Locally Covariant Quantum Field Theory

Locally covariant quantum field theory treats a theory as one coherent rule over many spacetimes, not as an unrelated algebra chosen separately on each background. Spacetimes and causal embeddings form a category; observable algebras and injective star-homomorphisms form another. A theory is a functor between them. Functoriality is the exact statement that an experiment embedded in a larger admissible spacetime keeps its identity and that two successive embeddings agree with their composite.

Required background. Flat-Spacetime Quantum Field Theory supplies smeared fields and observable algebras whose curved-background assignment is generalized here.

Helpful background. Partial Differential Equations and Green Functions provides causal propagators and their compatibility with causal embeddings; the necessary intertwining identity is stated below. Causality and Global Structure defines global hyperbolicity and causally convex images. Quantum Fields in Curved Spacetime provides the surrounding route from classical propagation to state and renormalization questions.

The spacetime category Loc

An object of $\mathbf{Loc}$ is a smooth, oriented, time-oriented, globally hyperbolic Lorentzian spacetime $(M,g)$ of fixed dimension. A morphism

$$\psi:M\longrightarrow N$$

is a smooth isometric embedding that preserves orientation and time orientation and whose image is causally convex. These geometric hypotheses are required for the free-field construction below. Some formulations also fix connectedness or admit finitely many components; such choices must be held fixed when comparing results.

Causal convexity is essential. If a causal curve could leave $\psi(M)$ and return, propagation calculated intrinsically in $M$ could disagree with propagation inherited from $N$. With a causally convex image, compactly supported test functions push forward by extension by zero. The correct Green identity uses restriction back to the embedded spacetime:

$$\psi^*\bigl(G_N^{\mathrm{ret/adv}}\psi_*f\bigr) =G_M^{\mathrm{ret/adv}}f.$$

The pullback $\psi^*$ here restricts a smooth solution to $\psi(M)$ and identifies it with a solution on $M$. It is generally wrong to extend $G_Mf$ by zero and claim equality on all of $N$: the propagated field need not vanish at the image boundary, and the target solution can continue outside $\psi(M)$. Uniqueness with causal support proves the restricted identity. Pairing it with another compact test function shows that the causal propagators and symplectic pairings agree. The category and its admissible morphisms are defined in Brunetti, Fredenhagen, and Verch 2003, §§2.1–2.2, pp. 5–8, PDF.

An isometric embedding of only a coordinate patch is not automatically a $\mathbf{Loc}$ morphism. It must preserve all declared structures, and its image must meet the causal condition. Likewise, conformally related metrics are not isometric; conformal covariance requires a different source category and field weights rather than silently broadening this one.

The algebra category Alg

Let $\mathbf{Alg}$ have unital star-algebras as objects and injective, unit-preserving star-homomorphisms as morphisms. One may instead choose $C^*$-algebras, topological star-algebras, or other completions. The choice is part of the formulation because continuity, states, and unbounded fields depend on it.

A locally covariant theory is a covariant functor

$$\mathcal A:\mathbf{Loc}\longrightarrow\mathbf{Alg}.$$

It assigns an algebra $\mathcal A(M)$ to every spacetime and a morphism $\mathcal A(\psi):\mathcal A(M)\to\mathcal A(N)$ to every admissible embedding. The functor laws are

$$\mathcal A(\operatorname{id}_M)=\operatorname{id}_{\mathcal A(M)}, \qquad \mathcal A(\chi\circ\psi) =\mathcal A(\chi)\circ\mathcal A(\psi).$$

The second equation is the mathematical content of consistent embedding. An observable prepared in $M$, embedded into $N$, and then into $L$ must be the same observable obtained by embedding $M$ directly into $L$. A rule that renormalizes anew using unrelated coordinates after each embedding would violate this equation even if every single algebra looked reasonable.

The Klein–Gordon functor

Fix once and for all the scalar mass $m$ and curvature coupling $\xi$. Changing either parameter gives a different functor, not another object of the same theory. For each $M$, let $\mathcal A(M)$ be generated by $\Phi_M(f)$, $f\in C_0^\infty(M)$, with

$$\Phi_M(P_Mh)=0, \qquad [\Phi_M(f),\Phi_M(g)]=iE_M(f,g)\mathbf 1,$$

where $E_M=G_{M,\mathrm{ret}}-G_{M,\mathrm{adv}}$. Define

$$\mathcal A(\psi)\Phi_M(f)=\Phi_N(\psi_*f).$$

This is well defined because the embedding intertwines $P$, $E$, reality, and the volume pairing. In particular,

$$E_N(\psi_*f,\psi_*g)=E_M(f,g),$$

so the commutator is preserved. Extension by zero composes, which proves the functor law on generators and hence on the generated algebra. The construction and its causal and time-slice properties are developed in Brunetti, Fredenhagen, and Verch 2003, §2.3, Theorem 2.2, pp. 8–9, PDF.

The functor uses no preferred state. A state $\omega_N$ on $\mathcal A(N)$ pulls back along $\mathcal A(\psi)$ to a state on $\mathcal A(M)$, but there need not be a canonical state assignment in the opposite direction. This asymmetry is physical: an embedding specifies how a local experiment sits in a larger spacetime, not how the exterior is prepared.

Locality and dynamics in functorial form

The net on one spacetime is recovered from the functor. For each admissible inclusion $\iota_O:O\hookrightarrow M$, define

$$\mathcal A_M(O) =\mathcal A(\iota_O)\bigl(\mathcal A(O)\bigr) \subset\mathcal A(M).$$

Functoriality immediately gives isotony. Einstein causality becomes a condition on pairs of morphisms with causally disjoint images: their image subalgebras commute. The time-slice property says that $\mathcal A(\psi)$ is an isomorphism whenever $\psi(M)$ contains a Cauchy surface of the target. These conditions appear together in the generally covariant locality principle Brunetti, Fredenhagen, and Verch 2003, §2.2, Definition 2.1, p. 7, PDF.

The time-slice condition turns agreement of metrics outside a compact region into comparison maps through past and future Cauchy neighborhoods. That construction leads to relative Cauchy evolution. Thus background response is not an extra structure pasted onto the theory: it can be built from the same functor and its dynamics.

Same theory versus same algebra

An algebra isomorphism $\mathcal A(M)\cong\mathcal B(M)$ on one spacetime does not prove that $\mathcal A$ and $\mathcal B$ are the same theory. One must compare their assignments and embedding maps on the full category. The appropriate equivalence is a natural isomorphism: a family $\eta_M:\mathcal A(M)\to\mathcal B(M)$ satisfying

$$\eta_N\circ\mathcal A(\psi) =\mathcal B(\psi)\circ\eta_M$$

for every $\psi:M\to N$. Without this commuting square, a family of objectwise isomorphisms may scramble the physical identification of local observables from one embedding to another.

Conversely, functoriality alone is not a complete physical theory. A trivial functor assigning the same scalar algebra to every spacetime can obey the formal composition law while ignoring dynamics. Causality, time-slice, scaling, field content, and suitable locality conditions supply additional discriminators. The later notion of dynamical locality asks whether the local content defined by embeddings agrees with the content detected by response to metric changes.

A map-by-map prescription that fails composition

Consider three nested admissible spacetimes $M\subset N\subset L$ and retain the usual Klein–Gordon algebra on each. For every proper inclusion, propose the map $\widetilde\alpha_\psi\Phi_M(f)=-\Phi_N(\psi_*f)$. Each individual map preserves the field equation, adjoint, and CCR: the two minus signs in a commutator cancel. Nevertheless, two successive proper inclusions give

$$\widetilde\alpha_\chi\widetilde\alpha_\psi\Phi_M(f) =+\Phi_L((\chi\circ\psi)_*f),$$

whereas the prescribed direct map gives the negative of that field. Thus good objectwise algebras and relation-preserving maps do not yet define a functor. Their compositions must also agree. A consistent geometric pushforward does satisfy this test, as do coordinate Jacobians used with their proper chain rule; coordinate dependence alone is not a proof of a composition failure.

Exercises

Let $\psi:M\to N$ and $\chi:N\to L$ be admissible embeddings. Prove the functor law for the free Klein–Gordon generator, and identify the support fact needed for extension by zero.

Solution

By definition,

$$\mathcal A(\chi)\mathcal A(\psi)\Phi_M(f) =\mathcal A(\chi)\Phi_N(\psi_*f) =\Phi_L(\chi_*\psi_*f).$$

Extension by zero is compatible with composition, so $\chi_*\psi_*f=(\chi\circ\psi)_*f$. Therefore the last expression is $\mathcal A(\chi\circ\psi)\Phi_M(f)$. The pushforward remains smooth at the image boundary because $f$ has compact support strictly inside $M$; its image is compactly contained in $\psi(M)$, so extension by zero crosses no nonzero boundary value. Causal convexity then ensures that the target causal propagator reproduces the source pairing, preserving the field relations.

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.