Equivalence, Uniqueness, and Comparison Notions

Two mathematical descriptions of a quantum field theory can share observables and still fail to be unitarily equivalent. The conclusion depends on what is compared—abstract algebras, represented algebras, states, local restrictions, categories, or only selected correlation functions. This page separates the standard comparison relations and applies them to quasifree canonical-commutation-relation representations in infinite volume.

Required background. Theorem-First Claim Records supplies the claim grammar, and QFT Frameworks, Object Classes, and Typed Maps fixes the objects on which comparison maps act. Helpful background. Representations, Intertwiners, and Invariants reviews representation theory; Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks supplies physical examples; Six-Dimensional Origins, Compactification, and Duality Frames and AGT and Exact-Correspondence Dictionaries: Status and Limits illustrate correspondences whose precise strength must be stated.

Comparison relations retain different data

Let $X$ and $Y$ denote objects in a declared class. The words in the first column below are not interchangeable synonyms.

Relation	Data required	What it licenses	What it need not license
Equality	$X$ and $Y$ are literally the same object in a fixed presentation	substitution without a comparison map	presentation independence
Isomorphism	an invertible structure-preserving map $F:X\to Y$	transport of every structure named in the object class	a preferred state or a unitary between representations
Unitary equivalence	a unitary $U:\mathcal H_1\to\mathcal H_2$ with $U\pi_1(A)U^*=\pi_2(A)$; add $U\Omega_1=\Omega_2$ when pointed representations are compared	equality of represented predictions after conjugation	equality of vectors or operators before applying $U$
Quasiequivalence	the representations have the same normal state folium, equivalently their generated von Neumann algebras are normally isomorphic in the representation-preserving way	the same density-matrix-type states and normal expectation functionals	a single unitary intertwiner in reducible representations
Local quasiequivalence	quasiequivalence after restricting to every specified local algebra $\mathcal A(O)$	agreement of locally normal sectors	global quasiequivalence or agreement of infrared sectors
Equivalence of categories	a functor that is full, faithful, and essentially surjective, or functors with natural isomorphisms to both identities	objects and morphisms correspond up to isomorphism	literal equality of objects or equality after forgetting structure
Morita equivalence	an equivalence of suitable module or representation categories, commonly implemented by an invertible bimodule	the same representation theory at the declared level	an isomorphism of the underlying algebras
Duality dictionary	a map between stated observables, parameters, backgrounds, and regimes, with whatever inverses and checks have actually been proved	only the entries and regime established	categorical equivalence, nonperturbative existence, or completeness by vocabulary alone
Matching observables	equality of a declared family $\{\langle O_i\rangle\}_{i\in I}$	equality of those numbers or distributions	equality of all observables, states, algebras, or theories

The object class is part of each assertion. An algebra isomorphism may fail to preserve a chosen state; a state-preserving isomorphism may fail to intertwine a spacetime action; a natural transformation may be an objectwise quasi-isomorphism without being an isomorphism of cochain complexes. Any use of “unique” therefore has the form “unique up to $R$,” where $R$ is one of these explicitly defined relations.

There is also a quantifier boundary. A theorem saying that for every bounded region $O$ there exists an intertwiner $U_O$ does not provide one global $U$ compatible with all inclusions. Conversely, a global algebra isomorphism need not preserve the chosen net $O\mapsto\mathcal A(O)$. Compatibility and naturality are conclusions that require their own hypotheses.

Quasifree states on a common Weyl algebra

Let $(\mathcal S,\sigma)$ be a real symplectic space. Its Weyl $C^*$-algebra is generated by symbols $W(f)$, $f\in\mathcal S$, subject to

$$W(f)^*=W(-f), \qquad W(f)W(g)=e^{-i\sigma(f,g)/2}W(f+g).$$

A real symmetric form $\mu$ satisfying

$$\lvert\sigma(f,g)\rvert^2 \leq 4\mu(f,f)\mu(g,g)$$

defines a centered quasifree state

$$\omega_\mu(W(f))=\exp[-\mu(f,f)/2].$$

The state has a GNS triple $(\mathcal H_\mu,\pi_\mu,\Omega_\mu)$. Consider two such forms $\mu_1$ and $\mu_2$ on the same $(\mathcal S,\sigma)$.

Equality of abstract algebras. Both states act on the same universal Weyl algebra because the generators and symplectic relations are identical. This says nothing yet about equality of states or representations. Indeed,

$$\omega_{\mu_1}(W(f))=\omega_{\mu_2}(W(f)) \quad\hbox{for all }f$$

holds exactly when the quadratic forms agree, by differentiating at the origin. The identity algebra map therefore need not be state preserving.

Quasiequivalence. Complete the quotient of the test-function space in a common Hilbert topology induced by the two covariances. Araki and Yamagami prove a necessary-and-sufficient criterion: the topologies induced by the two covariance forms must agree, and, after representing the two-point forms by positive operators $S_1,S_2$ in that common completion, $S_1^{1/2}-S_2^{1/2}$ must be Hilbert–Schmidt Araki and Yamagami 1982, § I, pp. 283–295. Failure of either condition permits disjoint representations even though the abstract CCR algebra is unchanged.

Unitary implementability. A symplectic transformation $T$ of the one-particle data induces an automorphism $W(f)\mapsto W(Tf)$. In a Fock representation, it is implemented by a unitary only when its positive/negative-frequency mixing satisfies the Hilbert–Schmidt condition; in an equivalent formulation, the antilinear Bogoliubov part is Hilbert–Schmidt Shale 1962, Theorem 4.1, pp. 163–165. Infinite volume makes this condition substantive: modewise finite transformations can accumulate an infinite particle number and cease to be unitarily implementable.

The two Hilbert–Schmidt statements answer different questions. The Araki–Yamagami criterion compares two quasifree GNS representations, while Shale’s theorem asks whether a specified symplectic automorphism is implementable in a selected Fock representation. Neither may be substituted for the other without identifying the relevant operators and completions.

Local agreement can coexist with global inequivalence

For the Klein–Gordon field on a globally hyperbolic spacetime, let $\mathcal W(O)$ be the Weyl subalgebra generated by test functions supported in a relatively compact region $O$. Two representations $\pi_1$ and $\pi_2$ are locally quasiequivalent on this net when

$$\pi_1\mathbin{\upharpoonright}_{\mathcal W(O)} \quad\hbox{and}\quad \pi_2\mathbin{\upharpoonright}_{\mathcal W(O)}$$

are quasiequivalent for every such $O$. Quasifree Hadamard states have this property: their GNS representations are locally quasiequivalent on arbitrary globally hyperbolic spacetimes Verch 1994, abstract, p. 507. The short-distance Hadamard condition controls the local comparison, while global infrared or topology-sensitive data can still differ. Brunetti, Fredenhagen, and Verch use this result to construct a locally quasiequivalent state space for the locally covariant Klein–Gordon theory Brunetti, Fredenhagen, and Verch 2003, Theorem 3.4, p. 52.

This distinction is central to later chapters of Mathematical QFT: local observables can be represented normally in the same way even when no global unitary identifies the vacua. It is therefore possible for a comparison to be adequate for every bounded experiment yet fail as a statement about the full infinite-volume representation.

First application: four decisions for two scalar representations

Take an infinite-dimensional real solution space $\mathcal S$ for a free scalar field and split its Hilbert completion as $K\oplus K^\perp$, where $K$ contains the modes probed by a chosen finite set of compactly supported smearings $f_1,\ldots,f_N$. Let $\mu_1$ and $\mu_2$ agree on $K$ but choose their covariance operators on $K^\perp$ so that the square-root difference is not Hilbert–Schmidt. Assume both remain positive and satisfy the CCR bound.

The four questions now have separate answers.

1. Abstract algebra: yes, the Weyl algebra is identical because $(\mathcal S,\sigma)$ is identical.
2. Selected correlators: yes, all quasifree correlators made only from $f_1,\ldots,f_N$ agree because the covariance agrees on their span.
3. Global quasiequivalence: no, provided the induced topologies are comparable but the Araki–Yamagami Hilbert–Schmidt condition fails; the representations are then not quasiequivalent.
4. Local quasiequivalence: undecided from this construction alone. It follows under an added theorem such as the quasifree Hadamard hypotheses above, not from agreement on $K$.

For pure quasifree representations, irreducibility can turn quasiequivalence into unitary equivalence, but that extra property must be proved. For general reducible GNS representations, quasiequivalence does not supply a distinguished unitary or a vacuum-preserving intertwiner.

Failure test: finitely many correlators do not determine a representation

The adversarial inference is

$$\bigl\langle\phi(f_{i_1})\cdots\phi(f_{i_n})\bigr\rangle_1 = \bigl\langle\phi(f_{i_1})\cdots\phi(f_{i_n})\bigr\rangle_2 \quad\text{for a finite tested family} \quad\Longrightarrow\quad \pi_1\simeq\pi_2.$$

It fails. Finite agreement determines only the restriction of the state to the algebra generated by the tested smearings. A unitary-equivalence conclusion still needs a cyclic construction from a determining family, completeness or continuity extending equality to the full algebra, an intertwiner defined on the entire represented algebra, and inverse or surjectivity data. If the vacuum vectors are part of the claim, the unitary must also carry $\Omega_1$ to $\Omega_2$.

The strongest surviving conclusion is exact agreement of the tested correlators, or equality of the restricted quasifree states when the tested subspace is closed under the relevant operations. No claim about global unitary equivalence, quasiequivalence, local quasiequivalence, or categorical equivalence survives without its own criterion.

Independent checks

Finite-dimensional check. On a finite-dimensional symplectic space every linear operator is Hilbert–Schmidt, and the regular irreducible CCR representation is unique up to unitary equivalence. The obstruction above must therefore enter through the infinite number of modes, as expected.

Particle-number check. For a Bogoliubov map with antilinear coefficients $\beta$, compute $\operatorname{Tr}(\beta^*\beta)$. A divergent trace means that the transformed vacuum would contain infinitely many original particles and no implementing Fock-space unitary exists.

Local-versus-global check. Restrict both covariances to test functions supported in a relatively compact $O$ and apply the relevant Hilbert–Schmidt criterion there. Passing every such local test does not produce a single compatible global unitary; that is precisely the distinction between local quasiequivalence and global equivalence.

Map check. For any claimed equivalence, verify both composites and every preserved structure: algebra product, adjoint, inclusions, covariance action, state, and, where relevant, naturality. An injective comparison is not an equivalence until essential surjectivity or an inverse up to the declared relation is proved.

Common pitfalls

Treating an algebra as its representation. The universal Weyl algebra can be the same while its GNS representations are disjoint. Always name the state and representation when the conclusion concerns Hilbert spaces or particles.

Calling a dictionary an equivalence. A useful observable map may be neither full nor faithful and may cover only a protected sector. State the indexed observable family, regime, and inverse data actually established.

Confusing local with global. Local normality controls bounded-region measurements. It does not remove infrared distinctions or select a unique global vacuum.

Exercises

1. Equality of quasifree states. Suppose $\omega_{\mu_1}(W(tf))=\omega_{\mu_2}(W(tf))$ for every real $t$ and every $f\in\mathcal S$. Prove that $\mu_1=\mu_2$.

Solution

The quasifree formula gives $e^{-t^2\mu_1(f,f)/2}=e^{-t^2\mu_2(f,f)/2}$. Differentiating twice at $t=0$ yields $\mu_1(f,f)=\mu_2(f,f)$ for all $f$. The real polarization identity,

$$4\mu(f,g)=\mu(f+g,f+g)-\mu(f-g,f-g),$$

then gives equality of the bilinear forms. Hence the characteristic functionals, and therefore the states on the Weyl algebra, agree.

2. Build finite agreement without global equivalence. Let $H=\ell^2(\mathbb N)$, let $K$ be the span of the first $N$ basis vectors, and define positive diagonal operators $C_1=I$ and $C_2=I+P_{K^\perp}$. Show that $C_1$ and $C_2$ agree on $K$ but $C_1^{1/2}-C_2^{1/2}$ is not Hilbert–Schmidt.

Solution

On $K$, $P_{K^\perp}=0$, so both operators equal the identity. On each basis vector $e_n$ with $n>N$, the square-root difference has eigenvalue $1-\sqrt2$. Its squared Hilbert–Schmidt norm would be

$$\sum_{n>N}\lvert1-\sqrt2\rvert^2,$$

which diverges. Thus finitely many modes can have identical covariance while the infinite tail violates the Hilbert–Schmidt condition. To make this a CCR example one must additionally check the positivity bound against the chosen symplectic form; the operator calculation isolates the comparison obstruction.

3. Separate categorical properties. Give examples showing that full faithfulness alone and essential surjectivity alone do not make a functor an equivalence.

Solution

The inclusion of the one-object full subcategory $\{\mathbb R\}$ into finite-dimensional real vector spaces is full and faithful but not essentially surjective: $\mathbb R^2$ is not isomorphic to its only image object. Conversely, the constant functor from a category with two distinct parallel arrows to the terminal category is essentially surjective but not faithful, because it identifies those arrows. An equivalence requires full faithfulness and essential surjectivity together, or equivalent quasi-inverse data.

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.
• 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.
• Shale, David. “Linear Symmetries of Free Boson Fields.” Transactions of the American Mathematical Society 103 (1962): 149–167. 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.