Structural Principles and Axiom Maps
Covariance, positive energy, Hilbert-space positivity, locality, and clustering are distinct statements. This chapter explains how they enter structural results such as spin–statistics and CPT, and how to identify which conclusion survives when an assumption changes.
Enter this chapter
Section titled “Enter this chapter”Begin with Poincaré Covariance and the Spectrum Condition and the following positivity and locality pages. The final failure-mode page is useful after those distinctions are clear.
Topics
Section titled “Topics”- Poincaré Covariance and the Spectrum Condition — Separates transformation laws from the restriction on energy–momentum support.
- Hilbert Positivity and Unitary Evolution — Distinguishes positive probabilities from self-adjoint time evolution.
- Microcausality and Relativistic Compatibility — States spacelike commutativity for observables and graded locality for fields.
- Clustering, Vacuum Assumptions, and Long-Range Correlations — Explains the assumptions controlling the decay of connected correlations.
- Wightman, Euclidean, Local-Algebraic, Constructive, and Perturbative Frameworks — Compares primitive objects and conditional bridges among major formulations.
- The Spin–Statistics Connection — Identifies the hypotheses relating spin and exchange statistics.
- CPT: Hypotheses, Content, and Limits — Separates the combined CPT conclusion from individual discrete symmetries.
- Structural Hypotheses and Failure Modes — Changes one assumption at a time to determine what remains valid.
How the ideas fit together
Section titled “How the ideas fit together”Microcausality restricts commutators, whereas clustering concerns large-separation correlations; neither is the definition of positive energy. Spin–statistics and CPT combine hypotheses about these structures within a stated dimension and field framework. Their free-field illustrations are checks, not proofs of unrestricted theorems.
The following comparison preserves the distinctions needed to read the later theorem pages. In particular, lower-dimensional exchange topology requires its own statistics statement; see Fewster and Rejzner 2019, § 8.3.
| 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 |
Review the chapter
Section titled “Review the chapter”If a massless theory has correlations that decay algebraically, must clustering fail?
Solution
No. Clustering asks whether the connected correlation tends to zero, not whether it decays exponentially. An algebraic tail can satisfy that limit. The decay rate and any converse relation to a mass gap need their own assumptions.
Where to go next
Section titled “Where to go next”Continue to Wightman Axioms and Reconstruction for precise theorem statements, or Interacting QFT: Definitions, Limits, and Handoffs for the limits of free-field constructions.