Causality and Propagators in QFT

Causality

Causality in QFT requires that events or measurements that are space-like separated (i.e., events that cannot influence each other) do not affect each other. This is formalized by ensuring that the commutators of local observables vanish for space-like separated points. Specifically, for two local observables O1(x)O_1(x)O1​(x) and O2(y)O_2(y)O2​(y) at spacelike separation, we have:

[O1(x),O2(y)]=0 for (x−y)2<0[O_1(x), O_2(y)] = 0 \text{ for } (x – y)^2 < 0[O1​(x),O2​(y)]=0 for (x−y)2<0

A local observable is an operator defined at a specific point in spacetime or within a local neighborhood. To check for causality, we calculate the commutator of the field operators $\varphi (x)$ and $\varphi (y)$:

$\Delta (x-y)=[\varphi (x),\varphi (y)]$

For free fields, this commutator can be expressed in terms of mode expansions and is shown to vanish for spacelike separations, ensuring causality​​.

Propagators

Propagators in QFT describe the probability amplitude for a particle to travel from one point to another in spacetime. The propagator $D(x-y)$ for a scalar field is defined as:

$D(x-y)=⟨0|\varphi (x)\varphi (y)|0⟩$

This expression gives the amplitude for a particle created at point $y$ to be annihilated at point $x$. Using mode expansions and vacuum expectation values, the propagator can be written as:

D(x−y)=∫d3p(2π)312Epe−ip⋅(x−y)D(x – y) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} e^{-ip \cdot (x – y)}D(x−y)=∫(2π)3d3p​2Ep​1​e−ip⋅(x−y)

Interestingly, $D(x-y)$ is non-zero even for spacelike separations, which seems to conflict with causality at first glance. However, this does not violate causality because the commutator of the fields $\varphi (x)$ and $\varphi (y)$ for spacelike separations vanishes due to a cancellation of contributions from particle and antiparticle processes. Specifically:

$[\varphi (x),\varphi (y)]=D(x-y)-D(y-x)$

This ensures that the net effect of any possible propagation between spacelike separated points is zero, preserving causality​​.

In summary, causality in QFT is maintained by ensuring that commutators of field operators vanish for spacelike separations, while propagators provide the amplitude for particle propagation between points in spacetime, contributing to the understanding of how fields and particles interact within the framework of QFT.