Propagator

A propagator is a two-point kernel describing propagation and, in a free theory, a Green function for the quadratic kinetic operator. Its boundary prescription determines its type: the time-ordered Feynman propagator uses an $iϵ$ prescription, while retarded and advanced propagators impose causal boundary conditions.

The Feynman propagator is the time-ordered Green function selected by the $iϵ$ prescription.

With ${\Delta }_F=⟨0|T\{\varphi (x)\varphi (y)\}|0⟩$, for a mode of positive energy $E>0$ the Feynman i-epsilon prescription puts poles $+E-i0$ and $-E+i0$ in opposite half-planes. The energy contour integral closes clockwise below for positive time separation and counterclockwise above for negative separation. The residue theorem gives $e^{-iE|t|}/(2E)$ in either case. Thus the four-dimensional Fourier integral has numerator $i$ and denominator $p^2-m^2+i0$. The integral is understood as a distribution limit, not an ordinary absolutely convergent integral; changing the definition to $i{\Delta }_F$ would change the displayed normalization.

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

The Feynman $iϵ$ prescription displaces poles or tilts an integration contour by an infinitesimal imaginary part. It selects vacuum boundary conditions and damps oscillatory integrals at infinite time.

A Wightman function is an unordered vacuum correlation function. For a free scalar mode $f_k(t)$, the positive-frequency two-point function is $G^{>}(t_1,t_2;k)=f_k(t_1)f_k^{\ast }(t_2)$.

The free Dirac propagator is $S_F(p)=i(/p+m)/(p^2-m^2+iϵ)$ and inverts the Dirac kinetic operator with Feynman boundary conditions.

For action $-\int \bar{\psi }(\gamma \cdot \partial +m)\psi$, choose gamma matrices with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$ and signature $(-,+,\dots ,+)$. The Fourier kernel of $D=\gamma \cdot \partial +m$ is $i\gamma \cdot p+m$. Its algebraic inverse is $(-i\gamma \cdot p+m)/(p^2+m^2)$. The Grassmann Gaussian integral for $e^{-i\bar{\psi }D\psi }$ has covariance $-i{D}_F^{-1}$, giving the displayed Dirac propagator. Its denominator has positive-energy poles below and negative-energy poles above the real energy axis by the Feynman i-epsilon prescription. Both the metric and the factor of $i$ in the derivative kernel must be specified when comparing propagator formulas.

In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

At non-null four-momentum, these complementary projectors satisfy $P_L^2=P_L$, $P_T^2=P_T$ and $P_L{P}_T=0$. The gauge-fixed Maxwell kinetic operator is $-k^2(P_T+{\xi }^{-1}{P}_L)$; its inverse gives propagator $-i(P_T+\xi P_L)/k^2$. Transversality here removes one four-dimensional longitudinal direction, leaving rank three; it does not by itself project onto the two physical photon polarizations. The pole requires the Feynman i-epsilon prescription.

Define the inclusive photon-field tensor for hadronic final states by $W_A^{\mu \nu }=2\pi {\rho }_h(s)(-g^{\mu \nu }+q^{\mu }{q}^{\nu }/s)$ for timelike $q$ with $s=q^2$ and $q^0>0$. Here each final state has relativistic normalization and the sum includes the four-momentum delta function. This hadronic photon spectral density has mass dimension $-2$. At leading electromagnetic order, ${\rho }_h=e^2{\rho }_J/s$, with ${\rho }_J$ the dimensionless hadronic electromagnetic-current spectral density. Stating the tensor convention prevents mismatched powers of $s$ or $e$ in an inclusive cross-section.

For $q=(q^0,\mathbf{q})$ with $\mathbf{q}\ne 0$, the Coulomb gauge propagator hasThe temporal component is instantaneous; the spatial part projects onto transverse photon polarizations. Although these components are not individually Lorentz covariant, their contraction with conserved external currents equals the covariant answer. Zero-spatial-momentum limits should be taken after combining the current contractions.

For $q\cdot J=q\cdot K=0$, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:The identity follows from $\mathbf{q}\cdot \mathbf{J}=q^0{J}^0$ and $q^2=(q^0{)}^2-|\mathbf{q}{|}^2$. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.

Time-ordering the canonical transverse photon field gives a propagator with numerator equal to the spatial transverse projector and denominator $k^2+i0$. Its two energy poles encode forward propagation of the positive-frequency term and backward propagation of the negative-frequency term. This is the reduced free-field propagator; it is distinct from the covariant four-potential propagator.