Haag–Kastler Nets and Locality

A Haag–Kastler theory replaces a single global operator algebra by a compatible assignment of observables to spacetime regions. The assignment records inclusion and causal separation before a Hilbert-space representation is chosen; a vacuum state then adds implementable spacetime symmetry and the positive-energy spectrum. The distinction between the abstract C*-net and its represented von Neumann net is essential: locality belongs to the assignment, whereas normality, commutants, and factor type depend on a representation.

Required background. Positivity, Spectrum, Covariance, and Locality Hypotheses separates the four logical inputs used below. Wightman Fields, Domains, and Axioms supplies the unbounded-field starting point, and The Wightman Reconstruction Theorem explains when vacuum distributions produce that field framework.

Helpful background. Operator Algebras and Positive Functionals supplies C*- and von Neumann-algebra language. Regions, Causal Complements, and Nets of Observables develops the subsystem interpretation. CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities is a useful comparison when locality is only approximate.

The local net and its axioms

Let $\mathcal K$ be a family of open regions in Minkowski spacetime, commonly bounded double cones; wedges and causally complete regions are also useful, but changing $\mathcal K$ changes the statement. An abstract local net is a map

$$O\longmapsto \mathcal A(O)$$

from $\mathcal K$ to unital C*-algebras, with compatible unital injective star homomorphisms. After identifying each algebra with its image, the basic conditions are:

• isotony: $O_1\subset O_2$ implies $\mathcal A(O_1)\subset\mathcal A(O_2)$;
• Einstein causality: $O_1\subset O_2'$ implies $[\mathcal A(O_1),\mathcal A(O_2)]=0$;
• covariance: an action $g\mapsto\alpha_g$ satisfies $\alpha_g(\mathcal A(O))=\mathcal A(gO)$;
• time-slice or primitive causality, when assumed: a neighborhood of a Cauchy surface generates the algebra of its causal development.

Here $O'$ is the interior of the causal complement. Locality gives an inclusion in a commutant, not automatically Haag duality. Additivity, duality, and the time-slice property are separate conditions and are compared later in this chapter.

A state $\omega_0$ on the quasilocal algebra is a vacuum state when it is Poincaré invariant and its GNS representation has a strongly continuous implementing representation $U_0$ with

$$U_0(g)\pi_0(A)U_0(g)^{-1}=\pi_0(\alpha_g(A)), \qquad U_0(g)\Omega_0=\Omega_0,$$

and joint translation spectrum in the closed forward cone. The represented local algebra is

$$\mathfrak A_0(O)=\pi_0(\mathcal A(O))''.$$

Taking the double commutant is representation-dependent. A different state can produce a nonisomorphic von Neumann closure even though the abstract net is unchanged. The original framework and its separation of locality, covariance, and state assumptions are set out in Haag and Kastler 1964, §§ 2–3, pp. 850–856; a modern construction with explicit free fields is given in Fewster and Rejzner 2020, § 4.1, pp. 13–15.

First application: the massive free-scalar Weyl net

Spacelike Compatibility and Local Observables supplies the free-field commutator whose bounded Weyl form is used here.

Let $P=\Box+m^2$ with $m>0$, let $E=E_{\mathrm{adv}}-E_{\mathrm{ret}}$, and write

$$\mathcal S=C_c^\infty(M,\mathbb R)/P C_c^\infty(M,\mathbb R), \qquad \mathcal S(O)=\{[f]\in\mathcal S:\operatorname{supp}f\subset O \text{ for some representative }f\},$$

and equip $\mathcal S$ with

$$\sigma([f],[g])=E(f,g).$$

The Weyl algebra is generated by unitaries $W([f])$ satisfying

$$W([f])^*=W(-[f]), \qquad W([f])W([g])=e^{-i\sigma([f],[g])/2}W([f+g]).$$

Define $\mathcal A(O)$ as the C*-subalgebra generated by $W([f])$ with $\operatorname{supp}f\subset O$. If $O_1\subset O_2$, the generating set for $O_1$ is contained in that for $O_2$, proving isotony. If $O_1$ and $O_2$ are causally disjoint, causal support of $E$ gives $E(f,g)=0$, so the Weyl relations give exact commutativity. Poincaré transformations act by $\alpha_g(W([f]))=W([g_*f])$ because $P$ and $E$ are covariant. These checks are construction-level identities, not assumptions about a chosen Fock space Fewster and Rejzner 2020, § 4.2, pp. 15–19.

The positive-frequency two-point function defines a Poincaré-invariant quasifree vacuum state. Its GNS representation is the usual bosonic Fock representation; translations have joint spectrum in $\overline V_+$ and the vacuum is the unique invariant vector for $m>0$. Local cyclicity is a further theorem: the spectrum condition, covariance, locality, cyclicity for the global algebra, and weak additivity yield the Reeh–Schlieder conclusion for every nonempty double cone Reeh and Schlieder 1961, pp. 1051–1068. It does not follow merely from the Weyl relations.

Failure test: imposed tensor factors

Suppose one assigns an independent tensor factor $\mathcal B(\mathcal H_O)$ to every bounded region. For nested regions $O_1\subset O_2$, independent factors provide no canonical injective map preserving the observables already assigned to $O_1$. For overlapping regions, two unrelated tensor decompositions generally disagree on the overlap; for causal complements, the tensor commutant may contain exactly the wrong operators. Thus the proposal fails before any continuum type-III claim is needed.

The surviving statement is weaker: a nonzero separation and a split inclusion may provide an intermediate type-I factor under additional phase-space hypotheses. It does not turn every sharp local algebra into an intrinsic tensor factor.

Independent checks

For a finite diagram choose $O_1\subset O_2$ and $O_3\subset O_2'$ and check, in this order,

$$\mathcal A(O_1)\subset\mathcal A(O_2), \qquad \mathcal A(O_3)\subset\mathcal A(O_2)', \qquad \alpha_g(\mathcal A(O_i))=\mathcal A(gO_i).$$

For the Weyl net these reduce respectively to support inclusion, $E(f_2,f_3)=0$, and covariance of $E$. After selecting the vacuum, separately verify that the Fourier support of the two-point function lies on the positive-energy mass shell. This order prevents a representation-specific spectral fact from being mistaken for an abstract-net axiom.

Exercises

Let $f$ and $g$ have causally disjoint supports. Show directly from the Weyl relations that $W([f])$ and $W([g])$ commute.

Solution

Causal support of the advanced-minus-retarded operator gives $E(f,g)=0$. Hence

$$W([f])W([g])=W([f+g])=W([g])W([f]),$$

because both Weyl phase factors are one. The conclusion uses the causal support theorem for $E$; disjoint support without causal disjointness would not suffice.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI; Open PDF.
• Haag, Rudolf, and Daniel Kastler. “An Algebraic Approach to Quantum Field Theory.” Journal of Mathematical Physics 5 (1964): 848–861. DOI.
• Reeh, Helmut, and Siegfried Schlieder. “Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten Feldern.” Il Nuovo Cimento 22 (1961): 1051–1068. DOI.