Local Normality, Quasiequivalence, and Folia

Local normality asks whether a state is normal after restriction to each bounded regional algebra; local quasiequivalence asks whether two representations have the same normal state space on every such region. These notions can hold even when the global representations are disjoint. They are therefore the right comparison for finite-region measurements, but they do not produce a global unitary or identify infrared sectors.

Required background. Quasilocal C*-Algebras and Inductive Limits supplies the common abstract algebra and its local subalgebras. States, GNS Representations, and Folia supplies normal states, folia, and disjointness.

Helpful background. Von Neumann Factors and Type-III Local Algebras prevents normality from being confused with an intrinsic local density matrix. Ground, KMS, and Symmetry-Selected States gives the physical state classes. Restricted States and Subregion Observables explains restriction without tensor factorization, and GNS Representations, Local Normality, and Local Quasiequivalence gives the curved-spacetime free-field comparison.

Regional normality and folia

Let $\mathcal A$ be the quasilocal algebra and $O\mapsto\mathcal A(O)$ a local net. A state $\omega'$ is locally normal relative to a representation $\pi$ when, for every admitted bounded $O$, there is a normal functional $\widetilde\omega'_O$ on

$$\mathfrak M_\pi(O)=\pi(\mathcal A(O))''$$

such that $\omega'|_{\mathcal A(O)}=\widetilde\omega'_O\circ\pi$. Equivalently, the restriction belongs to the folium of $\pi|_{\mathcal A(O)}$. The representing trace-class operator is on the ambient Hilbert space; it does not imply $\mathfrak M_\pi(O)=\mathcal B(\mathcal H_O)$.

Representations $\pi_1$ and $\pi_2$ are locally quasiequivalent when their restrictions to every $\mathcal A(O)$ are quasiequivalent. One useful equivalent statement is that

$$\pi_1(A)\longmapsto\pi_2(A), \qquad A\in\mathcal A(O),$$

extends to a normal star isomorphism $\mathfrak M_{\pi_1}(O)\to\mathfrak M_{\pi_2}(O)$. Local quasi-containment is the one-directional version: every state normal in the first restricted representation is normal in the second. Mutual quasi-containment gives quasiequivalence.

The word “local” quantifies over the stated region class. Agreement on all double cones does not automatically mean agreement on wedges, the whole spacetime, or an infrared algebra at infinity.

The quasifree comparison criterion

For quasifree CCR states with covariance forms $S$ and $S'$ on the same real symplectic test space $K$, the Araki–Yamagami theorem gives a necessary and sufficient global criterion. First, the norms induced by $S$ and $S'$ must define the same topology after null directions are removed. Second, on the common completion,

$$\widetilde S^{1/2}-\widetilde S'^{1/2}$$

must be Hilbert–Schmidt. Both conditions are necessary; comparing only pointwise two-point functions is insufficient Araki and Yamagami 1982, theorem in § 1, pp. 284–285, and necessity proof in § 8, pp. 322–331.

For local quasiequivalence, apply the criterion after restricting the test functions or Cauchy data to $O$. Smoothness of the difference of two Hadamard two-point functions improves the regional covariance difference. Under Verch’s hypotheses for the Klein–Gordon Weyl algebra on a globally hyperbolic spacetime, quasifree Hadamard GNS representations are locally quasiequivalent Verch 1994, Theorem 3.6, pp. 522–526. The theorem is local and model-specific; it is not a statement about arbitrary states or interacting nets.

First application: vacuum and thermal free fields

Infinite-Volume KMS States, Passivity, and Phase Multiplicity supplies the massive free-scalar KMS state and its thermodynamic interpretation.

Let $\omega_0$ be the Minkowski vacuum state and $\omega_\beta$ the translation-invariant quasifree KMS state at $0<\beta<\infty$ on the same massive-scalar Weyl algebra. Both two-point functions have the same Hadamard short-distance singularity; their difference is smooth. Therefore, on each bounded double cone, Verch’s theorem makes their GNS restrictions quasiequivalent. Every vacuum-normal regional preparation can be represented as a normal state in the thermal regional representation and conversely.

Globally, the Bose occupation $n_\beta(\mathbf p)$ is present at every spatial position. On the infinite-volume one-particle space, the corresponding nonzero translation-invariant covariance difference fails the Hilbert–Schmidt condition: its kernel carries an infinite volume factor. The global vacuum and extremal thermal representations are consequently disjoint in the standard massive free model. Local normal agreement and global disjointness are compatible because they test different completions.

Failure test: patching local equivalences into one unitary

Suppose for every bounded $O$ one chooses a unitary-like local intertwiner between the two restricted theories. Local quasiequivalence does not make those choices compatible under inclusions, and it does not control their behavior as $O\nearrow M$. A global unitary would force global quasiequivalence and hence the global Hilbert–Schmidt condition, which the thermal occupation violates.

The strongest valid conclusion is equality of regional folia on each tested $O$. It does not identify the global cyclic vectors, Hamiltonians, particle numbers, or infrared phases.

Independent checks

For a nested sequence $O_1\Subset O_2\Subset\cdots$, restrict both covariance forms and evaluate the two Araki–Yamagami conditions at each stage. Record the regional Hilbert–Schmidt norm rather than only whether it is finite. Then repeat on the global completion. In the free thermal example every fixed bounded stage passes under the Hadamard theorem, while the global norm diverges with volume. This change of domain, not a contradiction between theorems, is the diagnostic.

Exercises

Show that local quasiequivalence is transitive for a fixed region class.

Solution

For each region $O$, quasiequivalence means equality of the two restricted folia. If $\mathcal F(\pi_1|_O)=\mathcal F(\pi_2|_O)$ and $\mathcal F(\pi_2|_O)=\mathcal F(\pi_3|_O)$, then $\mathcal F(\pi_1|_O)=\mathcal F(\pi_3|_O)$. Since this holds for every $O$ in the same class, $\pi_1$ and $\pi_3$ are locally quasiequivalent. The conclusion says nothing about global folia.

References

• Araki, Huzihiro, and Shigeru Yamagami. “On Quasi-Equivalence of Quasifree States of the Canonical Commutation Relations.” Publications of the Research Institute for Mathematical Sciences 18 (1982): 283–338. DOI; Open PDF.
• Verch, Rainer. “Local Definiteness, Primarity and Quasiequivalence of Quasifree Hadamard Quantum States in Curved Spacetime.” Communications in Mathematical Physics 160 (1994): 507–536. DOI.