Structural Hypotheses and Failure Modes

There is no single failure switch for relativistic quantum field theory. Covariance, the spectrum condition, physical positivity, unitary evolution, locality, vacuum-sector assumptions, a mass gap, and asymptotic completeness are logically different inputs. If one is weakened, the conclusions that use it must be withdrawn or reformulated, while conclusions supported by the remaining inputs can survive. Most importantly, making a theorem inapplicable does not prove its conclusion false. This page applies that rule to a free massive scalar, a covariantly quantized free photon, a massless scalar, and a low-dimensional topology-sensitive sector.

Required background. Poincaré Covariance and the Spectrum Condition separates covariance from forward spectral support. Hilbert Positivity and Unitary Evolution separates a physical positive inner product from an indefinite auxiliary form and from norm-preserving dynamics. Microcausality and Relativistic Compatibility supplies observable locality, graded field locality, and their non-implications. Clustering, Vacuum Assumptions, and Long-Range Correlations supplies connected-correlation decay, phase selection, and the role of a mass gap.

Hypothesis loss and surviving conclusions

Write a theorem package schematically as

$$\mathsf H =\bigwedge_{i=1}^{n}\mathsf H_i, \qquad \mathsf H\Longrightarrow\mathsf C.$$

If one input $\mathsf H_k$ is absent, then this implication no longer licenses $\mathsf C$. It does not follow that $\neg\mathsf C$ is true:

$$\neg\mathsf H_k \quad\not\Longrightarrow\quad \neg\mathsf C.$$

The conclusion might still hold for an independent reason, in a weaker form, or under a different theorem. Conversely, retaining $\mathsf H_k$ does not supply the other inputs. Positive energy does not manufacture locality; locality does not manufacture a positive metric; and a mass gap does not manufacture scattering completeness.

Use the chapter’s structural hypothesis matrix as the baseline. For each case below, change one input and use a three-question test:

1. Which mathematical object changed? Distinguish a physical Hilbert space from a gauge-fixed auxiliary space, a local observable from a nonlocal representative, and a selected pure phase from a mixed state.
2. Which conclusion used the changed input? Name the exact theorem, estimate, probability statement, or scattering construction rather than a general slogan.
3. What remains independently supported? Preserve every conclusion whose own hypotheses still hold, and mark what is merely unknown rather than false.

This distinction separates three cases that are often conflated: a hypothesis can genuinely fail in the physical theory; it can fail only in an auxiliary description; or it can simply be unproved for the model under study. The mass gap, vacuum uniqueness or extremality, and asymptotic completeness appear in the shared matrix as additional hypotheses or non-implications, not as substitute structural rows. They must still be tested separately here.

Positivity must be tested on the physical space

The covariantly quantized free photon is the cleanest example. Its four-component auxiliary Fock representation has an indefinite form. A timelike one-photon polarization can have negative auxiliary norm, yet the covariant wave equation, momentum-space propagator, and Poincaré transformation law remain useful. None of those facts licenses a Born-rule interpretation on the auxiliary space.

In the free Gupta–Bleuler construction, the subsidiary condition selects $\mathcal V_{\mathrm{GB}}$. The restricted form is positive semidefinite, and the physical Hilbert space is obtained from the radical

$$\mathcal N =\mathcal V_{\mathrm{GB}} \cap\mathcal V_{\mathrm{GB}}^{\perp}, \qquad \mathcal H_{\mathrm{phys}} =\overline{\mathcal V_{\mathrm{GB}}/\mathcal N}.$$

Free evolution preserves the subsidiary subspace and its radical, so it descends to unitary evolution on $\mathcal H_{\mathrm{phys}}$. Appropriately smeared gauge-invariant field strengths act on that quotient and obey the physical locality statement. The potential $A_\mu$ is an auxiliary representative and need not itself define a physical observable there.

Thus the negative auxiliary norm is not a negative physical probability and not a counterexample to a positive-metric theorem. The theorem becomes applicable only after its physical space and observables have been identified. Steinmann 1989, p. 300 states this subsidiary-condition and null-quotient pattern; Covariant Free-Photon Quantization and Propagator develops the complete free calculation. This conclusion is bounded to the free Abelian theory, not a substitute for a general gauge or BRST construction.

Lorentz symmetry can be absent while Hamiltonian structure survives

A continuous-time spatial lattice scalar changes a different row. With nearest-neighbor spatial differences and lattice spacing $a$, its free one-particle dispersion in the first Brillouin zone, $p_j\in[-\pi/a,\pi/a)$, is

$$\omega_a(\mathbf p)^2 =m^2+\frac{4}{a^2} \sum_{j=1}^{3}\sin^2\!\left(\frac{ap_j}{2}\right).$$

At fixed $a$, the Hamiltonian can be self-adjoint and bounded below on a positive Hilbert space: its quadratic form is a sum of momentum squares, mass terms, and nearest-neighbor difference squares. The lattice couplings are local in the lattice sense. But spatial translations are discrete, rotations are reduced to the cubic group, and continuous boosts are absent. The Poincaré spectrum condition and four-dimensional Lorentz-field CPT or spin–statistics theorems are therefore not statements about this finite-$a$ model.

For $|ap_j|\ll1$,

$$\omega_a(\mathbf p)^2 =m^2+|\mathbf p|^2 -\frac{a^2}{12}\sum_j p_j^4 +O(a^4|\mathbf p|^6).$$

The relativistic dispersion is a continuum target, not an exact finite-cutoff symmetry. Recovering it in an interacting scaling limit requires evidence for the limit and restoration of the desired spacetime symmetries; the Taylor series alone proves neither. Zinn-Justin 2021, § 8.7, printed pp. 175–176, Eqs. (8.45)–(8.48) gives the hypercubic free propagator, its small-spacing expansion, and the continuum-limit qualification. Exact Symmetries, Broken Spacetime Symmetries, and Restoration develops that regulator-to-continuum question.

This example also sharpens the locality distinction. Finite-range lattice couplings give a well-defined lattice dynamics, but they are not by themselves the Minkowski microcausality hypothesis used in a relativistic theorem. Positivity and unitarity survive; exact Lorentz conclusions are withheld.

Locality can fail while covariance and positivity remain

A wrong-statistics free scalar supplies a controlled locality counterexample. Build a positive CAR Fock space with positive one-particle energy, but assign the scalar field odd grading. Its spacelike test is then the anticommutator. For $z=x-y$,

$$\{\phi(x),\phi(y)\} =\Delta_m^{(1)}(z)\mathbf 1, \qquad \Delta_m^{(1)}(z) =\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3 2E_{\mathbf p}} \left(e^{-ip\cdot z}+e^{+ip\cdot z}\right), \qquad E_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2},$$

which is nonzero at generic spacelike separation. For example, at equal time for $m>0$ and $r=|\mathbf r|>0$,

$$\Delta_m^{(1)}(0,\mathbf r) =\frac{m}{2\pi^2r}K_1(mr)>0.$$

Scalar covariance, positive energy, and the positive CAR Fock metric have not disappeared. What fails is the declared spacelike field bracket, and the associated local energy or observable construction becomes nonlocal. This model therefore lies outside the four-dimensional spin–locality theorem rather than refuting it Greenberg 1998, arXiv v2, manuscript pp. 5–6 (PDF).

The lesson is theorem-specific. Full microcausality may fail while the weaker Jost-point order relation needed by a particular CPT statement survives. Test the locality input of the conclusion actually being claimed rather than replacing every locality notion by a single switch.

A missing mass gap need not destroy clustering

For a free real scalar in 3+1 dimensions, the equal-time connected vacuum two-point function at separation $r=|\mathbf r|>0$ is

$$C_m(r)=\frac{m}{4\pi^2r}K_1(mr).$$

For $m>0$ it has the large-distance form

$$C_m(r) \sim \frac{\sqrt m}{4\sqrt2\,\pi^{3/2}} \frac{e^{-mr}}{r^{3/2}}.$$

The standard free massive scalar supplies the baseline package: a positive Fock space, unitary Poincaré covariance, forward spectral support, a unique vacuum in the chosen representation, microcausality, a mass gap $m$, and exponential clustering. In its free Fock space, the asymptotic theory is the theory itself and $S=I$.

Set $m=0$ instead. The vacuum is still Poincaré invariant, the physical Fock inner product remains positive, evolution remains unitary, and the commutator still has causal support. What disappears is the positive gap above the vacuum. Nevertheless,

$$C_0(r)=\frac{1}{4\pi^2r^2} \longrightarrow0 \qquad (r\to\infty).$$

The massless scalar clusters algebraically. Losing the gap withdraws the massive exponential estimate; it does not negate clustering. The Bessel form and massless limit follow from Weinberg 1995, § 5.2, p. 202, NIST DLMF, Eq. 10.30.2, and NIST DLMF, Eq. 10.40.2. Infrared behavior can be subtler in other dimensions, sectors, states, or for nonlocal operators; this one free four-dimensional calculation is not a universal massless theorem.

Several vacua do not force every phase to fail clustering

Vacuum uniqueness and a mass gap are different assumptions. Suppose two translation-invariant pure phases satisfy

$$\omega_\pm(\Phi)=\pm v$$

and each selected phase clusters. Then its raw two-point function can tend to $v^2$ while its connected part tends to zero. Now form the non-extremal mixture

$$\omega_{\mathrm{mix}} =\tfrac12(\omega_++\omega_-).$$

Its one-point function vanishes, but distant insertions retain the common phase label:

$$C_{\mathrm{mix}}(x-y) \longrightarrow v^2.$$

Thus the mixture fails clustering even though the selected pure phases may cluster. The decisive issue is phase selection or extremality, not the mere existence of several vacua. Nor does a nonzero raw two-point limit in one pure phase signal failure when it equals the disconnected one-point product. Weinberg 1996, § 19.1, pp. 163–167 develops the infinite-volume phase-selection distinction.

A theorem that assumes a unique invariant vacuum cannot simply be applied to the symmetric mixture. That does not show that no CPT map, spectrum condition, local algebra, or unitary dynamics exists; those claims have separate inputs.

Asymptotic completeness is stronger than unitary dynamics

Let $\Omega_{\mathrm{in}}$ and $\Omega_{\mathrm{out}}$ be isometric scattering wave operators from a declared asymptotic Hilbert space into a physical sector. In the simplest massive-particle setting, asymptotic completeness means

$$\operatorname{Ran}\Omega_{\mathrm{in}} =\operatorname{Ran}\Omega_{\mathrm{out}} =\mathcal H_{\mathrm{sector}}.$$

Self-adjoint time evolution does not imply these equalities. Neither does an isolated one-particle shell, a mass gap, locality, or the formal existence of an interaction. If the two wave-operator ranges agree on a proper scattering subspace, then

$$S=\Omega_{\mathrm{out}}^\dagger\Omega_{\mathrm{in}}$$

is unitary on the declared asymptotic Hilbert space even though other physical states lie outside the common range. Conversely, failure to prove completeness does not erase local observables, vacuum correlators, or spectral information already constructed. Dybalski and Gérard 2014, § 1, manuscript pp. 1–4 (PDF) states the conventional range condition and explains why superselection sectors and nonparticle states require a sector-sensitive formulation.

The free massive scalar is the controlled limiting case: its physical Fock space is already the asymptotic Fock space. Interacting massive theories, massless radiation, infraparticles, confined sectors, and topological sectors require distinct scattering notions and separate existence results. Asymptotic Completeness: Definitions and Known Models develops those variants.

Dimension and topology change the statistics question

In 3+1 dimensions, exchanging identical localized particles uses the permutation-group setting behind the ordinary Bose/Fermi alternative. In 2+1 dimensions, the configuration space can support braid-group exchange. Positive energy, a positive physical Hilbert space, covariance under the appropriate spacetime group, and locality of the observable net can all remain meaningful while charged sectors carry braid or anyonic statistics.

One precise massive charged-sector theorem uses an isolated mass $m>0$ separated from the rest of the sector’s spectrum by a gap, only finitely many particle types at that mass, and a common spin $s$. With Mund’s exchange orientation convention, its statistics phase obeys

$$\omega_\chi=e^{2\pi i s}.$$

This scoped phase relation does not determine a non-Abelian braid representation Mund 2009, § 1, manuscript pp. 1–2 (PDF).

That behavior does not contradict the four-dimensional point-field spin–statistics theorem: the dimensional and localization hypotheses have changed. Charged fields may also require cone or string localization even when compactly localized observables commute. Fewster and Rejzner 2019, arXiv v2, §§ 8.2–8.3, printed pp. 37–40 (PDF) give the permutation-versus-braid sector framework. Braided Sectors and Anyonic Statistics in Low-Dimensional Nets develops the theorem-level qualifications.

This is the topology-sensitive member of the four-example comparison. It changes the theorem package rather than supplying a pathology inside the four-dimensional one.

Four examples, one disciplined comparison

The compact table below applies the shared matrix without duplicating its full eight rows.

What changes and what survives in four controlled examples

Example	Changed or diagnostic input	What survives	What must be withheld or reformulated
Free massive scalar in 3+1 dimensions	Baseline: the displayed core inputs hold in the chosen vacuum Fock representation	Positive energy, physical positivity, unitary evolution, microcausality, exponential clustering, and trivial free scattering completeness	No general interacting theorem, regulator limit, or framework equivalence follows from the free example
Covariantly quantized free photon	Positivity fails on the auxiliary four-component space; the photon is also gapless	Covariant auxiliary calculations survive, and positive unitary physics survives after the subsidiary restriction and null quotient	No auxiliary Born probabilities; physical locality is stated for quotient-compatible gauge-invariant observables
Massless free scalar in 3+1 dimensions	The positive mass gap is absent	Poincaré covariance, positivity, unitary evolution, microcausality, and power-law clustering remain	The massive exponential cluster estimate and isolated-vacuum gap cannot be claimed
Localized sector in 2+1 dimensions	Dimension, exchange topology, and possibly charged-field localization differ	Positive energy, physical positivity, covariance, and local observable compatibility can remain	The four-dimensional two-sign field theorem is replaced by a sector- and braid-sensitive statement

The lattice and mixed-vacuum cases answer two further axes in the page’s question: exact Lorentz symmetry can be absent while positivity and unitary Hamiltonian dynamics remain, and vacuum non-extremality can obstruct clustering without erasing locality or the spectrum condition. Asymptotic completeness is yet another independent statement about the range of scattering constructions.

Common inference failures

Theorem inapplicable does not mean theorem false. A missing hypothesis removes one route to a conclusion. A genuine counterexample must satisfy the remaining advertised package while violating the conclusion.

Covariance is not the spectrum condition. Neither input implies locality, vacuum uniqueness, a gap, or asymptotic completeness.

An indefinite auxiliary form is not a physical negative probability. But an actually indefinite physical state space also cannot be described as an ordinary positive-probability QFT without a further construction.

Microcausality is not decorrelation. It neither makes Wightman functions vanish at spacelike separation nor gives causal support to every propagator. If the locality condition required by a theorem is absent, positivity and covariance do not supply it. Full microcausality may fail while a weaker Jost-point condition still licenses a CPT variant, so the theorem-specific locality input must be checked.

Several vacua do not automatically defeat clustering. Test a selected extremal phase and the connected correlator. A non-extremal mixture answers a different question.

No gap does not mean no clustering. Power-law decay can still tend to zero. Conversely, a gap is not an isolated particle in every operator channel and does not imply asymptotic completeness.

A unitary $S$-matrix on a scattering subspace need not be complete. Exact closed-system time evolution, scattering unitarity, and exhaustion of the physical sector are three different claims.

Low-dimensional or topological behavior is not a four-dimensional violation. Change the dimension, localization class, or exchange topology, and the theorem statement must change with it.

Check your understanding

Check 1: remove the gap

A four-dimensional free scalar has $C_0(r)=1/(4\pi^2r^2)$. Which structural statement fails and which long-distance statement survives?

Answer. A positive mass gap is absent, so the massive exponential bound is unavailable. The connected correlator still tends to zero, so clustering survives algebraically.

Check 2: locate photon positivity

Does a negative-norm timelike photon polarization disprove physical Hilbert positivity?

Answer. No. It belongs to the auxiliary covariant representation. The physical statement is made only after the Gupta–Bleuler restriction and the quotient by the radical; the resulting transverse sector has a positive physical inner product.

Check 3: distinguish unknown from false

Suppose an interacting theory has a self-adjoint Hamiltonian and a local observable algebra, but no proof of asymptotic completeness. What can be concluded?

Answer. Unitary time evolution and locality remain available under their own hypotheses. One may not claim that asymptotic particle states exhaust the physical sector; lack of that proof does not show that they fail to do so.

Where to continue

• Prove clustering under specified hypotheses: Clustering, Vacuum Uniqueness, and the Mass Gap develops theorem versions and their state and spectral assumptions.
• Test spacetime-symmetry restoration: Exact Symmetries, Broken Spacetime Symmetries, and Restoration separates exact lattice symmetries from continuum targets.
• Study scattering-range questions: Asymptotic Completeness: Definitions and Known Models distinguishes massive, massless, charged, and detector-based formulations.
• Change dimension and sector structure: Braided Sectors and Anyonic Statistics in Low-Dimensional Nets develops braid-sensitive statistics beyond the four-dimensional field slogan.
• Return to complete theorem packages: Wightman Fields, Reconstruction, and Structural Theorems develops precise domain, locality, positivity, spectrum, reconstruction, spin–statistics, and CPT statements.

References

• Dybalski, Wojciech, and Christian Gérard. “A Criterion for Asymptotic Completeness in Local Relativistic QFT.” Communications in Mathematical Physics 332 (2014), 1167–1202. DOI. Open manuscript PDF.
• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” arXiv:1904.04051v2, 2019; published in Progress and Visions in Quantum Theory in View of Gravity, edited by Felix Finster, Domenico Giulini, Johannes Kleiner, and Jürgen Tolksdorf, 1–61. Birkhäuser, 2020. DOI. Open manuscript PDF, v2.
• Greenberg, O. W. “Spin-Statistics, Spin-Locality, and TCP: Three Distinct Theorems.” Physics Letters B 416 (1998), 144–149. DOI. Open manuscript PDF, v2.
• Mund, Jens. “The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1.” Communications in Mathematical Physics 286 (2009), 1159–1180. DOI. Open manuscript PDF, v2.
• National Institute of Standards and Technology. Digital Library of Mathematical Functions, Chapter 10, “Bessel Functions,” Eqs. 10.30.2 and 10.40.2. Small-argument formula; large-argument formula.
• Steinmann, Othmar. “On the Characterization of Physical States in Gauge Theories.” Annales de l’Institut Henri Poincaré. Physique Théorique 51, no. 3 (1989), 299–321. NUMDAM.
• Weinberg, Steven. The Quantum Theory of Fields. Vol. I, Foundations. Cambridge University Press, 1995. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Vol. II, Modern Applications. Cambridge University Press, 1996. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford University Press, 2021. DOI.