How Can Signal Models Resolve the EPR Paradox in Special Relativity?

[JesseM]

I've just thought of a simpler proof: if we denote your expectation values for the sum of n trials as P_A(a,b), P_A(b,c), and P_A(a,c), and Bell's expectation values for a single trial as P_B(a,b), P_B(b,c), and P_B(a,c), then as long as there is no time-variation in the individual expectation values P_B(a,b) etc. from one trial to the next, it should just be the case that

P_A(a,b) = n*P_B(a,b)
P_A(b,c) = n*P_B(b,c)
P_A(a,c) = n*P_B(a,c)

In this case we can start with Bell's inequality:

1 + P_B(b,c) ≥ |P_B(a,b) - P_B(a,c)|

Since n is a positive integer, we can multiply both sides by n without changing the inequality:

n + n*P_B(b,c) ≥ |n*P_B(a,b) - n*P_B(a,c)|

And by substitution this gives:

n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|

Similarly if you wanted to label the results on each individual trial by ±(1/2) rather than ±1, then we would have the inequality you wrote in post #43:

(1/4)n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|

But in any case, are you claiming it would be possible to violate these inequalities in a local realistic model? If so, why do you believe that?

 

[Anamitra]

"Thirty years later John Stewart Bell responded with a paper which posited (paraphrased) that no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics (known as Bell's theorem)."

The above quotation is from the following Wikipedia link:
http://en.wikipedia.org/wiki/Principle_of_locality
Bell in his paper has given an example of such a contradiction using the inequality in section IV,Contradiction:
[the paper may be obtained from the first link of posting 23]

The inequality used is:
1+P(a,b)>= Mod[P(a,b)-P(a,c)]
Incidentally Bell has considered average values of P(a,b) etc. I would like to refer to equation (19) where the average value of P(a,b) has been considered.Incidentally this average value pertains to some particular value,if one uses the mean value theorem.

It does not take into consideration the cumulative effect of a short time interval,specifically a minimum time needed for measurement.
[This is similar to what Jesse said in posting in 28 and I answered in posting 29 and the subsequent ones]
With the consideration of the cumulative effect of measurement time we have

n+P(a,b)>=Mod[P(a,b)-P(a,c)]

P(a,b) ,P(a,c) and P(b,c) should have the interpretation as I have expounded in postings 47 and 50 . "n" is some positive number greater than one.
Now the violation shown in the example Bell's paper does not exist.Rather it can be avoided.[one may assign the value 2 or 3 to n]. We have got a certain amount of flexibility in the application of the inequality.

[a and b are unit vectors,though I have not used bold letters for them]

[Incidentally results like P(a,b)=-a.b are quantum mechanical results which have been used test Bell's inequality which comes basically from commonsense intuition]

 

[JesseM]

With the consideration of the cumulative effect of measurement time we have

n+P(a,b)>=Mod[P(a,b)-P(a,c)]

P(a,b) ,P(a,c) and P(b,c) should have the interpretation as I have expounded in postings 47 and 50 . "n" is some positive number greater than one.
Now the violation shown in the example Bell's paper does not exist.Rather it can be avoided.[one may assign the value 2 or 3 to n]. We have got a certain amount of flexibility in the application of the inequality.

No, you don't get any increased flexibility, because changing the value of n also changes the values of P(a,b) and P(b,c) and P(a,c) (when they are defined as you define them, as expectation values for the sum of each result over n trials), in such a way as to make the inequality equally impossible to violate under local realism. As I said before if we denote your cumulative expectation values as P_A(a,b), P_A(b,c) and P_A(a,c), while denoting Bell's single-trial expectation values as P_B(a,b), P_B(b,c) and P_B(a,c), then we have:

P_A(a,b) = n*P_B(a,b)
P_A(b,c) = n*P_B(b,c)
P_A(a,c) = n*P_B(a,c)

This should make it obvious that if this local realism always gives values of P_B(a,b), P_B(b,c) and P_B(a,c) that satisfy this inequality:

1 + P_B(b,c) ≥ |P_B(a,b) - P_B(a,c)|

Then it must also always give values of P_A(a,b), P_A(b,c) and P_A(a,c) that satisfy this one:

n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|

...since the second inequality is simply obtained by multiplying both sides of the first by n, obtaining n + n*P_B(b,c) ≥ |n*P_B(a,b) - n*P_B(a,c)|, and then performing the substitutions P_A(a,b) = n*P_B(a,b) etc.

 

[Anamitra]

Let us have a look at the following result:

n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|

If each value like n*P(a,b)<1 [n*P(a,c) and n*P(b,c)<1]even if "'n" is a suitable positive number larger than one[it may be 2,3, 5.9 etc], the testing of Bell's inequalities with Quantum Mechanical results like P(a,b)=-a.b will not produce any contradiction. You may consider the example in Bell's paper.
[The value of "n" definitely provides us a good amount of flexibility in the application of Bell's inequality,I mean the transformed one]

 

[JesseM]

Let us have a look at the following result:

n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|

If each value like n*P(a,b)<1 [n*P(a,c) and n*P(b,c)<1]even if "'n" is a suitable positive number larger than one[it may be 2,3, 5.9 etc], the testing of Bell's inequalities with Quantum Mechanical results like P(a,b)=-a.b will not produce any contradiction. You may consider the example in Bell's paper.

If the absolute value of each of them is smaller than 1 while n ≥ 3, then the inequality will be satisfied. But the point is that the inequality cannot be violated in local realism, whereas QM does violate this inequality (i.e. it violates your altered inequality above just like it violates Bell's original inequality). To disprove Bell's theorem you would have to find a local realistic model that allows for the inequality to be violated (under the test conditions Bell outlined), but Bell proved this is impossible.

 

[Anamitra]

You may consider the example in Bell's paper.
The inequality
n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|
is not violated by Quantum Mechanics by the values considered[a.c=0,a.b=b.c=1/[sqrt(2)],that is,P(a.c)=0;P(a.b)=P(b.c)= -1/[sqrt(2)] ]

This applies to local or non-local conditions.
But the inequality
n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|
gets violated by the choice of values considered above.

Incidentally P(a.b)=-a.b is a quantum mechanical result.

The effect of the hidden variables may be operative in the local or non local situation

The action of the hidden variable may be of a complicated nature. It may not depend on distance. It may depend on the nature of the signals received/emitted by the particles and not on the strength of the signals. Even if there is a dependence on the strength of the signals it may be argued that each particle has to take the full share of the signals emitted by the other. We are dealing with a closed system[isolated one] containing only two particles.

 

[SpectraCat]

You may consider the example in Bell's paper.
The inequality
n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|
is not violated by Quantum Mechanics by the values considered[a.c=0,a.b=b.c=1/[sqrt(2)],that is,P(a.c)=0;P(a.b)=P(b.c)= -1/[sqrt(2)] ]

This applies to local or non-local conditions.
But the inequality
n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|
gets violated by the choice of values considered above.

Incidentally P(a.b)=-a.b is a quantum mechanical result.

The effect of the hidden variables may be operative in the local or non local situation

The action of the hidden variable may be of a complicated nature. It may not depend on distance. It may depend on the nature of the signals received/emitted by the particles and not on the strength of the signals. Even if there is a dependence on the strength of the signals it may be argued that each particle has to take the full share of the signals emitted by the other. We are dealing with a closed system[isolated one] containing only two particles.

*sigh* You can't use the same values from the Bell paper in your re-derived inequality. If you pick a value for n, then you need to multiply each of the P(x,y) terms in the inequality by n to get the P_A(x,y) terms you use in your new inequality. As JesseM showed, you are just doing elementary mathematical manipulations with the original Bell inequalities. You have not added any new content.

 

[JesseM]

You may consider the example in Bell's paper.
The inequality
n + P_A(b,c) ≥ |P_A(a,b) - P_A(a,c)|
is not violated by Quantum Mechanics by the values considered[a.c=0,a.b=b.c=1/[sqrt(2)],that is,P(a.c)=0;P(a.b)=P(b.c)= -1/[sqrt(2)] ]

As SpectraCat said, the values in Bell's paper are the single-trial expectation values, i.e. there you have P_B(a.b) = P_B(b.c) = -1/[sqrt(2)]. These are different from the sum-over-cumulative-trials expectation values, which in this case would be P_A(a.b) = P_A(b.c) = -n/[sqrt(2)]

 

[Anamitra]

As SpectraCat said, the values in Bell's paper are the single-trial expectation values, i.e. there you have P_B(a.b) = P_B(b.c) = -1/[sqrt(2)]. These are different from the sum-over-cumulative-trials expectation values, which in this case would be P_A(a.b) = P_A(b.c) = -n/[sqrt(2)]

This is not a correct assertion. You simply cannot measure a single trial result. There may be one million trials in a measurement spanning across a short or an infinitesimally small interval of time.
The Quantum Mechanical Expectation[which may be obtained in a physical experiment] covers such a span of time which may include one million trials of the distribution function rho[lambda]

In any experiment we need a minimum time to get the correct measured value. Even a single click may incorporate a million trials

Quantum Mechanical expectation:

{P}_{QM}{(}{a}{,}{b}{)}{=}{-}{a}{.}{b}

{P}_{h}{(}{a}{,}{b}{)}{=}{\int}{n}{\rho}{(}{\lambda}{)}{A}{(}{a}{,}{\lambda}{)}{B}{(}{b}{,}{\lambda}{)}{d}{\lambda}{=}{-}{a}{.}{b}If one is to measure this result in an experiment, a single click might involve one million trials.

 

[JesseM]

This is not a correct assertion. You simply cannot measure a single trial result.

I wasn't measuring a single trial result when I gave P_A(a.b) = P_A(b.c) = -n/[sqrt(2)], I was talking about the expectation value for a sum of results over n trials, for example if I had n=4 trials my expectation value would be -4/[sqrt(2)], while if the actual trials gave results -1, -1, +1, -1 then my sum in this case would be -2.

There may be one million trials in a measurement spanning across a short or an infinitesimally small interval of time.

A "trial" is defined as a single recorded outcome, like a single "click" of the detector. It doesn't matter if at some hidden level unknown to us, the detector is really caused to click by a million brief interactions with a cloud of particles, it's still only a single trial if we only have one outcome. If you want the word "trial" to represent something else then this would once again change the meaning of P(a.b) and so forth, and there's no reason to expect the same inequality would be derived.

The Quantum Mechanical Expectation[which may be obtained in a physical experiment] covers such a span of time which may include one million trials of the distribution function rho[lambda]

That doesn't make much sense. What is a "trial of the distribution function" supposed to mean in physical terms? Physically the distribution function just tells you probability the hidden variables will take various values (each value of lambda represents a complete state of hidden variables), it's true these hidden variables could be rapidly changing during the measurement period, but in his more carefully worded paper La nouvelle cuisine he defined lambda to give the values of the hidden variables in at every point in space time in some complete cross-section of the past light cone of the region of spacetime where the measurement happened, like region "3" in the diagram at the top of this page. So in this case lambda isn't even meant to tell you the value of any hidden variables during the measurement period itself.

 

[Anamitra]

I wasn't measuring a single trial result when I gave P_A(a.b) = P_A(b.c) = -n/[sqrt(2)], I was talking about the expectation value for a sum of results over n trials, for example if I had n=4 trials my expectation value would be -4/[sqrt(2)], while if the actual trials gave results -1, -1, +1, -1 then my sum in this case would be -2.

A "trial" is defined as a single recorded outcome, like a single "click" of the detector. It doesn't matter if at some hidden level unknown to us, the detector is really caused to click by a million brief interactions with a cloud of particles, it's still only a single trial if we only have one outcome.

I have not used the word trial with the meaning of a single recorded outcome. It is simply the impact[or the influence] of the distribution function----and we are considering several such impacts---to get a single recorded outcome.

That doesn't make much sense. What is a "trial of the distribution function" supposed to mean in physical terms? Physically the distribution function just tells you probability the hidden variables will take various values (each value of lambda represents a complete state of hidden variables), it's true these hidden variables could be rapidly changing during the measurement period, but in his more carefully worded paper La nouvelle cuisine he defined lambda to give the values of the hidden variables in at every point in space time in some complete cross-section of the past light cone of the region of spacetime where the measurement happened, like region "3" in the diagram at the top of this page. So in this case lambda isn't even meant to tell you the value of any hidden variables during the measurement period itself.

What is a "trial of the distribution function" supposed to mean in physical terms?

We are considering the same normalized distribution function to be valid for each instant of time in the measuring interval.One may consider different distribution functions[normalized ones] for different instants. That will not alter the basic nature of my arguments and the conclusions following from them. To get a theoretical estimate of the result of measurement we have to consider the cumulative effect of these functions.I have considered this cumulative effect by using a weight denoted by "n".

 

[JesseM]

I have not used the word trial with the meaning of a single recorded outcome.

Then you can no longer say that your P_A(a.b) is equal to n*P_B(b.c), since Bell's P_B(b.c) is the result of a single recorded outcome. If you redefine "trial" then you must redefine the meaning of the "expectation value", and there is no reason to expect the same inequality would still apply, since that inequality was derived under the assumption we were talking about a trial as a recorded outcome.

It is simply the impact[or the influence] of the distribution function----and we are considering several such impacts---to get a single recorded outcome.

Your language is completely incomprehensible. How can a "distribution function" have multiple "impacts"? Have you ever heard someone say "Ouch! I've just been impacted in the head by a distribution function"? You need to explain your ideas in more physical terms, the distribution function is just mean to give the probability that lambda will take various values, where each value of lambda represents the state of some hidden variables (in Bell's argument it gives the value of these variables in a cross-section of the past light cone of a region of spacetime where a single measurement was performed).

What is a "trial of the distribution function" supposed to mean in physical terms?

We are considering the same normalized distribution function to be valid for each instant of time in the measuring interval.One may consider different distribution functions[normalized ones] for different instants.

That doesn't tell me what the distribution function is mean to represent a distribution of, in physical terms! The distribution is assigning probabilities, yes? So what is do you think it is assigning probabilities to, in physical terms?

Again, in Bell's terms the distribution function would give the probabilities that the hidden variables take different possible values in a cross-section of the past light cone of the region of spacetime where the measurement is performed. If you adopt this physical definition of the distribution function, it makes no sense at all to talk about it taking different values at different instants during measurement, because we are talking about the hidden variables in a fixed region of spacetime (region 3 in the diagram, please follow this link to look at the diagram I'm talking about), not the hidden variables from moment to moment during measurement.

 

[Anamitra]

Then you can no longer say that your P_A(a.b) is equal to n*P_{B(b.c), since Bell's PB(b.c) is the result of a single recorded outcome. If you redefine "trial" then you must redefine the meaning of the "expectation value", and there is no reason to expect the same inequality would still apply, since that inequality was derived under the assumption we were talking about a trial as a recorded outcome.}

The expectation value for each moment/instant may be denoted by P(a,b). This does not correspond to a single click. One may also divide a small click interval interval into even smaller[infinitesimally smaller intervals] and associate the P(a,b) with each such interval.

Your language is completely incomprehensible. How can a "distribution function" have multiple "impacts"? Have you ever heard someone say "Ouch! I've just been impacted in the head by a distribution function"? You need to explain your ideas in more physical terms, the distribution function is just mean to give the probability that lambda will take various values, where each value of lambda represents the state of some hidden variables (in Bell's argument it gives the value of these variables in a cross-section of the past light cone of a region of spacetime where a single measurement was performed).

The manner in which the variable lambda influences the expectation value for each moment/instant is governed by the nature of the distribution function.

It is nothing like a force or a ball out of somebody's bat that could hurt a person watching the game.
But this distribution function has a great power from the physical point of view in its capability of determining the component P(a,b) values that add up to produce the final expectation value that gets recorded in a measurement[a click]

You have talked of a region of spacetime where a single measurement is made. Such a region can have millions of time coordinates.At a single spatial point you may consider one million time instants----- corresponding to the interval of measurement.

 

[JesseM]

The expectation value for each moment/instant may be denoted by P(a,b).

Expectation value of what physical quantity, if not an observed "click"?

The manner in which the variable lambda

What is the physical meaning of "the variable lambda" in your mind, if it doesn't have the same meaning that Bell assigns to it

You have talked of a region of spacetime where a single measurement is made. Such a region can have millions of time coordinates.

Sure, but so what? In Bell's terminology lambda does not represent the value of any hidden variables at a single time coordinate in the measurement region. Rather a single value of lambda tells you the value of all hidden variables at all points in spacetime in another region that's in the past light cone of the measurement, like region 3 in the diagram. Region 3 is not the measurement region, that's region 1 in the diagram. Of course region 3 lasts an extended period of time in Bell's diagram too (though he could have made it just a single instantaneous spacelike cross-section of the past light cone), but that doesn't mean lambda is changing because lambda does not represent the value of the hidden variables at a single instant of time, rather a single value of lambda represents the values of the hidden variables at every point in region 3.

If you have trouble understanding this, suppose we have a hidden variable that at any time can be in two states A or B, and it can only change once every 5 seconds, at T=0, T=5, T=10 etc. Suppose we have a region of spacetime that goes from T=7 to T=13. Then to specify the value of this simple hidden variable in this region, we need to know both its value from T=7 to T=10, and its value from T=10 to T=13. So we could define a new variable lambda that can take 4 values, lambda=1, lambda=2, lambda=3, lambda=4, with the following physical meaning:

lambda=1 means hidden variable in state A from T=7 to T=10, in state A from T=10 to T=13

lambda=2 means hidden variable in state A from T=7 to T=10, in state B from T=10 to T=13

lambda=3 means hidden variable in state B from T=7 to T=10, in state A from T=10 to T=13

lambda=4 means hidden variable in state B from T=7 to T=10, in state B from T=10 to T=13

So if we specify the value of the variable lambda, we have specified the state of that specific hidden variable during both time intervals, we wouldn't say that lambda can "change" at T=10, although the hidden variable itself can. And obviously this could be generalized to a larger region of spacetime where the hidden variable could change multiple times, or to a region of spacetime where there were multiple hidden variables at any given moment in time. In either case, we could define the variable "lambda" in such a way that a single value of lambda tells us the value of arbitrarily many local hidden variables at arbitrarily many different times in whatever region of spacetime we're interested in (like region 3 in Bell's spacetime diagram). This is what Bell's lambda is supposed to represent, it isn't just telling you about a single moment in time. If you want to define lambda differently you need to explain what you mean by it physically, but be warned that any significant change will probably invalidate the derivation of the Bell inequality.

 

[Anamitra]

Bell's treatment/formulation is of a general type, intended to cover all possible situations concerning the hidden variable.

It is quite interesting to observe the valiant attempt of the Scientific Adviser to contradict this basic general nature of the paper.I have reasons to thank him--and he would find it very difficult to understand this.

 

[DrChinese]

Bell's treatment/formulation is of a general type, intended to cover all possible situations concerning the hidden variable.

It is quite interesting to observe the valiant attempt of the Scientific Adviser to contradict this basic general nature of the paper.I have reasons to thank him--and he would find it very difficult to understand this.

This is pretty funny. So far, nothing you have said makes any sense that I can see. It is a lot of nice looking formulae that goes nowhere, which is sort of your forte as I read your other posts. JesseM is quite knowledgeable, so I think you are misconstruing the situation greatly.

It would be helpful if you would ask specific questions rather than making general statements which have no specific connection to a technical issue. Nothing you have written remotely supports your brash statements that entanglement can be explained by light speed interactions.

 

[JesseM]

Bell's treatment/formulation is of a general type, intended to cover all possible situations concerning the hidden variable.

Not true, if you define lambda in arbitrary ways then you may not have a basis for claiming that the result A at one detector can be deduced in a deterministic way from only the detector setting a and lambda, in other words you may need to use a probabilistic function P(A|a,lambda) rather than a deterministic function A(a,lambda), and in fact Bell does use a probabilistic function in most of his later papers. But even if you use a probabilistic function, to derive a Bell inequality you still need a step like the on on p. 15 of this paper where you say P(A,B|a,b,lambda)=P(A|a,lambda)*P(B|b,lambda) which depends on the assumption that lambda "screens off" any statistical correlation between the result/setting A/a and the result/setting B/b due to influences from the region where the past light cones of both measurements overlap (because of the possibility of such influences, you could not say that that P(A,B|a,b)=P(A|a)*P(B|b), for example). If you don't make some assumption like treating lambda as telling you all hidden variables in a cross-section of the past light cone of the measurements this step may not be justifiable. And of course here we are defining "A" and "B" as the observable measurement outcomes, whereas you seem to be defining them differently yet you refuse to actually explain what physical quantity you are calculating an "expectation value" for if not the observable measurement outcome. In this case there is obviously no justification for either the claim that this "expectation value" is equal to an integral involving deterministic functions (A|a,lambda) and (B|b,lambda), or the probabilistic claim that P(A,B|a,b,lambda)=P(A|a,lambda)*P(B|b,lambda). Neither of these steps is justified on the basis of pure probability theory, they both depend on physical assumptions about the physical meaning of expectation values P, so if you change the meaning you can't justify these steps unless you provide a clear definition of what physical quantity you are computing an expectation value for.

It is quite interesting to observe the valiant attempt of the Scientific Adviser to contradict this basic general nature of the paper.

Certainly the paper is fairly general, but you only show your lack of comprehension if you think it's so general that you don't have to worry at all about the physical meaning of various terms like lambda and the expectation values P(a,b) etc. The paper involves multiple steps that can't be justified on the basis of abstract mathematics alone, you can't arbitrarily change the physical meaning of the symbols and expect it to still work.

 

[Anamitra]

We can consider three points G1,G2 and X at the time of measurement. G1 and G2 correspond to the gadget locations while X is a point inside the closed system where the particles are present. The intersection of the past light cones of (G1 , X) and (G2,X) with the exclusion of (G1,G2 )is considered at the time of measurement. The influence of the hidden variable can be explained by influences/signals from such regions.

We may always have formulations of the type:
{E}_{h}{(}{a}{,}{b}{)}{=}{\int}{n}{\rho}{(}{\lambda}{)}{P}{(}{A},{B}{\mid}{a}{,}{b}{,}{\lambda}{)}{A}{(}{a}{,}{\lambda}{)}{B}{(}{b}{,}{\lambda}{)}{d}{\lambda}
"n" takes care of the short time of measurement. {E}_{h} on Lhs: Expectation
Where,
{\int}{n}{\rho}{(}{\lambda}{)}{P}{(}{A},{B}{\mid}{a}{,}{b}{,}{\lambda}{)}{d}{\lambda}{=}{m}
[m is larger than or equal to two: one can make this quantity flexible following this condition]
But I would like to stress an important point here. If the separating particles are to influence each other by signals right from the time of their creation, we have to consider the past light cones of these particles[their intersection].
At the time of measurement we have to consider the intersection of the past light cones of these particles[Considering the time of their creation we have to consider truncated light cones[past] for interaction between the particles]. This will coincide with[precisely, be a subset of] the intersection of past light cones of the gadgets at the time of measurement. It is not necessary to exclude such a region[though it is conventionally excluded. In fact I have followed this exclusion in the initial part of this posting].

 

[Anamitra]

Not true, if you define lambda in arbitrary ways then you may not have a basis for claiming that the result A at one detector can be deduced in a deterministic way from only the detector setting a and lambda, in other words you may need to use a probabilistic function P(A|a,lambda) rather than a deterministic function A(a,lambda), and in fact Bell does use a probabilistic function in most of his later papers.

In the relation:

{P}_{h}{=}{\int}{n}{\rho}{(}{\lambda}{)}{A}{(}{a}{,}{\lambda}{)}{B}{(}{b}{,}{\lambda}{)}{d}{\lambda}

the functions A and B are not as deterministic as one might be tempted to think of. There is a lambda controlling these functions and this lambda in turn is being governed by the probability distribution function rho(lambda).

One may use other controlling functions like P(A,B/ a,b,lambda).This function establishes the association between the measurements A and B through the influence of lambda.In the other formula lambda works out this association by its presence in A and B. But the basic conclusion does not change----we can always evolve forms of Bell's Inequalities consistent with QM results/predictions