No-Natural-State Results and Covariant State Spaces

A locally covariant theory may carry a useful family of Hadamard, ground, or thermal states without selecting one state naturally on every spacetime. The no-natural-state theorem targets a single functorially compatible choice. Under dynamical locality, Reeh–Schlieder, faithfulness, and extended locality, such a choice would force a theory with nontrivial local observables to collapse.

Required background. States, GNS Representations, and Folia supplies states and faithful GNS representations. Globally Hyperbolic Spacetimes and the Loc Categories, Locally Covariant QFT as a Functor, and The Same-Physics-in-All-Spacetimes Principle supply the geometric and comparison framework.

Helpful background. Ground, KMS, and State-Selection Criteria, Vacua, States, and Representations, Vacuum Ambiguity and Observer Dependence, and State-Selection Failure Modes distinguish local regularity from preferred-state claims.

State spaces and natural sections

A state-space assignment is contravariant: to each $M$ it assigns a set $\mathcal S(M)$ of states on $\mathcal A(M)$, and to $\psi:M\to N$ it assigns pullback

$$\mathcal A(\psi)^*:\mathcal S(N)\longrightarrow\mathcal S(M), \qquad \omega_N\longmapsto\omega_N\circ\mathcal A(\psi).$$

A natural state is one selected state $\omega_M\in\mathcal S(M)$ for every object such that

$$\omega_M=\omega_N\circ\mathcal A(\psi)$$

for every Loc morphism. This is much stronger than requiring that pullback preserve a class such as Hadamard states. A class can be locally covariant even when it has no natural section.

The naturality equation also makes every selected state invariant under relative Cauchy evolution. Each of the four maps defining $\operatorname{rce}_M[h]$ is a Cauchy morphism, and repeated pullback along the commuting diagram gives

$$\omega_M\circ\operatorname{rce}_M[h]=\omega_M.$$

The no-natural-state theorem

Fewster and Verch’s precise result assumes a dynamically local theory with a natural state. Suppose there is a spacetime $M$ with noncompact Cauchy surfaces for which the GNS representation of $\omega_M$ is faithful and has the Reeh–Schlieder property. Then relative Cauchy evolution is trivial on $M$. If the theory also obeys extended locality—kinematic algebras of causally disjoint regions intersect only in scalars—dynamical locality propagates this triviality and the theory itself is trivial in the stated categorical sense Fewster and Verch 2012, Definition 6.12 and Theorem 6.13, pp. 42–43.

The proof has a transparent mechanism. State invariance lets $\operatorname{rce}[h]$ be unitarily implemented in the GNS representation with the cyclic vector fixed. Choose a region spacelike to $\operatorname{supp}h$; relative Cauchy evolution already acts identically on its algebra. Reeh–Schlieder makes the cyclic vector separating for the relevant commutant argument, so the implementer must be the identity. Faithfulness then returns trivial rce at the abstract-algebra level. Dynamical locality turns absence of background response into collapse of regional distinctions; extended locality removes any common non-scalar residue.

Ground states: the concrete failed construction

On each ultrastatic spacetime, spectral theory can select a ground state for the Klein–Gordon field. Attempting to extend “choose the ultrastatic ground state” to a natural state on every globally hyperbolic spacetime fails because an embedding and a compact relative-Cauchy-evolution loop force the chosen state to be invariant under arbitrary admissible local metric changes. For a nontrivial free field with the regularity and Reeh–Schlieder properties above, the theorem turns that invariance into the contradiction. The physical selection criteria and their domains are set out in Ground, KMS, and State-Selection Criteria.

This failure does not mean that Hadamard states do not exist, that ground states never exist on stationary backgrounds, or that covariant state spaces are impossible. It says there is generally no single locally covariant preferred state satisfying the theorem’s assumptions. Dropping faithfulness, Reeh–Schlieder, dynamical locality, noncompact-Cauchy-surface input, or extended locality removes a step and requires a new argument.

Why each hypothesis is visible

Faithfulness is what carries identity of represented automorphisms back to the abstract algebra. Reeh–Schlieder supplies the cyclic/separating leverage that turns local agreement and invariance of the GNS vector into identity of the implementer. Noncompact Cauchy surfaces provide causally disjoint regions suitable for the argument. Dynamical locality converts trivial response into a statement about the full regional subobjects, and extended locality reduces their common intersection to scalars. Omitting any one of these does not prove that a natural state exists; it only blocks this no-go proof.

Thermal equilibrium illustrates the domain issue. A KMS state is selected relative to a specified time-evolution automorphism, usually supplied by a stationary Killing flow. A generic globally hyperbolic spacetime has no such distinguished flow. Likewise, an ultrastatic ground-state construction uses the product time and cannot be transported through an arbitrary curved embedding. These are valuable state-selection rules on subcategories, not candidates for an unrestricted natural section on Loc.

As an independent check, ask whether the proposed selection survives a compact metric perturbation that leaves the geometry unchanged in the far past and future. If the rule produces different scattering data but naturality demands the same pullback state around the Cauchy loop, it fails before any no-go theorem is invoked.

Exercise

Derive $\omega_M\circ\operatorname{rce}_M[h]=\omega_M$ from naturality and the four Cauchy maps.

Solution

Naturality identifies the state on $M$ with the pullback of the state on $M[h]$ through either $\tau^+$ or $\tau^-$. Hence $\omega_M=\omega_{M[h]}\circ\tau^+=\omega_{M[h]}\circ\tau^-$. Composing the second equality with $(\tau^-)^{-1}\tau^+$ gives $\omega_M\circ\operatorname{rce}_M[h]=\omega_M$ for the convention used here.

References

• 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.