Probably the last of these.

In https://thephilosophyforum.com/discussion/9391/determinism-reversibility-decoherence-and-transaction I looked at transactional quantum mechanics, the idea that, in addition to the usual retarded mechanics we're more familiar with, advanced (moving backwards in time) waves might both limit the plurality of final states of a quantum system and be the future cause of particular events.

In https://thephilosophyforum.com/discussion/9506/cosmology-and-determinism I looked at universe geometries as a way of applying the first to curved spacetime, offering a fairly arbitrary example for illustration.

There was an interesting paper last year on superluminal frames of reference. A point of contact between advanced waves a superluminal frames... Advanced solutions to wave equations are waves in which the cause lies in the future of the effect. I gave the example before of photon transmission: an atom A spontaneously emits a photon which is probabilistically collapsed and deterministically absorbed by atom B. Time-reversed, this looks identical: emission events become absorption events and vice versa. The presence of both retarded and advanced wavefunctions in the transaction resolves the non-determinism of spontaneous emission and wavefunction collapse.

Similarly, superluminal frames of reference can act to reverse the causal order of events.



The central rule of relativity is that both frames must be describing the same physical system acting in the same way, merely seen from different points of view. An example I gave recently was of the magnetic attraction or repulsion of a moving charge to a current-carrying wire. There will exist a reference frame in which that charge is at rest and therefore cannot be subject to a magnetic force. However, in this frame there exists an uneven length contraction between the ions of the wire and its current-carrying electrons: its positively charged components and its negatively charged ones, yielding a net electric field that will either attract of repel the now stationary external charge.



A trivial solution to the ambiguous causality between subluminal and superluminal frames is that both directions of causality hold in both frames. When we detect that atom A has transitioned to a lower-energy state and atom B has transitioned to a higher-energy one, we conventionally say A caused B, but accepting and understanding the implication of superluminal frames, B also caused A. What we're seeing is self-consistency between A and B from a biased view point.

To accept superluminal frames of reference is to abandon the strict localism of special relativity. In fact, rejecting superluminal frames is precisely how Einstein established localism. But tests of the Bell inequalities suggest that either a realist or localist interpretation of quantum mechanics must be rejected: in other words, if reality is objective, it cannot be local in the way Einstein hoped.

The paper linked to above makes an unsatisfactory first attempt to show how superluminal frames give rise to particles following multiple trajectories. The case given is of a particle emitted from point A, reflected at point M, and absorbed at point B. Time-reversed as previously discussed, this would be an emission from B followed by absorption at A. From a superluminal perspective, it can be a spontaneous emission from M to both A and B simultaneously, with the absorber depending on the reference frame. (If M is moving in one direction, it will be B, if the other, it will be A.)

What is unsatisfactory about this is that it doesn't account for emission. If A absorbs, B must emit (it must describe the same system as the subliminal frame, in which energy is transmitted between A and B, and M has no permanent change bar some neglected recoil) and vice versa. However if we're a bit more rigorous about it, we can see another feature of quantum mechanics popping up.

Let's say M emits toward A and B and ultimately B absorbs. M must be unchanged overall (bar recoil), so where did M get the energy to transmit to B? Well A must emit if B absorbs, so the logical conclusion is that M "borrowed" the energy from future A to pay B. That is, the path from M to A does not only resolve to a possible absorption site (depending on reference frame) but is also a route to do the very opposite: to borrow energy (send net negative energy) to pay B. This is sounding a lot like the energy-time component of the uncertainty principle, and the presence of an energy-time uncertainty principle in special relativity automatically suggests a momentum-position one.

From this superluminal frame, we could also describe AM as an advanced wave: energy and momentum is transmitted from A to M backwards in time (a photon is it's own antiparticle, so will appear as a photon moving forward in time from that frame) in order for M to pay B. This also means that normal-looking causal chains in superluminal frames could look spontaneous in subluminal ones. This is quite useful as QM interactions that exploit the energy-time uncertainty principle are precisely the irreversible interactions that, in my first thread on this, did not seem to me to be amenable to a deterministic description. The acceptance of superluminal frames, as well as being consistent with that OP, looks like precisely the ingredient needed to widen determinism to all interactions in flat spacetime.

Take, for instance, spontaneous neutron decay, in which a neutron collapses into a proton, an electron, and a positron neutrino. I recall in my undergraduate years being corrected on the point that neutron decay is NOT the reverse of electron capture, in which an electron and a proton combine to produce a neutron and an electron neutrino. In both cases there is an energy barrier to overcome: in the latter, this is partly provided by the incoming electron which is the independent cause of the event; in the former, this is enabled probabilistically by the uncertainty principle alone: all of the energy to overcome the barrier must be "borrowed".



However, if we accept superluminal frames, there should exist a frame in which the positron is an input -- an advanced electron -- not an output of the decay process, i.e. the positron is the cause that lends the energy needed for decay. Self-consistency is again key: the same particle is a cause and effect of the decay, depending on the frame of reference. I expect that we could treat all fundamental irreversible processes in this way.

We expect, as I say, some superluminal frame in which neutron decay appears to be a regular chain of cause and effect. Likewise then, in some frame, electron capture will appear to be spontaneous. As a final consideration, we need to understand what we mean by "accepting" superluminal frames. Do we mean objects travelling faster than light? Tachyons? No, not necessarily. What we mean is that whatever physics we see of light and subluminal massive matter, a description must be possible in superluminal frames that is possible in subluminal ones.