Natural Transformations, Fields, and Subtheory Embeddings

The same observable formula on many spacetimes is not yet a locally covariant field. A field must intertwine every admissible embedding, and a map between two theories must do so component by component. Natural transformations encode exactly these commuting-square requirements and expose background-dependent choices that would otherwise be hidden.

Required background. QFT Frameworks: Object Classes and Maps supplies typed comparison claims. From Pointlike Fields to Nets and Affiliated Operators separates unbounded fields from bounded local algebras. Locally Covariant QFT as a Functor supplies the two functors compared below.

Helpful background. Categories, Functors, and Natural Transformations supplies the categorical definition. Conformal Transformations and Frame Changes illustrates why transformation weights and background data must be carried explicitly.

Fields as natural transformations

Let $\mathcal D:\mathsf{Loc}\to\mathsf{Vec}$ assign $C_c^\infty(M)$ to $\boldsymbol M$ and extension by zero $\psi_*$ to a Loc morphism. Regard each algebra $\mathcal A(\boldsymbol M)$ as a vector space when defining a linear field. A locally covariant field is a natural transformation

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

meaning that every $\psi:\boldsymbol M\to\boldsymbol N$ makes the square commute:

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

This equality has types: the left side first forms an element of $\mathcal A(M)$ and transports it; the right side first transports a test function and then evaluates the field on $N$. Brunetti, Fredenhagen, and Verch formulate fields this way in Brunetti, Fredenhagen, and Verch 2003, § 2.5, pp. 11–13.

For the Klein–Gordon CCR functor, $f\mapsto\Phi_M(f)$ is natural by construction. This realizes a concrete application of Local Covariance, Isometries, and Boundaries: the square commutes for every admissible embedding because extension by zero and the algebra homomorphism were defined together.

Theory morphisms and subtheories

If $\mathcal A,\mathcal B:\mathsf{Loc}\to\mathsf{Phys}$ are theories, a natural transformation $\zeta:\mathcal A\Rightarrow\mathcal B$ consists of Phys morphisms

$$\zeta_M:\mathcal A(M)\longrightarrow\mathcal B(M)$$

such that $\mathcal B(\psi)\zeta_M=\zeta_N\mathcal A(\psi)$ for every Loc morphism. If every component is monic, $\mathcal A$ is a subtheory of $\mathcal B$ in this sense. If every component is an isomorphism, $\zeta$ is a natural isomorphism and the two theories are equivalent within the chosen categories. An isomorphism on one spacetime is not yet a natural isomorphism; dynamical locality will later provide a theorem under additional hypotheses.

Natural fields can also be transported through a theory embedding: $\zeta\circ\Phi$ is a $\mathcal B$-valued natural field. The construction does not say that every local observable is generated by one distinguished list of fields, nor that two different field coordinatizations define different theories.

Renormalized fields and the covariance test

For Wick powers, local covariance sharply restricts finite renormalization. In four dimensions, for example, two admissible prescriptions for a real scalar field can differ schematically by

$$\widetilde{\Phi^2}_M(f)-\Phi^2_M(f) =\int_M f\,(c_1R_M+c_2m^2)\,\mathbf1\,d\mathrm{vol}_M,$$

with constants $c_1,c_2$ fixed across the category and with the appropriate scaling, smoothness, and microlocal hypotheses. Hollands and Wald state the local-covariant field condition and classify such ambiguities in Hollands and Wald 2001, Definition 3.2, p. 17, and Theorem 5.1, pp. 30–34.

Choosing $c_1$ separately on each spacetime may leave a perfectly meaningful formula on every object, but it generally violates the naturality square. For an embedding $\psi:M\to N$, the two sides then multiply the pulled-back curvature by different constants. This is the adversarial test: use one spacetime as a causally convex subspacetime of another and compare the two routes exactly.

An independent check uses composition. If the square holds for $\psi$ and $\chi$, then applying the two equalities must reproduce the square for $\chi\circ\psi$. A prescription that depends on an untransported coordinate choice, cutoff, or independently selected curvature coefficient fails this check.

Products, derivatives, and naturality

Natural transformations may be combined only when their products are defined in the target. For ordinary free fields, $\Phi_M(f)\Phi_M(g)$ is an algebra element, but the coincident-point symbol $\Phi_M(x)^2$ is not obtained by setting $f=g=\delta_x$. A locally covariant Wick square requires a microlocal extension to the diagonal plus the finite-renormalization conditions above. Naturality constrains the result after that analytic construction; it does not perform the extension.

Differential operators can act naturally as well. If $Q_M$ is built locally and covariantly from the metric and preserved background data, then $f\mapsto\Phi_M(Q_Mf)$ is natural because $Q_N\psi_*f=\psi_*Q_Mf$ on the embedded region. A coordinate derivative or a nontransported choice of frame lacks this intertwining relation. Testing the operator square before testing the field square is often the quickest diagnosis.

These examples separate three demands: existence of each component, analytic control of its products or domains, and covariance of the resulting family. Passing one demand does not supply the others.

Exercise

Let $c(M)$ be a real number assigned to each Loc object and define $\Psi_M(f)=\Phi_M(f)+c(M)\int_M f\,d\mathrm{vol}_M\,\mathbf1$. Find the condition on $c$ for $\Psi$ to be natural.

Solution

For every $\psi:M\to N$, naturality requires $c(M)\int_Mf=c(N)\int_N\psi_*f=c(N)\int_Mf$ for all $f$. Hence $c(M)=c(N)$ whenever a Loc morphism joins the objects. On a category connected by zigzags of morphisms this forces one common constant. Arbitrary object-by-object choices are not natural.

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.
• 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.