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Symbols and notation

Decode frequently used symbols and distinguish meanings that share a letter.

Symbols can have different meanings in different contexts; check the accompanying definition before using a formula.

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11entries

  • D_F — scalar Feynman correlator

    The vacuum time-ordered two-point function of the free real scalar.

    Applies to: The displayed definition and its stated source convention.

    DF(x−y)=⟨0∣Tϕ(x)ϕ(y)∣0⟩D_F(x-y)=\langle0|\mathrm T\phi(x)\phi(y)|0\rangle
    Dimensionality
    Position-space mass dimension d−2 for a canonically normalized field.
    Notation caution
    The delta-normalized inverse is G_F=iD_F; other authors can absorb this i into the name propagator.

    Full explanation: Scalar Propagators, Ordered Correlators, and Sources

  • J — bosonic source

    A prescribed classical source coupled with the displayed sign.

    Applies to: The displayed definition and its stated source convention.

    SJ=S+∫ddx J(x)ϕ(x)S_J=S+\int\mathrm d^d x\,J(x)\phi(x)
    Dimensionality
    Mass dimension (d+2)/2 for a canonical scalar.
    Notation caution
    A generating source is not the same object as a conserved current, often also denoted J or j.

    Full explanation: The Generating Functional

  • T — time ordering

    Roman T orders operator insertions in time; other orderings receive distinct labels.

    Applies to: Lorentzian time-ordered operator products and correlators.

    • Contour, radial, path, and thermal ordering are labeled separately.
    T\mathrm T
    Notation caution
    Italic T commonly denotes temperature; typography and prose must distinguish it from the time-ordering operator.

    Full explanation: Scalar Propagators, Ordered Correlators, and Sources

    Notation and checks
    Also called
    time ordering, \mathrm T

  • W[J] — connected generator

    The logarithmic generator of connected time-ordered correlation functions in the Lorentzian source convention.

    Applies to: The displayed definition and its stated source convention.

    W[J]=−ilog⁡Z[J]W[J]=-i\log Z[J]
    Dimensionality
    Dimensionless.
    Notation caution
    W can instead denote a Wightman function; its arguments and definition distinguish the two.

    Full explanation: Connected Correlators and Cumulants

  • Z[J] — source functional

    The normalized vacuum amplitude in the presence of the source; derivatives generate full time-ordered correlators.

    Applies to: The displayed definition and its stated source convention.

    Z[J]=⟨0∣Tei∫Jϕ∣0⟩,Z[0]=1Z[J]=\langle0|\mathrm T e^{i\int J\phi}|0\rangle,\qquad Z[0]=1
    Dimensionality
    Dimensionless.
    Notation caution
    The one-particle pole residue Z is a different quantity.

    Full explanation: The Generating Functional

  • β_g — beta function of a coupling

    For renormalization, beta_g is the scale derivative of the renormalized coupling at fixed bare data.

    Applies to: Renormalization-group flow of a coupling coordinate g.

    • Regulator dimension, subtraction scheme, coupling coordinates, anomalous-dimension sign, and flow direction are stated locally.
    βg\beta_g
    Notation caution
    Plain beta denotes inverse temperature in thermal scope; the coupling subscript and RG context disambiguate this record.

    Full explanation: Beta Functions, Running Masses, and Field Anomalous Dimensions

    Notation and checks
    Definition
    \beta_g(g)=\left.\mu\frac{\mathrm d g}{\mathrm d\mu}\right|_{\text{bare data fixed}}
    Also called
    beta function, \beta_g, RG beta function
    Conventions

  • Γ — 1PI effective action

    The Legendre transform generating one-particle-irreducible vertices, with phi_c=delta W/delta J.

    Applies to: The displayed definition and its stated source convention.

    Γ[ϕc]=W[J]−∫ddx Jϕc\Gamma[\phi_c]=W[J]-\int\mathrm d^d x\,J\phi_c
    Dimensionality
    Dimensionless in natural units.
    Notation caution
    The same letter Γ also denotes a decay width, which has mass dimension one.

    Full explanation: The 1PI Effective Action and Mean-Field Equations

  • μ — renormalization scale

    In renormalization contexts, mu denotes the renormalization scale unless the page declares another meaning.

    Applies to: Renormalized quantities and renormalization-group equations.

    • The subtraction scheme and flow convention are stated locally.
    μ\mu
    Notation caution
    Mu is heavily overloaded in physics; chemical potential, index labels, reduced mass, and measures require separate scoped records.

    Full explanation: Scale Independence and the Callan–Symanzik Equation

    Notation and checks
    Also called
    mu, \mu, renormalization scale
    Conventions

  • φ — scalar field

    A scalar field: a classical configuration or an operator-valued distribution, as specified by context.

    Applies to: The displayed definition and its stated source convention.

    ϕ(x)\phi(x)
    Dimensionality
    For a canonically normalized scalar, mass dimension (d−2)/2 in d spacetime dimensions.
    Notation caution
    A field configuration, field operator, and mean field are distinct objects.

    Full explanation: Quantum Fields as Operator-Valued Distributions

  • ℳ — invariant scattering amplitude

    The scattering amplitude with its overall momentum-conserving delta function removed.

    Applies to: The displayed definition and its stated source convention.

    ⟨f∣T∣i⟩=(2π)4δ(4)(Pf−Pi)Mfi,S=1+iT\langle f|T|i\rangle=(2\pi)^4\delta^{(4)}(P_f-P_i)\mathcal M_{fi},\qquad S=1+iT
    Dimensionality
    Mass dimension zero for a four-point amplitude of canonically normalized fields in four dimensions.
    Notation caution
    A Feynman-rule calculation usually produces iℳ; do not drop this distinction when checking interference signs.

    Full explanation: Cross Sections and Decay Rates