The Same-Physics-in-All-Spacetimes Principle

Agreement on one spacetime need not mean agreement everywhere. A locally covariant theory can be engineered to contain one number of field species on one class of spacetimes and another number elsewhere while remaining functorial. The same-physics-in-all-spacetimes principle becomes a theorem only after restricting to a class—such as dynamically local theories—in which one component isomorphism propagates naturally.

Required background. Counterexamples, Nonconverses, and Hypothesis Stress Tests supplies the directional claim discipline. Locally Covariant QFT as a Functor and Natural Transformations, Fields, and Subtheory Embeddings define theories and their embeddings. Dynamical Locality and Kinematic–Dynamic Nets gives the hypothesis used in the positive theorem.

Helpful background. Claims, Dictionaries, and Regimes gives a broader framework for regime-limited equivalence claims.

A categorical same-physics criterion

Let $\mathfrak T$ be a class of theories $\mathcal A:\mathsf{Loc}\to\mathsf{Phys}$. The class has the SPASs property if every natural transformation $\zeta:\mathcal A\Rightarrow\mathcal B$ between members of $\mathfrak T$ that is an isomorphism at one spacetime has isomorphism components at every spacetime. Equivalently, a partial equivalence compatible with all embeddings cannot become proper elsewhere. Fewster and Verch give the precise categorical definition in Fewster and Verch 2012, Definition 4.1, p. 20.

This is a property of a chosen class of theories, not an extra axiom attached to one functor. Nor does it claim that any two algebras that happen to be abstractly isomorphic at one object define the same theory: a natural transformation must already relate them.

Diagonal theories: the adversarial example

The unrestricted class of locally covariant theories does not have the property. Start with a theory $\mathcal A$ and allow an object-dependent theory label $\lambda(M)$ that is monotone along Loc morphisms. A diagonal construction can set

$$\mathcal B(M)=\mathcal A(M)^{\otimes\lambda(M)}$$

and use the allowed structure maps to append unit factors whenever $\lambda$ increases. With a categorical choice of $\lambda$ adapted to the existence of morphisms, $\mathcal B$ can agree with $\mathcal A$ on Minkowski spacetime and contain extra species on selected spacetimes. The transformation is an isomorphism at the former but not the latter. Fewster and Verch construct and analyze these diagonal counterexamples in Fewster and Verch 2012, § 4, pp. 20–28.

This is the concrete stress test for Local Covariance, Isometries, and Boundaries: construct a diagonal pair that agrees on Minkowski space but changes species number on a selected class, then check which additional condition excludes it. Functorial covariance alone does not.

Dynamical locality gives the positive theorem

Let $\mathcal A$ and $\mathcal B$ be dynamically local theories in a target category with the required subobjects, equalizers, and unions. If $\zeta:\mathcal B\Rightarrow\mathcal A$ is a natural transformation and $\zeta_M$ is an isomorphism for one spacetime $M$, then every component is an isomorphism: $\zeta$ is a natural isomorphism Fewster and Verch 2012, Theorem 6.10, p. 41.

The proof mechanism explains the hypothesis. Naturality first transports the isomorphism through Cauchy morphisms. Dynamical locality identifies each regional subobject with the part detected by relative Cauchy evolution. Regional compatibility and additivity then propagate the component result along suitable spacetime deformations and inclusions. A diagonal extra species cannot remain invisible on one spacetime yet appear dynamically in another without breaking kinematic–dynamic equality.

The conclusion is directional. Dynamical locality is sufficient for this SPASs theorem within the stated setting; the theorem does not show that every physically acceptable theory is dynamically local, nor that every possible formulation of “same physics” reduces to natural isomorphism of observable functors.

There is also no claim that isomorphism of bare global algebras identifies local physics. Infinite systems often have abstractly isomorphic global algebras with inequivalent regional assignments or states. The natural transformation keeps the embedding maps in view, and dynamical locality keeps localization in view. Both are used before an isomorphism is interpreted physically.

For massive free scalar theories, the earlier dynamical-locality theorem supplies a nonempty class to which the result applies. For the unreduced massless scalar, the locally constant mode prevents that route; one must first pass to the current theory in the dimensions where its own theorem holds.

Independent check

Given a proposed equivalence, test three levels separately: one component $\zeta_M$; naturality for every $\psi:M\to N$; and invertibility of every $\zeta_N$. The inverse of one component cannot be inserted into another component’s square. If a proof uses only the Minkowski component and never invokes dynamical locality or a deformation chain, the diagonal theory remains an explicit counterexample.

Exercise

Why does tensoring a theory with an extra factor on only one spacetime usually fail before the SPASs question is reached?

Solution

Morphisms into and out of that spacetime require algebra homomorphisms compatible with composition. An arbitrary one-object modification supplies no such coherent maps. Diagonal counterexamples use a functorial object-dependent label and structure maps precisely to preserve composition; once that work is done, they show that functoriality alone still does not give the SPASs property.

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.