Propagators and causal influence

Keep $\hbar=c=1$, metric $(+---)$, and the free real scalar field. Define $\Delta_F(x-y)=\langle0|T\phi(x)\phi(y)|0\rangle$, including the factor of $i$ in its momentum-space representation below. For bosons, time ordering places the later operator on the left. This is a vacuum expectation of an ordered product, not the expectation of the field produced after an experimenter turns on a source.

The infinitesimal $i\epsilon$ prescription tells us how the energy integration passes the poles. It implements the vacuum time-ordering boundary condition and cannot be discarded before integration. Different pole prescriptions solve related differential equations with different boundary conditions. Calling all of them “the inverse wave operator” omits information that is as necessary as boundary conditions were for the classical initial-value problem.

For spacelike separation, the free scalar commutator vanishes although $\Delta_F$ generally does not. This is not a contradiction. Two observables can be correlated in a shared state without either allowing a local agent to change the other’s outcome distribution instantaneously. In local relativistic QFT, commuting spacelike observables encode this operational separation. A vacuum correlation measures the state’s structure; it does not on its own specify an intervention or a response.

A useful response object is $G_R(z)=i\theta(z^0)\langle[\phi(z),\phi(0)]\rangle$ in our convention. For a source added as $+J\phi$ to the Lagrangian, it solves the classical retarded Green-function problem and determines the linear induced mean field. The step function removes response before the source; the spacelike commutator removes response outside the light cone. These two restrictions address different parts of causal support.

For two events separated by $(\Delta t,\Delta\mathbf x)$, calculate the interval $\Delta t^2-|\Delta\mathbf x|^2$ before drawing causal conclusions. A spacelike interval stays spacelike under a Lorentz transformation, but the time order of its endpoints can reverse. The vanishing commutator makes the time-ordered bosonic correlation consistent across such frames: exchanging the two spacelike operators changes nothing. This connects an algebraic condition to relativity’s geometry.

For timelike separations, the later event can lie in the earlier event’s future, but the amount of response still depends on the field equation and the source. A light cone describes possible causal support, not a guarantee that every allowed response is nonzero. Likewise, a Feynman diagram’s internal line is an integration kernel in an amplitude calculation, not an observed particle path threading the apparatus. Keep the question, operator ordering, and boundary condition attached to every propagator.