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Stochastic Mechanics Without Ad Hoc Quantization: Theory and Applications to Semiclassical Gravity

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Stochastic Mechanics Without Ad Hoc Quantization: Theory and Applications to Semiclassical Gravity Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity arXiv:1804.01394v1 [quant-ph] 31 Mar 2018 Stochastic Mechanics Without Ad Hoc Quantization: Theory And Applications To Semiclassical Gravity Stochastische mechanica zonder ad hoc quantisatie: theorie en toepassingen op semi-klassieke zwaartekracht (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. G. J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op donderdag 19 oktober 2017 des middags te 12.45 uur door Maaneli Derakhshani geboren op 3 oktober 1985 te Rancho Palos Verdes, Verenigde Staten van Amerika Promotoren: Prof. dr. R. B. Mann Prof. dr. L. T. G. Theunissen Copromotor: Dr. G. Bacciagaluppi Papers and Publications Chapter 2 • A Suggested Answer To Wallstrom’s Criticism: Zitterbewegung Stochastic Mechanics I Maaneli Derakhshani https://arxiv.org/abs/1510.06391 Chapter 3 • A Suggested Answer To Wallstrom’s Criticism: Zitterbewegung Stochastic Mechanics II Maaneli Derakhshani https://arxiv.org/abs/1607.08838 Chapter 4 • Semiclassical Newtonian Field Theories Based On Stochastic Mechanics I Maaneli Derakhshani https://arxiv.org/abs/1701.06893 Chapter 5 • Semiclassical Newtonian Field Theories Based On Stochastic Mechanics II Maaneli Derakhshani https://arxiv.org/abs/1702.02472 Work which is not directly part of this dissertation (but is referenced throughout): Probing Gravitational Cat States In Canonical Quantum Theory vs. Objective Collapse • Theories Maaneli Derakhshani https://arxiv.org/abs/1609.01711 Probing a Gravitational Cat State: Experimental Possibilities • Maaneli Derakhshani Journal of Physics: Conference Series, Volume 1, conference 1 (2016) https://arxiv.org/abs/1603.04430 i Newtonian Semiclassical Gravity in the Ghirardi-Rimini-Weber Theory with Matter Den- • sity Ontology Maaneli Derakhshani Physics Letters A, Volume 378, Issue 14-15 (2014) https://arxiv.org/abs/1304.0471 The Newtonian Limit Of Stochastic Gravity Doesn’t Entail The Many-Body Stochastic • Schr¨odinger -Newton Equations Or Gravitationally-Induced Wavefunction Collapse Maaneli Derakhshani (In preparation) On Multi-Time Correlations And Effective Collapse In Stochastic Mechanics • Maaneli Derakhshani and Guido Bacciagaluppi (In preparation) ii Contents 1 Introduction 11 2 A Suggested Answer To Wallstrom’s Criticism: ZSM I 19 2.1 Introduction.................................... 19 2.2 Nelson-Yasue Stochastic Mechanics . ........ 23 2.3 Wallstrom’sCriticism . 31 2.4 Classical Model of Constrained Zitterbewegung Motion . ............. 34 2.4.1 Onefreeparticle ............................... 35 2.4.2 Classical Hamilton-Jacobi statistical mechanics for one free particle . 37 2.4.3 One particle interacting with external fields . ......... 39 2.4.4 Classical Hamilton-Jacobi statistical mechanics for one particle interact- ingwithexternalfields. .. .. .. .. .. .. .. .. 43 2.5 Zitterbewegung Stochastic Mechanics . ......... 45 2.5.1 Onefreeparticle ............................... 45 2.5.2 One particle interacting with external fields . ......... 56 2.5.3 The central potential revisited . ..... 65 2.6 Conclusion ...................................... 66 2.7 Acknowledgments................................. 66 3 A Suggested Answer To Wallstrom’s Criticism: ZSM II 67 3.1 Introduction.................................... 67 3.2 Nelson-Yasue Stochastic Mechanics for Many Particles . ............. 68 3.3 Interpretation of the Nelson-Yasue wavefunction . ............ 78 3.4 Classical Model of Constrained Zitterbewegung Motion for Many Particles . 83 3.4.1 Free zbw particles .............................. 83 3.4.2 Classical Hamilton-Jacobi statistical mechanics for free zbw particles . 84 3.4.3 External fields interacting with zbw particles ............... 86 3.4.4 Classical Hamilton-Jacobi statistical mechanics for zbw particles inter- actingwithexternalfields . 89 3.4.5 Classically interacting zbw particles .................... 91 iii Contents 3.4.6 Classical Hamilton-Jacobi statistical mechanics for two interacting zbw particles ................................... 96 3.4.7 Remarks on close-range interactions . 101 3.5 Zitterbewegung Stochastic Mechanics . ......... 102 3.5.1 Free zbw particles .............................. 102 3.5.2 Classical fields interacting with zbw particles . 119 3.5.3 Remark on on close-range interactions . 132 3.6 Plausibility of the Zitterbewegung Hypothesis . ............ 132 3.7 ComparisontoOtherAnswers. 137 3.8 Conclusion ...................................... 141 3.9 Acknowledgments................................. 141 4 Semiclassical Newtonian Field Theories Based On Stochastic Mechanics I 143 4.1 Introduction.................................... 143 4.2 Overview of ZSM for many free particles . ....... 147 4.3 ZSM-Newton/Coulomb: Basic equations . ....... 159 4.4 Schr¨odinger-Newton/Coulomb equations as mean-field theories . 176 4.5 Comparison to other semiclassical Newtonian field theories ........... 190 4.5.1 Comparison to non-hidden-variable approaches . ......... 190 4.5.2 Comparison to alternative hidden-variable approaches .......... 194 4.6 Conclusion ...................................... 207 4.7 Acknowledgments................................. 208 5 Semiclassical Newtonian Field Theories Based On Stochastic Mechanics II 209 5.1 Introduction.................................... 209 5.2 Large N center-of-mass approximation in ZSM-Newton/Coulomb . 211 5.2.1 Generalapproach............................... 211 5.3 Classical nonlinear Schr¨odinger equation for large N center-of-mass motion . 224 5.3.1 Oriols et al.’s derivation . 224 5.3.2 Conditional Madelung equations . 225 5.3.3 Comments on the classical nonlinear Schr¨odinger equation . 230 5.4 Recovering classical Newtonian gravity for many macro particles . 230 5.5 Recovering classical Vlasov-Poisson mean-field theory .............. 233 5.6 Incorporating environmental decoherence . .......... 235 5.7 Conclusion ...................................... 236 iv Contents 6 Appendix to Chapter 2 239 6.1 Proof of the 1-particle Stochastic Variational Principle.............. 239 6.2 Classical Zitterbewegung in the Central Potential ................. 240 7 Appendix to Chapter 3 243 7.1 Proof of the N -particle Stochastic Variational Principle . 243 8 Summary and Outlook 245 9 Samenvatting 247 10 About the author 249 Bibliography 251 v Preface This thesis is the culmination of a nearly decade-long journey. The journey started when I was a sophomore undergraduate at Stony Brook University (SBU), excitedly learning about Nelson’s stochastic mechanics and the “Wallstrom criticism” thereof from the papers of Dr. Guido Bacciagaluppi, and from discussions with Prof. Sheldon (Shelly) Goldstein at Rutgers University. Having found stochastic mechanics a compelling way to make sense of quantum mechanics, I was determined to find an answer to the criticism. I explored various possibilities to no avail until, one day, out of sheer luck, I stumbled upon a dusty, yellow-paged book entitled “Observation and Interpretation: A Symposium of Philosophers and Physicists”, in the bookcase of Dr. John No´e, the Director of the Laser Teaching Center at SBU. (Thank you for the book, John.) This book, published in 1957, had among its contributions a little-known paper by David Bohm entitled “A proposed explanation of quantum theory in terms of hidden variables at a sub-quantum-mechanical level”. In this paper, Bohm sketches a model - which he credits to Louis de Broglie (in fact, Bohm’s model was just a slight reformulation of an idea suggested by de Broglie in the latter’s Ph.D thesis, an idea which also happens to be the precursor to the de Broglie-Bohm pilot-wave theory) - of an elementary particle as a localized periodic process of fixed frequency in a hypothetical sub-quantum medium. Invoking the Lorentz transformation of special relativity, Bohm shows how the model recovers a quantization condition of Bohr- Sommerfeld type for the phase of the periodic process, and how this quantization condition is related to the single-valuedness condition on wavefunctions in quantum mechanics. In fact, the quantization condition obtained in the model of de Broglie and Bohm is precisely what’s needed for stochastic mechanical theories to recover the Schr¨odinger equation of non-relativistic quantum mechanics (as I will explain in the thesis). And it was precisely the lack of justifi- cation for this quantization condition in stochastic mechanics that Wallstrom emphasized and criticized in the late 80’s and early 90’s. So it became clear to me how to answer the criticism - reformulate Nelson’s stochastic mechanics so as to consistently incorporate the model of de Broglie and Bohm. Throughout my circuitous path through graduate school, I persistently worked on this prob- lem, hoping to base my Ph.D thesis on it. Eventually I would find the opportunity to do so under the supervision of Guido (mentioned above), Prof. Robb Mann (at the University of Waterloo), Prof. Bert Theunissen (at Utrecht University), and the flagship of Utrecht Uni- vii Contents versity. Along the way, I extended the reformulated stochastic mechanics to the domain of semiclassical Newtonian gravity, finding that it has advantages over other approaches to semi- classical Newtonian gravity (e.g., the Schr¨odinger-Newton equation). The result is the thesis before you. Maaneli (Max) Derakhshani Utrecht,
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