Propagators and correlators

A two-point formula is usable only after its state, ordering, and boundary condition are known. This guide collects the free real-scalar vacuum functions in flat $d$-dimensional spacetime, with $m>0$, and the Gaussian identities built from them. The conventions fix Fourier phases and source signs; each equality below is an equality of distributions, with fields smeared when operator expressions are used.

Helpful background. Quantizing the Real Scalar Field supplies the mode expansion and vacuum normalization. For the derivations of the kernels, use Scalar Propagators, Ordered Correlators, and Sources.

Vacuum two-point functions

Let $z=x-y$, $P=\Box+m^2$, and $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. The positive-frequency Wightman function is

$$W^+(z)=\langle0|\phi(x)\phi(y)|0\rangle =\int\frac{\mathrm d^{d-1}\mathbf p}{(2\pi)^{d-1}2E_{\mathbf p}}\,e^{-ip\cdot z}, \qquad p^0=E_{\mathbf p}.$$

Define $W^-(z)=W^+(-z)$ and $C(z)=W^+(z)-W^-(z)$. In the free theory, $[\phi(x),\phi(y)]=C(z)\mathbf1$. The Wightman functions and commutator solve the homogeneous equation $PW^\pm=PC=0$.

Time ordering gives the Feynman correlator:

$$\begin{aligned} D_F(z)&=\theta(z^0)W^+(z)+\theta(-z^0)W^-(z),\\ \widetilde D_F(p)&=\frac{i}{p^2-m^2+i0},\\ PD_F&=-i\delta^{(d)}. \end{aligned}$$

The $+i0$ places the positive-energy pole below the real energy axis and the negative-energy pole above it. It is part of the distribution, not a small parameter to discard before integration. The vacuum contraction and contour result are derived in Schwartz 2014, § 6.2, pp. 75–77.

Green inverses and causal response

If a Green kernel is defined by $PG=\delta^{(d)}$, its normalization differs from the raw time-ordered correlator. In particular,

$$\begin{aligned} G_F&=iD_F,\\ G_{\mathrm{ret}}(z)&=i\theta(z^0)C(z),\\ G_{\mathrm{adv}}(z)&=-i\theta(-z^0)C(z). \end{aligned}$$

All three are delta-normalized inverses. The retarded kernel has future causal support, the advanced kernel has past causal support, and the Feynman kernel obeys vacuum time-ordering boundary conditions. For a source with no incoming disturbance, $P\phi_J=J$ is solved by $\phi_J=G_{\mathrm{ret}}*J$ plus any chosen homogeneous solution.

Normalization check. The equal-time commutator gives $C(0,\mathbf z)=0$ and $\partial_{z^0}C(0,\mathbf z)=-i\delta^{(d-1)}(\mathbf z)$. Differentiating $i\theta(z^0)C(z)$ therefore produces exactly one positive delta source. This check detects a missing factor of $i$ without evaluating a contour integral.

Spacelike vacuum correlations can be nonzero even though the commutator vanishes there. Correlation and causal response are different objects; a nonzero $W^+$ outside the light cone is not a signal propagating faster than light.

Gaussian sources and Wick contractions

For the centered free vacuum and a plus-sign source coupling, the normalized source functional is

$$\begin{aligned} Z_0[J]&=\exp\!\left[-\frac12\int\mathrm d^dx\,\mathrm d^dy\, J(x)D_F(x-y)J(y)\right],\\ \langle\mathrm T\phi_1\cdots\phi_n\rangle &=\left.(-i)^n\frac{\delta^n Z_0}{\delta J_1\cdots\delta J_n}\right|_{J=0}. \end{aligned}$$

Here $\phi_i$ denotes $\phi(x_i)$ and $J_i=J(x_i)$. Differentiating four times, with $D_{ij}=D_F(x_i-x_j)$, gives

$$\langle\mathrm T\phi_1\phi_2\phi_3\phi_4\rangle =D_{12}D_{34}+D_{13}D_{24}+D_{14}D_{23}.$$

There are three pairings: field 1 can be paired with any of fields 2, 3, or 4, leaving a unique final pair. More generally, $2n$ centered bosonic fields have $(2n-1)!!$ complete pairings; odd moments vanish. See Schwartz 2014, § 7.A, pp. 100–103; § 14.3.2, p. 263 and the full Wick-theorem treatment.

Connected functions and limits

For a centered field, the connected four-point function is the full four-point function minus the three products of two-point functions displayed above. It vanishes in this Gaussian vacuum. A nonzero connected four-point function measures information absent from the two-point function, although a vanishing fourth cumulant alone does not prove that every higher cumulant vanishes.

These pairing formulas do not apply to an arbitrary state simply because its field equation is free. Fermionic contractions require permutation signs, and coincident composite products require a separate short-distance prescription. Continue with Connected Correlators and Cumulants or Free Wick Products and Point Splitting when either issue matters.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.