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Theorems and mechanisms

Read useful structural results together with the hypotheses that make them applicable.

Selected results with explicit hypotheses; the linked Library treatments supply derivations and fuller statements.

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6entries

  • Goldstone's theorem

    A nonzero broken variation of a scalar operator, together with an exactly conserved internal-symmetry current, forces massless spin-zero spectral weight in the relativistic vacuum.

    Applies to: Spontaneously broken continuous internal global symmetry in a Lorentz-invariant vacuum with positive-energy physical spectrum and a controlled infinite-volume current limit.

    • The symmetry is global rather than a gauge redundancy.
    • The current is exactly conserved, local, and free of boundary leakage; the broken variation is nonzero in the selected infinite-volume vacuum.
    Distinguish
    The statement establishes massless spectral weight. Identifying an ordinary one-particle pole or counting modes needs the corresponding additional hypotheses.
    Hypotheses
    A positive-energy physical spectrum, Lorentz-invariant vacuum, locality, a controlled infinite-volume partial-charge limit, and no explicit or anomalous breaking of the current are required.
    Failure warning
    Gauge symmetries, finite volume, low-dimensional obstructions, spacetime-symmetry breaking, and nonrelativistic systems require separate statements.

    Full explanation: Goldstone's Theorem: Hypotheses and Pole Argument

    Notation and checks
    Also called
    Goldstone theorem, Nambu–Goldstone theorem

  • Källén–Lehmann representation

    The vacuum two-point function of a Hermitian scalar operator has a nonnegative invariant-mass spectral measure. Vacuum time ordering gives a superposition of free Feynman kernels, with local extension terms included when needed.

    Applies to: A vacuum-subtracted scalar in a positive physical Hilbert space; Poincaré-invariant vacuum, unitary positive-energy translations, completeness, and distributional regularity.

    D~F(p)=∫[0,∞)ρ(dμ2)ip2−μ2+i0\widetilde D_F(p)=\int_{[0,\infty)}\rho(\mathrm d\mu^2)\frac{i}{p^2-\mu^2+i0}
    Failure warning
    The displayed unsubtracted formula requires convergence and no additional local term. Gauge-variant propagators in indefinite spaces do not automatically have positive spectral measures. Positivity alone does not imply unit total weight.

    Full explanation: The Källén–Lehmann Representation

  • LSZ reduction

    Scattering amplitudes are obtained from the simultaneous external one-particle pole residues of time-ordered correlators, after division by each external field's overlap with the normalized asymptotic state.

    Applies to: Stable external particles with isolated one-particle poles, nonzero interpolating-field overlap, and well-defined in/out states.

    Failure warning
    An unstable resonance or infraparticle cannot simply be inserted as an ordinary stable external LSZ leg. Amputation alone does not supply the external overlap factors.

    Full explanation: LSZ Reduction: Poles, Residues, and Stable External States

  • Optical theorem

    Unitarity relates the absorptive forward scattering amplitude to a sum over physical intermediate states. With S=1+iT, the underlying operator identity is −i(T−T†)=T†T.

    Applies to: A unitary S-matrix on the physical asymptotic-state space, with consistent state normalization and inclusive sums when required.

    −i(T−T†)=T†T-i(T-T^\dagger)=T^\dagger T
    Failure warning
    The intermediate-state sum contains phase-space integrals and identical-particle factors. It is not a sum of unconstrained off-shell Feynman graphs; massless infrared problems need additional care.

    Full explanation: The Optical Theorem and Cut Interpretation

  • Ward–Takahashi identities

    A localized change of variables expresses a quantum symmetry as relations between a current divergence and the transformations of operator insertions. Contact terms encode the action of the symmetry on those insertions.

    Applies to: A well-defined regulated transformation of action, measure, sources, and insertions, with boundary terms included.

    Failure warning
    Discarding contact terms changes the identity. An anomalous measure or explicit breaking adds a breaking term; classical invariance alone is insufficient.

    Full explanation: Localized Transformations and Ward–Takahashi Identities

  • Wick's theorem

    For free fields in a Gaussian state, ordered products expand into normal-ordered products and contractions. In a centered Gaussian state, even vacuum moments are sums over complete pairings and odd moments vanish.

    Applies to: Free fields with a declared Gaussian state and normal-ordering prescription.

    ⟨Tϕ1ϕ2ϕ3ϕ4⟩=D12D34+D13D24+D14D23\langle\mathrm T\phi_1\phi_2\phi_3\phi_4\rangle=D_{12}D_{34}+D_{13}D_{24}+D_{14}D_{23}
    Variables
    For the displayed centered bosonic Gaussian formula, phi_i=phi(x_i) and D_ij is the time-ordered two-point expectation value of phi_i and phi_j in that same free state.
    Failure warning
    Fermionic pairings carry permutation signs. An interacting or non-Gaussian state need not factorize; coincident composite operators require a separate definition.

    Full explanation: Wick’s Theorem and Free Gaussian Factorization