Can retrocausality resolve the paradoxes of quantum mechanics? The Copenhagen interpretation presumes something has no property until it is measured, but retrocausal interpretations explain that away by claiming that thing has a definite property before measurement, but the choice of which property is affected by the configuration of the measuring apparatus in its future. The philosopher Huw Price has written a lot about this. Nonlocal entanglements are also explained away because entangled pairs are connected by going to the past, then back up to the future. Or going to the future, then back to the past without travelling faster than light at any time.
Not "resolve". Give an interesting account, perhaps. This kind of interpretation suffers from the usual problem for interpretations, that it doesn't add enough that is useful enough for many Physicists to have wanted to use it, at least in the relatively elaborated form proposed by Huw Price. On the other hand, understanding this kind of interpretation and its position relative to other interpretations of quantum theory is essential to anyone working on the Philosophy or Foundations of Physics. It's also notable that backward trajectories do appear in some readings of Feynman diagrams, which could be said to make this kind of interpretation rather mainstream, however I would say that Physicists' use of this kind of language is not quite enough to justify calling it an interpretation.
Explaining "nonlocality" by introducing "backward propagation" could well be said to be no explanation at all. Neither concept is exactly classically intuitive, for example. Nonetheless, we can hope that understanding that a tradeoff can be made between nonlocality and backward propagation is worthwhile.
initial state: a pair of electron and a positron moving toward each other
A pair of electron and a positron have certain probability of annihilating to each other in certain amount of time, the resulting photon amplitudes are described by some Feynman diagram (the sum of all Feynman diagram with the same In and Out states of course)
In this case we could interpret the electron as bouncing from a backward propagating photon, and becoming a backward propagating electron (a positron), but such interpretation is pointless because of conditional probabilities and entanglement
Let me explain that last phrase above: the initial state assumes that i've already make measurements which couple me (the observer) to a definite state at time $t_0$, so, if i accept that any retrocausal interpretation needs to produce consistent histories (i.e: if i measure A at $t_0$, i always will measure A at $t_0$, regardless of what happens before or after) then all probabilities can only define what happens at other times that are not $t_0$, and such probabilities are given by propagators and a well known state at $t_0$.
so, i hope to have made clear from the above paragraph that before we can speak consistently of any "retrocausality", we need an alternative way to represent dynamics evolutions to the usual propagator which assumes 100% knowledge of state at $t_0$ and produces amplitude probabilities of states at $t_1$. So we need some formalism that might take a partial state at $t_0$, compute amplitudes at another partial state at $t_1$, then from making measurements at both times, and measure correlations (which could be interpreted in principle as entanglement correlations nonetheless) one could infer that a definite result at the measurement in $t_1$ changed part of the state at $t_0$ that was unknown to us, but it obviously had to be measured by a separate apparatus in order for us to measure such a correlation
It should not be a surprise that even if we had such a formalism, it would only differ from the Propagator formalism in the interpretation. I could still assume that the state at $t_0$ was perfectly known, only by different apparatuses, and the propagator produced the observed amplitudes at $t_1$
the whole "retrocausality" concept becomes non-trivial only if there are fate conditions, that is, boundary conditions in the future that are unavoidable (for example, imagine a universe with a entirely time simmetrical big-crunch, where the universe seems to have an arrow of time in the first half, but when approaching the half life of the universe, matter slowly starts becoming antimatter by seemingly random fluctuations). Our current universe does not manifest any such flagrant violations of the second law, so the concept of retrocausality is not useful to model physics.
The ordinary Retrocausal interpretation of quantum mechanics, was Cramers transactional interpretation, where an event is formed as a single point in space time, where an incoming wavefunction from the past, and an incoming (advanced wave) wavefunction from the future meet. The theory has a fixed past and future points, which can then predict observation in the time between. It remains mathematical correct, but of course is hardly useful for predictions.
Interesting to me, is between an transactional interpretation with one fated future, and a many worlds interpretation with all possible futures, are there any conditions which would allow us to removed possible futures, or truncate the calculation of probability amplitudes at low levels of likelihood.
Retrocausal interpretations may explain Bell inequality violations for entangled particles and delayed-choice experiments, but can retrocausality explain how quantum computers can factorize large integers or compute discrete logarithms using Shor's algorithm? Even if retrocausal effects exist, they can't constitute a complete account of quantum mechanics.