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 R_M[/tex]. In addition, specify that the measure of the empty set is zero, and the measure of the entire set is 1.<br /> But, because of the properties that were ascribed to signal locality above it&#039;s easy to show that this prospective measure is indeed well-defined, and a probability measure.<br /> Now, because this is physics, it&#039;s important to discuss whether the mathematical and physical properties match up:<br /> The notion that was described as signal locality is, technically, stronger than signal locality since it&#039;s concievable that multiple measurements could occur local to each other, but that concern can be addressed by narowing down the notion of measurement.<br /> 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]\lambdacan 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.