Fields as Natural Transformations

A locally covariant field is not a separate operator-valued distribution chosen on every background. It is one family of smear-to-observable maps that commutes with every admissible spacetime embedding. Category theory packages that requirement as a natural transformation. The naturality square is the test that a proposed field has the same physical meaning whether it is constructed in a small spacetime first or after that spacetime is embedded in a larger one.

Required background. Flat-Spacetime Quantum Field Theory supplies operator-valued distributions and smearing, which are reformulated here without a preferred representation.

Helpful background. Partial Differential Equations and Green Functions provides the field equation and propagator identities respected by embedding; the relevant quotient is restated below. Causality and Global Structure defines the admissible causal embeddings. Quantum Fields in Curved Spacetime provides the volume-wide context in which locally covariant Wick fields and stress–energy will later be constructed.

Test functions as a functor

Let $\mathcal D$ assign to each spacetime $M$ its compactly supported smooth test functions,

$$\mathcal D(M)=C_0^\infty(M),$$

and to each admissible embedding $\psi:M\to N$ the pushforward by extension by zero, $\mathcal D(\psi)f=\psi_*f$. Compact support ensures that this extension is smooth. Identity embeddings act as identities and pushforwards compose, so $\mathcal D$ is a covariant functor from $\mathbf{Loc}$ to vector spaces.

Tensor and spinor fields require the appropriate test-section bundles and pushforwards. A scalar density convention can move volume factors between the test object and the field; either choice is acceptable if used consistently. What is not acceptable is to regard coordinate components as geometric test sections without specifying their transformation.

The observable assignment is another functor, $\mathcal A:\mathbf{Loc}\to\mathbf{Alg}$. A field must connect these two functors rather than merely produce an element of each algebra. This functorial formulation is introduced in Brunetti, Fredenhagen, and Verch 2003, §2.5, Definition 2.4, pp. 12–15, PDF.

The naturality square

A locally covariant scalar field is a natural transformation, after viewing the observable algebras as their underlying vector spaces:

$$\Phi:\mathcal D\Longrightarrow\mathcal A.$$

More formally the target is $U\circ\mathcal A$, where $U$ forgets algebra multiplication but retains the vector-space structure. The smearing map is linear, not an algebra homomorphism from test functions. For every spacetime $M$ it has a component $\Phi_M:\mathcal D(M)\to\mathcal A(M)$. For every admissible embedding $\psi:M\to N$, naturality requires

$$\mathcal A(\psi)\bigl(\Phi_M(f)\bigr) =\Phi_N(\psi_*f), \qquad f\in\mathcal D(M).$$

Equivalently, the square

$$\begin{array}{ccc} \mathcal D(M) & \xrightarrow{\ \Phi_M\ } & \mathcal A(M)\\ \downarrow\scriptstyle{\psi_*} && \downarrow\scriptstyle{\mathcal A(\psi)}\\ \mathcal D(N) & \xrightarrow{\ \Phi_N\ } & \mathcal A(N) \end{array}$$

commutes. Following the top then right means “construct the observable in $M$, then embed it.” Following the left then bottom means “embed its smearing data, then construct it in $N$.” Equality says that the ambient spacetime does not alter the observable inside the embedded region.

Naturality is stronger than writing an expression with contracted indices. It tests compatibility under all morphisms, including embeddings between different global geometries. It is also weaker than invariance: the field is allowed to transform, but only through the declared functor maps.

The Klein–Gordon field

For fixed $m$ and $\xi$, the smeared free field obeys

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

with $E_M=G_{M,\mathrm{ret}}-G_{M,\mathrm{adv}}$. If $\psi:M\to N$ is admissible, the differential operator and volume form are geometric:

$$P_N\psi_*h=\psi_*P_Mh, \qquad E_N(\psi_*f,\psi_*g)=E_M(f,g).$$

Therefore $\Phi_N(\psi_*f)$ satisfies exactly the same field-equation, reality, and commutator relations as the image $\mathcal A(\psi)\Phi_M(f)$. In the universal algebra construction this identifies the two and proves naturality.

The equation can itself be read as an identity between natural transformations: $\Phi\circ P=0$, where $P$ is a natural differential operator on test functions. Derivatives and curvature couplings are allowed when formed locally from the background fields. A term that refers to a preferred coordinate origin, a global mode basis, or data outside the causal embedding is not locally covariant.

Composite fields and finite freedom

Products such as $\Phi_M(x)^2$ are singular and cannot be defined by pointwise multiplication. After Hadamard subtraction one seeks a Wick square $\Phi_M^2(f)$ that is again natural. Local covariance requires that the subtraction be built from the metric, curvature, mass, coupling, and a locally specified scale—not from a vacuum selected on one favored spacetime. Hollands and Wald formulate these fields and their covariance conditions in Hollands and Wald 2001, §§2–3, pp. 293–306.

Naturality does not make the answer unique. In four dimensions two acceptable Wick squares can differ schematically by

$$\widetilde{\Phi^2}_M =\Phi_M^2+(c_1m^2+c_2R_M)\mathbf 1,$$

with constants constrained further by scaling and other axioms. The curvature scalar is itself natural: $\psi^*R_N=R_M$. Thus the finite term respects the naturality square. Local covariance classifies the ambiguity; it does not remove the renormalization conditions needed to fix its constants.

Normal ordering with respect to the Minkowski vacuum illustrates the opposite behavior. It defines a useful field on Minkowski spacetime, but a generic curved spacetime has no corresponding preferred vacuum. Transporting that prescription through embeddings can depend on the chosen ambient state, so it does not define a natural field over $\mathbf{Loc}$.

Fields are not natural states

A natural field and a preferred natural state are different structures. The field gives compatible algebra elements for compatible test data. It does not assign expectation values. A state $\omega_N$ pulls back to $M$ by $\omega_N\circ\mathcal A(\psi)$, but choosing one state $\omega_M$ on every spacetime and demanding compatibility in all directions is a much stronger condition.

This distinction blocks a tempting but false inference: because the Klein–Gordon field is locally covariant, its vacuum must also be locally covariant. The field equation and commutator depend only on local geometric data, whereas positive-frequency splitting depends on global symmetry or asymptotic information. Under suitable dynamical hypotheses, a universal natural state would force a theory toward triviality; distinguished stationary or thermal states on particular backgrounds remain possible.

The same logic separates local observables from global particle number. A Wick polynomial can be natural even though the mode number operator used in a chosen Fock representation is not. Covariance organizes what every local observer can compare; it does not manufacture a global notion of particles.

A failed naturality square

Define on each spacetime a field

$$\Psi_M(f)=\Phi_M(f)+c_M\int_M f\,dV_M\,\mathbf 1,$$

where $c_M$ is a fixed dimensionally appropriate constant $c\ne0$ when $M$ has finite total four-volume, and is zero otherwise. Each $\Psi_M$ is a linear, continuous smear-to-observable map and preserves the commutator. It need not satisfy the homogeneous Klein–Gordon equation; it is an example of an attempted scalar observable field, not a second free-field solution. Embed a finite-volume diamond $M$ into all of Minkowski spacetime $N$, and choose a test function with $I_f=\int_M f\,dV_M\ne0$. Then

$$\mathcal A(\psi)\Psi_M(f) =\Phi_N(\psi_*f)+c I_f\mathbf 1 \ne \Phi_N(\psi_*f) =\Psi_N(\psi_*f).$$

The failure occurs even when the metrics agree throughout $\psi(M)$. Replacing $c_M$ by a local curvature polynomial with the appropriate smearing can restore naturality, provided dimensions and tensor type agree. This dropped-locality test is the practical content of the abstract square. It is a discriminating counterexample: an objectwise field definition need not be a locally covariant field.

Exercises

Show that $M\mapsto R_M\mathbf 1$ defines a locally covariant scalar field when smeared as $\mathcal R_M(f)=\int_M R_M f\,d\operatorname{vol}_M\, \mathbf 1$. State the embedding hypotheses used.

Solution

For an orientation-preserving isometric embedding $\psi:M\to N$, curvature is preserved on the image and the volume form pulls forward. Hence

$$\int_N R_N\,\psi_*f\,d\operatorname{vol}_N =\int_M R_M f\,d\operatorname{vol}_M.$$

The algebra morphism preserves the unit, so

$$\mathcal A(\psi)\mathcal R_M(f) =\mathcal R_N(\psi_*f).$$

We used isometry to preserve $R$, orientation to preserve the integration convention, and compact support to make extension by zero smooth and the integrals local. No state or global volume enters.

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.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI. Open PDF.