I’ve been hearing a lot about Bell’s theorem lately in the news. Is it possible that spacetime at the quantum level has only two dimensions? Quantum entangled systems don’t recognize spatial separation. At least that would preserve local realism…
That’s what’s called the holographic principle. The simplest forms of it have been experimentally ruled out by the GEO gravitational wave detector in Germany, which turns out to be a much better holographic-universe detector than it is a gravitational-wave detector. For a while there, it had some unexplained noise that looked just like what you’d expect from a holographic universe, which was quite exciting, but then they eventually managed to find mundane causes for that noise and correct them. But of course, there’s always a more complicated version of any theory that can be adapted to fit experimental facts, so not everyone has given up on the holographic principle.
That said, while a holographic universe would in some sense be nonlocal, I don’t know whether it would have the right sort of nonlocality to account for Bell’s Theorem.
It depends what you mean, the basic dimensional requirement for Bell’s theorem is not in physical space or spacetime, but in Hilbert space and that requirement is the dimension of the Hilbert space is at least 4. Generally a system even in 1+1 D spacetime will be described by an infinite dimensional Hilbert space and therefore Bell’s theorem applies.
If I understand it, the Holographic principle developed from the macro perspective to explain how information isn’t lost at the event horizon of a black hole. I’m trying to get some intuition into Bell’s inequality. From the quantum perspective a 2-d quantum system is being observed from a 3-d reference frame. How would a 2-d system recognize the spatial dimension? Shouldn’t local realism be based on the entangled system’s reference frame?
Causal influences need time to propagate spatially, which happens with the separation of the entangled pairs.
The wave function collapsing can happen FTL without violating the speed of causality as there is no information conveyed.
The speed of light, or more correctly the speed of causation is not a upper bound for all effects, and there is no violation of relativistic causality.
While not quite this simple is is primarily a problem with human assumptions.
Note local realism is rejected in QFT, and this view is not incompatible with General Relativity.
If you look at Bell’s inequality there is no assumption as to spatial dimension, it is not relevant.
Here is a cite to one study that shows how many local realism theories were ruled out over the past few years.
Experimental demonstration of a quantum shutter closing two slits simultaneously
I thought Bell’s inequality suggests that Einstein was wrong; there are no hidden variables. This seems to introduce a spatial component. ‘Spooky action at a distance’
Einstein’s spooky action at a distance, or the EPR paradox is really what John Bell resolved in 1964.
Entanglement allows one particle to instantaneously influence another but does not send classical information faster than light.
The EPR paradox is only a “paradox” because classical intuitions do not correspond to physical reality.
As the dope doesn’t support images or MathTex I am going to link to a PBS Spacetime video, which are made to be accessible without without much math.
It will probably provide a better explanation.
OK, so there’s a couple of questions here. First, Bell’s theorem: basically, if you get down to the math, Bell’s theorem really is just a statement about joint probability distributions. A joint probability distribution of two events is essentially just something you can take and compute the probability of either event happening from. So, for instance, if you flip two coins, then the outcomes are (H=heads, T=tails) (HH), (HT), (TH), (TT), where the positioning distinguishes the two coins. For two fair coins, the probability of each of these occurring is 25%. This gives you the joint probability distribution.
How do you get the probability that the second coin lands tails? Easy, you just sum up the cases in which it does, since the probability of the second coin landing tails is the probability of the first coin doing whatever, and the second coin landing tails. So we have P(HT) + P(TT), which yields 50%.
Bell’s inequalities (equivalent forms of which were discovered almost 100 years earlier by George Boole) are then nothing but necessary conditions for some set of events to have a joint probability distribution. What does it mean that a set of events doesn’t have a joint PD? Well, it’s easy: imagine you have three coins, C[sub]1[/sub], C[sub]2[/sub], C[sub]3[/sub]. Each of which lands H or T with 50% probability. When you throw two of them simultaneously, you observe the following:
[ul]
[li]If C[sub]1[/sub] and C[sub]2[/sub] are thrown together, the result is (HH) or (TT) with 50% probability[/li][li]If C[sub]2[/sub] and C[sub]3[/sub] are thrown together, the result is again (HH) or (TT) with 50% probability[/li][li]If C[sub]1[/sub] and C[sub]3[/sub] are thrown together, the result is (HT) or (TH) with 50% probability[/li][/ul]
These three do not possess a joint probability distribution. Indeed, trying to throw all three coins at once runs into an inconsistency: if C[sub]1[/sub] comes up (H), so must C[sub]2[/sub], and consequently, C[sub]3[/sub]; but due to the anticorrelation between C[sub]1[/sub] and C[sub]3[/sub], the latter must come up T.
This mirrors the situation in quantum mechanics: certain observations cannot be made simultaneously (as with Heisenberg’s famous uncertainty relation). Likewise, you can’t throw all three coins together—but you can throw any pair as often as you want, and thus, generate statistics. These statistics, however, will not yield a joint probability distribution—after all, the probability that all three come up (HHT), say, is just ill defined—this is not an event that actually might occur.
Bell (and Boole before him) now derived inequalities that are necessary conditions for a set of events to have a joint probability distribution. If all such inequalities for a given scenario are obeyed by the experimental statistics, they could have been jointly sampled from a single distribution, or a physical realization thereof—say, an urn with the right distribution of colored balls inside.
In quantum mechanics, Bell’s inequalities may be violated. Hence, there exists events (measurement outcomes) such that they do not have a joint probability distribution!
Well, so what? What’s any of that got to do with realism or locality?
The question really is: when would we expect for a set of measurement outcomes to possess a joint probability distribution? And the answer is, basically, whenever we could create some urn model from which we can draw little colored balls with the appropriate frequency. That is, there must be some ensemble from which objects with pre-defined properties are drawn (which gives you realism—even unmeasured properties have definite values), in such a way that the ensemble is not influenced by our choice of what to measure (which gives you locality—if it were possible that, say, throwing a coin yielding heads influences the probability for another coin to yield tails, we wouldn’t have a joint PD either; to ensure the absence of such interactions, in Bell tests, systems are placed sufficiently far away from each other in order to be unable to influence one another via signals sent at light speed).
So, Bell’s theorem tells us that we can’t create urn models of quantum experiments. That is, either locality or realism must fail—but we don’t know which! (And hence, it’s wrong to say that Bell experiments prove action at a distance: that’s just one of the two possibilities, the other being value indefiniteness—i.e. that a given property doesn’t have a pre-determined value).
So, this doesn’t have anything to do, as such, with spatial separation, or the dimensionality of space-time.
(I was going to write something more about entanglement, but I think the post is already plenty long enough to kill this thread… But if anyone’s interested, I’d be happy to write a couple of paragraphs!)