Bell's Inequality Untwisted

Bell’s Inequality Untwisted

Copyright © 2014 by James Spinosa

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ISBN: 978-0-359-48657-1

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Dedicated to Steven G. Spinosa

The Spinner cares

Introduction

In 1964, the science magazine Physics published a paper by John S. Bell entitled, On the Einstein Podolsky Rosen Paradox. John S. Bell’s article is concise. The page count for his article is only six, but they are densely written pages. These pages present the argument for what would later be designated as Bell’s theorem. Since the final mathematical formulation in his argument is an inequality, it is also known as Bell’s inequality. John S. Bell’s eponymic inequality holds a very important place in the development of quantum mechanics. Hagiographic books and articles have gathered around Bell’s inequality. Aimed at a wider audience, these texts serve as a kind of incoherent explanation of an uncanny aspect of quantum mechanics.

Because of the short length of John S. Bell’s paper, it is possible to present a detailed critique of it, which is not excessively long. In the following chapters the entire article is reproduced along with the necessary, explanatory material to make Bell’s inequality accessible to a wider audience.

A significant step in making Bell’s inequality accessible to a wider audience would be to present a simple model, which could replicate the behavior of paired spin one-half particles. Finding such a model turns out to be a difficult task. In fact, a way of approaching Bell’s inequality is to conclude that there cannot be any simple, mechanistic model for the behavior of paired particles.

Let’s start by examining the flipping of two fair coins. We will attempt to build a model of the behavior of paired particles around their behavior. Let’s say we flipped a pair of coins simultaneously and noted the results. Next, we did this experiment repeatedly until we had done it 10,000 times.

If every result was either that both coins were heads or that both coins were tails, the correlation between the coins would be +1.

If every result was either that when coin A was heads, then coin B was tails or that when coin A was tails, then coin B was heads, the correlation between the coins would be –1.

If every result was either that when coin A was heads, then coin B was heads for half of the results and tails for the other half or that when coin A was tails, then coin B was heads for half of the results and tails for the other half, the correlation between the coins would be 0. The correlation between the two coins ranges from –1 to +1 with 0 indicating there is no correlation between the two coins.

If every time coin A is heads, then coin B is heads and if every time coin A is tails, then coin B is tails, the correlation between the two coins is +1. The correlation is positive because the coins A and B display the same face.

If every time coin A is heads, then coin B is tails and if every time coin A is tails, then coin B is heads, the correlation between the two coins is –1. The correlation is negative because coins A and B display opposite faces.

When we are flipping two fair coins, we are likely to obtain a correlation of 0. But, through the operation of chance, we could obtain any positive correlation that is greater than 0 and less than or equal to 1. Also, through the operation of chance, we could obtain any negative correlation that is less than 0 and greater than or equal to –1.

For instance, we could obtain a positive correlation of +.5. The following equation is called the Pearson product-moment correlation. It is named for its inventor, Karl Pearson. We can use this equation to determine the correlation between two coins: , where rxy is the correlation between the two coins, n is the number of trials, which is to say the number of times the experiment is done, x represents coin A and y represents coin B. The term ∑ indicates a summation. The book, Statistics for People Who (Think They) Hate Statistics, by Neil J. Salkind describes the circumstances for which the use of the Pearson correlation is valid. The Pearson correlation coefficient examines the relationship between two variables, but both those variables are continuous in nature. In other words, they are variables that can assume any value along some underlying continuum, such as height, age, test score, or income. But there is a host of other variables that are not continuous. They’re called discrete or categorical variables, such as race, social class, and political affiliation. You need to use other correlational techniques such as the point-biserial correlation in these cases.1

Ron Larson and Betsy Farber’s Elementary Statistics: Picturing the World gives us their requirements for the valid use of the Pearson correlation coefficient. Two requirements for the Pearson correlation coefficient are that the variables are linearly related and that the population represented by each variable is normally distributed. If these requirements cannot be met, you can examine the relationship between two variables using the nonparametric equivalent to the Pearson correlation coefficient — the Spearman rank coefficient.2

The random variables that represent the results we get when we make numerous trials, which consist of flipping two coins simultaneously, seem discrete and not normally distributed. The random variables that represent the results we get when we make numerous trials, which consist of measuring a certain spin component of paired one-half spin particles, seem discrete and not normally distributed. Yet, in both instances the Pearson correlation coefficient might be used to measure the correlation between the random variables. In practice there may be a certain amount of flexibility in deciding whether two sets of random variables meet the requirements for the valid use of the Pearson correlation coefficient.

To use the Pearson correlation coefficient to measure the correlation between numerous trials consisting of two coins flipped simultaneously, we arbitrarily assign heads the value of +1 and tails the value of –1. To obtain a correlation of precisely .5 we must satisfy the following prerequisites. We must flip the coins at least 24 times. We must make sure that the number of instances that coin A is heads is equal to the number of instances that it is tails. We also must make sure that the number of instances that coin B is heads is equal to the number of instances that it is tails. We can meet these prerequisites by discarding individual experimental results that don’t fulfill these requirements. This method is valid since we are merely trying to create a concise example in this situation.

A brief outline of the steps involved in producing a correlation of .5 between coin A and coin B will be helpful. The value of heads is +1 and the value of tails is –1. Coin A is represented by x, and coin B is represented by y. The number of instances coin A is heads is equal to the number of instances it is tails. The number of instances coin B is heads is equal to the number of instances it is tails. These considerations allow us to conclude the∑ x = 0 and ∑ y = 0. From these results we obtain (∑ x)² = 0 and (∑ y)² = 0. The number of instances we do the experiment is 24 so n = 24. Since (+1)² =1 and (–1)² =1 and since n = 24, ∑ x² = 24 and ∑ y² =24. Let’s examine an idealized sequence of coin flipping trials. Since an eBook achieves its best appearance if there is an unimpeded flow of the text, all the charts, tables and graphs present in the original text have been reworked as text only.

For the flips 1 through 9, coin A is heads and coin B is heads for every trial. For the flips 10 through 18, coin A is tails and coin B is tails for every trial. For the flips 19 through 21, coin A is heads and coin B is tails for every trial. For the flips 22 through 24, coin A is tails and coin B is heads for every trial.

From this sequence we obtain ∑ xy = 12. The term ∑ xy indicates that we are to calculate the sum of the products of x multiplied by y. In our example, we have 24 products consisting of x multiplied by y, and the sum of those 24 products is 12. When we insert these values into the Pearson correlation coefficient equation, we obtain r =.5. For a slightly expanded discussion of these calculations see the analysis of Eq. (19) in chapter four.

If we wanted to obtain r =.5 or any other correlation, there is a simple way in which we could obtain it. We would use the Pearson correlation coefficient equation and other information to calculate the type of sequence we would need to obtain the correlation we desired. With that information at hand, for every flip, we would note whether coin A was heads or tails. Then we would reach out either adjusting coin B or not adjusting coin B so that it fit our prearranged sequence.

This method may seem crude and strange. It is not a tenable model for the behavior of paired particles, but we can imagine a more sophisticated method.

We would need to flip the coins on a special table. The special table would need to be equipped with hidden cameras that could report the outcomes of the flip of coin A and the flip of coin B to a computer hidden in the table. The computer would need to determine a proper sequence for any correlation. You could enter into the computer the correlation you desired by pressing some hidden keys. Whenever the coins were flipped, the computer would note whether coin A was heads or tails. Then the computer would note whether coin B was heads or tails. To arrive at a desired correlation, coin B may need to be adjusted to match the predetermined sequence. A very thin metal pin would rise very quickly under coin B and flip it into its proper position. These thin metal pins would need to be hidden throughout the surface of the table. This entire sequence must occur so quickly and seamlessly that it would appear as though the flipped coins landed without any interference of any kind.

This sophisticated method is dependent on the transfer of information from coin A to the computer and then from the computer to coin