Samuel Bonaya Buya
ON THE FOUNDATIONS OF A COMPLETE HIDDEN LOCAL CAUSAL VARIABLE THEORY OF QUANTUM MECHANICS First Author and sole author: Samuel Bonaya Buya, Article Type: Original Scientific Research Suggested Reviewers: Shariar S. Afshar B. sc. Physics Professor, Physics and Astronomy, Rowan University afshar@rowan.edu His research interest in deterministic model of the atom Ernst Knoesel PhD in Physical Chemistry Professor, Physics and Astronomy, Rowan University knoesel@rowan.edu His main area of interest is wave- particle duality and Bohr's principle of complementarity. He has worked with Professor Shahriar S. Afshar on the Afshar experiment. His has interest in examination of the foundations of quantum mechanics. Antony Valentini PhD Physics Professor, Physics, Imperial College, London a.valentini@imperial.ac.uk His research interest in hidden variable theories ABSTRACT I present a hidden local causal variable theory of quantum mechanics. I show that of a Local Realistic interpretation of quantum mechanics is perfectly possible. To meet the requirement of Bell’s theory requires I will postulate a complete state or rather a hidden vector to accompany a state vector. The composite system under consideration will be the multi-electron atom. The constituents of the composite system under test consideration will be the electrons spatially spaced in the atom. The complete state will be formulated to enable it simultaneously predict the position, momentum and energy of a particle whenever a measurement is made irrespective of the measurement procedures. The realism aspect of the complete vector will be founded on the assumption that the test particle has particle nature co-existing with its wave nature all the time. Thus in the hidden vector the spatio-temporal trajectories of the test particle will be represented by an orbit number defined in relationship to the velocity of the particle, the orbit radius and also in relationship to an integral principle quantum number and an effective nuclear charge. Additionally, complete state postulated will conform to special relativity. I identify a biasing parameter in Bell test experiment the inclusion of which makes hidden local causal variable theories viable. I show how measurements of spins entangled of photons can be derived without recourse to instantaneous communication between entangled photons. I show that the peculiarities of quantum mechanics can be removed in our view of the universe. REQUEST: This ten page document comprehensively touches the key issues on my investigation on the foundations of quantum mechanics. Given the sensitivity of the subject matter in this paper I request a team of unbiased scientists to review this paper and give a report on it. The reason for asking in many of the journals I have sent my work the final review comment is that the work is out of the scope of the journal. I suggest the above reviewers and other objective reviewers to give an independent report on this paper. KEY WORDS: Complete state; hidden vectors; uniquely determined outcomes; identical systems; principle of reality; principle of locality; state vector; multi-electron atom; orbit number; principal quantum number; effective nuclear change; composite system; Bell’s theory; spatio-temporal trajectories; radius of orbit; Bell test experiment; local causal chain; photon spin bias; dynamical bias in coin spin ; Bell test experiment; spin angular momentum vector; interaction vector of the measuring apparatus; spookiness; instantaneous distant interaction PACS: 03.65.-w; 03.65.Ud; 32.00.00; 32.30.-r; 03.67.Mn; 06.20.Jr INTRODUCTION There have been different views on the issue concerning the statistical nature of quantum mechanics. The statistical nature of quantum mechanics is due to the observation that there are no unique outcomes in quantum mechanics measurement. Instead there is a possible range of outcomes in quantum mechanics measurement so that it seemingly looks impossible to make confident prediction on these measurements. The dominant interpretation has been that the statistical of quantum mechanics is due random or non-deterministic nature of the universe. There has been a feeling still among others that if a quantum mechanical system is completely described a deterministic framework can be obtained. I in this paper establish a complete theoretical framework founded both on the principles of reality and locality that will unlock the hidden deeper reality beneath quantum mechanics. Two identical systems A and B represented by the same state vector can give different measurements of the same property because the two systems have different hidden vectors. Through the complete state representation I will show that outcomes of measurements on systems are uniquely determined. The state vector description in quantum mechanics is not complete” because it ignores element of reality and locality necessary for showing the unique character of each outcome of measurement. A hidden deeper reality beneath quantum mechanics can predict outcomes of experiments. A COMPLETE IN QUANTUM MECHANICS “It has been argued that quantum mechanics is not locally causal and cannot be embedded in a locally causal theory ---“. Quote John Stewart Bell One way of embedding quantum mechanics into a local causal theory is to complete the state vector by incorporating into it elements of reality and locality. I will call the complete state vector with elements of reality embedded in it a hidden vector. It is necessary for the hidden vector defined here to accompany the state vector of quantum mechanics of to account for outcomes of quantum mechanics observation. Formalism of movable boundary between the quantum system and the classical apparatus will not be needed in the theory in this paper since the hidden vector will take care of any interaction with the measuring apparatus. Hidden variable theory I will present will assign a value to measured quantities regardless of the measuring procedure. I postulate a hidden vector carrying within it the elements of reality and locality and quantization given by:
H=e^((-ic〖tR〗_H)/n^2 )---------------- 1
The various elements of the hidden vector defined by equation 2 and the consequent equation 3 below.
n^2=(α^2 c^2)/v^2 =N^2/((Z-a)^2 )=r/(Zr_B ) --------2
n = orbit number and can take fractional or decimal values. The quantization aspect of the hidden vector is further hidden and defined in n. N = the principal quantum state of the electron Z = atomic number v = velocity of the electron α = fine structure constant Z-a = effective charge for the given quantum state and orbit number. t = time r = radius of the orbit r_B= Bohr’s radius The hidden element of the spatio-temporal trajectories the electron are defined in equation 2 so that electron orbit radius is given by: r=n^2 Zr_B=(N^2 Zr_B)/〖(Z-a)〗^2 ----------------3 The energy in electron-volts energy of an electron orbit can be derived through the action of the appropriate operator and is given by: E=-1/q_e ih/H ∂H/∂t=-1/q_e R_H hc/n^2 =-1/q_e R_H hc 〖(Z-a)〗^2/N^2 ------- 4 q_e = charge of the electron Equation 4, a derivative of the complete state of equation 1 can derive the total energy of an electron in the allowed orbits of Bohr’s model of the hydrogen atom on the assumption that Z=1 and a=0 The complete state postulate is therefore physical. The hidden vector accompanied the definitive equations 2 - 4 will assign values to all measurements together with the local causal relationships of each value assigned. If the out come of measurement of property A is say x_1 then x_1will be associated to a unique orbit n_1 and the unique orbit number n_1 will also be associate with a unique outcome of property B a unique outcome of property C and so forth. If the out come of measurement of property A is say x_2 then x_2 will be associated to a unique orbit n_2 and the unique orbit number n_2 will also be associate with a unique outcome of property B a unique outcome of property C and so forth. Thus a measurement of property A can produce outcomes x_1,x_2,x_3,x_4---x_N and these outcomes are associated with unique orbit numbers n_1,n_2,n_3,n_4---n_N and each of these orbits is associated to a unique outcome of property B, property C and so forth. Thus outcomes of measurement of property A can be used to calculate outcomes of properties B, C, D and so forth through a causal chain of relationship. Thus each outcome of measurement of a property is unique in that it can be used to calculate the magnitude of another property on the basis of its value. Thus each outcome of measurement is unique in sense of it enables determination of value of another property. Quantum mechanics lacks the causal chain of relationship between properties because its state vector is incomplete. It therefore assumes indefiniteness of properties. A complete Hidden Local Causal Variable theory establishes the concept of definiteness through its local causal chain of relationships it possesses. This means a complete hidden local causal variable theory has the mechanisms to establish deterministic relationship between properties. Quantum mechanics through its incomplete state vector formulation lacks this mechanism and assumes Heisenberg’s Uncertainty Principle. The Copenhagen interpretation assumes that nature lacks deterministic mechanism at fundamental level. This treatise shows that it not nature that lacks the element of determinism at fundamental level but it is an element that is lacking in quantum mechanics formulation.
If your notice the spin property has not been included in the local causal chain of property relationship – that is equation 2. I will examine the issue of spin separately and how it can be integrated in the causal chain of equation 2.
Note in equation 1 we could make the time co-ordinate look like a space coordinates by treating time as imaginary as in the case of Einstein’s special relativity so that: x_4=ict -------------- 5 Additionally if in equation 1: y=n^2/R_H -------------- 6 If equations 5 and 6 are carefully integrated into equation1, the complete state can be postulated to assign probabilities to possible results of measurements. However, a complete state vector postulated on determining outcomes a needful in a hidden local causal variable theory in order for such a theory to establish the legal framework behind quantum mechanics observables. Notice according to equation 2, a given principal quantum number can have several orbit numbers to account for eigenvalues returned by measurements related to a particular principal quantum number. For a given composite system the eigenvalue returned on measurement will be associated with a specific orbit number. The orbit number will in turn be associated a specific electron momentum, a specific energy of electron orbit and a specific orbit radius. If the composite system under consideration is the hydrogen atom and on measurement the position of the electron is 4.7626×〖10〗^(-10)m from nucleus of the atom, then the measurement will be associated with orbit number n = 3 as per equation 3, electron momentum = 6.64284×〖10〗^(-25) kgms^(-1) the energy of electron in the orbit = -1.5117441 eV. If the composite system under test is the uranium atom and the measurement of the position of an electron the nucleus is 0.002 nm, then such a measurement is related to the orbit number 0.02027 as per equation 3, electron momentum = 9.83156×〖10〗^(-23) kgms^(-1) the energy of the electron in the orbit is -33.119 KeV. If measurements are made on several entangled spatially separated composite atomic systems each eigenvalue returned by each composite system will be associated to an orbit number which can be used as per the prescribed equations other particle properties as prescribed by equations 2, 3 and 4. I will shortly show that equations 2 and 3 have observational support by using them to derive Coulombs law. OTHER PHYSICAL APPLICATION OF THE ABOVE FORMULATION If equation 2 is substituted into equation 3 we obtain the relationship: r_n v^2=α^2 c^2 r_B Z ----------- 7 Dividing both sides of equation 7 by r^2and multiplying by the rest mass of an electron we obtain the Coulomb law relation: F_C=(m_e v^2)/r=(m_e α^2 c^2 r_B Z)/r^2 =(Ze^2)/(4πε_o r^2 ) --------- 8 Note that m_e α^2 c^2 r_B=e^2/(4πε_o ) --------- 9 We note that equations 2 and 3 that form the definitive basis of the state vector derive Coulombs law. Equations 2 and 3 are therefore physical. We also note from equation 9: α=h/(2πm_e r_B c) ------------- 10 Using equation 10 it can be shown further that: r_B=(h^2 ε_o)/(πm_e e^2 )=h/(2πm_e αc) ------------ 11 Using equations 11 and 3 it can be shown further that the radius of an orbit is given by: r=hcαZ/(2πm_e v^2 ) ------------- 12 Equations 12 and 2 can be used to further establish a classical quantization rule given by: m_e vr=nhZ/2π=NhZ/(2π(Z-a)) -------------- 13 THE PHYSICAL FRAMEWORK FOR PARTICLE SPIN IN RELATIONSHIP TO OTHER PROPERTIES; ISSUES IN QUANTUM ENTANGLEMENT I will first propose state vector representing electron spin is given by: m_s=〖1/2 e〗^(-iπN_sp ) ----------- 14 m_s = Spin quantum number N_sp= Principal spin quantum number The principal spin quantum number can either be 1 or 2. When N_sp=1 m_s=-1/2 . When N_sp=2 m_s=+1/2 . From equation 14 2s=e^(-iπN_sp ) ----------- 15 To infuse spin in the local causal chain relating electron properties the exponents of equation 1 and 15 on the basis that 2s=H with the time taken as spin time , t_sp so that:
n^2=(cR_H t_sp)/(πN_sp )=(α^2 c^2)/v^2 =N^2/((Z-a)^2 )=r/(Zr_B ) ---------16
Equation 16 completes the local causal chain of relationship of properties.
Thus by equation 16 an orbit number can be associated with to Principal spin numbers each with its own spin time. If n=1 and N_sp=1 then t_sp=9.550282×〖10〗^(-16) s. If n=1 and N_sp=2 then t_sp=1.91005641×〖10〗^(-15) s.
The spin angular momentum of the electron was worked out to be h/4π . We can say that the probability of an unbiased orbit number having a particular Spin Principal Quantum number is 1/2 . The Stern- Gerlach experiment can be used to establish this. The sodium atom has one unpaired electron. A beam of neutral sodium atoms is split into two components angle so that one half is in one direction and the other half in the other direction when it passes through a magnetic field at right angles. The two split components have opposite spin from physical principles. From the above treatise electrons having the same momentum will have the same orbit number. Consequently the above probability follows. There are theoretical issues that were not given adequate consideration in Bell’s theory. In the conducting of an optical Bell test experiment if the photon encounters the polarizer at an angle there is are possibility of the counts to be biased. My context of count bias is that in measurements involving two equiprobable outcomes, counts are biased when the counts of one type of spin exceed significantly and consistently the other type of spin in a given measuring arrangement. In events of such biases the degrees of freedom that balance observed outcomes are tipped to give more weight to one type of observation. If with a slight change of the measuring arrangement the count bias is removed and the outcomes consistently show both outcomes with equal probability then the degrees of freedom influencing outcomes are evenly distributed between the two outcomes. If change in measuring arrangement involves change the angle of orientation between the measuring apparatus and the object on which measurements is being made and if additionally there is an interaction between the measuring apparatus there is a possibility for such interactions under special measuring conditions to bias spin outcomes by tipping the evenly distributed degrees of freedom influencing outcomes to weigh on the side of the most dominant outcome. If in a Bell’s test experiment the angle of orientation of the polarizer is say θ° and a significant 10 count test registered by the analyzer oriented at this angle are (- + - + + + + + - +) we can say the counts are biased due oblique interaction (measurement in this case involves an interaction). If the orientation angle of the other polarizer is say 0° and the counts registered are say (- + - + - + - + - +) the counts are not biased in this case. The quantum correlation in these two sets of measurement is E=((1+1+1+1-1+1-1+1+1+1))/10=0.6 Changing the orientation has the effect of biasing the spin counts due to oblique interaction between the analyzer and the photon. The bias created projects and an appearance of an unfair sampling loophole but it is not as we will shortly discuss. To illustrate further, if the detector settings of Alice were not change but the detector settings of Bob were changed so that the outcomes were distributed between different spins as in the table below, it should be possible to work out the correlation between the outcomes of Alice and Bob as in the table below: Outcomes of Alice Outcomes of Bob Correlation between outcomes of Bob and Alice worked out using Bobs results Negative/ Positive
spins Positive /
Negative spins Negative/ positive spins (x) Positive/ negative spins (y) C= (±2x)/100=(±(100-y))/100 50% 50% 50% 50% ±1 50% 50% 40% 60% ±0.8 50% 50% 30% 70% ±0.6 50% 50% 20% 80% ±0.4 50% 50% 10% 90% ±0.3 It seems from the above table the relationship between the orientation of Bob’s detector and Bob’s results is given by: 2θ=〖cos〗^(-1) C=〖cos〗^(-1) (±2x)/100=〖cos〗^(-1) (±(100-y))/100 ------- 17 That is to say: E=cos2θ=(±x)/50=(±(100-y))/100 ---------- 18
Thus when the test angle is 22.5° x=35.36% and y=64.64% When the test angle is 45° x=0% and y=100% When the test angle is 67.5° x=35.36% and y=64.64% The results obtained by Bob are purely dependent on detector settings. The orientation of Bob’s detector is the local causal variable behind the spin bias in his results. Bell’s experiment when viewed from a hidden local causal variable theory view point is a setting to investigate relation ship between spin bias and detector setting and the correlation that exists between unbiased spins and bias spins of entangled photons. Thus the results of quantum entanglement can be derived and accounted for without recourse to instantaneous communication between entangled photons. Equation 18 relates the expectation with Bobs results. In Bob’s results when x=36% E=±0.72 A Complete Hidden Local Causal Variable theory breaks the straight line limit placed in Bell’s theory for entangled particles. Bell’s no go theorem just as von Neumann’s no hidden variable proof is flawed. Any test against hidden variable theory based on Bell’s theory is flawed because the limit was derived on erroneous assumptions. Suppose further the source producing entangled pairs of photons only sent photons to Bob’s detector and Bob records a hundred counts in each detector settings. Bob still notices that by changing his detector settings the spins are still biased and the same pattern of results are obtained for his results as in the table above. One is forced to conclude that it is purely the influence of the detector settings that biases the outcomes of Bob and that in the previous experiment conducted with Alice there was no instantaneous distant interaction, no spookiness involved. To clear the doubt of a possible interpretation that might arise, the source can be made to produce photons which are not necessarily entangled and sent to Bob who records a hundred counts under different detector settings. The photon spins are still biased and same pattern of results is obtained. This means there is no spookiness involved. The measurement we make on one particle does not instantaneously impart momentum one the other particle. No spooky action at a distance implies that particles have well defined properties after all. Copenhagen interpretation is wrong. In the four separate subexperiments in a Bell test experiment with Bell test angles: 0°,45°,22.5°,67.5° the numerical value of the test static will exceed 2 if the biasing of the spins due to oblique interaction is given consideration in a hidden local causal variable theory. The CHSH (Clauser- Home-Shimony-Holt) inequality is therefore violated by a hidden local causal variable theory that puts into consideration bias in spins created by oblique interactions. Suppose we had an experimental set up in which we had a field or mechanism to spin a coin in very specific ways to bias outcomes of coin flip. Suppose the experimental set up had a mechanism to change the bias of the coin through changing the angle between the normal to the coin and the angular momentum vector. If one coin is sent to Bob and the other is sent to Alice. Suppose further that the apparatus of Bob is aligned such that it registers 70% head counts and 30% tail counts and the apparatus of Alice is aligned such that it registers 50% head counts. Suppose further in recording the counts a head is assigned + and a tail is assigned - and in working the correlations + is assigned +1 and negative is assigned -1. If Bob and Alice made simultaneous measurements and in a ten count observation the following pairs of results were returned: (+, +), (-, -), (+, +), (-, -), (+, +), (-, -), (+, +), (-, +), (+, +), (-, +) the correlation between the two results would be worked as: E = ((1+1+1+1-1+1-1+1+1+1))/10=0.6 For more information on dynamically biased coins refer to a significantly important paper headed: DYNAMICAL BIAS IN THE COIN SPIN by Persi Diaconis Susan Holmes Richard Montgomery Department of Mathematics, Department of statistics, Department of Mathematics and Statistics Sequioa Hall University of California Stanford University Stanford University Santa Cruz http://comtop.stanford.ed/priprints/heads.pdf Biasing of photon spin in Bell test experiment needs to be included in a hidden local causal variable theory to validate it. The biasing dynamics could be worked comprehensively through a model of elastic collision involving interaction of the measuring device and the photon whose spin direction is being measured. Mathematics has laid the foundation for working this bias.
CONCLUSION A complete local hidden variable theory can derive quantum mechanics observables. Particle properties, atomic spectra and observations involving entangled can be derived using a local hidden variable theory. Bell in his investigation of the foundations of quantum mechanics failed to include the biasing effect on direction of photon spin through interaction of the measuring apparatus and the photon on his in inequality. One through use of Bell test experiments therefore cannot claim to rule out Local realistic theories on the basis of the failure of the CHSH inequality to make spin bias considerations in working the expectations of a hidden variable theory in the test angles under consideration. When the expectations of the inequality are worked out putting spin bias into consideration the claim is easily falsified. Bell’s no go theorem just as von Neumann’s no hidden variable proof is flawed. The settings of Bell test experiment can be used to falsify instantaneous distance interaction when the biasing effects of detector settings are taken into consideration. The EPR paper is confirmed and experimentally significant. A complete hidden local causal hidden variable theory can successfully account for outcomes of quantum observations. A Bell test experiment can meet the measurement conditions of the EPR paper when: there is a non – spin bias interaction between the photon the measuring apparatus and the photon The correlation coefficient of the spin outcomes of the two observers is +1 Quantum mechanics can be founded on the basis of: definite objectiveness of properties determinacy of measurement outcomes Locality. There is one framework underlying quantum mechanics and relativity and that the traditional Laplacean identification of causality holds in these two branches of physics. REFERENCES Can Quantum Mechanical Description of Physical Reality Be Considered Complete? , Journal: Phys. Rev.} Volume 47 Pages 777 – 780 by A. Einstein, B. Podolsky, and N. Rosen}, Year {1935}} On the Constitution of Atoms and Molecules, Part1 by N. Bohr, Philosophical magazine Volume 26 Pages 1-24 Year 1913 N. Bohr Can quantum mechanical description of physical reality be considered complete? By N. Bohr Phys. Rev. 696-702 Volume 48 Year 1935 Bell's theorem and delayed determinism by Franson Journal, Physical review D Volume vol. 31, No. 10D Pages 2529-2532 Year may 1985 Heisenberg and the Wave-particle Duality, Journal: Studies in History and Philosophy of Modern Physics by Camilleri, K. Volume 37, issue 2 Pages 298-315 Year 2006 The Philosophy of Niels Bohr. The Framework of Complementarity by Folse, H., Journal Review of modern Physics Volume Vol. 39, No. 1 pages 78 - 124 Year January 1967 On the Einstein-Podolsky-Rosen Paradox by J.S. Bell Journal: Physics Volume Pages 195-200 Year 1964 John Bell, 'Free Variables and Local Causality', 'Epistemological Letters', 15, 1977 Bell, John S, The Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Dynamical Bias in Coin Spin by Persi Diaconis Susan Holmes Richard Montgomery Department of Mathematics, Department of statistics, Department of Mathematics and Statistics Sequioa Hall University of California Stanford University Stanford University Santa Cruz This is copyrighted with http://www.copyrightdeposit.com/rep17/0030333.html (Samuel Bonaya Buya (talk) 15:12, 9 August 2010 (UTC))