# Does a nonlocal deterministic hidden variable theory imply retrocausality?

I've had this idea for a while, and recently I stumbled upon a short paper from N. Gisin that formulated this idea, but I could not find a meaningful discussion on the problem. The paper that I found was https://arxiv.org/pdf/1002.1390.pdf, and Gisin argues that a simple argument can be formulated against nonlocal hidden variable theories (such as Bohmian mechanics).

To summarize the paper, if some nonlocal deterministic hidden variable theory exists - such as Bohmian mechanics, then for a typical EPR-Bohm experiment procedure where spacelike separated Alice and Bob measure entangled particles. Since they are spacelike separated, there will be a reference frame where Alice measures her particle first, and thus

$$p(\alpha | a, b, \lambda) = p(\alpha | a, \lambda)$$

Where $$\alpha$$ is the measurement result of Alice and $$a$$, $$b$$ are the experiment settings of Alice and Bob. However, there must also be a reference frame where Bob measures his particle first - thus giving $$p(\beta | a, b, \lambda) = p(\beta | b, \lambda)$$. Since these two reference frames are both equally plausible, Gisin argues that any covariant nonlocal hidden variable theory defines a local model in the sense of Bell - which is ruled out by Bell's theorem.

Some ways I think this problem can be avoided are 1) drop determinism and think of a stochastic model (since Bohmian mechanics do exist, dropping determinism altogether seems not to be the solution) 2) drop relativity and assume a universal reference frame (which is obviously not favorable) 3) drop free will (as in superdeterminism) or 4) assume retrocausality is possible (which seems to me - if superdeterminism is not assumed, the most favorable choice).

I'm still not sure if non-retrocausality implies $$p(\beta | a, b, \lambda) = p(\beta | b, \lambda)$$ - this condition is equivalent to parameter independence (or locality, in deterministic theories). Then does parameter independence hold when there is a clear time order? I.e., if both the settings of Alice affect the outcome of Bob and vise versa, then since either Alice or Bob must have performed the experiment one before the other, then some sort of retrocausality must have happened. This seems to be the problem here.

I also believe this issue can be extended - in special relativity if some particle travels faster than the speed of light, it can experience backward travel in time. Similarly, can nonlocality in quantum mechanics imply some sort of retrocausality?

• Bohmian mechanics. as formulated originally, violates relativity in a trivial fashion because the non-relativistic SE that it is based on violates relativity. Which relativistic version of Bohmian mechanics are we talking about that can reproduce the standard model? And what, exactly, do we need non-locality for? The universe seems to be perfectly local based on every actual theory that reproduces real measurements. May 19 at 0:40

Great questions! I suspect Gisin's view may be a lot more nuanced now after attending this conference a year ago.

Your Solution #1 doesn't work; indeterminism and stochasticity don't help with a local resolution to the dilemma of Bell's Theorem. The secret to Bohmian mechanics is the nonlocal wavefunction, linking spacelike separated events, combined with your solution #2.

I'm not sure Solution #2 is a solution at all; certainly you lose Lorentz covariance. If there were a preferred reference frame, yes, you could have physical mediators between Alice and Bob, but they would travel faster than light (FTL) in every reference frame, which would still be non-local by Bell's lights. (The mediators would have not only have to be FTL, but unmeasureable, unblockable, incapable of carrying a signal, and in some reference frames would be FTL-retrocausal.)

Skipping over #3, which I've found to lead to generally fruitless discussions, that brings us to your main question, option #4.

First of all, all retrocausal models explicitly violate the parameter independence equations; they violate $$P(\lambda|a,b)=P(\lambda)$$. The idea of a (non-FTL) retrocausal model is that something hidden in the past ($$\lambda$$) depends on the future settings (a and b) in the future light cone of ($$\lambda$$). So this equation wouldn't hold true in a retrocausal model. If it did hold true, it wouldn't be retrocausal.

The answer to your original question, in the title, is "no". If you had meant to write "does a local hidden variable model imply retrocausality", and you mean "local" in the sense of having Lorentz-covariant physical mediators to explain the correlations in entanglement experiments, then yes, I think it's fair to say that retrocausality is the only option. Support for this claim can be found in this Rev. Mod. Phys. paper.

• Thanks for the answer - so does this mean that any Lorentz covariant hidden variable theory must be local - i.e., any nonlocal hidden variable theory like Bohmian mechanics cannot satisfy Lorentz covariance - thus if we were to try and recover Lorentz covariance, due to Bell's theorem, parameter independence must be violated and the theory must be retrocausal? May 21 at 14:00
• I think that's fair to say, depending of course on what precisely and exactly you mean by the word "local". Some people might argue that there are "flash ontologies" which are Lorentz-coviarant alternatives to retrocausal models, but if you define locality carefully (see the Rev Mod Phys paper I quoted), any flash ontology which could handle entanglement would need some non-locality to get around Bell's arguments. May 24 at 19:37