Restricted States and Subregion Observables

Restricting a QFT state to a region means restricting its expectation-value functional to the region’s represented observable algebra. It is not, in general, a partial trace. A density operator describes the restriction only when a suitable type-I representation or regulator has supplied a tensor factor and a trace.

Required background. Use the definitions of normal states and operator algebras, observable matrix elements, and the local net. Helpful background. Hilbert-space completion and duality clarify which topology is being used.

Restriction is an algebraic operation

A state on a unital $C^*$-algebra $\mathfrak A$ is a positive normalized linear functional $\omega$. For an inclusion $\mathfrak N\subset\mathfrak A$, its restriction is simply

$$\omega_{\mathfrak N}(A)=\omega(A), \qquad A\in\mathfrak N.$$

No Hilbert-space decomposition is needed. The restriction preserves positivity and normalization, and it contains exactly the statistics of measurements drawn from $\mathfrak N$.

In a representation $\mathfrak N\subset\mathcal B(\mathcal H)$, normality is additional structure: $\omega_{\mathfrak N}$ is normal when it is continuous under monotone increasing limits of positive operators, equivalently when it lies in the predual $\mathfrak N_*$. A trace-class operator $\rho$ on $\mathcal H$ defines a normal functional by $\omega(A)=\operatorname{Tr}(\rho A)$, but the same functional need not be represented by a density matrix inside $\mathfrak N$.

For a genuine factorization $\mathcal H=\mathcal H_A\otimes\mathcal H_B$ with $\mathfrak N=\mathcal B(\mathcal H_A)\otimes 1$, restriction is equivalent to

$$\omega_{\mathfrak N}(A\otimes 1) =\operatorname{Tr}_{\mathcal H_A}(\rho_A A), \qquad \rho_A=\operatorname{Tr}_{\mathcal H_B}\rho.$$

This familiar formula is a special representation theorem, not the definition of restriction.

The structural map places Restricted States and Subregion Observables among sharp local algebras, split inclusions, and regulated or operational substitutes.

A causally complete region determines a sharp local algebra, usually type III. A nonzero split collar or an explicit cutoff, mode selection, or probe model can instead supply a type-I realization; these alternatives enable ordinary density matrices but retain different physical approximations. Schematic, not to scale.

A regulated interval and its continuum target

Consider a scalar field regulated as a finite harmonic chain. For sites in an interval $A$, the canonical variables generate $\mathcal B(\mathcal H_A)$, and the vacuum or one-particle density matrix can be partially traced to $\rho_A$. Local expectation values computed from $\rho_A$ agree with the global state restricted to operators supported on those sites.

As the lattice spacing tends to zero at fixed physical interval, the local continuum algebra is generally type III. The sequence $\rho_A(a)$ remains useful for regulated entropies and correlators, but there need not be a limiting trace-class operator $\rho_A(0)$ belonging to the continuum local algebra. The stable comparison is instead

$$\lim_{a\to0}\operatorname{Tr}\!\bigl[\rho_A(a)A_a\bigr] =\omega_O(A)$$

for a controlled family of observables $A_a$ approximating $A\in\mathfrak A(O)$. The observable family, state convergence, and topology must all be stated.

Local indistinguishability

If two global states $\omega$ and $\varphi$ satisfy $\omega|_{\mathfrak N}=\varphi|_{\mathfrak N}$, no measurement whose effects lie in $\mathfrak N$ distinguishes them. Conversely, any $A\in\mathfrak N$ with $\omega(A)\ne\varphi(A)$ is a local witness. This makes the choice of algebra part of every distinguishability claim.

A common error is to enlarge the observable set during tomography. Reconstructing a regulated covariance matrix, for example, licenses claims about the regulated mode algebra that was measured; it does not automatically determine the state on a sharper continuum algebra.

Before applying this result, use the validity map to keep the algebra, state, operation class, resources, and approximation fixed.

Every local-information claim must specify the represented algebra and state, the allowed operations and resource support, and any split collar or regulator. The dashed lower boxes show what fails when the algebra, protocol, or limiting prescription is left implicit. Schematic.

Further reading

• Haag, Rudolf, and Daniel Kastler. “An Algebraic Approach to Quantum Field Theory.” Journal of Mathematical Physics 5 (1964): 848–861. DOI.
• Takesaki, Masamichi. Theory of Operator Algebras I. Berlin: Springer, 1979. DOI.