Why do you think it's not possible? Each particle measurement is based on a single discrete "click" at some detector, so you can just count the number of clicks, we aren't talking about measuring a continuously-varying quantity like temperature.Anamitra said:Any measurement process usually involves a huge number of trials/impacts. It is not possible to know the exact number of such trials..
Impacts of what? A "function" isn't a physical entity that can impact with anything. In any case, each "trial" consists of a single click which is recorded as +1 or -1 (or +1/2 and -1/2 if you prefer), regardless of what is really going on physically in each click.Anamitra said:A discrete click may involve a million impacts[the reception of a million impacts--impacts of the distribution function rho]
I don't get it, what does it mean to "consider" a function several million times? Do you mean considering several million possible values of lambda which are each assigned probabilities (or probability densities) by the distribution function, or something else?Anamitra said:In a single click of your gadget,you may have to consider the same distribution function[the normalized rho] several million times--the theoretical analysis of a click[what it measures]--involves such a consideration.
Huh? This is a totally vague answer, you didn't address my specific questions above about the meaning of "considering" the distribution function several million times. And of course in a hidden variables theory the variables might causally influence the result in a complicated way, but that doesn't change the fact that the result itself is only recorded as one of two possible outcomes, a click at one detector or a click at another. It is the probabilities or expectation values for these recorded outcomes that Bell inequalities are dealing with, an inequality involving hidden variables would be useless since we wouldn't have a way to measure these variables so there'd be no way to test whether the inequality was respected or violated.Anamitra said:What happens "inside" the click is important.We have "hidden variables" that can exist physically and contribute to the result of the click. The result of a click is not a magical thing that you are possibly inclined to believe in. If it is not magical ,how does it occur[I mean how the result of the measurement takes place]? We need to investigate this and the hidden variables can play an important role, a definite role--in a manner I have indicated.
By the mechanism provided in the postings 27 and 42 one will obtain:JesseM said:Of course it does, the inequality would just be modified if you use different numbers to label "spin-up" and "spin-down" measurements. Look at the step at the top of p. 406 (p. 4 of the pdf) of Bell's derivation, he invokes equation (1) which says that A(b,λ) = ±1 in order to justify equating the integrand [A(a,λ)*A(c,λ) - A(a,λ)*A(b,λ)] to the integrand A(a,λ)*A(b,λ)*[A(b,λ)*A(c,λ) - 1]...in other words, he's using the fact that A(b,λ)*A(b,λ)=+1. If you instead labeled the results with +(1/2) and -(1/2), then A(b,λ)*A(b,λ)=+(1/4), so that step wouldn't work. Instead the second integrand would have to be A(a,λ)*A(b,λ)*[4*A(b,λ)*A(c,λ) - 1].
Continuing on with steps analogous to the ones in Bell's paper, we can then apply the triangle inequality (which works as well for integrals as it does for discrete sums) to show that if P(a,b) - P(a,c) = \int d\lambda \rho(\lambda) A(a,\lambda)A(b,\lambda)[4A(b,\lambda)A(c,\lambda) - 1], that implies \mid P(a,b) - P(a,c) \mid \le \int d\lambda \mid \rho(\lambda) A(a,\lambda)A(b,\lambda)[4A(b,\lambda)A(c,\lambda) - 1] \mid. Since \rho(\lambda) is always positive and A(a,λ)*A(b,λ)=±(1/4) that becomes \mid P(a,b) - P(a,c) \mid \le \int d\lambda \rho(\lambda) (1/4) \mid [4A(b,\lambda)A(c,\lambda) - 1] \mid, then since 4*A(b,λ)*A(c,λ) - 1 is always equal to -2 or 0, its absolute value is always equal to 1 - 4*A(b,λ)*A(c,λ) so we now have \mid P(a,b) - P(a,c) \mid \le \int d\lambda \rho(\lambda) (1/4) [1 - 4A(b,\lambda)A(c,\lambda)] or \mid P(a,b) - P(a,c) \mid \le \int d\lambda \rho(\lambda) [(1/4) - A(b,\lambda)A(c,\lambda)]. Bell then notes that \int d\lambda \rho(\lambda) [-A(b,\lambda)A(c,\lambda)] = P(b,c) and since \int d\lambda \rho(\lambda) = 1 this gives an inequality equivalent to the one Bell derives:
(1/4) + P(b,c) ≥ |P(a,b) - P(a,c)|
With the measurement results labeled +(1/2) or -(1/2), there's no way this inequality can be violated in a realist theory that respects relativity.
Are you sure? Terms like P(b,c) and P(a,b) represent expectation values for a single trial, if you want them to represent an expectation value for a sum of results over many trials that would probably change aspects of the derivation, I'm not sure you would actually end up with the equation above. You need to show your work, give the steps in deriving the final inequality like I did.Anamitra said:By the mechanism provided in the postings 27 and 42 one will obtain:
(1/4)n + P(b,c) ≥ |P(a,b) - P(a,c)|
Anamitra said:But one may consider "hidden variables" to be responsible for the association between the two spin values.
I understood that, but the point is that this is what P(a,b) and such meant in the original formula, if you want to change the meaning of the terms you need to provide a new derivation of an inequality involving the terms with revised meanings.Anamitra said:Response to Posting 45
I have not meant the expectation values of a single trial.
Yes, I understood that as well, which is why I said "if you want them to represent an expectation value for a sum of results over many trials that would probably change aspects of the derivation". My point is that you haven't provided any justification for believing that your modified formula is actually correct under your new definitions, you need to show your work and provide a derivation of that formula.Anamitra said:In the revised formula I have meant the cumulative expectation values over the time intervals concerned.
JesseM said:I understood that, but the point is that this is what P(a,b) and such meant in the original formula, if you want to change the meaning of the terms you need to provide a new derivation of an inequality involving the terms with revised meanings.
Yes, I understood that as well, which is why I said "if you want them to represent an expectation value for a sum of results over many trials that would probably change aspects of the derivation". My point is that you haven't provided any justification for believing that your modified formula is actually correct under your new definitions, you need to show your work and provide a derivation of that formula.
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 PA(a,b), PA(b,c) and PA(a,c), while denoting Bell's single-trial expectation values as PB(a,b), PB(b,c) and PB(a,c), then we have:Anamitra said: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.
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 said:Let us have a look at the following result:
n + PA(b,c) ≥ |PA(a,b) - PA(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.
Anamitra said:You may consider the example in Bell's paper.
The inequality
n + PA(b,c) ≥ |PA(a,b) - PA(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 + PA(b,c) ≥ |PA(a,b) - PA(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.
As SpectraCat said, the values in Bell's paper are the single-trial expectation values, i.e. there you have PB(a.b) = PB(b.c) = -1/[sqrt(2)]. These are different from the sum-over-cumulative-trials expectation values, which in this case would be PA(a.b) = PA(b.c) = -n/[sqrt(2)]Anamitra said:You may consider the example in Bell's paper.
The inequality
n + PA(b,c) ≥ |PA(a,b) - PA(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)] ]
JesseM said:As SpectraCat said, the values in Bell's paper are the single-trial expectation values, i.e. there you have PB(a.b) = PB(b.c) = -1/[sqrt(2)]. These are different from the sum-over-cumulative-trials expectation values, which in this case would be PA(a.b) = PA(b.c) = -n/[sqrt(2)]
I wasn't measuring a single trial result when I gave PA(a.b) = PA(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.Anamitra said:This is not a correct assertion. You simply cannot measure a single trial result.
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.Anamitra said:There may be one million trials in a measurement spanning across a short or an infinitesimally small interval of time.
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 said: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]