Structural Principles and Axiom Maps

This chapter is organized as a map from structural hypotheses to conditional conclusions. Begin with covariance and the spectrum condition, physical positivity and unitary evolution, and microcausality; then branch to clustering, framework comparison, spin–statistics, charge conjugation, parity, and time reversal (CPT), or a systematic failure test. None of these inputs silently supplies the others. Every conclusion retains its spacetime dimension, field class, state, locality, positivity, and spectral assumptions. The chapter gives physical definitions, free-field tests, and proof maps; theorem-level constructions and equivalence results belong to Mathematical QFT.

From structural inputs to physical conclusions

The chapter’s first discipline is to keep similarly named claims separate. With the site’s translation convention,

$$U(a)=e^{iP\cdot a}, \qquad \operatorname{sp}P\subset\overline V_+ =\{p:p^0\ge0,\ p^2\ge0\}.$$

The first relation describes a unitary representation of translations. The second is the spectrum condition. Covariance does not imply positive energy, and the forward cone includes massless support. Likewise,

$$\omega(A^*A)\ge0 \quad\Longrightarrow\quad \langle[A],[A]\rangle_\omega=\omega(A^*A)\ge0, \qquad \|e^{-iHt}\psi\|=\|\psi\| \quad(H=H^\dagger).$$

The first input is positivity of an algebraic state. It induces the Gelfand–Naimark–Segal (GNS) positive semidefinite form shown above and, after quotienting null vectors and completing, a Hilbert space. The second input is norm-preserving evolution generated by a self-adjoint Hamiltonian. Neither statement alone says that $H$ is bounded below or that a scattering matrix (S-matrix) exists.

Locality is different again. For spacelike-separated regions,

$$[\mathcal A(\mathcal O_1),\mathcal A(\mathcal O_2)]=0 \qquad (\mathcal O_1\perp\mathcal O_2)$$

is an observable-algebra form of microcausality. It does not make vacuum correlations vanish. A graded field algebra needs a declared parity and graded locality before commutators are replaced by graded commutators. Fewster and Rejzner develop positive states, local algebras, covariance, the spectrum condition, and persistent vacuum correlations in Fewster and Rejzner 2019, arXiv v2, §§ 2.2–2.3, 4.1–4.2, and 5.1–5.2, printed pp. 5–9, 13–20, and 24–27 (PDF).

Clustering adds vacuum-sector and long-distance information. Under suitable uniqueness and spectral assumptions one asks whether

$$\langle\Omega|A_1\alpha_x(A_2)|\Omega\rangle \longrightarrow \langle\Omega|A_1|\Omega\rangle \langle\Omega|A_2|\Omega\rangle$$

as $x$ becomes large and spacelike. A mass gap can strengthen the decay rate; massless algebraic decay need not violate clustering. This is not the same statement as cluster decomposition of scattering amplitudes.

Spin–statistics and CPT then use selected intersections of these earlier inputs, together with dimension, field, domain, and analyticity hypotheses. They are not consequences of the word “relativistic.” Framework comparison is horizontal rather than hierarchical: it identifies primitive objects and theorem-backed bridges without declaring Wightman, Euclidean, algebraic, constructive, and perturbative formulations universally interchangeable.

Check your entry point

This chapter has no global prerequisite. Use the observable tasks below to choose an entry rather than treating them as a gate.

Can you already…	Page you can approach	Hard page inputs still required	If the capability is missing, repair with…
distinguish a Poincaré particle representation from a finite-dimensional field representation and identify a future mass shell	covariance and spectrum	one-particle states	Lorentz Field and Poincaré Particle Representations and spectral calculus
distinguish positive norm, self-adjoint time evolution, and scattering-amplitude unitarity	positivity and evolution	vacua, states, and representations plus the canonical algebra	Bilinear and Hermitian Forms and Self-Adjointness and Unitary Evolution
explain why a spacelike commutator can vanish while a Wightman function remains nonzero	microcausality	spacelike compatibility and local observables	Hyperbolic Equations and Causal Propagators
distinguish vacuum connected-correlation clustering from multiparticle or scattering organization	clustering	connected correlators plus multiparticle states	revisit those two hard inputs before entering the page
name the primitive data in Lorentzian field, Euclidean, and local-algebraic descriptions	framework comparison	What Is a QFT?, Schwinger functions, and microcausality	Operator Algebras and Positive Functionals
quantize the free Dirac field and track its grading separately from Hilbert positivity	spin–statistics	microcausality, positivity and evolution, and free Dirac quantization	use the relativity, Lorentz, and spin repair and close the three hard inputs
state an antiunitary CPT convention for scalar and Dirac fields without assuming separate C, P, or T invariance	CPT hypotheses and limits	microcausality, covariance and spectrum, and The Dirac Field	use the relativity, Lorentz, and spin repair and close the three hard inputs

Choose a route

Arrows in this table mean a suggested reading order. The chapter guide below states the actual page-level preparation.

Physical question	Close these hard inputs first	Suggested linked route	Result at the exit
Which assumptions control energy–momentum support and physical probabilities?	one-particle states; vacua and representations; canonical algebra; local observables	covariance and spectrum → positivity and evolution → microcausality	separate transformation, spectral, Hilbert, dynamical, and locality claims
What changes in a massless or long-range vacuum sector?	connected correlators and multiparticle states	clustering and long-range correlations	distinguish failure of a mass gap from failure of clustering
Why are statistics tied to a theorem package?	microcausality, positivity and evolution, and free Dirac quantization	the spin–statistics connection	name the dimension, field class, locality, positivity, and spectrum assumptions used
What does a CPT theorem actually require and conclude?	microcausality, covariance and spectrum, and The Dirac Field	CPT hypotheses, content, and limits	distinguish the combined antiunitary statement from separate C, P, or T symmetries
How do major formulations relate?	What Is a QFT?, Schwinger functions, and microcausality	framework comparison	identify theorem-backed bridges and unresolved construction steps
Which conclusion survives when one assumption is removed?	complete covariance and spectrum, positivity and evolution, microcausality, and clustering	structural hypotheses and failure modes	change one hypothesis at a time and preserve the conclusions that remain licensed
Where are the precise theorem and domain statements?	first complete the hard inputs of the relevant physical page	Wightman axioms and reconstruction or theorem-first claim grammar	move from a physical orientation map to a specified theorem package

Structural hypothesis matrix

The table is a comparison aid, not an implication diagram. Each row names one input, the conclusion it directly controls, and a nearby overclaim to avoid. Spin–statistics and CPT depend on combinations of several rows; they are not extra axioms smuggled into one cell. Structural Hypotheses and Failure Modes applies the same matrix to examples in which one input is weakened or absent.

Structural inputs, controlled conclusions, and non-implications

Structural input	What it controls	Additional data or hypotheses	What does not follow	Framework and rigorous continuation
Poincaré covariance	Covariant transformation of regions, observables, states, fields, and correlators under the declared Poincaré action	A continuous Poincaré action appropriate to the formulation—automorphisms on a net or unitary implementers in a Hilbert representation—with field intertwiners and common domains when fields are used	Positive energy, locality, a unique vacuum, or a mass gap	Representation theory, Wightman fields, or covariant local nets; continue to Wightman fields and axioms
Spectrum condition	Joint translation spectrum confined to the closed forward cone	Unitary translations with self-adjoint joint generators and a declared physical sector	A spectral gap, an isolated particle shell, or vacuum uniqueness	Wightman and algebraic QFT; continue to Wightman spectral support
Hilbert positivity	Nonnegative GNS or Hilbert norm squares and probabilistic interpretation after the physical null quotient or restriction	A positive algebraic state or positive inner product, the physical subspace or quotient, the adjoint operation, and the observable algebra	Reflection positivity, positive energy, or S-matrix unitarity; gauge-fixed auxiliary spaces may be indefinite	Hilbert-space, Wightman, and algebraic formulations; continue to axioms and reconstruction
Unitary evolution	Norm-preserving time evolution and reversible closed-system dynamics	A self-adjoint generator with its domain and a specified representation	A lower spectral bound, asymptotic completeness, or amplitude unitarity without scattering states	Canonical and Hilbert-space formulations; continue to self-adjoint evolution and later scattering theory
Microcausality	Compatibility of spacelike-separated observables, or graded locality for declared field parity	Localized observables or fields, a spacelike relation, domains, and the applicable grading	Vanishing spacelike correlations, causal support of every propagator, or a complete operational no-signaling theorem	Wightman fields and local nets; continue to algebraic and locally covariant QFT
Clustering	Factorization of vacuum correlations, or decay of connected correlations, at large separation	A specified vacuum sector, translation action, separation regime, and infrared or spectral assumptions	Zero finite-distance entanglement, exponential decay without a gap, or scattering cluster decomposition	Vacuum correlation and local-net formulations; continue to algebraic QFT or to Scattering for amplitude clustering
Dimensional setting	Available Lorentz representations, exchange topology, and the form of spin–statistics and CPT statements	Spacetime dimension, signature, orientation and time orientation, and any global assumptions	A dimension-independent statistics or CPT theorem; low-dimensional braid statistics need different statements; see Fewster and Rejzner 2019, arXiv v2, § 8.3, printed pp. 38–40 (PDF)	Theorem-specific Wightman, algebraic, and topological settings; continue to spin–statistics variants
Field assumptions	Adjoints, spin and charge content, domains, grading, transformation matrices, and which objects are observable	Field class, locality notion, physical versus auxiliary space, regularity, and a precise CPT convention	That graded locality proves spin–statistics, or that CPT implies separate C, P, and T symmetries	Wightman, Becchi–Rouet–Stora–Tyutin (BRST), and local-algebraic formulations; continue to CPT theorem variants

The matrix deliberately leaves theorem names in the last column. For example, Euclidean reflection positivity is only one member of a corrected Osterwalder–Schrader hypothesis package; it is not itself the reconstruction theorem. Osterwalder and Schrader’s 1975 paper explicitly corrected the sufficiency claim made with the original 1973 package Osterwalder and Schrader 1973, § 3, pp. 87–90 (Open PDF), Osterwalder and Schrader 1975, introduction, pp. 281–283, and § IV.1, pp. 287–288 (Open PDF).

The eight pages in order

1. Poincaré Covariance and the Spectrum Condition. This page separates field covariance from particle representations and positive-energy spectral support. It requires one-particle states; Lorentz representations and spectral calculus are helpful. Its free-scalar anchor is support on the forward mass shell. Continue to Wightman spectral support for the theorem-level treatment.

2. Hilbert Positivity and Unitary Evolution. This page distinguishes physical positive norm, self-adjoint dynamics, and other uses of “unitarity.” It requires vacua, states, and representations and the canonical algebra; Hermitian forms and self-adjointness are helpful. Its anchor contrasts the free scalar physical Hilbert space with the auxiliary space of the covariantly quantized photon. Continue to S-Matrix Unitarity only after scattering states are available.

3. Microcausality and Relativistic Compatibility. This page states locality for observables and qualified graded locality for fields. It requires spacelike compatibility and local observables; causal propagators are helpful. Its free-scalar anchor is $\Delta_m(z)=0$ for $z^2<0$, not a vanishing Wightman function. Schwartz gives the free scalar and spinor locality checks in Schwartz 2014, § 12.6, pp. 219–223.

4. Clustering, Vacuum Assumptions, and Long-Range Correlations. This page asks when connected vacuum correlations decay and which uniqueness, spectral, and infrared assumptions determine the rate. It requires connected correlators and multiparticle states. Its anchor contrasts massive exponential estimates with massless algebraic behavior. Weinberg’s scattering cluster principle is a separate use of the word, developed in Weinberg 1995, § 4.3, pp. 177–181.

5. Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks. This page compares primitive data, claims, and theorem-backed bridges without declaring universal equivalence. It requires What Is a Quantum Field Theory?, Euclidean Schwinger functions, and this chapter’s microcausality page; operator algebras and nuclear spaces are helpful. The free scalar is located in each framework, but no interacting construction is inferred.

6. The Spin–Statistics Connection. This page maps the assumptions connecting spin, locality, positivity, and statistics. It requires microcausality, Hilbert positivity, and free Dirac quantization. Its anchor compares scalar commutators with Dirac anticommutators. A free-field construction is not the general theorem; continue to spin–statistics theorems and failure modes.

7. CPT: Hypotheses, Content, and Limits. This page asks what a CPT theorem actually assumes and concludes, with scalar and Dirac examples under one explicit antiunitary convention. It requires microcausality, covariance and spectrum, and The Dirac Field. CPT does not imply separate C, P, or T invariance. Weinberg’s local Lorentz-invariant field argument is bounded to its setting in Weinberg 1995, § 5.8, pp. 244–246; continue to CPT theorem variants.

8. Structural Hypotheses and Failure Modes. This synthesis page starts from the first four leaves and removes one assumption at a time. Its examples compare the free scalar, the covariantly quantized photon, a massless scalar, and topological settings. It deepens the structural matrix above instead of treating every failure as the same pathology.

Conventions and recurrent hazards

• Forward cone. The site uses $(+---)$, so $p^2\ge0$ and $p^0\ge0$ describe the closed forward causal cone. A source with the opposite signature must translate both the invariant and the Fourier phases.
• Translations versus Schrödinger evolution. With $H=P^0$, the site convention gives $U(t,\mathbf 0)=e^{iHt}$ and therefore $e^{-iHt}=U(-t,\mathbf 0)$. The same generator appears with opposite translation parameters; the signs must be derived from the declared action rather than merged by mnemonic.
• Physical versus auxiliary space. Hilbert positivity belongs to the physical state space. Covariant gauge-fixed fields may live in an indefinite auxiliary space before a constraint, cohomology, or quotient is imposed.
• Locality versus correlation. Commuting spacelike observables can remain entangled and correlated. Microcausality is neither a statement that every propagator vanishes outside the cone nor, by itself, a complete measurement-theoretic no-signaling proof.
• Two cluster statements. Vacuum-correlation clustering concerns large separations in a state. Scattering cluster decomposition concerns the organization of amplitudes for distant experiments.
• Theorem scope. Spin–statistics and CPT change with dimension, field class, grading, regularity, and locality framework. A slogan without these inputs is not a theorem statement.
• Framework arrows. Analytic continuation, reconstruction, scaling limits, and perturbative comparison each require a named theorem package. A shared free example does not make two formulations universally equivalent.

Free fields are structural tests, not general proofs

The free scalar supplies one coherent check across the core pages: its positive-energy two-point measure lies on the forward mass shell; its Fock space is positive; its Hamiltonian is self-adjoint on a controlled domain; and

$$[\phi(x),\phi(y)]=i\Delta_m(x-y)\mathbf1, \qquad \Delta_m(z)=0\quad\text{for }z^2<0.$$

The massive and massless scalar then separate exponential from algebraic long-distance behavior. The free Dirac field supplies the corresponding graded-locality, spin–statistics, and CPT sign checks. The covariantly quantized photon is useful precisely because it forces a distinction between the auxiliary gauge-fixed space and the physical state space.

These examples test normalization, signs, supports, and missing assumptions. They do not prove the general spin–statistics or CPT theorems, construct an interacting model, or show that every QFT framework is equivalent. Schwartz explicitly separates unitary particle representations from finite-dimensional covariant field representations in Schwartz 2014, § 8.1, pp. 111–113, while its positive two-point spectral support is developed in Schwartz 2014, § 24.2.1, pp. 467–468.

Misconception clinic

“Lorentz covariance guarantees positive energy.” Covariance fixes transformation behavior. The forward-spectrum condition is an additional spectral hypothesis.

“A self-adjoint Hamiltonian is automatically stable.” Self-adjointness generates unitary evolution. Stability needs a lower spectral bound, and scattering needs still more structure.

“Spacelike commutativity means no correlations.” It means compatible local observable algebras at spacelike separation. Vacuum correlations and entanglement can remain nonzero.

“Massless fields do not cluster.” The rate may become algebraic and infrared hypotheses matter. That is not the same as a universal failure of clustering.

“Graded locality proves spin–statistics.” Assigning a grading and verifying a graded commutator is an input or a model check. A theorem must derive the allowed relation under its complete hypothesis package.

“CPT means C, P, and T separately.” The combined antiunitary statement can hold when one or more separate transformations fail to be symmetries.

“All rigorous frameworks are equivalent descriptions.” Some arrows are reconstruction or comparison theorems under specific hypotheses; others are perturbative, model-dependent, or open.

What the chapter establishes

Result	Established here	Not established here
structural core	distinct definitions of covariance, spectrum, physical positivity, unitary evolution, locality, and clustering	that any one input implies all the others
free-field tests	mass-shell support, positive Fock norms, causal commutator support, and massive/massless clustering contrasts	an interacting existence theorem
theorem maps	explicit hypothesis categories for spin–statistics and CPT	full proofs or dimension-independent slogans
framework comparison	primitive objects and named theorem-backed bridges	universal equivalence of formulations
failure diagnosis	the conclusion that survives after one hypothesis changes	one universal remedy for gauge, infrared, topological, or representation failures

Review the chapter

Review task	Prompt	Successful answer criterion	Repair
retrieval	Name the structural input that constrains the joint translation spectrum	identifies the spectrum condition, not covariance alone	return to Poincaré covariance and spectrum
explanation	Distinguish Hilbert positivity, unitary time evolution, reflection positivity, and amplitude unitarity	assigns each claim a different domain and required object	review Hilbert positivity and reflection positivity
counterexample	Give a quantity that can remain nonzero at spacelike separation without violating microcausality	names a Wightman function, connected correlation, or entanglement measure rather than a commutator of local observables	review microcausality
hypothesis reconstruction	State what must be added before a spin–statistics or CPT slogan becomes a theorem claim	includes dimension, field class, locality, positivity, spectrum, vacuum/domain, and proof setting	follow the spin–statistics and CPT routes
convention translation	Translate a mostly-plus forward cone into the site convention	flips the invariant sign while preserving positive energy and the physical support set	review covariance and spectrum
comparison	Explain why reflection positivity alone is not an equivalence between Euclidean and Lorentzian QFT	names the rest of a corrected reconstruction package and its domain assumptions	review framework comparison
transfer	A massless correlator decays algebraically. Decide whether clustering failed	says the limit and state assumptions must be tested; exponential decay was not universally required	review clustering
synthesis	Remove physical positivity from a covariantly gauge-fixed description and state what remains	preserves covariance and auxiliary correlator calculations while withholding physical probability claims until restriction or quotient	use structural failure modes

Where to continue

• Apply the same hypothesis discipline to interacting fields: Interacting QFT: Definitions, Limits, and Handoffs begins with representations, regulators, asymptotic fields, and Haag-type obstructions without treating them as one slogan.
• Enter theorem-level axioms and reconstruction: Wightman Fields, Reconstruction, and Structural Theorems develops domains, spectral support, analyticity, reconstruction, spin–statistics, and CPT proofs.
• Compare proof frameworks precisely: Theorem-First Frameworks and Claim Grammar states what a theorem claim must name before frameworks are compared.
• Separate structural from scattering unitarity: S-Matrix Unitarity develops the amplitude statement after asymptotic states are specified.
• Use a guided physical route: Physical Foundations for Mathematicians connects the prerequisite, physical, and theorem-level reading routes.
• Return to the volume map: Foundations places this chapter beside the free-field, functional, spectral, and interacting-field routes.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” arXiv:1904.04051v2, 2019, 47 pp.; 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.

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI. Open PDF.

• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions II (with an Appendix by Stephen Summers).” Communications in Mathematical Physics 42 (1975): 281–305. DOI. Open PDF.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.

• Weinberg, Steven. The Quantum Theory of Fields. Vol. I, Foundations. Cambridge University Press, 1995. DOI.