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Thanks for your question – it’s good to have a chance to try to explain in informal language.
We’re trying to see what QM per se can tell us about causality. For example, where in the algebra is retrocausality ruled out (if it is)? We know the joke about a physicist being asked to come up with a theory about horses in a paddock, the natural starting point is to assume the horses are spheres and the paddock is a frictionless plane. So we begin by specifying a simple scenario – the “neutral causal background” (NCB).
First of all in classical terms. The rules are Alice flips a fair coin and sends the result (H or T) to Bob. Alice can’t send a message this way. She could if she used an unfair coin – we rule out that sort of strategy by requiring the initial state to be maximally mixed. Alice could message by not always sending on the result (eg retaining some or all H results) – we deal with that strategy by ruling out sub-ensemble selection – all runs must be used. There is a correlation between Alice’s and Bob’s results – when Alice gets H, so does Bob. For the NCB we stop this by introducing C which randomises Alice’s result (eg toss a different coin and send it to Bob or change Alice’s result in 50% of cases, etc). Now Alice and Bob just get H or T half the time and there is no correlation between them. This is the NCB.
In the QM case Alice and Bob don’t have to measure in the same basis but otherwise the scenario is the same.
Now let C be any operation. Can C change the probabilities of Alice’s and Bob’s measurements? If so, are there any restrictions imposed on the effects of C by QM?
If there is a C which can change Bob’s probabilities, we say C is a cause.
If there is a C which can change Alice’s probabilities, we say C is a retrocause.
If C changes the joint probabilities of Alice and Bob without changing the probabilities of Alice or Bob, we say it is a correlator.
If C changes the joint probabilities of Alice and Bob and changes the probabilities of Alice or Bob, we would say it is a correlator and cause (Bob’s) or correlator and retrocause (Alice’s).
Note in the above, I have used “message” here and then morphed to “cause”. This bears further discussion elsewhere. In the NCB scenario, the two ideas are the same?/similar?. Given the restriction to a maximally mixed preparation state and no sub-ensemble selection, Bob can never “feel” a cause from Alice (without C). Therefore the link between them is better thought of as a correlation. [Obviously, in the back of our minds is the EPRB experiment.] By the way, Matt Farr calls sub-ensemble selection “artificial” causation which I think is an excellent term.
It turns out that C as a cause or a correlator is easy, even unitary operations (in a larger Hilbert space) do the trick.
The surprising thing is that C as a retrocause does not appear to be logically impossible according to QM. It’s certainly technically impossible. It seems that starting from the postulates of Barnett et al and Pegg et al, the algebra of QM is flexible enough to accommodate retrocausation proper (as distinct from correlation where the “retro” issue doesn’t arise – there is no basis for saying (pure) correlation acts ‘forwards” rather then “backwards”).