Bells Theorem and Hidden VariableTheories

Alhun Aydn

January 3, 2011

Department of Physics, College of Sciences,Koc University, Istanbul, 34450, Turkey

Phys 490: Independent Study II - Fall 2010

Supervisor: Prof. Tekin Dereli

Abstract

Although standard quantum mechanics is compatible with experiments,it could not clarify some important unsolved problems like quantum realityand measurement process. Hidden variable theories provide solutions tothese conceptual problems of standard quantum mechanics. We examineBells theorem, one of the most essential theorems about foundations ofquantum mechanics, and Bell-test experiments which empirically rules outlocal hidden variable theories. Then we discuss pilot wave theory which isa non-local hidden variable theory and one of the most significant possiblesuccessors of standard quantum mechanics. Moreover, by touching uponalso philosophical issues, we give the shape of the possible interpretation ofquantum mechanics.

Contents

1 Introduction 21.1 Why do we need another interpretation? . . . . . . . . . . . . 3

2 Bells theorem 52.1 EPR, local hidden variables and Bells theorem . . . . . . . . 52.2 Bells inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 CHSH inequality . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Kochen-Specker theorem . . . . . . . . . . . . . . . . . . . . . 11

3 Bell test experiments 123.1 History of experiments . . . . . . . . . . . . . . . . . . . . . . 123.2 An example of Bell test experiment . . . . . . . . . . . . . . . 123.3 Loopholes vs developments . . . . . . . . . . . . . . . . . . . 14

4 Non-local Hidden Variable Theories 164.1 Pilot wave theory . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Wave and particle . . . . . . . . . . . . . . . . . . . . 174.1.2 Quantum potential . . . . . . . . . . . . . . . . . . . . 184.1.3 Quantum equilibrium . . . . . . . . . . . . . . . . . . 204.1.4 Measurement process . . . . . . . . . . . . . . . . . . . 20

4.2 Leggett-type inequalities . . . . . . . . . . . . . . . . . . . . . 214.3 Experimental tests of non-local hidden variable theories . . . 21

5 Discussion 235.1 Locality vs Realism . . . . . . . . . . . . . . . . . . . . . . . . 235.2 What should new interpretation looks like? . . . . . . . . . . 23

6 Conclusion 256.1 Future of hidden variable theories . . . . . . . . . . . . . . . . 25

Bibliography 29

1

Chapter 1

Introduction

Before quantum mechanics, who would have thought that the difference be-tween microscopic and macroscopic world can be that much and the rules ofmicroscopic world can be such controversial. Postulates of quantum theorywere so weird that people could not get together in a common denominatorespecially about philosophical fundamentals of it. Hence, physicists triedto interpret quantum mechanics and approach to its problems accordingto their own philosophical beliefs as realism, orthodoxy, ontology, locality,determinism or agnosticism.

There are several interpretations of quantum mechanics (QM), sincethere is no consensus on several issues among physicists. However, oneof them became more popular and the more accepted one, Copenhagen in-terpretation, which is the interpretation of standard quantum mechanics,considered by majority of the physicists as the best valid interpretation. Onthe other hand, there are other popular interpretations of QM with theirdazzling propositions. As a very brief review; for example, according to theMany-worlds interpretation, there are infinite number of parallel universesfor every possible event. Many-minds interpretation put the mind into theaccount by downing the distinction of parallel universes to the consciouslevel. Consistent histories proposing only classically consistent alternativehistories in self-consistent framework. Penrose interpretation tries to reduceeverything into gravitation and says that gravity gives reality to the mat-ter. Transactional interpretation explains quantum interactions by standingwaves. Relational interpretations says that quantum states relative to theobserver. Finally, de Broglie-Bohm interpretation (pilot wave theory), whichis the only relatively popular non-local hidden variable theory that couldntbe refuted, proposes deterministic, realistic and ontological view that allparticles have guiding waves determining their trajectories in space-time.In brief, people looking at same picture see fairly different things and dodifferent explanations, whereas there must be one thing with one properexplanation if there is one picture and everyone has same eye structure.

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1.1 Why do we need another interpretation?

To understand the necessity of another interpretation instead of Copen-hagen, we should first examine what is the problem about Copenhagen andstandard quantum mechanics (SQM).

According to the Copenhagen interpretation, formulated by Niels Bohrand Werner Heisenberg, a state of a particle or a system of particles canbe described completely by its wave function. Distinctively from classicalmechanics, SQM has purely probabilistic nature, and this probability isdefined by the absolute value square of the wave function. Since its natureis completely probabilistic and random, it is not a deterministic theory.Measurement of an observer forces system to collapse and choose one of itsprobable states. It is meaningless to ask in which state the particle beforethe measurement, since measurement creates the reality of that state.

The biggest controversy about QM emerged from the Double-Slit Ex-periment (DSE). In DSE, there are two slits and a screen behind. When wesend a beam of particles say electrons to the slits, an interference patternappears on the screen. Interference pattern also appears if we send electronsone by one. This is odd, because even if electrons behave like waves, an elec-tron would have to pass through both slits and interfere with itself to forman interference pattern. The idea of the superposition was roughly emergedfrom that. According to the superposition principle a particle occupies si-multaneously all of its possible states. That means a particle or a state ofa particle could be in two or more positions at the same time. Although, itis actually a mathematical property of linear systems, physically it was andstill is a very brave idea. However it is not satisfactory for realists to thinkone thing can be everywhere at the same time.

The identity of matter questioned in DSE, but matter sometimes behaveas particles and sometimes as waves. Bohr introduced complementarityprinciple to explain the wave-particle duality in the experiments. Accordingto this principle, matter can behave like wave or particle, but not at thesame time. But why? Is that the only possible solution to the question ofthe identity of matter? One can also think that maybe there is both waveand particle or maybe there is another identity that we couldnt know as yet.Maybe when we are measuring we reveal the particle and not measuring wesee the effects of its wave.

According to Bohr, measurement creates reality. For example, for anelectron with spin 1/2 system; before the measurement, spin state of elec-tron is in a superposition of two possible spin values, means electron is inboth states simultaneously. The act of measurement defines the electronsspin and after the measurement electron chooses randomly (with the properprobabilities obtained by the wave function) one of the possible spin states.So, before the measurement, electrons state is not a physical reality. Thething that creating the reality is the measurement of an observer. Also

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to identify the process this phenomenon (transition from superposition todefinite position) was called wave function collapse. However, so far thereis no evidence of such a collapse. In fact the wave function collapse is incontradiction with the linear dynamics of the SQM, since it is a non-linearprocess. Quantum measurement was trying to be explained also with quan-tum decoherence that says environmental effects causes collapse, which isalso controversial.

According to the realist view, reality exists independently from ob-servers. Therefore observer dependent realism of standard quantum me-chanics should be incomplete and revised by adding hidden variables. How-ever, SQM was built on orthodox view that says our observation createsthe reality. Actually there is one more view called agnostic view, but letsignore it. After all they are not interested in to find any answer and ignoreeverything.

Some physicists keenly opposed to Copenhagen interpretation, especiallyEinstein. He said I think that a particle must have a separate realityindependent of the measurements. I like to think that the moon is thereeven if I am not looking at it. Even though his views were fabulouslylogical, the mathematical formulation of Copenhagen interpretation was soconsistent and useful that no one interested in the problems that can beconsidered as philosophical. Besides, all unchallenged experiments up todate are compatible with SQM.

Nevertheless, by progressing on mathematical formulation without know-ing the true logic behind of phenomena, we may reach the wrong directions.After all just for adapting one can change the philosophy without chang-ing the mathematics of it and in fact there are mathematically consistentsystems that are not physical.

Physics laws should be independent from any external consciousness,if they are the same laws that are creating that consciousness. Otherwisephysics laws will be either consistent but incomplete or complete but incon-sistent. [1] Thinking that physics become to reality by human observationsis almost same as thinking the earth is the center of the universe and theuniverse with all physics rules is there just for human-beings, which is theidea from middle ages mentality.

To summarize, we do need another interpretation of quantum mechan-ics, as SQM cannot explain (the worse is do not intend to explain) thebefore measurement and during measurement process of systems, which isthe heart of quantum mechanics. The nature of measurement, complemen-tarity, superposition principle, correspondence principle and wave functioncollapse is not known. But probably, the most importantly, SQM is basedon subjective observations of conscious observers which such a subjectivityshouldnt be in a physical law. Therefore, it seems like quantum mechanicsneeds better interpretation.

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Chapter 2

Bells theorem

2.1 EPR, local hidden variables and Bells theo-rem

In 1935, Einstein, Podolsky and Rosen (EPR) attempted to show, by indi-cating non-local behaviours between distanced particles, that quantum me-chanics would be incomplete if its not violating locality. [2] They thoughtthere might be some variables in addition to wave function, determiningthe statistical behaviour of the particles, or Schrdingers wave function wasentirely wrong. Hidden variables are the variables which for completing thequantum theory, establishing physical reality and to give physics laws anobserver-free mechanism. These variables are called hidden because sofar we have no idea about their values or what they are look like, if theyexist. (So its not because theyre really hidden, but theyre hidden tous). However, in 1964 John S. Bell proved that if a theory is local, therewould not be hidden variables in it and vice versa. [3] This is called Bellstheorem. Therefore, physical reality or counterfactual definiteness (definite-ness of the properties of objects even if they are not being measured) andlocality cannot be true at the same time in any theory. One of them at leastmust be violated. That means Einsteins local realism dream came to theend.

Local hidden variable theories say that there are no non-local effectsbetween any objects. Everything should be compatible with the localityof special relativity. But now we know that non-local effects between ob-jects like quantum entanglement do not violate special relativity, since thesuperluminal effects cannot be using in any classical context like sendingclassical information by superluminal signals. [4] Non-local effects are inquantum level and it is known from experiments that there is no way toturn a quantum information to a classical information without disturbingit. [5, 6]

Although Bells theorem and Bell test experiments showed that there

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were no local hidden variable theories, there are some objections that ar-gues the probability of loopholes in the Bells theorem and the experiments.[7, 8, 9] The objections are generally about the experimental errors and fi-delity. However, in 2009, A. D. Boozer showed that Bells theorem might notrule out all local hidden variable theories, and there are consistent hiddenvariable theories that violates only one of the four postulates; symmetry,locality, causality, independency. [31] Some people are also attacked Bellstheorem theoretically by using superdeterminism, which is an abstract ideaand less likely to be proven with the current knowledge of human-being.[10, 11]

2.2 Bells inequality

John S. Bell showed the impossibility of local hidden variables by an elegantwork and introduced an inequality. [3, 9, 12] Lets consider the decay ofthe neutral pi meson into an electron and a positron as in EPR-Bohmsgedankenexperiment: [13]

0 e + e+ (2.1)

After the decay, from the conservation of angular momentum, there will bea correlation called entanglement between electron and positron. Since thepion has spin zero, electron and positron will be in a singlet configuration:

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(+ +) (2.2)

Figure 2.1: Demonstration of the correlation of electron-positron pair [13]

Now, lets detect spins and record them to show the predictions of stan-dard quantum mechanics. According to the Bohms version of the EPRexperiment, we can orient electron and positron detectors independently.Let one detector measures the component of electrons spin in the direc-tion of the unit vector a and the other detector measures the component ofpositrons spin in the direction of the unit vector b. If detector shows +1as the value of spin along that direction, that means particle is spin up andfor the value 1 spin down. (We take ~/2 as a unit, since it is same for

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spin 1/2 states.) If the detectors are parallel (a = b), that means if one de-tector measures spin up, than the other measures spin down and vice versa.Eventually their products will be always 1;

P (a, a) = 1 or P (b, b) = 1 (2.3)

When detectors are oriented orthogonal (a = b), then two detectors willmeasure the same spin value. (Note that in this case also electron andpositron have opposite spin values, but the detectors directions are orthog-onal to each other.) So their products will be always +1;

P (a,a) = +1 or P (b, b) = +1 (2.4)

Then, for arbitrary orientations of detectors, prediction of QM will be:

P (a, b) = a b (2.5)

This result is like the mathematical formulation of non-locality. Lets lookwhether it is compatible with local hidden variables. If there are hidden vari-ables (), then they would determine the spin values of electron and positron.To take locality into account, consider we take far away the positron in sucha manner that it cannot affect electron by any subluminal signal. So wemay measure the positrons spin without affecting the electron; in that caseoutcome of the electron is totally independent from the orientation b of thepositron detector. Then, we can constitute that A(a, ) will give the re-sult of electron measurement and B(b, ) will give the result of positronmeasurement with values +1 or 1;

A(a, ) = 1 and B(b, ) = 1 (2.6)

For any hidden variable the result will be:

A(a, ) = B(a, ) (2.7)

Lets now define the probability density () for the hidden variable. Itwould be positive and

()d = 1, since it is probability. Then we write

average value of the product of the measurements with probability density;

P (a, b) =

()A(a, )B(b, )d (2.8)

From equation (2.7), equation (2.8) becomes;

P (a, b) = ()A(a, )A(b, )d (2.9)

Now, we will define another unit vector c, and subtract its probability fromprobability product to change unit vector b. The reason why we are doing

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this is that we want to see how the probability product will respond to achange on b. Therefore, we take into account that hidden variable becomeresponsible from that change;

P (a, b) P (a, c) = ()[A(a, )A(b, )A(a, )A(c, )]d (2.10)

Since [A(b, )]2 = 1, we can put it into (2.10) and write it as:

P (a, b)P (a, c) = ()[A(a, )A(b, )A(a, )A(c, )A(b, )A(b, )]d

(2.11)By taking A(a, )A(b, ) brackets in (2.11), we get:

P (a, b) P (a, c) = ()A(a, )A(b, )[1A(b, )A(c, )]d (2.12)

Since from equation (2.6) we can say that 1 [A(a, )A(b, )] +1, so()[1A(b, )A(c, )] 0. Then,

|P (a, b) P (a, c)| ()[1A(b, )A(c, )]d (2.13)

By arranging this, we would reach to the Bell inequality;

|P (a, b) P (a, c)| 1 + P (b, c) (2.14)

This theorem is true, independent from the nature or number of the hiddenvariables and their distribution.

Lets now look at whether it is compatible with quantum mechanicspredictions or not. a and b are the unit vectors that are orthogonal and cis another vector in between them. Let be the angle between a and c andcos gives respective orientations between two detectors.

Figure 2.2: Simple plane representation of detectors

For demonstration of violation of Bells inequality, lets test for example = 45;

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P (a, b) = cos90 = 0, P (a, c) = P (b, c) = cos45 = 0.707

by putting them into (2.14), we can see that they are inconsistent withBells inequality;

0.707 1 0.707 0.707 0.293

In fact, regardless of the degree of angle between two orthogonal vectorsand c, it is easy to see that Bells inequality violated by QM, that meanslocal hidden variable theories incompatible with QM predictions.

The theorem clearly shows that there is a non-local correlation betweenentangled particles, which is considered by many physicists as a thing thatEinstein would never wanted. However these correlations are not conflictingwith Einsteins relativity theories since they would not allow superluminalcommunication and causality remains preserved. In entanglement, informa-tion stuck in quantum levels and one cannot turn it in to classical informa-tion so that there is no way we can use entanglement to send superluminalsignals to anywhere. Hence, implying that, Einstein would never want togive up locality and he would have prefer locality from locality-reality con-tradiction from Bells theorem, might not be correct.

Although mathematical proof is quite simple, there is even more simplerexplanation of what Bells theorem says. [14] Lets loosely visualize Bellstheorem;

For a system with properties A, B and C, Bells inequality says that;the number of objects with property A, but not property B, plus numberof objects with property B, but not property C is greater than or equal tothe number of objects with property A, but not property C. Note that eachproperty has two possible states. (Such as being A or not)

X(A,not B) + Y (B,not C) Z(A,not C)

where X, Y and Z are the functions of number of objects.Lets assume our system be a classroom and the properties be the fol-

lowing;

A: Gender (Male or Female)B: Major (Physics or Mathematics)C: Eye color (Blue or Green)

Independent of the size or characteristics of the people in the classroom,according to the Bells theorem, it would be always true that: The numberof male mathematics students plus the number of green eyed mathematicsstudents is greater than or equal to the number of green eyed male students.

To verify this, we can simply use sets. There are 8 different sets:

1- Blue eyed, male, mathematics students2- Green eyed, male, mathematics students

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3- Blue eyed, male, physics students4- Green eyed, male, physics students5- Blue eyed, female, mathematics students6- Green eyed, female, mathematics students7- Blue eyed, female, physics students8- Green eyed, female, physics students

Therefore Bells inequality becomes;

(Set 1 + Set 2) + (Set 4 + Set 8) (Set 2 + Set 4)

(Set 1 + Set 8) 0

Since we cannot have negative number of students of objects, Bells the-orem holds. Of course this is just a visualization of the inequality.

2.3 CHSH inequality

Bells inequalities attracted some theoretical attention and after his papermore people started to work on that. John Clauser, Michael Horne, Ab-ner Shimony and Richard Holt (CHSH) generalized Bells inequality andproposed an experiment to test local hidden variables, in 1969. [15] Theirversion is the more generalized version of the original Bells inequality. Letslook at Bells derivation of CHSH inequalities in 1971 [9], by considering twospin 1/2 particles again.

In the original inequality hidden variables considered as the local depen-dencies of the instruments. But instruments themselves also could containhidden variables which may affect the results in an indeterministic way (soin this case system does not have to be deterministic in contrast to Bellsoriginal derivation). Thus, to generalize the formulation, we can considerthat:

|A| 1 and |B | 1 (2.15)

instead of A(a, ) = 1 and B(b, ) = 1 in the original derivation. Aand B are the averages which are independent from b and a respectively(of course they may depend on a and b respectively). Then the quantummechanical probabilistic prediction yields;

P (a, b) =

()A(a, )B(b, )d (2.16)

To recover Bells experimentally unrealistic restriction we will define a andb, and p(b, b) = 1 u where 0 < u < 1 instead of u = 0. Because in

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experimental tests we find u close to zero but not equal. [15] Then we have,

P (a, b) P (a, b) =()[A(a, )B(b, ) A(a, )B(b, )]d

=

()[A(a, )B(b, )(1 A(a, )B(b, )]d

()[A(a, )B(b, )(1 A(a, )B(b, )]d

(2.17)

By restricting the conditions (2.15),

|P (a, b) P (a, b)| ()(1 A(a, )B(b, )]d

+

()(1 A(a, )B(b, )]d

(2.18)

then sum,|P (a, b) P (a, b)| 2 (P (a, b) + P (a, b)) (2.19)

and by arranging,

|P (a, b) P (a, b)| + |P (a, b) P (a, b)| 2 (2.20)

This is called CHSH inequality. Note that for a = b, from P (b, b) = 1(2.20) become,

|P (a, b) P (a, b)| 1 + P (b, b) (2.21)

The most beautiful thing of CHSH inequality is its testability by experi-ments. After the proposition of experiments by CHSH, many experimentshas been done. [9, 19] Not surprisingly, all experiments were consistent withtheory.

2.4 Kochen-Specker theorem

In addition to Bells and CHSHs inequalities there is also one other impor-tant no-go theorem that stated in 1967 by Simon Kochen and Ernst Specker.[16] They proved that both definiteness of all hidden variables of a quantumsystem at any given time and independency of these hidden variables fromthe device that measuring them, cannot be true at the same time. Thistheorem eliminates the hidden variable theories which are requiring non-contextual physical reality. In other words, a hidden variable theory mustcontains contextuality. It means hidden variables of an observed systemhave to be dependent also on observers interaction.

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Chapter 3

Bell test experiments

3.1 History of experiments

After Bells profound paper, a lot of experiments have been done to testBells inequalities. The first actual Bell test was done by Freedman andClauser by using CHSH inequality, in 1972 [17], and by Clauser and Hornein 1974. [8] The other significant and relatively most popular tests were doneby Aspect et al., in 1981 and 1982. [18, 19, 20] In 1997, non-local quantumcorrelations were demonstrated experimentally over than 10 km, by Tittelet al. [21] That means distance does not destroy quantum entanglement.In 1998, Weihs and Innsbruck group filled the loophole that Aspects 1982experiment has, with concreting the locality by using random-setting andultra-fast analyzers. [22] In 2008, Salart et al. overcame another loopholeby separating detectors far away so that there could not be any informationtransfer between the detectors before the measurements. [23]

3.2 An example of Bell test experiment

One of the first proper and elegant Bell test experiments is the experimentthat was done by Aspect et al. in 1982. [19] Lets look at that experimentcloser. This experiment based on Einstein-Podolsky-Rosen-Bohm gedanken-experiment, but with using two photon atomic transitions instead of piondecay.

Before this experiment, there was no direct test of inequalities and thedesign of the optical devices were not efficient enough to test the inequalitiesaccurately. Single-channel analyzers had been used in previous experiments,which lead to low efficiency in the detection of coincidences because orthog-onal polarization measurements could not be detected. In previous experi-ments Bell inequalities also violated, but the results were rather criticized.[8, 9, 18]

In Aspects 1982 experiment, they used two-channel analyzers. In the

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Figure 3.1: Single channel Bell test setup [25]

experimental setup, there are two polarimeters with orientations a and b toperform measurements on incoming photons, just like the measurements onStern-Gerlach experiments for spin 1/2 particles. Polarimeter is a compositeoptical device that contains both polarizer and analyzer, to measure rota-tion angle of the polarized light. Polarimeters can rotate around the axisof the incident beam respectively and separate two orthogonal linear polar-izations. To be able to detect the coincidences of individual photons, fourphotomultipliers has been used in the experiment. Photomultiplier multi-plies the weak flux of a light, produced by one or several photons, aroundten million times, so that flux can be detected by the coincidence detectors.To generate two photons in a singlet state, selectively excited calcium-40has been used.

Figure 3.2: Double channel Bell test setup [26]

Similar to EPR-Bohms gedankenexperiment, two photons generated andsent to polarimeters to determine their polarization directions. Polarizers(polarizing cubes) transmit the light which is parallel polarized and reflectsthe perpendicularly polarized. Then detectors count the coincidences andthe singles.

According to the CHSH inequality,

2 S 2 (3.1)

whereS = P (a, b) P (a, b) + P (a, b) + P (a, b) (3.2)

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Figure 3.3: Bell test photon analyzer [27]

and

P (a, b) =C++(a, b) + C(a, b) C+(a, b) C+(a, b)C++(a, b) + C(a, b) + C+(a, b) + C+(a, b)

(3.3)

In Aspects experiment, they found,

Se = 2.70 0.05 (3.4)

which leads clear violation of (3.1).Although there were some imperfections in experiment, the results are

way clear than any imperfection can chop.

3.3 Loopholes vs developments

It has been argued since Pearle (1970) that there might be some experi-mental problems that affects the results of the Bell-test experiments. [7]Two most encountered problems are fair sampling and efficiency. Mostof the problems were minimized by the developments of technology andexperimental methods. Some of the most mentioned possible loopholes inexperiments are the imperfections of light source, optical polarizer and si-multaneous detections of detectors. [8, 9, 19]

Developments in optics will of course provide us better experimentalfacilities in the future. Most physicists expected that these loopholes willbe filled up and violation of Bell inequalities will be an unquestionable fact.[24]

However there is one thing that can be considered as a theoretical loop-hole, superdeterminism suggested by Gerard t Hooft. [10, 11] If the universeis superdetermined that means there is no free will and independent choice,hence, all the choices, events and relations are already predetermined by

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physics laws, then Bells theorem might not be true and local hidden vari-ables can produce the predictions of QM. In absolute determinism everythingis completely dependent to other things and one cannot arrange detectorsand make measurements independently. It is meaningless to talk about in-dependency. But since superdeterminism cannot be tested at least in thenear future, it can be considered as deep physical or philosophical question.

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Chapter 4

Non-local Hidden VariableTheories

We discussed that local realistic hidden variable theories do not agree withthe predictions of quantum mechanics which are tested by experiments. ButBells theorem mentions nothing about non-local hidden variable theories. Infact, from an angle it says, only non-local theories can fill the reality hole ofstandard quantum mechanics. Then there is a possibility that these theoriesmight give the correct and complete interpretation of quantum mechanics.Although there are few non-local HVTs, the most well-known one is thepilot wave theory (PWT). It is also known as de Broglie- Bohm theory(attributed to them), casual interpretation (from Bohms first definition),Bohmian mechanics (more about mathematical formulation) or ontologicalinterpretation (from Bohms last definition).

4.1 Pilot wave theory

Pilot wave theory was first presented in 1927 at the Solvay Conference byLouis de Broglie, to provide a realistic interpretation of quantum mechan-ics. The theory had been constituting a serious opposition to the actualCopenhagen interpretation and largely criticized by the audience especiallyby Wolfgang Pauli. Moreover, in 1932 John von Neumann claimed in hisarticle that all hidden variable theories are impossible. (In fact his articleturned out to be wrong.) [9] Then Copenhagen interpretation became themost popular and widely accepted interpretation of QM and precipitated deBroglie to give up his theory reluctantly.

Two decades after, in 1952, David Bohm raked up de Broglies mireddown pilot wave theory and developed his approach, which can be consideredfor him as a sharp U turn from his one year old orthodox positioned Quan-tum Theory textbook. Since 1952, PWT has been debated among thephysicists who are interested in foundations of quantum mechanics. Along-

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side a lot of articles on PWT, The Undivided Universe by D. Bohm and B.J. Hiley and The Quantum Theory of Motion by Peter Holland representcomprehensive work about PWT, both published in 1993.

4.1.1 Wave and particle

Bohrs Copenhagen interpretation had complementarity principle which saysmatter is wave or particle. Pilot wave theory came up with different idea;wave and particle. According to this theory, every particle guided by awave in their trajectory in space-time. [28, 29] That pilot wave (guidingwave) controls and orients the movement of the particle.

Figure 4.1: Reality of both wave and particle in pilot wave theory [30]

Wave function provides only partial description of a quantum systemand the actual positions of the particles are determined by pilot wave. So,unlike standard quantum mechanics, this theory is deterministic at all times.Although we cannot figure out, particle is really somewhere before the mea-surement. The thing that we cannot find out, does not have to be unreal,so in PWT, there is a physical, objective reality of the particle even beforethe measurement. This objective reality clear away the subjective observersof SQM which forces system to collapse in a state. Since particles have de-terminate trajectories, there is no superposition of particles, but pilot wavecan be in superposition. Therefore in Double-slit experiment in account ofPWT, the particle does not pass through both slits, it passes through oneof the slits but its wave passes through both slits. When particle encoun-ters with two slit obstacle, its pilot wave changes its shape and develops aninterference pattern.

(,x)Quantum mechanics Pilot wave theory

The form of the pilot wave depends on the positions of every particlein the universe. By that, PWT represents an undivided universe that allthings somehow connected to each other. Non-locality is in true nature ofPWT, in that it defines a universal wave function that represents the timeevolution of all particles or the configuration of all fields. From that, for N

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Figure 4.2: Bohmian trajectories of particles in double-slit experiment [42]

particle universe, particle trajectories are governed by guiding equation thatdetermines velocities of particles in terms of the wave function:

dxidt

=~mi

Imtit

tt(x(t)) (4.1)

where is the wave function of the universe and x(t) = (x1(t), ...,xN (t))sare the positions of particles which are the hidden variables in that caseand with the Schrdinger equation they completely define particles in PWT.Note that behaviour of each particle depends on the configuration of otherparticles independent from the distances between particles. So it is ensuredthat the hidden variable in PWT is non-local. [32]

4.1.2 Quantum potential

Bohms formulation of pilot wave theory uses Hamilton-Jacobi formulationof classical mechanics as a basis. It differs from classical law of motionby a term called quantum potential. Thus, Newtonian mechanics can bederived from quantum mechanics according to the PWT. It is important foridentifying the limit between quantum and classical mechanics. Quantumpotential can be considered as potential energy of the pilot wave and todefine trajectories it creates a quantum force on particles. Quantum force

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determines accelerations of particle. Quantum potential represents particlesenergy in the wave field at a specific position. It has the information (calledactive information) of all particles which actually forms its shape. [33, 34]Lets consider Schrdinger equation;

i~

t=( ~

2

2m2 + V

) (4.2)

and write the wave function in polar form as;

= ReiS/~ (4.3)

where R = R(x, t) is a real amplitude function and S = S(x, t) is a realphase function. By inserting wave function into the Schrdinger equationand multiplying by eiS/~, we obtain;

i~[R

t+

(iR

~

)S

t

]= ~

2

2m

{2R

(R

~2

)(S)2

+ i

[(2

~

)R S +

(R

~

)2S

]}+ V R

(4.4)

by dividing its real part with R (R 6= 0), its real part gives Hamilton-Jacobiequation with an additional term which is quantum potential;

S

t+

(S)2

2m+ V ~

2

2m

2RR

= 0 (4.5)

and by dividing its imaginary part by ~/2R, its imaginary part gives conti-nuity equation;

R2

t+

(R2Sm

)= 0 (4.6)

and particle velocities in that terms (same as guiding equation);

v =Sm

(4.7)

It is important that, unlike other (classical) potentials, particle is indepen-dent from the intensity or amplitude of the quantum potential, rather itdepends on its form. [33, 34] The motion of particles are affected by thatinternal (Q) and external potentials (V );

S

t+

(S)2

2m+ V +Q = 0 (4.8)

where Q is the quantum potential,

Q = ~2

2m

2RR

(4.9)

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Although quantum potential is useful for spinless particles and it is equiva-lent to the guiding equation (4.1) for particles without spin, mostly guidingequation has been used when describing systems that particles have spins.[32]

PWT considers spin as the property of the wave function, instead ofintrinsic property of the particle. [33] Spin is also related to the motion ofthe particle, so can be also defined in guiding equation.

4.1.3 Quantum equilibrium

Pilot wave theory can reproduce all predictions of SQM provides and pre-dictions of PWT and SQM must be equivalent experimentally according tothe quantum equilibrium hypothesis, because both theory uses Schrdingerequation in terms of evolution of the wave function. [32] (Remember PWThas additional guiding function). Besides, since guiding function is hidden,we are experimenting and defining the results of the measurements as thesolutions of Schrdinger equation.

Statistical distribution of wave function in quantum mechanics (||2) isthe quantum equilibrium distribution in PWT. [32] Hence, PWT representsmore wider theory than SQM, in fact PWT includes SQM. In quantum equi-librium PWT approximates into SQM. Universe is in quantum equilibriumand because of that SQM works well in experiments.

As a matter of fact, because of that empirical equivalence, if there is anexperiment that refutes PWT, it should also refute SQM and vice versa.Also it is impossible to know or control the positions beyond quantum equi-librium distribution (||2), so that Heisenbergs uncertainty principle is pre-served for a universe in quantum equilibrium. [32]

4.1.4 Measurement process

Pilot wave theory claims to solve also the measurement problem. [32, 33, 34]It approaches to process by taking the measured system and the measure-ment device as a subsystem of a closed system (like our universe). Thus mea-surement process can be analyzed within the subsystem with a conditionalwave function, which is different from the wave function of the universe thatis used in guiding equation (4.1). This conditional wave function includesthe configuration and the environment of the subsystem. Consequently, bymeasuring the particle, conditional wave function changes because config-uration and environment changes. This change usually called as quantumdecoherence, or simply decoherence.

Quantum decoherence is the disturbance of a quantum system by en-vironment. The concept was first introduced by Bohm in 1952. [29] Sincequantum systems are so small and so open to the effects of the environment,their superposition state is not so durable and turns into a single measured

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state. It can be interpreted as loss of information, like work loss in the heattransfer of the thermodynamically irreversible thermal contact of hot andcold reservoirs. [35] While we trying to find the information of a system, weunwittingly disrupt its information.

Nevertheless, it is not only a property of PWT, but also adapted by someother interpretations like Copenhagen (contemporary) and Many-worlds in-terpretations. [36] Although these interpretations present decoherence asthe solution of the measurement problem in quantum mechanics, it is hardto say that it actually solves anything; after all the definition of interactionof environment and measurement are not precise and the structure of themeasurement mechanism is still not known exactly. Besides, it could notexplain why measured system arrive into only one of the probabilities.

There is no actual collapse in PWT. Measurement affects the conditionalwave function of the particle thus, we measure particle on a definite statethat is defined by guiding equation.

Briefly, pilot wave theory is a theory of motion, describing particlesguided by waves. [37] As we see, PWT explains almost all the phenom-ena that SQM cannot explain. As one of the last survived non-local hiddenvariable theories, pilot wave theory can be considered as the most challeng-ing opponent of the standard quantum mechanics, in the league of realism.

4.2 Leggett-type inequalities

In 2003, Anthony James Leggett discovered another inequalities that rulesout certain kind of non-local hidden variable theories by showing their in-compatibility with realism. [38] These non-local theories are called crypto-non-local. According to him, in these theories the behaviour of the emissionof entangled photon pairs is just like the behaviour of the emission of singlephoton, although there are non-local effects.

There are also experimental tests of Leggetts inequalities which showsthat crypto-non-local theories are incompatible with the predictions of QMand they could not shelter realism on their grounds. [39, 40, 41] However,Leggetts inequalities related only with some class of non-local HWTs andother kind of non-local HVTs such as PWT, are still unrefuted. [37]

4.3 Experimental tests of non-local hidden vari-able theories

Bells theorem cleared away locality, Kochen-Specker theorem eliminatednon-contextuality and finally Leggett inequalities ruled out certain type ofnon-local hidden variable theories and all of them tested by experiments.[16, 19, 39] All experiments are in favour of standard quantum mechanicsso far. But is that mean non-local hidden variable theories are stranded?

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It might be harsh to say that, since some theories, like pilot wave theory,make same predictions (with a different theory) on testable issues.

According to the quantum equilibrium hypothesis, we cannot distinguishstandard quantum mechanics and pilot wave theory with any experiment,in quantum equilibrium conditions. There is no experiment that demon-strates empirically different results between pilot wave theory and standardquantum mechanics, so far.

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Chapter 5

Discussion

5.1 Locality vs Realism

Bells theorem restricted us to make a choice between locality and realism.We couldnt choose both of them according to his inequalities. But whichone should we prefer and why?

Firstly, nature is already non-local. There are a lot of things that prop-agates superluminal; shadows, phase velocities, quantum information in en-tanglement process and so on. But none of them violates causality. In factBells theorem proves the non-local structure of universe for any formulation.So, the more proper comparison would be between causality and realism.Thus the question is; can we find an (or is there any) interpretation thatinvolves both causality and realism, without conflicting with the predictionsof quantum mechanics that are tested with experiments?

If we want to obtain realism, we must first get rid of superposition prin-ciple and clear up before the measurement process. To certify causality,we must clarify the role of time and the action-reaction (or effect-cause)mechanism in QM. However, we cannot achieve these without giving up thepostulates of SQM. Thats why we are searching for a new interpretation.

5.2 What should new interpretation looks like?

First of all, a valid interpretation would have to agree with the experimentsand since experiments are on the side of SQM in many cases, the new in-terpretation should give the same results on those cases. It should bringsatisfactory explanations to the problems of SQM such as superposition,wave function collapse and so on.

On the philosophical side of the theory; causality have to be protectedin case of any paradoxes as stated by relativity. It should contain objectiverealism, since it would be better to think that universe can also live with-out humans. Indeed, causality and realism eventually leads determinism, so

23

theory would be deterministic. (Actually the main purpose of hidden vari-ables is to make the theory deterministic.) It should be contextual which isalready ensured by non-locality.

In a more advance way, quantum mechanics with gravitation (quantumgravity) would have to be demystified, by that classical and quantum worldshould be unified in the same laws. Hence, new interpretation might alsoopen new gates for the search of theory of everything.

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Chapter 6

Conclusion

6.1 Future of hidden variable theories

Problems of standard quantum mechanics and hidden variable attempts tosolve and complete the quantum theory has been discussed up until now.Is it possible to replace quantum mechanics with a realistic theory? Arethere really hidden variables? Can we calculate or measure hidden vari-ables? These are the questions that we have to answer in order to advancea quantum theory.

Problems about the reality and objectivity in measurement theory isquite severe in QM. I really dont think an electron cares too much aboutour information about itself. There is also very nice quote about that byJohn S. Bell; Was the wave function of the world waiting to jump forthousands of millions of years until a single-celled living creature appeared?Or did it have to wait a little longer, for some better qualified system ...with a Ph.D.? [9]

The main purpose of hidden variable theories, as well as other interpre-tations of QM, is to present a complete and actual definition of the motionin microscopic scale. Another interpretation does not mean QM was en-tirely wrong. At least from experiments it is shown that SQM is a good wayto explain the motion in microscopic scale. But the possibility of a largerconcept or theory is still wide open.

Even though there is no such hidden variable theory, dealing with thesesubjects would lead us to more satisfactory explanations and evidences onfoundations of quantum mechanics. After all, even John S. Bell had in-spired from hidden variable theories and found his Bells theorem while hewas actually trying to prove the existence of hidden variables. He also ad-mired Bohms work and considered pilot wave theory as one of the possibleconsistent interpretations of quantum mechanics.

Status quo may attract people but if there is even a single mathematicallyand experimentally consistent physical objection to a theory, it must be

25

considered and current theory must be queried. Science is not dogmatic.Therefore, we should always approach to problems with a grain of salt, onlywithout ditching mathematical, physical and logical consistency.

26

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