The source calculation is Fewster and Verch 2015, §4, Eqs. (19)–(20), pp. 14–15, PDF; its opposite-signature metric perturbation is $h_{\mathrm{source}}=-h$, so its stress-tensor smearing also reverses sign in this translation.

This identity does not make the metric a quantum observable. It differentiates the matter algebra with respect to an externally specified metric change. Coupling $\langle T_{ab}\rangle$ back into an equation for $g_{ab}$ belongs to semiclassical gravity and requires a state, renormalized gravitational couplings, and control of the approximation.

Diffeomorphisms and conservation

Let $X^a$ be compactly supported and let $\psi_s$ be its flow. The induced metric variation is

$$h_{ab}(s)=\psi_s^*g_{ab}-g_{ab}, \qquad \dot h_{ab}(0)=\mathcal L_Xg_{ab}=2\nabla_{(a}X_{b)}.$$

Because this perturbation is a compactly supported relabeling of the same geometry, covariance gives $\operatorname{rce}_M[h(s)]=\operatorname{id}$. Differentiating and using the stress-tensor formula yields

$$[T_{\mathrm{ren}}(\mathcal L_Xg),A]=0.$$

Formally integrating by parts,

$$T_{\mathrm{ren}}(\mathcal L_Xg) =-2\int_M (\nabla_aT_{\mathrm{ren}}^{ab})X_b \,d\operatorname{vol}_g.$$

Hence stress–energy is conserved as a derivation-valued distribution; if the commutator is inner, any remaining divergence lies in the center. The result is a quantum counterpart of the diffeomorphism Noether identity, obtained without choosing a preferred state.

Checks and limitations

There is a simple check on the stress-tensor sign. On an ultrastatic spacetime with compact spatial section, perturb the lapse to $N_s(t)=1+s n(t)$, where $n$ has compact time support. Then $h_{00}=-2n$ and the extra elapsed proper time is $s\Delta\tau$, $\Delta\tau=\int n(t)dt$. The declared future-to-past comparison advances the future field by $s\Delta\tau$. Meanwhile $T(h)=-2\Delta\tau H$, so

$$-\frac i2[T(h),\Phi(t)] =i\Delta\tau[H,\Phi(t)] =\Delta\tau\,\partial_t\Phi(t),$$

in agreement with that time advance. Compactness of the spatial section makes this a compactly supported metric perturbation, as required. It is not generated by a compactly supported diffeomorphism if $\Delta\tau\ne0$, since the future and past time coordinates are shifted relative to one another.

Several further tests prevent overinterpretation:

• Causal support: an observable causally disjoint from $K$ must have zero first variation.
• Pure gauge: $h=\mathcal L_Xg$ must give the identity automorphism and the conservation identity.
• Convention reversal: replacing rce by its inverse reverses the derivative; replacing covariant by inverse-metric variation also changes the stress-tensor sign package.
• Finite response: the first derivative determines a linear response, not the full automorphism for a large perturbation.
• Background status: no claim about gravitational backreaction follows until a metric equation and a controlled state-dependent approximation are supplied.

Topological gauge sectors and theories without the time-slice property can require modifications. Even for the scalar field, differentiability depends on the topology used on the algebra or on a chosen state space. The response formula should therefore be stated with its field, algebra, and differentiable domain rather than as a universal operator identity.

Exercises

Assume the rce derivative is generated by $T_{\mathrm{ren}}$ as above. Show that compactly supported diffeomorphism covariance implies conservation of stress–energy in its commutator action.

Solution

For $h(s)=\psi_s^*g-g$, covariance gives $\operatorname{rce}[h(s)]=\operatorname{id}$, so its derivative vanishes. Because $\dot h(0)=\mathcal L_Xg$,

$$0=\frac{1}{2i}[T_{\mathrm{ren}}(\mathcal L_Xg),A].$$

Symmetry of $T^{ab}$ and compact support of $X$ give

$$T_{\mathrm{ren}}(\mathcal L_Xg) =2\int T^{ab}\nabla_aX_b\,d\operatorname{vol}_g =-2\int(\nabla_aT^{ab})X_b\,d\operatorname{vol}_g.$$

Thus the last smeared expression commutes with every $A$. Even when the center consists only of scalar multiples of the identity, this proves it is a scalar, not that the scalar vanishes. Exact conservation of the stress tensor requires the corresponding renormalization condition or Ward identity. Relative Cauchy evolution by itself determines only its commutator action and cannot detect additive geometric multiples of the identity.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Field Theory.” Communications in Mathematical Physics 237 (2003): 31–68. DOI. Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Algebraic Quantum Field Theory in Curved Spacetimes.” In Advances in Algebraic Quantum Field Theory, edited by Romeo Brunetti et al., 125–189. Cham: Springer, 2015. DOI. Open PDF.