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Decoherence in Infinite Quantum Systems Decoherence in Infinite Quantum Systems von Mario Hellmich Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urPhysik der Universit¨atBielefeld Januar 2009 iii 1. Gutachter: Prof. Dr. Philippe Blanchard (Bielefeld) 2. Gutachter: Prof. Dr. Robert Olkiewicz (Breslau) 3. Gutachter: Prof. Dr. Gianfausto Dell'Antonio (Rom) Tag der Disputation: 30. Januar 2009 iv Acknowledgments First and foremost I would like to express my heartfelt thanks to Prof. Dr. Philippe Blanchard for supervising this thesis. Through his enthusiasm for the subject and through many spirited discussions (not only about physics or mathematics) he had a great influence on this work, and on many occasions he has put me on the right track. His constant encouragement and support in scientific as well as organizatorial and fundraising matters have been a great help to me without which this thesis would not have come to completion. I would like to thank Prof. Dr. Robert Olkiewicz for his fruitful collab- oration and for sharing many of his ideas with me. It was always very inspiring and rewarding to discuss with him. His hospitality in Wroc law and his immediate willingness to write a report on this thesis are greatly appreciated. Prof. Dr. Dell'Antonio provided the third report on this thesis, I would like to thank him for his interest and his efforts. I am grateful to the International Graduate School \GRK 881" Biele- feld{Paris for financial support. Moreover, thanks are due to Evangelisches Studienwerk Villigst e. V. for funding a large part of this work and for the many experiences and opportunities beyond the financial support. I would like to express my sincere thanks to our secretary Hannelore Litschewsky for the many favors she has done to me during the last few years and for always finding time for a chat. Many people have influenced me, and hence this work, during its mak- ing, and I would like to thank them for what they have done for me. Special mention deserve my parents, without their support and encour- agement I would have been lost on many occasions. Last, but not least, I am grateful to Hanna-Lena for cheering me up in difficult moments. Contents Introduction 1 1. Quantum Probability and Open Systems 5 1.1. Mathematical Description of Physical Systems . 5 1.1.1. States and Observables . 5 1.1.2. C*-Algebraic Description of Physical Systems . 7 1.1.3. C*-Algebras and von Neumann Algebras . 10 1.2. Quantum Probability and Physical Systems . 14 1.2.1. Algebraic Classical Probability . 15 1.2.2. The Quantum Generalization . 17 1.2.3. Classical Physical Systems . 18 1.2.4. Quantum Systems . 20 1.2.5. The Representation Problem . 22 1.2.6. Infinite Systems . 27 1.2.7. Differences Between Classical and Quantum Systems 29 1.3. Open Systems and Dynamics . 33 1.3.1. Dynamics of Subsystems and Conditional Expecta- tions . 33 1.3.2. Reduced Dynamics and Dynamical Semigroups . 36 1.4. Fixed Points and Invariant States . 39 1.4.1. Fixed Points . 39 1.4.2. Almost Periodicity . 40 1.4.3. Amenability and Mean Ergodicity . 44 1.4.4. Invariant and Subinvariant States . 49 2. Decoherence : Technical Tools 51 2.1. Mathematical Preliminaries . 51 2.1.1. Tomita{Takesaki Theory . 51 2.1.2. The KMS Condition . 53 2.2. Asymptotic Splittings . 54 2.2.1. Isometric-Sweeping Splitting . 54 2.2.2. Jacobs{de Leeuw{Glicksberg Splitting . 59 2.2.3. An Example . 67 v vi Contents 2.3. Detailed Balance . 69 2.3.1. Definition and Properties . 69 2.3.2. Equivalent Characterization of Detailed Balance . 74 2.3.3. Sufficient Conditions for Detailed Balance . 80 2.3.4. Detailed Balance and Approach to Equilibrium . 82 3. Decoherence : Physical Interpretation 85 3.1. Decoherence in the Algebraic Formulation . 85 3.1.1. Introduction . 85 3.1.2. The Notion Decoherence in Algebraic Formulation . 88 3.2. Some General Theorems on Decoherence . 90 3.2.1. Sufficient Conditions . 90 3.2.2. Structure of the Algebra of Effective Observables . 93 3.2.3. Scenarios of Decoherence . 97 3.2.4. Mesurements . 99 4. Irreversible Evolutions on the CCR 103 4.1. Quasifree Maps . 104 4.2. Perturbed Convolution Semigroups . 105 4.2.1. Definition . 105 4.2.2. Construction of Perturbed Convolution Semigroups . 106 4.2.3. General Representation of a Perturbed Convolution Semigroup . 108 4.3. Dynamical Semigroups on A(S; σ) . 110 4.3.1. Symplectic Groups . 110 4.3.2. Nonsymplectic Semigroups . 111 4.4. Extension to Representations of the CCR Algebra . 119 4.4.1. The Extension Theorem . 119 4.4.2. A Generalization of the Extension Theorem . 122 4.5. Generators . 124 4.5.1. The Gaussian Case . 125 4.5.2. The Poisson Case . 130 4.5.3. The Dirac Case . 131 4.6. Normal Invariant States . 132 4.7. Detailed Balance for Quasifree Semigroups . 135 4.7.1. General Condition . 135 4.7.2. Example . 136 5. Conclusion and Open Problems 139 5.1. Conclusion . 139 5.2. Open Problems . 141 Contents vii A. The CCR Algebra 145 A.1. Definition and Properties . 145 A.1.1. Definition . 145 A.1.2. Properties . 147 A.1.3. Generating Functionals . 150 A.1.4. Automorphisms . 151 A.2. Regular and Quasifree Representations . 152 A.2.1. Regular Representations . 153 A.2.2. Quasifree States and Representations . 158 B. Extension of Semigroups 161 B.1. Extension of Semigroups on Banach Spaces . 162 B.2. Extension of Semigroups on von Neumann Algebras . 166 C. Nuclear Spaces and Maps 169 C.1. Nuclear Maps . 169 C.2. Nuclear Spaces . 172 D. Cylindrical Measures 177 D.1. Cylindrical Measures and Functions . 177 D.1.1. Cylinder Sets . 177 D.1.2. Cylindrical Measures . 179 D.2. The Generalized Minlos{Sazonov Theorem . 181 Bibliography 183 Summary 191 Introduction The theory of environmental decoherence starts from the question why the objects surrounding us obey the laws of classical physics, despite the fact that our most fundamental physical theory|quantum theory|results in contradictions to experimental and everyday experience when directly applied to these objects. This is an embarrassing situation since from its inception in the 1920's until today quantum theory has seen a remarkable success and an ever increasing range of applicability. Thus the question how to reconcile quantum theory with classical physics is a fundamental one, and efforts to find answers to it persisted throughout its history. Although there is still no general consensus how an answer can be achieved, environmental decoherence today is the most promising one and the one most widely discussed. The answer offered by the program of environmental decoherence to the question posed above is that quantum theory is universally valid, in partic- ular also in the macroscopic domain, but that one has to take into account the fact that macroscopic objects are strongly interacting with their en- vironment, and that precisely this interaction is the origin of classicality in the physical world. Thus classicality is a dynamically emergent phe- nomenon due to the interaction of quantum systems with other quantum systems surrounding them. In the past twenty years decoherence has been the subject of intensive theoretical and experimental research and since then has generated an enormous amount of literature. A search in the Science Citation Index for \decoherence" yields well over 5 000 hits. Despite this massive progress there is still a lot of discussion about the meaning, prospects and possibil- ities of decoherence going on. In order to clarify the status of decoherence and to provide a rigorous definition, Ph. Blanchard and R. Olkiewicz sug- gested a notion of decoherence formulated in the algebraic framework of quantum physics in [46], drawing on earlier work of R. Olkiewicz [105]. The algebraic framework is an alternative mathematical formulation of quantum physics besides the traditional textbook formulation, which is contained in the algebraic framework as a special case. However, it is more powerful since it can also accommodate classical systems, provides an el- egant formulation of superselection rules and can accommodate systems 1 2 Introduction with infinitely many degrees of freedom rigorously, which is important in order to discuss systems in the thermodynamical limit and in quantum field theory. Thus it is particularly well suited for the discussion of the dynamical emergence of classical properties. Recently the algebraic for- mulation becomes increasingly popular in the discussion of foundational and philosophical problems of quantum physics, see e. g. [90], and the work of Ph. Blanchard and R. Olkiewicz on decoherence was appraised by N. P. Landsman who declared that it \should provide important motivation for specialists in the foundations of quantum theory to learn the theory of operator algebras" [91]. To advertise the algebraic framework, especially for our purpose, we start in Chapter 1 by explaining briefly its key features; in particular, we emphasize that the formalism of quantum theory may fruitfully be viewed as a generalization of classical Kolmogorovian probability theory. In fact, as R. F. Streater contends, one should view \quantum theory is a generalisation of probability, rather than a modification of the laws of mechanics" [123]. In Chapter 1 we also introduce the necessary math- ematical tools to describe the dynamics of open systems, in particular quantum dynamical semigroups, which are fundamental for decoherence, as well as a number of useful mathematical tools. In Chapter 2 we develop the mathematical machinery which will allow us to prove that a given quantum dynamical semigroup displays the deco- herence effect. Besides giving a new proof for a special case of a theorem ofLugiewiczand Olkiewicz [94] we will also discuss the Jacobs{de Leeuw{ Glicksberg splitting for quantum dynamical semigroups as well as a notion of detailed balance which will lead to a sufficient condition for a quantum dynamical semigroup to display decoherence in the sense of Blanchard and Olkiewicz.
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