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Formula index

Retrieve convention-complete formula checkpoints with variables, assumptions, dimensional checks, validity limits, and links to derivations.

This index is a concise projection, not an independent explanation. Each record keeps the scope needed for use and links to its source page.

Coverage notes
  • Formula records are explicitly curated checkpoints and are never extracted from arbitrary display equations.
  • Every expression retains its variables, assumptions, applicable conventions, source page, and scope.

Coverage note: records included through 2026-08-12

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  • Formula

    Canonically normalized real-scalar action

    The reference checkpoint uses the mostly-minus metric, canonical kinetic normalization, a mass term, and a locally defined interaction potential.

    Expression
    S=∫ddx [12∂μϕ ∂μϕ−12m2ϕ2−Vint(ϕ)]S=\int \mathrm d^d x\,\left[\frac12\partial_\mu\phi\,\partial^\mu\phi-\frac12m^2\phi^2-\mathcal V_{\mathrm{int}}(\phi)\right]
    Variables
    phi is a real scalar field, m is its mass parameter, and V_int is the interaction potential.
    Dimensional check
    For canonical normalization, [phi]=(d-2)/2 and the Lagrangian density has mass dimension d.
    Limitation
    This checkpoint is not an assertion that every theory admits a canonical scalar description.
    Aliases
    real scalar action, Klein–Gordon action, canonical scalar action
    Scope
    A canonically normalized real scalar field in flat d-dimensional Lorentzian spacetime.
    Assumptions
    Boundary conditions justify the integrations by parts used by the owning derivation.; The interaction potential and its stability or EFT interpretation are specified locally.
  • Formula

    Connected and one-particle-irreducible generating functionals

    With the default source sign, W is minus i log Z and the Legendre transform gives delta Gamma over delta phi_c equal to minus J.

    Expression
    Z[J]=∫Dϕ eiS[ϕ]+i∫ddx Jϕ,W[J]=−ilog⁡Z[J],ϕc=δWδJ,Γ[ϕc]=W[J]−∫ddx JϕcZ[J]=\int\mathcal D\phi\,e^{iS[\phi]+i\int \mathrm d^d x\,J\phi},\quad W[J]=-i\log Z[J],\quad \phi_{\mathrm c}=\frac{\delta W}{\delta J},\quad \Gamma[\phi_{\mathrm c}]=W[J]-\int \mathrm d^d x\,J\phi_{\mathrm c}
    Identity
    δΓδϕc=−J\frac{\delta\Gamma}{\delta\phi_{\mathrm c}}=-J
    Variables
    J is a bosonic source, phi_c is the source-dependent classical field, W generates connected correlators, and Gamma generates one-particle-irreducible vertices.
    Dimensional check
    The source term integral and action are dimensionless in natural units.
    Aliases
    generating functional, effective action Legendre transform, W and Gamma
    Scope
    Lorentzian bosonic source functional with the declared plus source coupling.
    Assumptions
    The functional Legendre transform exists in the bounded sense used by the owning treatment.; Fermionic, Euclidean, background-field, and closed-time-path variants declare their signs locally.
  • Formula

    Free-scalar Feynman propagator

    The time-ordered free-scalar two-point function carries the Feynman plus-i-zero prescription in the site's Fourier and metric conventions.

    Expression
    ⟨0∣Tϕ(x)ϕ(0)∣0⟩=∫ddp(2π)di e−ip⋅xp2−m2+i0\langle0|\mathrm T\phi(x)\phi(0)|0\rangle=\int\frac{\mathrm d^d p}{(2\pi)^d}\frac{i\,e^{-ip\cdot x}}{p^2-m^2+i0}
    Variables
    x is the separation, p is momentum, m is the scalar mass, and i0 fixes the Feynman contour.
    Dimensional check
    The propagator has mass dimension d-2 for a canonically normalized scalar.
    Limitation
    Retarded, advanced, Wightman, Euclidean, in-in, and in-out objects are not interchangeable with this record.
    Aliases
    Feynman propagator, scalar propagator, time-ordered two-point function
    Scope
    The vacuum two-point function of a canonically normalized free real scalar field in flat Lorentzian spacetime.
    Assumptions
    The state is the free vacuum and the product is time ordered.; The plus-i-zero prescription specifies the Feynman boundary value.
  • Formula

    Renormalization-group beta function

    The beta function is the scale derivative of a renormalized coupling with the bare data held fixed.

    Expression
    βg(g)=μdgdμ∣bare data fixed\beta_g(g)=\left.\mu\frac{\mathrm d g}{\mathrm d\mu}\right|_{\text{bare data fixed}}
    Variables
    g is a renormalized coupling and mu is the renormalization scale.
    Dimensional check
    For a dimensionless coupling, beta_g is dimensionless; dimensionful couplings require the declared coordinate convention.
    Limitation
    Beta-function coefficients and even coordinates can be scheme dependent beyond invariant leading information.
    Aliases
    beta function definition, RG flow of a coupling
    Scope
    A renormalized coupling coordinate g and renormalization scale mu.
    Assumptions
    The regulator dimension, subtraction scheme, coupling coordinate, anomalous-dimension sign, and flow direction are declared.
  • Formula

    Spacetime Fourier transform pair

    The site-wide continuous spacetime transform pair fixes phases, the inverse measure, and the momentum representation of derivatives.

    Expression
    f~(p)=∫ddx e+ip⋅xf(x),f(x)=∫ddp(2π)de−ip⋅xf~(p)\widetilde f(p)=\int \mathrm d^d x\,e^{+ip\cdot x}f(x),\qquad f(x)=\int\frac{\mathrm d^d p}{(2\pi)^d}e^{-ip\cdot x}\widetilde f(p)
    Variables
    x is position, p is momentum, d is spacetime dimension, and f tilde is the transform of f.
    Dimensional check
    The measure and dimensions of f tilde compensate so the inverse transform has the dimension of f.
    Application
    Under this convention, partial_mu maps to -i p_mu.
    Aliases
    Fourier transform pair, momentum transform
    Scope
    Continuous functions or distributions on d-dimensional spacetime for which the transforms are defined.
    Assumptions
    Distributional transforms use an appropriate test-function or tempered-distribution setting.; Specialized transforms state their own measure and phases.