Retrocausality

A retrocausal model allows variables associated with an earlier region to depend on later boundary conditions or measurement settings. This is an ontological or dynamical claim. It is stronger than ordinary retrodiction from later evidence, and different from the ability to send a controllable message to the past. Here the focus is the causal structure; Retrocausal Approaches compares interpretive programs.

Retrodiction does not establish a reverse influence

After observing a later result $b$, Bayes’ rule changes the distribution assigned to an earlier event $a$:

$$p(a\mid b)=\frac{p(b\mid a)p(a)}{p(b)}.$$

This happens in classical statistics without any future event physically changing the past. Quantum postselection likewise gives conditional distributions that differ from unconditioned ones. A retrocausal theory must add more than the fact that those distributions differ.

One possible additional claim is that a microscopic variable $\lambda$ is fixed by an all-at-once boundary-value problem involving both past preparation and future settings. Such a construction need not describe a chronological algorithm that first chooses $\lambda$ and later overwrites it.

Which Bell assumption changes?

A Bell-local model with measurement independence has

$$p(a,b\mid x,y)= \int d\lambda\,\rho(\lambda)\, p(a\mid x,\lambda)p(b\mid y,\lambda).$$

A future-setting-dependent model instead uses $\rho(\lambda\mid x,y)$. This evades the measurement-independence premise of the standard Bell derivation, even if the outcome response factors conditional on $\lambda$. Whether that dependence represents a past common cause restricting settings or a dependence on future boundary conditions is a question about the causal model, not the probability notation alone.

Wharton and Argaman review locally mediated proposals that relax a forward-only causal assumption. Their existence as toy models does not automatically supply a complete interacting relativistic theory.

Setting dependence alone explains too much

For any target distribution $p_{\rm tar}(a,b\mid x,y)$, define $\lambda=(a_*,b_*)$ with

$$\rho(\lambda\mid x,y)=p_{\rm tar}(a_*,b_*\mid x,y),$$

and deterministic outputs $a=a_*$, $b=b_*$. This formally reproduces the target with factorized response functions. It reproduces signaling distributions as readily as quantum ones.

The construction is therefore a demonstration of lost constraint, not a satisfactory physical explanation. A substantive model needs a dynamical or boundary principle determining $\rho$, consistency across experiments, an account of interventions, and a reason for observed no-signaling. Reproducing one correlation table is insufficient.

No-signaling and boundary consistency

A viable model must ensure that averaging inaccessible variables yields the observed setting-independent marginals where required. Future dependence at a hidden level does not by itself imply a communication protocol, but it does not automatically prevent one either. The restriction needs to be shown.

Nor do delayed-choice experiments force this interpretation: ordinary quantum dynamics and conditioning already reproduce their statistics. Retrocausal models remain candidate explanatory structures whose merits depend on completeness, constraints, and empirical adequacy rather than the appeal of reversing an arrow in a diagram.

Exercise

Use the target-distribution construction for $a=y$ with certainty and an unbiased $b$. Does the response factorization prevent signaling from Bob to Alice?

Solution

No. The setting-dependent distribution places all weight on $\lambda$ with $a_*=y$, so Alice reads Bob’s setting directly. Factorization conditioned on $\lambda$ does not restore no-signaling after measurement independence has been abandoned. Additional structural constraints are essential.

References

• Wharton, K. B., and Argaman, N., “Colloquium: Bell’s theorem and locally mediated reformulations of quantum mechanics,” Reviews of Modern Physics 92, 021002 (2020), arXiv:1906.04313 — future-setting-dependent locally mediated models and their scope.