 |
| Not quite |
"ANY serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves."
A. Einstein in EPR.
Physical Theory
To ever element of a physical theory there corresponds an element of objective reality.
Guiding principle of this blog
It is the assertion of this blog that J. S. Bell's construct used to derive his inequality does not satisfy the requirement of a physical theory; there is an element of the construct that does not correspond to an element of objective reality. As such, much of what has been written (thousands of papers) about the consequence of violating Bell's inequality is, in truth, not correct.
To discover the non physical element in Bell's construct we start with the assumption that Bell's construct is a physical theory and then show that assumption to be false. As a physical theory the elements A(a,λ) and B(b,λ) correspond to measurement devices that exist in space and time and whose outputs are ±1.
Bell defines the correlation between A and B as
(1)
P(a,b) = ∫dλρ(λ)A(a,λ)B(b,λ)
The interpretation of (1) is that at some time a hidden variable λ of unspecified structure is created by a source (not shown), split into 2 parts and each of those parts then travels to the locations of A and B where they impinge on those detectors whose spatial orientations are 'a' and 'b', respectively. The product of the outcomes of detection (+1 or -1) is then integrated over all possible values of λ weighted by its probability density ρ(λ). The result is a number that ranges between -1 and +1 (the correlation between A and B). That correlation is a function of both 'a', and 'b' which takes a cosine form for spin-1/2 particles (and also for photons).
He then forms the difference between two correlations
(2)
P(a,b) - P(a,c)
Let's stop right there and consider the meaning of that expression. P(a,b) is determined when A takes orientation 'a' and B takes orientation 'b'. Now P(a,c) is determined when A takes orientation 'a' and B takes orientation 'c'. It is very important to notice that P(a,b) and P(a,c) can not be formed at the same time simply because detector B must be in two different orientations. Macroscopic measurement devices can not have two different orientations at the same time. They are restricted to take an orientation at one time and change that orientation at a different time. So implicit in (2) is the assumption of sequential measurement (sampling).
By sample it is meant a collection of observations. The sample size, N, specifies the number of observations taken in the sample. Note that (1) can not be realized physically and is the limiting case for large sample sizes. The sample estimator of (1) is written as
(1.1)
P(a,b) = 1/N ∑n=0N-1An(a)Bn(b)
where An , Bn are the nth observations for detectors A and B. For large N it is assumed that the hidden variable space is adequately sampled.
Bell's next step is to expand (2) in terms of (1). However, he also combines that with the 'perfect anti correlation' condition predicted by quantum mechanics when the two devices take the same orientation, B(a,λ) = -A(a,λ). That condition is not needed to show where Bell's derivation is non physical. Write (2) in terms of (1) directly as
Bell's next step is to expand (2) in terms of (1). However, he also combines that with the 'perfect anti correlation' condition predicted by quantum mechanics when the two devices take the same orientation, B(a,λ) = -A(a,λ). That condition is not needed to show where Bell's derivation is non physical. Write (2) in terms of (1) directly as
(2.1)
P(a,b) - P(a,c) = ∫dλ[ρ(λ)A(a,λ)B(b,λ) - A(a,λ)B(c,λ)]
Note well that when written in this form the hidden variable λ is common to all factors. One need not write that form. Instead one could write
(2.1')
P(a,b) - P(a,c) = ∫dλρ(λ)A(a,λ)B(b,λ) - ∫dλ'ρ(λ')A(a,λ')B(c,λ')
where the correlation for the first term is taken with different hidden variables from those used in the second term. Expression (2.1') is actually the correct way to write the form but Bell uses expression (2.1). To see why (2.1') is preferred (necessary), consider the estimators of (2.1) and (2.1').
(2.1e)
P(a,b) - P(a,c) = 1/N ∑n=0N-1[An(a)Bn(b) - An(a)Bn(c)]
and
(2.1'e)
P(a,b) - P(a,c) = 1/N ∑n=0N-1An(a)Bn(b) - 1/N ∑n=0N-1An+T(a)Bn+T(c)]
In (2.1e) it is assumed that all observations occur at the same time (n). In (2.1'e) it is assumed that for the second term the observations occur at a time T after the end of the sampling for the first term.
Consider the consequence of these two forms. In (2) we have stated that for physical measurement devices the sampling must be sequential in order for B to take on two different orientations. However, in the expression used by Bell (2.1) he has made the assumption that all observations occur simultaneously (with the same value λ). The estimated correlations (2.1e) and (2.1e') bring out that distinction clearly. Hence, Bell has a conflict, an inconsistency in his specification. He is asking for sequential and simultaneous measurements in the same expression.
The non physical element in his derivation is obvious in the expression
(3)
A(a,λ)B(b,λ) - A(a,λ)B(c,λ) = A(a,λ)[B(b,λ) - B(c,λ)]
where one has the difference of two quantities that can not be physically realized
(3.1)
B(b,λ) - B(c,λ)
They can not be realized because the common hidden variable λ forces the single measurement device B to have two different orientations 'b' and 'c' in the same instance.
If one ignores the physical world and treats (3.1) as nothing more than a mathematical expression then, as Bell says in his book "Speakable and unspeakable in quantum mechanics (p 65), the quantities B(b,λ) and B(c,λ) are the same function with different arguments. But it is precisely on that point where Bell violates physical theory. One can not realize (3.1) physically. If Bell is purporting to give a physical model of local hidden variable theories then he fails. If he is simply manipulating mathematics then he has lost the connection to the physical world and his effort has no significance as a model of local hidden variable theories.
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
Alice in Wonderland.
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
Alice in Wonderland.
