A proof that all signal local theories have local interpretations.

[NateTG]

This post is motivated by some of the discussion on the "Is QM inherently non-local" thread. The goal is to demonstrate that there is a local hidden interpretation possible for any signal local theory. This interpretation will be Bell local, but not Bell realistic.
So, let's say we have a set up $S$ on which there is a possible list of
individual measurements, indexed by some completely ordered set I. $$M=\{m_i \forall i \in I\}$$ and, that for each measurement $m_i$, there is a range of possible results $R_i$.
And, there is a set of possible experiments $$E \subset P(M)$$ which consist of subsets of the list of possible measurments with the additional property that for any $e \in E$, $P(e) \subset E$.
Then, for each experiment $e \in E$, there is a result space $R_e$ which is the cartesian product of the ranges of the individual measurements in $e=\{m_{i_1},m_{i_2},m_{i_3}...\}$
And, a theory $T$ is a set of probability measures for the result spaces of each of the experiments.
Now, since the experiments are all elements of $P(M)$ there is a natural partial order $\prec$ based on set inclusion, so if we have two experiments $e_1$ and $e_2$, and the set of measurements in $e_1$ is a subset of the measurements in $e_2$ then $e_1 \prec e_2$ and, I will refer to this as $e_2$ is a stricter experiment than $e_1$.
Then, a theory is 'signal local' if for any $e_1 \prec e_2$, the probability probability measure that the theory gives for $R_{e_1}$ is the same as the natural probability subspace generated by restricting $R_{e_2}$ i.e. that for any sub set $s \subset R_{e_1}$ the subset $s \times \prod_{m \in e_2, m \notin e_1} R_m$ has the same measure in $R_{e_2}$
And, a theory has a local hidden interpretation if there is some probability measur
e on $R_M$ (the cartesian product of the ranges of all individiually possible measurements) that gives the same probability measure as the theory if restricted to $R_e$ for any experiment in $E$.
Now let's say we have a signal local theory $L$, then for every experiment $e \in E$ there is a probability measure on $R_e$. Now, for every measurable subset $s \subset R_e$ we assign the same value to the subset $s \times \prod_{m \notin e} R_m$ in our prospective measure on [itex]R_M[/tex]. In addition, specify that the measure of the empty set is zero, and the measure of the entire set is 1.
But, because of the properties that were ascribed to signal locality above it's easy to show that this prospective measure is indeed well-defined, and a probability measure.
Now, because this is physics, it's important to discuss whether the mathematical and physical properties match up:
The notion that was described as signal locality is, technically, stronger than signal locality since it's concievable that multiple measurements could occur local to each other, but that concern can be addressed by narowing down the notion of measurement.
The notion that was described as a local hidden interpretation is quite easy - let $\lambda$ be an element of $R_M$ then, since $\lambda$ describes the results of all possible measurements, clearly [\itex]\lambda[itex] can be used as the local hidden state for any particle if it is assigned with the appropriate probability distribution.

 

[Hurkyl]

I'm suspicious of your definition of "signal local" because you appear to make no reference whatsoever to any sort of geometry.

 

[NateTG]

I'm suspicious of your definition of "signal local" because you appear to make no reference whatsoever to any sort of geometry.

The whole thing does need cleaning up -- I'll try to put together a better (and more legible) version later tonight.

Thanks for taking the time to read it though.

 

[NateTG]

Second Draft

Let's say we have a (repeatable) setup $S$ and a set of individually possible measurements $M$.
Then let $E$ be the set of all co-measureable subsets of $M$. So $e \in E$ means that $e \subset M$ and all the measurements in $e$ can be performed on a single iteration of $S$
Let $R_m$ be the set of possible results for a particular measurement $m \in M$, and for any $\mu \subset \M$ let $R_\mu=\prod_{m \in \mu} R_m$ - where $\prod$ is the cartesian product.
Then a theory $T$ is a function that assigns a probability distribution $T(e)$ on $R_e$ to each element of $e\in E$.
Now, let $V$ be the set of all possible measurement histories. Then a signal local theory is a theory with the property that the result probabilities for any particular measurement are determined by the measurement history preceeding that measurement.
For each $v \in V$ let $M_v$ be the set of all measurements that can occur with the particular measurement history $v$, and let $E_v$ be the set of all subsets of $M_v$ that can co-occur, with all having measurement history $v$. Clearly this implies that $\forall e\in vE_v$, $\forall m_i,m_j \in e$ implies that $m_i$ and $m_j$ are not in the future or past of each other.
And, a local interpretation $L_T$ of a theory is a function that assigns a probability distribution on $R_{M_v}$ for each $v \in V$ so that the probability distributions for each element in $E$ can be generated from the probability distributions in the local interpretation.
Now, for a singal local theory, measurements that co-occur with the same measurement history do not affect the probability distribution of each other's results. As a consequence, if we have $e_1,e_2 \in E_v$ and $e_1 \subset e_2$ then, if the theory assigns a measure to some set $s \subset R_{e_1}$ in $R_{e_1}$, then the theory assigns the same probability to the set $s \times \prod_{m \in e_2, m \notin e_1}R_m \subset R_{e_2}$ in $R_{e_2}$
Now, for each measurement history $v$ we can construct $L_T(v)$ by assigning the measure $0$ to the empty set, and, for any set $s$ that has measure in some $R_e$ for some $e \in E_v$ we assign the same measure to $s \cross \prod_{m \in M_v, m \notin e} m$ in $R_{M_v}$.
Now, it is necessary that this so called measure on $R_{M_v}$ is indeed a well-defined measure, but it is quite easy to see that, since the individual measures it is constructed from are $\sigma$-algebras, this measure must also be a $\sigma$-algrebra, and independance propery provided by signal locality shows that the measure is well-defined. Finally, it is also clear that the values that are assigned are in $[0,1]$ since the measures on the $R_e$ are probability measures.
Moreover it is quite clear that this probability distribution on $R_{M_v}$ will generate the appropriate probability distributions for $R_e, e \in E_v$.
Therefore, these generated probability distributions on $R_{M_v}$ constitute a local interpretation.