User:Tnorsen/Sandbox/Bell's concept of local causality

Bell's concept of local causality refers to a criterion formulated by the physicist John Stewart Bell that constitutes a necessary condition for a given physical theory to be locally causal. Informally, Bell says that a theory is locally causal when, according to that theory, the effects of a given event are inside its future light cone and its causes are inside its past light cone (so that the theory does not allow super-luminal causation nor backwards in time causation). Bell's criterion is formulated in terms of conditional probabilities involving the beables of the theory. Bell formulated his concept of local causality so that he could present a rigorous proof of his celebrated theorem, according to which no locally causal theory can account for the correlations predicted by quantum theory for certain experimental setups.

Correlations between spacelike separated events
Consider spacelike separated regions 1, 2 of spacetime as illustrated in the picture below. Clearly, the mere existence of correlations between events in 1 and in 2 is no indication of a violation of local causality, since such correlations can in general be explained in terms of some common cause that is in the intersection of the past light cones of 1 and of 2.



A very simple example of a (locally explicable) correlation between spacelike separated events appears in the story of the glove left at home: if I notice that I took with me the righthand glove then I know immediately that the glove left at home is lefthanded. Stating this in terms of local beables: if $$A$$ is a local beable associated to spacetime region 1 and $$B$$ is a local beable associated to spacetime region 2 then, in a locally causal theory, it is quite possible that the specification of the value of $$B$$ influences the probabilities for the value of $$A$$; in symbols:

$$P(A)\ne P(A|B)$$,

where $$P$$ stands for "probability". In the story of the glove left at home, the specification of the local beable $$B$$ (the fact that the glove that I took with me is righthanded) determines the value of the local beable $$A$$ (the fact that the glove left at home is lefthanded), i.e.:

$$P(A=\text{lefthanded}|B=\text{righthanded})=1$$

while, if B is not given (and assuming that both gloves are equally likely to have been forgotten) then:

$$P(A=\text{lefthanded})=1/2$$.

Formal definition of local causality
Now consider the situation depicted in the picture below:



Again, 1 and 2 denote spacelike separated regions of spacetime; region 3 completely "shields" region 1 from possible common causes of events in 1 and in 2 (more formally: if $$x$$ is an event in the intersection of the past light cones of 1 and 2 then any causal curve connecting $$x$$ with an event in region 1 must cross region 3). In the words of Bell (Speakable and Unspeakable in Quantum Mechanics, pg. 239):

"A theory will be said to be locally causal if the probabilities attached to values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a space-time region 3."

In symbols, if $$A$$ denotes a local beable associated to region 1, $$B$$ denotes a local beable associated to region 2 and if $$\lambda$$ is a full specification of local beables associated to region 3 then:

$$P(A|B,\,\lambda)=P(A|\lambda)$$.

In the story of the glove left at home, the beable $$\lambda$$ would already contain information that determines whether the glove left at home is righthanded or lefthanded; thus:

$$P(A=\text{lefthanded}|B=\text{righthanded},\,\lambda)=1$$,

but also:

$$P(A=\text{lefthanded}|\lambda)=1$$.

When analyzing the story of the glove left at home, it is important to have in mind that gloves are already lefthanded or righthanded when not looked at, i.e., the process of looking at a glove merely informs the observer about whether the glove is lefthanded or righthanded. Looking at a glove is an experiment that is very different from tossing a coin: the coin is neither "heads" nor "tails" before the tossing. In terms of local beables: the handedness of a glove is stored in a local beable associated to the region of spacetime in which the glove is; on the other hand, there is no local beable associated to the region of spacetime that contains a coin that says whether the coin is "heads" or "tails" (the result of the experiment of tossing a coin depends on very peculiar characteristics of the way that the coin is tossed). If the handedness of a glove were not stored in some local beable associated to the region of spacetime containing the glove (i.e., if, like coins, gloves where neither righthanded nor lefthanded when not looked at) then the variable $$\lambda$$ containing the full specification of local beables associated to region 3 wouldn't determine the value of $$A$$ and in this case we would have:

$$P(A=\text{lefthanded}|B=\text{righthanded},\,\lambda)=1$$,

but:

$$P(A=\text{lefthanded}|\lambda)<1$$.

This would constitute a violation of local causality. In the words of Bell (Speakable and Unspeakable in Quantum Mechanics, pg. 241):

"If I find that I have brought only one glove, and that it is right-handed, then I predict confidently that the one still at home will be seen to be left-handed. But suppose we had been told, on good authority, that gloves are neither right- nor left-handed when not looked at. Then that, by looking at one, we could predetermine the result of looking at the other, at some remote place, would be remarkable. Finding that this is so in practice, we would very soon invent the idea that gloves are already one thing or the other even when not looked at. And we would begin to doubt the authorities that had assured us otherwise."

Deterministic and Stochastic theories
Bell's concept of local causality can perhaps be better appreciated by considering its meaning separately for deterministic and stochastic theories. In a deterministic locally causal theory, the full specification $$\lambda$$ of all local beables associated to spacetime region 3 completely determines the value of any local beable $$A$$ associated to spacetime region 1, i.e., $$A$$ is expressed by the theory as a function $$A=f(\lambda)$$. In such case, we have $$P(A=f(\lambda)|\lambda)=1$$ and the specification of any local beable $$B$$ associated to spacetime region 2 is clearly redundant, i.e., $$P(A=f(\lambda)|B,\,\lambda)=1$$. If the theory is stochastic, then $$A$$ in general won't be expressible as a function of $$\lambda$$, i.e., the conditional probabilities $$P(A|\lambda)$$ can assume values other than 0 and 1. Nevertheless, in a locally causal theory, the specification of the beable $$B$$ should be redundant once $$\lambda$$ is given and this is formalized by the equality $$P(A|B,\,\lambda)=P(A|\lambda)$$. Another interesting exercise that helps understanding Bell's concept of local causality is the consideration of spacetime region 3' depicted in the figure below:



Like region 3, region 3' has the property that it intersects every causal curve that starts in region 1 and extends indefinitely into the past; but unlike region 3, region 3' does not "shield" region 1 from the common causes of 1 and 2, i.e., we have an event $$x$$ that is in the intersection of the past light cones of regions 1 and 2 and there is a causal curve connecting x with region 1 that does not cross region 3'. Let, as before, $$A$$ denote the value of a local beable associated to region 1, $$B$$ denote the value of a local beable associated to region 2 and now let $$\lambda'$$ denote a full specification of local beables associated to region 3'. In a deterministic locally causal theory the value of $$A$$ is completely determined by $$\lambda'$$ (i.e., $$A$$ is a function of $$\lambda'$$) and therefore the equality $$P(A|B,\,\lambda')=P(A|\lambda')$$ holds. On the other hand, in a stochastic locally causal theory, such equality may not hold. Namely, the event $$x$$ can influence the value of $$B$$, so that specification of the value of $$B$$ can give us information about $$x$$. The event $$x$$ can also influence the value of $$A$$ and, since the theory is stochastic, $$x$$ doesn't have to be determined by $$\lambda'$$ and therefore the value of $$B$$ can give us information about the value of $$A$$ that is not already contained in $$\lambda'$$. This illustrates that, in the definition of local causality, it is essential that region 3 completely shields region 1 from the common causes of 1 and 2.

Implications for Alice and Bob experiments
We now consider the implications of the hypothesis of local causality for Alice and Bob experiments. Alice performs an experiment in region 1; such experiment has only two possible outcomes and such outcome is stored in a local beable $$A$$ that can assume values 1 and -1 and is associated to spacetime region 1. The experimental apparatus used by Alice has a knob whose setting is stored in a local beable a associated to spacetime region 1. Similarly, Bob performs an experiment in region 2, the outcome of Bob's experiment is stored in a local beable $$B$$ (that assumes values 1, -1) associated to region 2 and Bob's experimental apparatus has a knob whose setting is stored in a local beable b associated to spacetime region 2. Consider the strip illustrated as region 3 in the figure below:



We denote by c the local beables associated to region 3 and that are relevant for determining the settings a, b of the knobs of the experimental apparatus. We denote by $$\lambda$$ a full specification of the other local beables associated to region 3.

By a standard rule for probabilities, we have:

$$P(A,\,B|a,\,b,\,c,\,\lambda)=P(A|a,\,b,\,c,\,\lambda,\,B)P(B|a,\,b,\,c,\,\lambda)$$.

Local causality then implies:

$$P(A|a,\,b,\,c,\,\lambda,\,B)=P(A|a,\,c,\,\lambda)$$

and

$$P(B|a,\,b,\,c,\,\lambda)=P(B|b,\,c,\,\lambda)$$,

so that:

$$P(A,\,B|a,\,b,\,c,\,\lambda)=P(A|a,\,c,\,\lambda)P(B|b,\,c,\,\lambda)$$.