Results and structures
Structural results explain why broad classes of calculations have the same organization. Their usefulness depends on matching the statement to the physical setting: a Gaussian vacuum, a positive Hilbert space, a stable asymptotic particle, or an exactly conserved current each supplies a different premise.
Results and their hypotheses
Section titled “Results and their hypotheses”The theorems and mechanisms guide gives compact statements of Wick’s theorem, the Källén–Lehmann representation, LSZ reduction, the optical theorem, Ward–Takahashi identities, and Goldstone’s theorem. Each statement identifies its essential hypotheses and links to a full derivation.
| Question | Result to use | First condition to check |
|---|---|---|
| Can a higher moment be reduced to pairings? | Wick’s theorem | The state is Gaussian; specify normal ordering and fermionic signs |
| What can a two-point function reveal about particles? | Källén–Lehmann representation | The operator acts in a positive physical Hilbert space with the stated vacuum and spectrum |
| How does a correlator yield a scattering amplitude? | LSZ reduction | External particles have isolated stable poles and well-defined asymptotic states |
| What constrains the imaginary part of forward scattering? | Optical theorem | The physical scattering operator is unitary and the intermediate-state sum is complete |
| How does a symmetry constrain correlators? | Ward–Takahashi identities | Include transformed insertions, contact terms, and possible anomalous contributions |
| When must symmetry breaking produce a gapless sector? | Goldstone’s theorem | The declared global-current, vacuum, locality, and infinite-volume hypotheses hold |
These entries are deliberately narrower than a claim that every formulation of every QFT satisfies the same theorem. For example, a resonance is not an ordinary stable LSZ external state, and a gauge-fixed propagator need not inherit a positive physical spectral density.
Operators and symmetry
Section titled “Operators and symmetry”The operators, defects, and symmetries guide distinguishes local operator insertions from extended Wilson lines and the charged objects of higher-form symmetry. The important data include the support, transformation law, representation or charge, and admissible endpoints.
A theory’s action does not by itself settle every operator question. Global gauge-group data and boundary conditions can change which lines are genuine observables. The Models atlas organizes particular theories; the Library’s Symmetry and Gauge Structure develops the general principles behind these distinctions.