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.

Schrödinger's plates

The Schrödinger's plates image has been around since at least 2016, as here, but it's lately been having still another renaissance. Philip Ball, on Twitter, suggests that it's "Possibly the worst Schrödinger's cat analogy ever," but I think it has some merit as a way into seeing how classical physics can ask the same kinds of noncommutative questions as quantum physics asks. That's enough to make an unconstrained classical physics as good for constructing models as quantum physics.

Here it is:

Classical physics might perhaps ask just one question, such as "Do the plates break when you open the door?", but it can also ask a multitude of questions such as "Can the plates be saved?", or, with much more detail, "If with a few friends we make sure the door doesn't vibrate when we remove the adjacent pane of glass, how likely is it that we can save the plates?"

There is a continuum of such yes/no questions that can all be asked by a classical physicist, any of which can be turned into a question about likelihood. We can ask whether we can save the plates if we smash the adjacent plane of glass with a stone of a given size or shape, or we can ask how likely it is that we can save the plates if we use a glass cutter to cut holes of particular sizes and positions in one or more of the panes of glass.

One tricky part is that it looks like we can't run the experiment more than once, because it looks difficult to get the plates into that position many times to find out what approach works best. At the heart of statistical physics, however, is the act of creating very elaborate initial conditions, making as sure as we can that each time is as close to the same as we can engineer them to be, and seeing what happens when we ask different questions. So imagine that we set up the conditions in the Schrödinger's plates image over and over again as multiple trials and record the results of many different questions, each many, many times. In physics experiments there may be as many as millions of trials per second, each engineered to be as close as possible to every other.

Crucially, for each trial we can ask many yes/no questions, but very often we can ask only one and not another for any one trial: we can ask "Do the plates break if we open the door?" but we cannot for the same trial also ask "Do the plates break if we smash the adjacent pane of glass?" On the other hand, we can ask a multitude of questions jointly of the same trial: "Does more than one plate break?" is compatible with "Can we find any shards further than 3 meters away?" We can imagine a continuum of both incompatible and compatible questions even though we only actually ask a small number of them.

With many trials, we can ask the same question over and over and discover that for some fraction the answer is yes: we can call that fraction the observed likelihood. When we ask only slightly different questions of different sets of trials, we expect that the observed likelihood will also only change slightly, whether the different questions are compatible or incompatible. We can, incompatibly, use a glass cutter to cut holes of different particular sizes and positions in one of the panes of glass, which we would not expect to give different observed likelihoods if the two sets of holes differed only by a millimeter in size and hardly at all in position.

When we ask compatible questions, we can ask what the joint observed likelihood is for the answers to different questions: we can ask for the same trial both "Does more than one plate break?" and "Can we find any shards further than 3 meters away?" because they can be jointly observed. Crucially, there is no necessity for the observed likelihoods for incompatible questions to allow a joint observed likelihood for the answers to questions that are not and cannot be asked jointly: if we ask "Do the plates break if we open the door?" we cannot jointly ask "Do the plates break if we smash the adjacent pane of glass?" In incompatible cases, we can still assign numbers, which it is best not to call jointly observed likelihoods because (1) they are not jointly observed and (2) they may for the sake of consistency and continuity have to be not fractions between 0 and 1, but this mathematically natural but apparently strange behavior as much happens for classical mechanics as for quantum mechanics.

This is no more than to picture what happens in physics experiments, or indeed to reimagine Schrödinger's cat. One can invent many examples of this kind of thought experiment, and there have been academic papers on contextuality for decades (see "Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity," for example). More such examples will not compel anybody to think about the relationship between classical and quantum mechanics any differently. The mathematics of An algebraic approach to Koopman classical mechanics (or, in Annals of Physics 2020), however, and the Hilbert space mathematics it allows us to construct, supports an idea that classical mechanics can be as capable as quantum mechanics for modeling any physical situation. Classical models are significantly different, but they are as capable. Understanding the relationship between classical mechanics and quantum mechanics frees us from the distraction of thinking they are so very different.

EDIT: I use Facebook rather extensively as a whiteboard. It's quite public, so the embarrassment of making a mistake can be great, but I tell myself that it's worth it because one remembers mistakes that one is embarrassed about. One never knows what flights of fancy or of folly some comment will lead to. In any case, a comment on a post on the Facebook Group Quantum Mechanics & Theoretical Physics led me to look more carefully for how the Schrödinger's cat thought experiment is like the Schrödinger's plates thought experiment:

"There is one thing that is the same, if we can open the box at a time of our own choosing. In that case, the probability that the cat is alive begins at 1, then the probability at some rate decreases. If we more want the cat to live, we should open the box more quickly. The measurement we make determines the probability of different results.

If we open the box at some early time, we cannot also open the box at some later time, without starting a new trial. Those are incompatible measurements. On the other hand, we can on the same trial measure whether the cat died and whether the cat died of panic and a heart attack or of failure of the lungs, ..., which are compatible measurements that can be jointly measured."