Lorentz invariant propagators in 3+1-dimensions

I'm reading an interesting quantum field theory textbook, by Thanu Padmanabhan, "Quantum Field Theory: The Why, What and How", Springer 2016, which goes through hoops to interpret Lorentz invariant propagators in terms of particles. [My sense, though from only a brief acquaintance, is that it's a thoughtful and good book.]

As always, my view is that we should pause for a decade or two before we mention particles again. When they emerge, carefully, it will be in terms of global measurements that have an integer-valued sample space (or an integer-valued spectrum, if we're talking about operators that correspond to those measurements.)

So for now, talking only about a field theory, one way to separate Lorentz invariant propagators for the Klein-Gordon equation (and solutions of the Klein-Gordon equation) into two kinds is as:

• The retarded and advanced propagators (and the Pauli-Jordan, "commutator", solution of the Klein-Gordon equation), which are zero outside the forward and backward light cones (and zero outside both the forward and backward light cones) respectively: in other words, these are non-zero only at time-like separation.
  
• The Feynman propagator (and the positive and negative frequency solutions of the Klein-Gordon equation), which are all non-zero at both space-like and time-like separation.

If we follow everyone else's thinking about particles, the second kind of propagator and solutions (which are the ones that matter in quantum field theory and for random fields) look as if there are faster than light effects, because they are non-zero at space-like separation, but if we think in terms of fields solutions that are non-zero at space-like separation can seem fairly natural, if we take it that when we change the universe or a model of the universe, we ease in a change of the past that is consistent with a new present and future — we do not hammer in a new present and future with no attention given to the past.

From a PDF that I constructed quite a few years ago, with the retarded and advanced propagators and the commutator solution down the right and the Feynman propagator and the positive and negative frequency solutions down the left:

[I hope I've got all the many signs right here, but long experience shows that I don't always, despite knowing that I get them wrong and consequently spending hours where I imagine geniuses spend minutes.]

The "solutions" of the Klein-Gordon equation are just that, whereas the propagators satisfy the inhomogeneous Klein-Gordon equation, with the Dirac delta function on the right-hand side. The J, K, and Y are Bessel functions, together with a Dirac delta function component. [That Dirac delta function component isn't there in 1+1-dimensions; in 1+1-dimensions there's also a lower degree Bessel function that only diverges logarithmically as a function of small space-like or time-like separations (instead of as the inverse square in 1+3-dimensions), which is why perturbation theory in quantum field theory is so much more straightforward in 1+1-dimensions.]

Which solution or propagator one should use in a model (or which linear combination of solutions and propagators) depends on the boundary conditions. A more intuitive idea, however, is that

• if we unilaterally hammer in a delta function change to the present at a certain point, then it's not unreasonable for only the future to be affected, only within the forward light cone, which we can model by using the retarded propagator;
• if we act more circumspectly, however, the present doesn't just appear from nothing, it has to be prepared: we have to change the past so that it causes a new solution of the Klein-Gordon equation to be as we see it. For such a change, we cannot just change only the backward and forward light-cones: any change to the backward light-cone will propagate to space-like separation.
  

For both quantum fields and random fields I have found it useful to think of our actions as applying modulations to the vacuum state (which we take to be a noisy broad-band carrier signal), which then resonate in a Lorentz invariant way with whatever measurements we perform in other parts of the space-time. As an effective model for a probabilistic resonance, quantum field theory uses a positive semi-definite bilinear form (which gives us the inner product of the Hilbert space) which pushes us firmly towards the positive and negative frequency solutions. Quantum field theory uses the positive frequency solution because for quantum theory positive frequency is associated with positive energy. In other words, we can say that, broadly speaking, quantum field theory acts circumspectly rather than unilaterally: quantum field theory blends the changes we make into the universe.

Because of my near obsession with the relationship between random fields and quantum fields, here I also have to mention that the difference from the point of view of solutions and propagators for the Klein-Gordon field is quite straightforward: for a quantum field we use just the positive frequency solution, whereas for a random field we use a sum of the positive frequency solution and the negative frequency solution. The imaginary component of the random field commutator at time-like separation cancels out, so that the commutator is zero everywhere, whereas the quantum field commutator is non-zero at time-like separation. This difference of course has consequences for the mathematics, which I usually summarize by saying that quantum field theory is an analytic form of the random field theory, because the quantum field theoretical restriction to only positive frequencies results in many extremely useful analytic properties.