Local Nets and Einstein Causality

A local net assigns an algebra of observables to each admissible spacetime region and makes causal separation an algebraic statement. Its central requirement is simple: observables supported in causally disjoint regions commute. This does not erase vacuum correlations; it says that an operation in one region cannot change the statistics of an observable in the other. The construction below uses the real Klein–Gordon field, but the distinction between localization, correlation, and signaling is general.

Required background. Flat-Spacetime Quantum Field Theory supplies the smeared-field and canonical-commutator ideas used below.

Helpful background. Partial Differential Equations and Green Functions supplies the causal-support fact for the propagator, which is also restated below. Causality and Global Structure defines causal disjointness and causally convex regions. Quantum Fields in Curved Spacetime provides the longer route from classical fields to this representation-free formulation and then to modular theory.

Regions and their observable algebras

Let $(M,g)$ be an oriented, time-oriented, globally hyperbolic spacetime. An admissible region $O\subset M$ will mean a nonempty open, causally convex, globally hyperbolic subspacetime with the induced structures. Causal convexity means that a causal curve in $M$ whose endpoints lie in $O$ never leaves $O$. This condition prevents the algebra assigned to $O$ from silently depending on propagation through an omitted part of spacetime.

A net is an assignment

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

of unital star-algebras, together with injective unital star-homomorphisms $\iota_{O_2O_1}:\mathcal A(O_1)\to\mathcal A(O_2)$ whenever $O_1\subset O_2$. The maps obey identity and composition laws. It is common to identify each algebra with its image in a common ambient algebra $\mathcal A(M)$; then the first axiom becomes isotony,

$$O_1\subset O_2 \quad\Longrightarrow\quad \mathcal A(O_1)\subset\mathcal A(O_2).$$

Isotony is not the assertion that a smaller laboratory can measure everything available in a larger one. It says only that an observable constructible in the smaller region remains the same observable when that region is regarded as part of the larger experiment. This categorical form of the net axioms is set out in Brunetti, Fredenhagen, and Verch 2003, §2.4, Proposition 2.3, pp. 10–12, PDF.

A second structural condition is additivity. If $O$ is covered by a family $\{O_i\}$, the algebra $\mathcal A(O)$ should be generated by the subalgebras $\mathcal A(O_i)$, subject to the precise cover and completion used by the theory. Additivity prevents observables from appearing only when one names a large region, despite being locally constructible throughout it. It is logically distinct from isotony and from the time-slice property: a theory can satisfy one without automatically satisfying the others.

Einstein causality as commutativity

Two regions $O_1$ and $O_2$ are causally disjoint when no future- or past-directed causal curve connects them. Einstein causality requires

$$[A_1,A_2]=0, \qquad A_i\in\mathcal A(O_i), \qquad O_1\perp O_2.$$

The notation $O_1\perp O_2$ records causal, not merely spatial, separation. Two regions lying on one Cauchy surface may fail to be causally disjoint if their time extensions overlap each other’s light cones. Conversely, curved geometry changes which pairs are disjoint without changing the algebraic meaning of the condition.

For fermionic field generators, one first has graded commutativity: odd generators anticommute at causal separation. Physical observables belong to the even subalgebra and commute in the usual sense. Gauge theories may also carry centers or topological observables, so a local-potential description can be too small; none of those qualifications weakens the required commutativity of genuinely local bosonic observables.

The formulation is deliberately representation independent. No Hilbert space, vacuum vector, or particle basis is needed to state it. A state is introduced later as a positive functional on the algebra. This order matters in curved spacetime, where a preferred Fock representation generally does not exist.

The free scalar net from causal support

For the Klein–Gordon operator $P=\Box-m^2-\xi R$, define the causal propagator with the convention

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}.$$

For test functions $f,g\in C_0^\infty(M)$, the bilinear pairing

$$E(f,g)=\int_M f(x)(Eg)(x)\,d\operatorname{vol}_g(x)$$

descends to the quotient $C_0^\infty(M)/P C_0^\infty(M)$. The smeared field algebra is generated by $\Phi(f)$ subject to linearity, reality, the field equation $\Phi(Ph)=0$, and

$$[\Phi(f),\Phi(g)]=iE(f,g)\mathbf 1.$$

The local algebra $\mathcal A(O)$ is generated by $\Phi(f)$ with $\operatorname{supp}f\subset O$. If the supports of $f$ and $g$ are causally disjoint, the support property of $E$ gives $E(f,g)=0$, so the generators commute. Polynomial combinations then commute term by term. Thus Einstein causality is not imposed by an unrelated rule in this example; it is the quantum expression of finite propagation speed. The complete quotient and CCR construction is developed in Bär, Ginoux, and Pfäffle 2007, Chapter 4.

The Weyl form makes the same point without relying on unbounded field operators. With

$$W([f])W([g]) =e^{-iE(f,g)/2}W([f+g]),$$

causally disjoint smearing functions give $W([f])W([g])=W([g])W([f])$. A change from $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ to the opposite convention changes both the symplectic form and the phase; changing only one would create a false sign disagreement.

Correlation is not signaling

Commutativity does not imply statistical independence. A state $\omega$ may have

$$\omega(A_1A_2)\ne\omega(A_1)\omega(A_2)$$

even when $[A_1,A_2]=0$. The Minkowski vacuum already has spacelike correlations, and later chapters explain why local continuum algebras make those correlations especially rich.

What commutativity controls is intervention. Suppose a unitary $U_1$ belongs to $\mathcal A(O_1)$ and $B_2\in\mathcal A(O_2)$ with $O_1\perp O_2$. After the local operation, the expectation of $B_2$ is

$$\omega_{U_1}(B_2) =\omega(U_1^*B_2U_1) =\omega(B_2U_1^*U_1) =\omega(B_2).$$

The remote marginal is unchanged, although joint correlations can be revealed after ordinary causal communication. The same check extends to an unselected local operation $B\mapsto\sum_jK_j^*BK_j$, with $K_j\in\mathcal A(O_1)$ and $\sum_jK_j^*K_j=\mathbf1$: commute $B_2$ through every $K_j$ to obtain $\sum_jK_j^*B_2K_j=B_2$. Conditioning on a particular outcome can change a conditional expectation at $O_2$, but identifying that subensemble requires classical communication. Causality constrains the unconditioned statistics. The causality and timeslice axioms are compared explicitly in Brunetti, Fredenhagen, and Verch 2003, §2.2, Definition 2.1, pp. 6–8, PDF.

Einstein causality is therefore weaker than Haag duality, the split property, or a tensor-factor decomposition. It also says nothing by itself about state preparation costs, detector switching, or energy inequalities. Treating any of those stronger conclusions as a synonym for commutativity hides the very questions the algebraic framework is designed to separate.

Exercises

Let $f$ and $g$ have causally disjoint compact supports. Use the Weyl relations to show directly that $W([f])$ and $W([g])$ commute. Then explain why this result is compatible with a correlated state.

Solution

Causal disjointness and finite propagation give $E(f,g)=0$. Antisymmetry also gives $E(g,f)=0$, so

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

This is an operator identity and is independent of the state. A state can still satisfy $\omega(W([f])W([g]))\ne \omega(W([f]))\omega(W([g]))$ because commutativity does not force a product state. If a local unitary generated by $W([f])$ is applied, however, it commutes through $W([g])$ and leaves the latter’s expectation unchanged.

References

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