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Conventions and normalizations

Check the metric, units, Fourier phases, source signs, gauge normalization, curvature sign, and other site-wide defaults before using a formula.

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
  • The index states site defaults and bounded cautions, not a judgment that other conventions are wrong.
  • An alternate-convention conversion appears only after its assumptions and round-trip check are explicitly registered.

Coverage note: records included through 2026-08-12

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  • Convention

    Hermitian gauge generators and covariant derivative

    The default Lie-algebra basis is Hermitian, with D_mu = partial_mu - i g A_mu and T(F) = 1/2 for the fundamental of SU(N).

    Algebra
    [Ta,Tb]=ifabcTc[T^a,T^b]=if^{abc}T^c
    Trace
    tr⁡R(TaTb)=T(R)δab\operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}
    Covariant derivative
    Dμ=∂μ−igAμD_\mu=\partial_\mu-igA_\mu
    Field strength
    Fμν=∂μAν−∂νAμ−ig[Aμ,Aν]F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu]
    Aliases
    Hermitian-generator convention, gauge covariant derivative convention
    Scope
    Gauge fields written as A_mu = A_mu^a T^a in a Hermitian generator basis.
    Assumptions
    Global form, representation, charge normalization, trace convention, bundle sector, and coupling placement remain local when material.; Anti-Hermitian-generator sources require an explicit translation.
  • Convention

    Lorentzian metric signature

    QFT.org uses the mostly-minus Lorentzian metric, with one positive time direction and negative spatial directions.

    Convention choice
    \eta_{\mu\nu}=\operatorname{diag}(+1,-1,\ldots,-1)
    Four-dimensional signature
    +---
    Use with care
    Changing signature without translating contractions, propagators, gamma matrices, or curvature formulas can change intermediate signs.
    Aliases
    mostly minus, +---, West Coast metric
    Scope
    Lorentzian spacetime in d dimensions; four-dimensional shorthand is (+---).
    Assumptions
    A Euclidean page declares its positive-definite metric separately.
  • Convention

    Lorentzian path-integral weight and source sign

    QFT.org uses the Lorentzian weight exp(iS) and a plus sign for the default bosonic source coupling.

    Weight
    eiSe^{iS}
    Source term
    +i∫ddx Jϕ+i\int \mathrm d^d x\,J\phi
    Euclidean weight
    e−SEe^{-S_E}
    Canonical treatment
    Gaussian Fields and Sources
    Aliases
    path-integral convention, source convention
    Scope
    Lorentzian bosonic functional integrals with the default source convention.
    Assumptions
    Fermionic, closed-time-path, background-field, and Euclidean functionals state ordering and signs locally.
  • Convention

    Natural units

    Unless dimensions or comparison require restoration, QFT.org sets Planck's constant, the speed of light, and Boltzmann's constant to one.

    Convention choice
    \hbar=c=k_{\mathrm B}=1
    Dimensional equivalence
    Mass, energy, inverse length, inverse time, and temperature share one dimension.
    Aliases
    natural units, hbar equals c equals kB equals one
    Scope
    Analytic expressions on substantive QFT.org pages.
    Assumptions
    Numerical results carry units or are explicitly dimensionless.; Units are restored for experiments, observations, engineering, datasets, community comparisons, or dimensional checks.
  • Convention

    Riemann curvature sign

    QFT.org fixes the Riemann tensor by the commutator of torsion-free covariant derivatives acting on a vector.

    Commutator
    [∇μ,∇ν]Vρ=RρσμνVσ[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma
    Ricci contraction
    Rσν=RρσρνR_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}
    Aliases
    curvature convention, Riemann sign convention
    Scope
    Torsion-free connections with the displayed index placement.
    Assumptions
    Torsion, nonmetricity, spin connections, extrinsic curvature, Euclidean gravity, and other Riemann signs require a local extension or translation.
  • Convention

    Spacetime Fourier transform

    The forward transform carries a positive phase and the inverse transform a negative phase with the full momentum measure divided by (2 pi)^d.

    Forward map
    f~(p)=∫ddx e+ip⋅xf(x)\widetilde f(p)=\int \mathrm d^d x\,e^{+ip\cdot x}f(x)
    Inverse map
    f(x)=∫ddp(2π)d e−ip⋅xf~(p)f(x)=\int \frac{\mathrm d^d p}{(2\pi)^d}\,e^{-ip\cdot x}\widetilde f(p)
    Derivative map
    ∂μ↦−ipμ\partial_\mu\mapsto -ip_\mu
    Aliases
    Fourier convention, momentum-space convention
    Scope
    Continuous d-dimensional spacetime Fourier transforms.
    Assumptions
    Spatial-only, thermal, finite-volume, lattice, Mellin, Laplace, and discrete transforms declare their own signs and measures.