Propagator Table

The word “propagator” does not fix a source normalization or a boundary condition. This table first states the operator and its delta source, then the Fourier prescription. It applies to free fields in Minkowski spacetime; external backgrounds require the two-point operator calculus of Relativistic Green Functions.

Required background. Relativistic Green Functions distinguishes source inverses from evolution. Helpful background. Klein–Gordon Propagators and Dirac Propagators derive the scalar and spinor prescriptions.

Source equations fix the kernel normalization

Use $\hbar=c=1$, $\eta=(+---)$, $z=x-y$, and

$$K(z)=\int\frac{d^4p}{(2\pi)^4} e^{-ip\cdot z}\widetilde K(p).$$

Define the free operators

$$\begin{aligned} L&=\Box+m^2,\\ D&=i\gamma^\mu\partial_\mu-m,\\ D_+&=i\gamma^\mu\partial_\mu+m, \qquad DD_+=-L I_4. \end{aligned}$$

All derivatives act on $x$. In the following table $X$ denotes the same retarded, advanced, or Feynman boundary choice on both sides of an identity.

Object	Source equation	Relation to a unit-source inverse
Scalar inverse $G_X$	$LG_X=\delta^4(z)$	Unit source by definition
Scalar time-ordered correlator $D_F$	$LD_F=-i\delta^4(z)$	$D_F=-iG_F$
Dirac inverse $K_X$	$DK_X=\delta^4(z)I_4$	$K_X=-D_+G_X$
Dirac time-ordered correlator $S_F$	$DS_F=+i\delta^4(z)I_4$	$S_F=iK_F=D_+D_F$

The scalar and spinor correlators have opposite source signs in this convention. They are consistent because $DD_+=-L I_4$. For the scalar correlator use $D_F(z)=\langle0|T\phi(x)\phi^\dagger(y)|0\rangle$; for a real field $\phi^\dagger=\phi$. For the spinor use $S_F(z)=\langle0|T\psi(x)\bar\psi(y)|0\rangle$, with fermionic time ordering. The correlator bridge explains the operator meaning.

Fourier denominators specify the boundary choice

Write $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. The three scalar multipliers are

Boundary choice	$\widetilde G_X(p)$
Retarded	$-[(p^0+i0)^2-E_{\mathbf p}^2]^{-1}$
Advanced	$-[(p^0-i0)^2-E_{\mathbf p}^2]^{-1}$
Feynman	$-[p^2-m^2+i0]^{-1}$

Retarded poles are both below the real axis; advanced poles are both above. Feynman places the positive-energy pole below and the negative-energy pole above.

Here $i0$ specifies a distributional boundary value, not a small finite parameter to be retained in the source equation. In particular, retarded and Feynman denominators are different. For the spinor inverse replace each scalar numerator $-1$ by $p\!\!\!/+m$ while keeping its denominator. Thus

$$\begin{aligned} \widetilde D_F(p)&=\frac{i}{p^2-m^2+i0},\\ \widetilde K_F(p)&=\frac{p\!\!\!/+m}{p^2-m^2+i0},\\ \widetilde S_F(p)&=\frac{i(p\!\!\!/+m)}{p^2-m^2+i0}. \end{aligned}$$

The pole displacements in the quadratic Feynman expression are understood in the limiting sense. At nonzero regulator, the shifted square root and the two linearized prescriptions are not identical finite-parameter functions. See Retarded, Advanced, and Feynman Propagators for their boundary-value comparison.

Causal response and homogeneous correlations

For the free positive-frequency Wightman function set

$$W(z)=\int\frac{d^3p}{(2\pi)^3\,2E_{\mathbf p}} e^{-ip\cdot z},\qquad p^0=E_{\mathbf p}.$$

It is a homogeneous solution, $LW=0$, and is not a unit-source inverse. Define the commutator distribution without an extra factor of $i$: $C(z)=W(z)-W(-z)$. Then

$$\begin{aligned} G_R(z)&=i\theta(z^0)C(z),\\ G_A(z)&=-i\theta(-z^0)C(z),\\ G_F(z)&=i\{\theta(z^0)W(z) +\theta(-z^0)W(-z)\}. \end{aligned}$$

The scalar commutator vanishes outside the light cone. Consequently $G_R$ has future-causal support and $G_A$ has past-causal support. $W$ and $G_F$ need not vanish at spacelike separation. Their nonzero spacelike correlations do not constitute a causal signal.

For the free Dirac anticommutator kernel $\mathcal A(z)=D_+C(z)$,

$$\begin{aligned} K_R(z)&=-i\theta(z^0)\mathcal A(z),\\ K_A(z)&=+i\theta(-z^0)\mathcal A(z). \end{aligned}$$

At equal time, $\mathcal A(0,\mathbf z)=\gamma^0\delta^3(\mathbf z)$. The corresponding jump $K_X(0^+)-K_X(0^-)=-i\gamma^0\delta^3$ gives the stated Dirac unit source. The spinor causal kernel includes both energy sectors; retaining only positive energy changes its source and support.

Task	Appropriate object	What must also be specified
Response to a source switched on after initial data	$G_R$ or $K_R$	Source coupling and initial homogeneous solution
A final boundary condition with no later response	$G_A$ or $K_A$	Advanced boundary prescription
Vacuum time-ordered perturbation theory	$D_F$ or $S_F$	Field normalization and interaction factors
Vacuum fluctuations or detector correlations	$W$ or the relevant spinor two-point function	Operator ordering, state and detector coupling
Hamiltonian evolution of Dirac initial data	$U(t,t')$	Equal-time Hilbert space and initial state

In a prescribed background the last row is related to a retarded Dirac inverse by

$$K_R(t,t')=-i\theta(t-t')U(t,t')\gamma^0,$$

where spatial kernels or operator actions are implicit. The $\gamma^0$ acts on the right because $D=\gamma^0(i\partial_t-H)$. This relation survives time-dependent backgrounds under the evolution assumptions of the Green-function owner. The free formula $K_X=-D_+G_X$ does not extend by simply replacing partial derivatives with covariant derivatives: their commutator produces a spin–field term.

References

• Bjorken, James D., and Sidney D. Drell. Relativistic Quantum Mechanics. McGraw–Hill, 1964. Relativistic Green functions and boundary prescriptions.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940. Scalar and fermionic time-ordered propagators.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2 and 5. Scalar fields and the Dirac field.