Locally Covariant QFT as a Functor

A locally covariant QFT is a covariant functor from geometric backgrounds to physical systems. The functor transports observables along admissible spacetime embeddings; Einstein causality constrains pairs of disjoint embeddings, while the time-slice axiom upgrades Cauchy embeddings to isomorphisms. These are independent requirements.

Required background. QFT Frameworks: Object Classes and Maps fixes the object–map–claim grammar. Haag–Kastler Nets and Locality supplies local algebras and causal commutation. Globally Hyperbolic Spacetimes and the Loc Categories defines the source category used here.

Helpful background. Categories, Functors, and Natural Transformations reviews functorial composition. Local Field Algebras, Causality, and the Time-Slice Property and Local Covariance, Isometries, and Boundaries give physical examples.

The functor and its two causal axioms

Let Phys be a category whose objects are unital $*$-algebras and whose morphisms are injective unit-preserving $*$-homomorphisms. A theory is a covariant functor

$$\mathcal A:\mathsf{Loc}\longrightarrow\mathsf{Phys}, \qquad \psi\longmapsto\mathcal A(\psi),$$

so that $\mathcal A(\mathrm{id}_{\boldsymbol M})=\mathrm{id}_{\mathcal A(\boldsymbol M)}$ and

$$\mathcal A(\chi\circ\psi)=\mathcal A(\chi)\circ\mathcal A(\psi).$$

Thus $\mathcal A(\boldsymbol M)$ is the algebra on a whole background and $\mathcal A(\psi)$ is the specified transport map—not an arbitrary comparison between two isomorphic algebras.

Einstein causality says that if $\psi_i:\boldsymbol M_i\to\boldsymbol N$ have causally disjoint images, then

$$[\mathcal A(\psi_1)(A_1),\mathcal A(\psi_2)(A_2)]=0$$

for all $A_i\in\mathcal A(\boldsymbol M_i)$. The time-slice axiom says that $\mathcal A(\psi)$ is an isomorphism whenever $\psi$ is Cauchy. Causality alone does not imply time-slice, and time-slice alone does not imply commutation. This functorial formulation and its free-field example are given in Brunetti, Fredenhagen, and Verch 2003, § 2.2, pp. 5–9.

Construction: the Klein–Gordon CCR functor

For $P_M=\Box_M+m^2+\xi R_M$, let $E_M=E_M^- -E_M^+$ be the causal propagator, with advanced and retarded Green operators defined consistently on every object. Generate a unital $*$-algebra by symbols $\Phi_M(f)$, $f\in C_c^\infty(M)$, subject to linearity and

$$\Phi_M(P_Mf)=0, \qquad \Phi_M(f)^*=\Phi_M(\overline f), \qquad [\Phi_M(f),\Phi_M(g)]=iE_M(f,g)\mathbf1.$$

For a Loc morphism $\psi:\boldsymbol M\to\boldsymbol N$, extension by zero gives $\psi_*f\in C_c^\infty(N)$, and one sets

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

Admissibility ensures that the Klein–Gordon operator and Green operators intertwine on the relevant supports. Therefore the field equation and commutator ideal are preserved. Pushforwards compose, so the maps are functorial. Causal support of $E_N$ gives Einstein causality. Uniqueness of the Cauchy problem gives the time-slice property. This constructs the concrete application requested by Local Covariance, Isometries, and Boundaries: each globally hyperbolic spacetime is sent to its causal-propagator algebra, and each Loc embedding to the induced homomorphism.

Checks and failure boundary

There are two fast independent checks. First, evaluate a composable pair on a generator:

$$\mathcal A(\chi)\mathcal A(\psi)\Phi_M(f) =\Phi_L((\chi\circ\psi)_*f).$$

Second, compare commutators before and after transport; one must obtain $E_N(\psi_*f,\psi_*g)=E_M(f,g)$. If an embedding is not causally convex, the target Green solution can leave and re-enter its image. The equality can fail, so the proposed generator map need not descend to the CCR quotient. An isometric embedding by itself is therefore insufficient.

The construction also has a clear scope boundary. It gives an algebraic free scalar theory, not a preferred state, Hilbert-space representation, or interacting continuum model. Those require additional inputs.

Functor covariance versus covariance on one spacetime

For an automorphism $\psi:M\to M$, the map $\mathcal A(\psi)$ is an algebra automorphism and resembles the usual action of a spacetime symmetry. LCQFT asks more: the same functor must also act on proper embeddings $M\to N$ and on morphisms between backgrounds with different curvature and topology. Poincaré covariance of a Minkowski theory therefore verifies only a small subcategory of the required diagrams.

Conversely, the fixed-spacetime net is recovered from the functor by restricting to inclusions $\iota_O:M|_O\to M$. Then

$$\mathcal A_M(O)=\mathcal A(\iota_O)\bigl(\mathcal A(M|_O)\bigr)$$

is isotonic because inclusions compose. This observation independently checks the direction of every map: smaller-region algebras enter the whole-spacetime algebra. It does not prove additivity, Haag duality, or that a net on one background uniquely extends to a functor on all of Loc.

The target category matters as well. Injective morphisms make regional embeddings faithful. If one instead permits arbitrary homomorphisms, a topological observable might be killed by an embedding and isotony in the usual sense is lost. Gauge theories force this issue explicitly rather than allowing it to be hidden in notation.

One final consistency test is the unit. Every $\mathcal A(\psi)$ must send $\mathbf1_M$ to $\mathbf1_N$; otherwise two spacelike subtheories would acquire incompatible identities inside the target algebra. Together with preservation of $*$ and multiplication, this confirms that the transport maps are physical-system morphisms rather than merely linear maps between vector spaces.

Exercise

Assume $\psi_1(M_1)$ and $\psi_2(M_2)$ are causally disjoint in $N$. Prove Einstein causality for the Klein–Gordon generators.

Solution

For $f_i\in C_c^\infty(M_i)$, no point of $\operatorname{supp}\psi_{1*}f_1$ is causally related to a point of $\operatorname{supp}\psi_{2*}f_2$. The support property of the causal propagator therefore gives $E_N(\psi_{1*}f_1,\psi_{2*}f_2)=0$. The CCR then makes the two transported generators commute. Polynomial algebras generated by them commute as well.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. Zürich: European Mathematical Society, 2007. DOI.
• 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.