# I Mathematical formulation of local non-realism

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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