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Backward Causation in Weak Measurements

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Backward Causation in Weak Measurements Universidad de Los Andes Undergraduate Thesis Backward Causation in Weak Measurements Author: Supervisor: Sebasti´an Murgueitio Alonso Botero, Ph.D Ram´ırez A thesis submitted in fulfilment of the requirements for the degree of Physicist in the Department of Physics August 6, 2014 Declaration of Authorship I, Sebasti´an Murgueitio Ram´ırez, declare that this thesis titled, 'Backward Causation in Weak Measurements' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: i \What is time? If no one asks me, I know; but if I were desirous to explain it to one that should ask me, plainly I know not ". Saint Augustine UNIVERSIDAD DE LOS ANDES Abstract Faculty of science Department of Physics Physicist Backward Causation in Weak Measurements by Sebasti´an Murgueitio Ram´ırez In this thesis I argue that the weak values revealed in weak measurements are to be understood as processes that involve backward causation. In the first part of the the- sis a philosophical examination of the concepts of causation and backward causation is provided. In this part the main theories of causation are examined, and the principal objections against the possibility of backward causation are addressed. In the second part of the work the theory of weak measurements is presented, together with the prob- lem of how these weak measurements are to be interpreted. In the last part of the work, the dependence of the weak value on the future post-selection of the system is carefully examined. Different positions around this dependence on the future are analyzed and it is argued that the only satisfactory way of explaining the mentioned dependence is by appealing to backward causation. At the end of the work an optical experiment, which is a modified Bell-type experiment, is proposed. This experiment will not only stress the allegedly \backward in time" nature of the weak values but will also throw new lights into the old Einstein-Bohr debate around the completeness of the quantum description of nature. Acknowledgements I would like to thank my advisor Alonso Botero who introduced me to the fascinating field of weak measurements, and who also gave me great advice during this semester. Alonso encouraged me to pursue a project that kept my main interests close, namely, physics and philosophy. It certainly was a fruitful (and demanding) experience. I would also like to thank Veronica, with whom I shared countless discussions about time and causation. Without her help this project would have been more difficult to carry out. In addition, I would like to thank Alejandra Valencia, who helped me plan the experiment proposed in the last chapter of the present work. I am very grateful to Pedro and Leonardo as well, with them I had many interesting and helpful debates about quantum mechanics. I also want to thank my friends Jose, Camilo, Manuelito, Laco, Maria P. who have made these past years very nice and unforgettable. Last but not least, I want to thank my lovely family for their constant support and their help, they encouraged me to study physics and philosophy and provided me both the material means and the emotional support needed to pursuit my studies. iv Contents Declaration of Authorshipi Abstract iii Acknowledgements iv List of Figures viii List of Tablesx 1 Introduction1 1.1 Thesis......................................2 1.2 Structure of the work.............................3 I Backward Causation5 2 Causation6 2.1 What is causation?...............................6 2.1.1 Regularist theories of causation....................7 2.1.2 Counterfactual theories of causation.................8 2.1.3 Probabilistic theories of causation..................8 2.1.4 Manipulabilists theories........................9 2.1.5 Process theories of causation..................... 10 3 Backward causation 12 3.1 Historical approach............................... 12 3.1.1 Theories of causation dealing with retro causation......... 15 3.2 Objections.................................... 17 3.2.1 The Bilking Argument......................... 17 3.2.2 A brief discussion of the arrow of time................ 18 3.2.2.1 The symmetry of the second law.............. 18 3.2.2.2 A subjective arrow of time................. 21 3.2.2.3 The Block Universe...................... 22 3.3 Backward causation in physics........................ 25 v Contents vi 3.3.1 The Wheeler-Feynman absorber theory of radiation........ 26 3.3.2 Crammer's transactional interpretation of quantum mechanics.. 29 3.3.3 Wheeler delayed choice experiment.................. 32 3.4 Broad overview of the chapter......................... 34 II Weak Measurements 35 4 Indirect Measurements 36 4.1 Indirect or ancilla measurement........................ 36 4.1.1 Interaction between the system and the pointer........... 37 4.1.2 Reading the meter........................... 41 4.2 The Von Neumann protocol.......................... 43 4.3 Some results................................... 45 4.4 Example..................................... 48 5 Weak Measurements 51 5.1 An intuitive approach............................. 51 5.1.1 Definition of the weak value...................... 55 5.2 Mechanical interpretation of weak values................... 56 5.2.1 Playing with pre and post-selected ensembles............ 56 5.2.2 The action-reaction picture...................... 59 5.2.3 General pointer variable statistics.................. 61 5.2.4 Comments................................ 64 5.3 Example 2.................................... 66 5.4 Final Remarks of the chapter......................... 70 III Backward causation revealed in weak measurements 71 6 On the physical meaning of weak values 72 6.1 Two possible objections............................ 73 6.1.1 Against the postulates of quantum mechanics?........... 73 6.1.2 Weak values can be complex or very eccentric............ 73 6.2 Two Stern-Gerlach Experiment........................ 75 6.2.1 Predictions in the weak regimen................... 77 6.3 More objections................................. 79 6.3.1 The Error View (EV)......................... 79 6.3.2 The coincidence view (CV)...................... 79 6.3.3 The no dependence on the future view (NDFV)........... 81 7 Backward causation in weak measurements 83 7.1 An experiment................................. 83 7.2 Hidden variables, again............................ 88 7.2.1 The independence assumption (IA).................. 89 7.2.2 A common cause in the future.................... 91 7.3 The Two-State Vector Formalism (TSVF).................. 93 7.3.1 The ABL rule.............................. 93 Contents vii 7.3.2 The Two States............................. 94 7.3.3 TSVF and weak measurements.................... 96 7.3.4 A last objection............................ 98 7.4 Summary of the chapter............................ 100 8 Conclusions 103 A Derivation of some results 106 A.1 Derivation of Eq. 4.13............................. 106 A.2 Derivation of Eq. 4.16............................. 106 A.3 Derivation of Eq. 4.22............................. 107 B Details of the experiment 108 Bibliography 116 List of Figures 3.1 Set up of Wheeler's delayed choice experiment............... 33 4.1 Pointer's distribution for an hypothetical case in which jα0j2 > jβ0j2, ∆Q = 0:01 and g = 2=~............................. 49 5.1 Pointer's distribution for ∆Q^ = 1....................... 52 5.2 Pointer's distribution for ∆Q^ = 10....................... 53 5.3 Probability density for m, g = 20=~ ...................... 53 5.4 g = 0:002=~ ................................... 54 5.5 Pointer's distribution for the pre and post selected states of the example 2. The dispersion is ∆Q = 20, which corresponds to the weak regimen (it is much bigger than the difference between the eigenvalues). I have drawn a line centered around the peak of the distribution so that the reader can easily see that this value is approximated equal to the weak value computed with 5.38............................ 68 5.6 Pointer's distribution for the observable S^z with the condition of the pre and post selected states of example 2. I have set ∆Q = 0:1, which corre- sponds to the strong regimen. Note that we have sharp peaks, centered around the eigenvalues 1 and −1. The right peak is considerable higher, as we expect from the ABL rule applied to this particular example (see footnote in the present or previous page)................... 69 6.1 Two Stern-Gerlach............................... 75 6.2 Screen Two Strong measurements....................... 76 6.3 Weakscreen................................... 77 7.1 Set up of the proposed experiment...................... 84 7.2 List of position per photon.......................... 85 7.3 Table after post-selection........................... 86 7.4 Schematic representation of the outcomes in the case we post-select the final state
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