I Mathematical formulation of local non-realism

Tags:
1. Mar 1, 2017

greypilgrim

Hi.

Bell formulated local realism as follows: The probability of a coincidence between separated measurements of particles with correlated (e.g. identical or opposite) orientation properties can be written as
$$P(a,b)=\int{d\lambda\cdot \rho(\lambda)\cdot p_A(a,\lambda)\cdot p_B(b,\lambda)}\enspace .$$

To get a better understanding of the terms "local" and "realistic", I'm trying to adapt this formula. So I'd say a theory that realistic, but not necessarily local, would satisfy
$$P(a,b)=\int{d\lambda\cdot \rho(\lambda)\cdot p_{AB}(a,b,\lambda)}\enspace ,$$
i.e. $p_{AB}(a,b,\lambda)$ is not necessarily a product distribution. As far as I can see quantum expectation values satisfy this probability distribution.

How would this formula look like for a nonrealistic (or not necessarily realistic), but local theory? Or is local realism not something that can be split up into locality and realism?

2. Mar 1, 2017

Demystifier

If $a$ and $b$ are spatially separated, then, according to the local non-realistic interpretation, there is no such thing as $P(a,b)$. Namely, there is no single observer who can measure $P(a,b)$, and things which nobody measures don't exist according to non-realistic interpretations.

If $a$ and $b$ are not spatially separated and a single observer measures both $a$ and $b$, then, according to the same interpretation,
$$P(a,b)=p_{AB}(a,b)$$
which is almost a tautology.

Last edited: Mar 1, 2017
3. Mar 1, 2017

greypilgrim

So in this interpretation it's not allowed that both observers make individual, spatially separated measurements and then construct $P(a,b)$ by comparing their results locally at a later time?

Do both observers need to assume the other one stays in a superposition until they compare their results over a classical channel?

4. Mar 2, 2017

Demystifier

It's allowed, but then the observables that are really compared are no longer spatially separated. According to non-realistic interpretations, there is no correlation until one observes the correlation.

In non-realistic interpretations (I am not a proponent of such interpretations, I just explain what such interpretations are), you don't assume anything about things which you don't observe.

5. Mar 2, 2017

greypilgrim

But in order to agree with experimentally verifiable QM predictions, observer $A$ needs a way to compute the correlations $P(a,b)$ that $A$ and $B$ will find when they later compare their measurements locally. So will he describe everything on $B$'s side as a unitary time evolution (i.e. with local Hamiltonians) and only use projective measurements when they meet (or talk over a classical channel)?

If I was a hardcore non-realist, would I need to assume I'm the only one in the whole universe capable of making QM measurements and everything else evolutes unitarily?

6. Mar 2, 2017

Demystifier

Yes, exactly.

That would be a kind of solipsism, and yes, I also think that hardcore non-realism leads to solipsism. See also
http://lanl.arxiv.org/abs/1112.2034