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Bell's Inequality Untwisted
Bell's Inequality Untwisted
Bell's Inequality Untwisted
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Bell's Inequality Untwisted

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“Bell’s Inequality Untwisted” is a unique book. My aim is to explain in detail all the equations and statements in John S. Bell’s ground-breaking paper, published in 1964, “On the Einstein Podolsky Rosen Paradox.” I attempt an in depth explanation of Bell’s paper in a way that is understandable to a wide audience. In the process of explaining every equation in detail, it becomes clear that Bell’s 1964 paper is a series of incoherent equations, and his famous eponymic inequality is an unjustified assertion. Bell’s elliptic style of writing calls out for comprehensive explanations of his equations. Explanations that fill the lacunae in his mathematical system are an imperative. When the gaps in his system are filled, the inconsistent and illogical nature of his reasoning becomes clear. Bell’s inequality holds a very important place in the development of quantum mechanics. The fact that the profound shortcomings of Bell’s paper are not widely known is telling.
I have found trying to promote my e-books to be quite difficult. Trying to promote “Bell’s Inequality Untwisted” has made me think about Murphy’s Law, i.e., everything that can go wrong will go wrong. It took a good deal of effort to give the many equations in the text a professional appearance. In a technical sense they are all photographs, which is why they rise slightly above the line of text in which they have been placed. It is also difficult to understand exactly how my original e-book description came to have a second paragraph that was essentially a recapitulation of the first paragraph. My supposition is that such an unprofessional error dissuaded many potential readers from downloading my e-book. From a certain point of view, it seems Murphy’s Law is a cloak to hide the operation of sinister forces. Why would a rational person assume that everything that can go wrong will go wrong? Gremlins seem to be relegated to “Bugs Bunny” cartoons, a famous “Twilight Zone” zone episode certain movies, but are they a hidden factor in our everyday lives?
It is difficult to know the appropriate message to give to a potential reader. Anyone reading this book description should download my e-book; the complex math is explained in detail, and the details are understandable. My e-book is an honest and straightforward analysis of Bell’s inequality; it is an attempt to cut through the miasma surrounding one of the most influential theories in quantum mechanics. The crux of my refutation of Bell’s inequality is presented in chapter 4 “A Tree at Night.” The reader may begin reading at that point if he thinks he would find the introductory material tedious and repetitious. The mathematical errors that John Bell makes concern basic algebra, if indeed they are errors and not deliberate attempts to deceive. I refreshed my understanding of basic algebra by watching the DVDs put out by a group of educators going under the name of the Standard Deviants. I watched such titles “Algebra Adventure” and “Pre-Algebra Power.” They were both entertaining and informative. One of the mistakes John Bell makes is an error in solving an inequality that contains terms within an absolute value sign. The correct method for solving an inequality containing terms within an absolute value sign is covered in one of the Standard Deviants DVDs and will likely be contained in any algebra textbook. The correct method is also discussed in my e-book.
It is difficult to know what tone to take with this extended promotional blurb. I turned to the “Collected Fictions” of Jorge Luis Borges for inspiration, but to no avail. Perhaps, I should have consulted a collection of Latin phrases.

LanguageEnglish
PublisherJim Spinosa
Release dateSep 24, 2014
ISBN9781370575152
Bell's Inequality Untwisted
Author

Jim Spinosa

Born in 1955,Jim Spinosa remembers,as a youngster,being entranced by the science fiction novels heperused in a small,corner bookstore in Denville,NJ. The cramped confines of that store had claimedto contain the largest selection of books in Northern New Jersey. His penchant for science fiction engendered an interest in physics. Often daunted by the difficulty of physics textbooks,hequestioned whether physics could be presented as clearly and concisely as science fiction,without sustaining any loss in depth Nuts and Bolts:TakingApart Special Relativity is an attempt to answer that question.

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    Bell's Inequality Untwisted - Jim Spinosa

    Bell’s Inequality Untwisted

    By Jim Spinosa

    Published by Jim Spinosa at Smashwords

    Copyright © 2014 Jim Spinosa

    All rights reserved

    ISBN: 9781370575152

    Smashwords Edition, License Notes: This e-book is licensed for your personal enjoyment only. This e-book may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each person. If you are reading this book and did not purchase it, or it was not purchased for your use only, then please return to Smashwords.com and purchase your own copy. Thank you for respecting the hard work of this author.

    Dedicated to Steven G. Spinosa

    The Spinner cares

    CONTENTS

    Introduction

    1. Juggler in Quicksand

    2. Puddles before the Rain

    3. Icarus Rain

    4. A Tree at Night

    5. Sawdust Doesn’t Float

    6. Daedalus’ Kaleidoscope

    Epilogue

    Final Notes

    About the Author

    Endnotes

    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

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