Skip to content
QFT .org

Reference

Conventions and normalizations

Check signs, Fourier transforms, units, gauge normalization, and common convention translations.

These are the site defaults. A different source convention is equally valid when all dependent signs and normalizations are translated together.

Browse and look up

13entries

  • Anti-Hermitian gauge-connection translation

    An anti-Hermitian geometric connection is an exact repackaging of the site’s Hermitian gauge field only when the curvature, coupling, trace, and Euclidean action are translated together.

    Applies to: Matrix-valued gauge connections related to the site field by mathcal A_mu = -i g A_mu.

    • The site generators are Hermitian and Tr_F(T^a T^b) = delta^{ab}/2.
    • The ordinary matrix trace is negative on squares of anti-Hermitian matrices.
    Connection map
    Aμ=−igAμ,Dμ=∂μ+Aμ\mathcal A_\mu=-igA_\mu,\qquad D_\mu=\partial_\mu+\mathcal A_\mu
    Curvature map
    Fμν=∂μAν−∂νAμ+[Aμ,Aν]=−igFμν\mathcal F_{\mu\nu}=\partial_\mu\mathcal A_\nu-\partial_\nu\mathcal A_\mu+[\mathcal A_\mu,\mathcal A_\nu]=-igF_{\mu\nu}
    Euclidean action
    SE=−12g2∫ddx Tr⁡F(FμνFμν)S_E=-\frac{1}{2g^2}\int\mathrm d^d x\,\operatorname{Tr}_F(\mathcal F_{\mu\nu}\mathcal F_{\mu\nu})

    Full explanation: Gauge Fields, Redundancy, and Observable Content

    Notation and checks
    Also called
    geometric gauge normalization, anti-Hermitian connection, mathematical gauge convention

  • Charged matter, gauge current, and Wilson transport

    The charge phase, covariant derivative, interaction current, Maxwell equation, and Wilson-line endpoint phases form one sign package.

    Applies to: Abelian charged matter in the inherited Hermitian gauge convention; the transporter formula also fixes the non-Abelian endpoint ordering.

    • The path runs from x_i to x_f.
    • An external source written with -A_mu J_source^mu obeys J_source^mu = -j_q^mu.
    Covariant derivative
    Dμ=∂μ−iqAμD_\mu=\partial_\mu-iqA_\mu
    Gauge transformation
    ψ↦e+iqαψ,Aμ↦Aμ+∂μα\psi\mapsto e^{+iq\alpha}\psi,\qquad A_\mu\mapsto A_\mu+\partial_\mu\alpha
    Gauge current
    jbilμ=ψˉγμψ,jqμ=qjbilμ=δSL,m/δAμj_{\mathrm{bil}}^\mu=\bar\psi\gamma^\mu\psi,\qquad j_q^\mu=qj_{\mathrm{bil}}^\mu=\delta S_{L,\mathrm m}/\delta A_\mu
    Interaction
    LL,int=+Aμjqμ\mathcal L_{L,\mathrm{int}}=+A_\mu j_q^\mu
    Gauge-field equation
    eA−2∂μFμν+jqν=0e_A^{-2}\partial_\mu F^{\mu\nu}+j_q^\nu=0
    Maxwell normalization
    LL=−14eA2FμνFμν+Aμjqμ\mathcal L_L=-\frac{1}{4e_A^2}F_{\mu\nu}F^{\mu\nu}+A_\mu j_q^\mu
    Wilson transporter
    Uγ(xf,xi)=exp⁡ ⁣(iq∫xixfA)U_\gamma(x_f,x_i)=\exp\!\left(iq\int_{x_i}^{x_f}A\right)
    Transporter endpoint law
    Uγ↦G(xf)UγG(xi)−1,G(x)=eiqα(x) in the Abelian caseU_\gamma\mapsto G(x_f)U_\gamma G(x_i)^{-1},\qquad G(x)=e^{iq\alpha(x)}\ \text{in the Abelian case}
    Line actions by signature
    SL,int=q∫γA,SE,int=−iq∫γAS_{L,\mathrm{int}}=q\int_\gamma A,\qquad S_{E,\mathrm{int}}=-iq\int_\gamma A

    Full explanation: Gauge Fields, Redundancy, and Observable Content

    Notation and checks
    Also called
    charged-matter convention, gauge-current convention, Wilson-line phase convention

  • Feynman plus-i-zero boundary value

    The Feynman propagator is the positive-imaginary boundary value; the plus-i-zero is retained whenever it fixes poles, contours, states, or discontinuities.

    Applies to: Lorentzian time-ordered propagators in the inherited metric and Fourier conventions.

    • Retarded, advanced, Wightman, in-in, and Euclidean objects retain their own state and contour data.
    ip2−m2+i0\frac{i}{p^2-m^2+i0}

    Full explanation: Scalar Propagators, Ordered Correlators, and Sources

    Notation and checks
    Also called
    +i0, Feynman prescription, causal boundary value

  • Fixed-bare renormalization flow

    Beta functions and field and mass anomalous dimensions are defined by varying the subtraction scale while holding the complete bare data fixed.

    Applies to: Multiplicatively renormalized fields, running mass squared, connected functions, and proper vertices.

    • Regulator dimension, subtraction scheme, coupling coordinates, composite-operator basis, and Wilsonian flow parameter remain local.
    Beta function
    βg=μdgdμ∣0\beta_g=\left.\mu\frac{\mathrm dg}{\mathrm d\mu}\right|_0
    Field anomalous dimension
    γr=12μdln⁡Zrdμ∣0\gamma_r=\left.\frac12\mu\frac{\mathrm d\ln Z_r}{\mathrm d\mu}\right|_0
    Mass anomalous dimension
    γm2=−μdln⁡m2dμ∣0\gamma_{m^2}=-\left.\mu\frac{\mathrm d\ln m^2}{\mathrm d\mu}\right|_0
    Fixed-bare flow operator
    D=μ∂∂μ+βg∂∂g−γm2m2∂∂m2\mathcal D=\mu\frac{\partial}{\partial\mu}+\beta_g\frac{\partial}{\partial g}-\gamma_{m^2}m^2\frac{\partial}{\partial m^2}
    Connected-function equation
    (D+nγr)GR(n)=0(\mathcal D+n\gamma_r)G_R^{(n)}=0
    Proper-vertex equation
    (D−nγr)ΓR(n)=0(\mathcal D-n\gamma_r)\Gamma_R^{(n)}=0

    Full explanation: Scale Independence and the Callan–Symanzik Equation

    Notation and checks
    Also called
    Callan-Symanzik convention, anomalous-dimension signs, fixed-bare RG
    Conventions

  • Four-dimensional Lorentzian orientation

    In oriented Minkowski coordinates the contravariant alternating tensor has epsilon^{0123} = +1 and lowering all indices gives epsilon_{0123} = -1.

    Applies to: Oriented four-dimensional Lorentzian coordinates with the mostly-minus metric.

    • Other dimensions and manifolds declare their orientation locally.
    • A page distinguishes alternating symbols, tensors, densities, and differential forms when material.
    ϵ0123=+1,ϵ0123=−1\epsilon^{0123}=+1,\qquad\epsilon_{0123}=-1

    Full explanation: Differential Forms, Integration, Orientation, and Stokes Theorem

    Notation and checks
    Also called
    epsilon convention, Levi-Civita orientation, Minkowski orientation

  • Four-dimensional Lorentzian spinors

    The four-dimensional Clifford algebra, Dirac adjoint, chirality matrix, and Lorentz generators are fixed as one mostly-minus package.

    Applies to: Four-dimensional Lorentzian spinors in the mostly-minus metric.

    • No explicit gamma-matrix representation is inherited.
    • Other dimensions and Euclidean signatures restate chirality, reality, charge conjugation, and adjoints.
    Clifford algebra
    {γμ,γν}=2ημν\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}
    Adjoint
    ψˉ=ψ†γ0\bar\psi=\psi^\dagger\gamma^0
    Chirality
    γ5=iγ0γ1γ2γ3\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3
    Lorentz generators
    σμν=i2[γμ,γν],Σμν=12σμν\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu],\qquad\Sigma^{\mu\nu}=\frac12\sigma^{\mu\nu}

    Full explanation: The Dirac Field

    Notation and checks
    Also called
    gamma-matrix convention, Dirac adjoint convention, gamma5 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).

    Applies to: Gauge fields written as A_mu = A_mu^a T^a in a Hermitian generator basis.

    • Global form, representation, charge normalization, trace convention, bundle sector, and coupling placement remain local when material.
    • Anti-Hermitian-generator sources require an explicit translation.
    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]

    Full explanation: Gauge Fields, Redundancy, and Observable Content

    Notation and checks
    Also called
    Hermitian-generator convention, gauge covariant derivative convention

  • Lorentzian metric signature

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

    Applies to: Lorentzian spacetime in d dimensions; four-dimensional shorthand is (+---).

    • A Euclidean page declares its positive-definite metric separately.
    Four-dimensional signature
    +---
    Use with care
    Changing signature without translating contractions, propagators, gamma matrices, or curvature formulas can change intermediate signs.

    Full explanation: Fields, Configurations, Dimensions, and Local Dynamics

    Notation and checks
    Convention choice
    \eta_{\mu\nu}=\operatorname{diag}(+1,-1,\ldots,-1)
    Also called
    mostly minus, +---, West Coast metric

  • 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.

    Applies to: Lorentzian bosonic functional integrals with the default source convention.

    • Fermionic, closed-time-path, background-field, and Euclidean functionals state ordering and signs locally.
    Weight
    eiSe^{iS}
    Source term
    +i∫ddx Jϕ+i\int \mathrm d^d x\,J\phi
    Euclidean weight
    e−SEe^{-S_E}
    Connected generator
    WL[J]=−ilog⁡ZL[J]W_L[J]=-i\log Z_L[J]
    Legendre transform
    ΓL[ϕc]=WL[J]−∫Jϕc,δΓL/δϕc=−J\Gamma_L[\phi_{\mathrm c}]=W_L[J]-\int J\phi_{\mathrm c},\qquad\delta\Gamma_L/\delta\phi_{\mathrm c}=-J

    Full explanation: Gaussian Fields and Sources

    Notation and checks
    Also called
    path-integral convention, source 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.

    Applies to: Analytic expressions on substantive QFT.org pages.

    • Numerical results carry units or are explicitly dimensionless.
    • Units are restored for experiments, observations, engineering, datasets, community comparisons, or dimensional checks.
    Dimensional equivalence
    Mass, energy, inverse length, inverse time, and temperature share one dimension.

    Full explanation: Fields, Configurations, Dimensions, and Local Dynamics

    Notation and checks
    Convention choice
    \hbar=c=k_{\mathrm B}=1
    Also called
    natural units, hbar equals c equals kB equals one

  • Reusable 1+1-dimensional Lorentzian extension

    Pages may opt into a shared two-dimensional metric, orientation, chirality, current-duality, and light-cone component package instead of redefining it piecemeal.

    Applies to: Lorentzian 1+1-dimensional spinors and light-cone coordinates x^plus = x^0 + x^1 and x^minus = x^0 - x^1.

    • This extension is not inherited automatically; a page opts into it explicitly.
    • An oriented light-cone measure declares its coordinate ordering.
    • A_+ and A_- are the coordinate components of the one-form A=A_mu dx^mu in the x^plus,x^minus chart.
    Orientation
    ημν=diag⁡(+,−),ϵ01=+1,ϵ01=−1\eta_{\mu\nu}=\operatorname{diag}(+,-),\qquad\epsilon^{01}=+1,\qquad\epsilon_{01}=-1
    Adjoint
    ψˉ=ψ†γ0\bar\psi=\psi^\dagger\gamma^0
    Chirality
    γ∗=γ0γ1,P±=12(1±γ∗),γμγ∗=−ϵμνγν\gamma_*=\gamma^0\gamma^1,\qquad P_\pm=\frac12(1\pm\gamma_*),\qquad \gamma^\mu\gamma_*=-\epsilon^{\mu\nu}\gamma_\nu
    Current duality
    jμ=ψˉγμψ,j5μ=ψˉγμγ∗ψ=−ϵμνjνj^\mu=\bar\psi\gamma^\mu\psi,\qquad j_5^\mu=\bar\psi\gamma^\mu\gamma_*\psi=-\epsilon^{\mu\nu}j_\nu
    Light-cone coordinates
    x±=x0±x1,∂±=12(∂0±∂1)x^\pm=x^0\pm x^1,\qquad\partial_\pm=\frac12(\partial_0\pm\partial_1)
    Component map
    A±=12(A0±A1),A^±=2A±A_\pm=\frac12(A_0\pm A_1),\qquad\widehat A_\pm=2A_\pm
    Measure Jacobian
    ∣∂(x0,x1)∂(x+,x−)∣=12\left|\frac{\partial(x^0,x^1)}{\partial(x^+,x^-)}\right|=\frac12

    Full explanation: Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities

    Notation and checks
    Also called
    1+1-dimensional convention, two-dimensional chirality convention, light-cone component convention

  • Riemann curvature sign

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

    Applies to: Torsion-free connections with the displayed index placement.

    • Torsion, nonmetricity, spin connections, extrinsic curvature, Euclidean gravity, and other Riemann signs require a local extension or translation.
    • A Fourier-phase reversal does not by itself reverse a two-derivative linearized-curvature formula.
    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}

    Full explanation: Levi–Civita Connections, Geodesics, and Riemann Curvature

    Notation and checks
    Also called
    curvature convention, Riemann sign 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.

    Applies to: Continuous d-dimensional spacetime Fourier transforms.

    • Spatial-only, thermal, finite-volume, lattice, Mellin, Laplace, and discrete transforms declare their own signs and measures.
    • A source using the opposite full spacetime phase maps every momentum argument by p_source = -p_site.
    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
    Opposite-phase map
    f−(p)=f~site(−p),(∂μf)−(p)=+ipμf−(p)f_{-}(p)=\widetilde f_{\mathrm{site}}(-p),\qquad (\partial_\mu f)_{-}(p)=+ip_\mu f_{-}(p)

    Full explanation: Fourier Series, Fourier Transforms, and Plancherel Theory

    Notation and checks
    Also called
    Fourier convention, momentum-space convention