Classical Systems in Quantum Mechanics

Classical Systems in Quantum Mechanics Pavel Bóna, e-mails: [email protected] [email protected] Home web-page: http://www.st.fmph.uniba.sk/ bona1/ ∼ Department of Theoretical Physics, Comenius University SK-842 48 Bratislava, Slovakia July 21, 2020 arXiv:1911.02081v2 [math-ph] 20 Jul 2020 2 Preface The work contains a description and an analysis of two different approaches determining the connections between quantal and classical theories. The first approach associates with any quantum-mechanical system with finite number of degrees of freedom a classical Hamiltonian system ‘living’ in projective Hilbert space P ( ), and it is called here the ‘classical projection’. H The second approach deals with ‘large’ quantal (= quantum mechanical) systems in the limit of infinite number of degrees of freedom and with their corresponding ‘macroscopic limits’ described as classical Hamiltonian systems of the system’s global (intensive) quantum observables. The last part of this work contains a series of models describing interactions of the “small” physical (micro)systems with the “macroscopic” ones, in which these interactions lead to a (macroscopic) change of some “classical” parameters of the large systems. These models con- nect, in a specific way, the two classes of the systems considered earlier in this work by modeling their mutual interactions leading to striking (i.e. theoretically impossible in the framework of finite quantum systems) results. The projective space P ( ) of any complex Hilbert space is endowed with a natural H H symplectic structure, which allows us to rewrite the quantum mechanics of systems with finite number of degrees of freedom in terms of a classical Hamiltonian dynamics. If a quantum- mechanical system is associated with a continuous unitary representation U(G) of a connected Lie group G on , the orbits (possibly factorized in a natural way) of the projected action H of U(G) in P ( ) are naturally mapped onto orbits of the coadjoint representation Ad∗(G) of H G. These coadjoint orbits have a canonical symplectic structure which coincides with the one induced from the structure of P ( ). For important classes of physical systems these symplectic spaces are either symplectomorphicH to the ‘corresponding’ classical phase spaces, or they are some extensions of them (describing, e.g. particles with ‘classical spin’). Quantal dynamics is projected onto these phase spaces in a natural way, leading to classical Hamiltonian dynamical systems without any limit of Planck constant ~ 0. → For a large (infinite) quantal system an automorphic group action of G on the C∗-algebra A of its bounded observables enables us to define a macroscopic subsystem being a classical Hamiltonian system of the same type as we obtained in the case of finite number of degrees of freedom. There is a difference, however, between the interpretations of ‘classical projections’ and of these ‘macroscopic limits’: The classical (mechanical) projection describes classical mechanics of expectation values of quantal observables whereas the macroscopic limit describes a quantal subsystem with classical properties - its observables are elements of a subalgebra MG of the center Z of the double dual A∗∗ of A. Any state ω on A has a unique ’macroscopic limit’ pM ω which is represented by a probability measure on the corresponding (generalized) classical phase space. This offers us a possibility of deriving a classical (macroscopic) time evolution 3 (which is, in general, in a certain sense stochastic, cf. [29, Sec.III.G]) from the underlying reversible quantal dynamics. A scheme of ‘macroscopic quantization’ is outlined, according to which a (nonunique) re- construction of the infinite quantal system (A; σG) from its macroscopic limit is possible. By determining a classical Hamiltonian function in the macroscopic limit of (A; σG) we can define a ‘mean-field’ time evolution in the infinite system (A; σG). Our definition of the ’mean-field’ evolutions extends the usual ones. The schemes and results developed in the work are applica- ble to models in the statistical mechanics as well as in gauge-theories (in the ‘large N limit’). They might be relevant also in general considerations of ’quantizations’ and of foundations of quantum theory. The last Chapter of this work is devoted to the description of several models of interacting ‘microsystems’ with ‘macrosystems’, mimicking a description of the ‘process of measurement in QM’. In these models, certain ‘quantal properties’ of the system, namely a (coherent) su- perposition of specific vector states (eigenstates of a ’measured’ observable), transform by the unitary continuous time evolutions (for t ) into the corresponding ‘proper mixtures’ of → ∞ macroscopically different states of the ‘macrosystems’ occurring in the models. In this connection we shall shortly discuss the old ‘quantum measurement problem’, which however, in the light of certain experiments performed in the last decades and suggesting the possibilities of quantum-mechanical interference of several macroscopically different states of a macroscopic system, need not be at all a fundamental theoretical problem; this might mean that the often discussed ‘measurement process’ can be included into the presently widely ac- cepted model of quantum theory. Acknowledgemets: This work is a revised and completed version of the unpublished text: “Classical Projections and Macroscopic Limits of Quantum Mechanical Systems” written roughly in the years 1985 - 1986. The author is indebted to Klaus Hepp for his stimulations and the kind help with correcting many formulations of the original text. During the work on the old text, encouragements by Elliott Lieb and by the late Walter Thirring were also stimulating, and the discussions with the colleagues, the late Ivan Korec, Milan Noga, Peter Preˇsnajder and in particular with Jiˇr´ıTolar, were useful and the author expresses his gratitude to all of them. Thankfulness for several times repeated encouragements to publish the old text should be expressed to Nicolaas P. Landsman. Author’s thanks belong also, last but not least, to his colleague Vladim´ırBalek for his kind help with final formulation of important parts of this book. Technical notes: (a) This book contains several technical concepts which are not introduced here in details. The readers needing to get a brief acquaintance with some additional elementary concepts and facts of topology, differential geometry (also in infinite dimensions), group theory, or theory of Hilbert space operators and theory of operator algebras, could consult e.g. the appendices of the freely accessible publication [37], and the literature cited in our 4 Bibliography. Due to connections of many places in the text of this book with the content of the work [37] it is recommended to keep the cited [37] as a handbook. The separate complementary text of [37, Textbook] might be also helpful. The frequent citations from [37] contain usually references to specific places of the cited work. (b) Two kinds of quotation marks are used: Either the ones which stress some “standard expressions”, or those which indicate ‘intuitive denotations’. The difference between these two is not, however, very sharp. (c) Many symbols appearing in mathematical formulas are introduced in various places of the text and repeatedly used in the rest of the book. For easier revealing of their meanings, they are included into Index and their first appearance in the text is stressed, sometimes in a not quite usual manner, by boldface form of a part of the surrounding text. Contents Preface ......................................... 2 1 Introduction 7 1.1 MotivationandSummary .............................. 7 1.2 QuantumMechanics ................................. 12 1.3 Classical Hamiltonian mechanics . 16 1.4 Quantumtheoryoflargesystems . ... 20 2 Geometry of the state space of quantum mechanics 25 2.1 Manifold structure of P ( ) ............................. 25 H 2.2 Symplecticstructure ................................ 29 2.3 Quantum mechanics as a classical Hamiltonian field theory . ..... 31 3 Classical mechanical projections of QM 37 3.1 Orbits of Lie group actions on P ( ) ........................ 37 H 3.2 Classical phase spaces from the quantal state space . ......... 40 3.3 Classical mechanical projections of quantal dynamics . ........ 43 4 Examples of classical mechanical projections 50 4.1 TheHeisenberggroup(CCR) . .. .. 50 4.2 ExtensionofCCRbyaquadraticgenerator . .... 57 4.3 Notesonotherexamples .............................. 62 5 Macroscopic limits 73 5.1 Multiplesystems ................................... 73 5.2 Generalized macroscopic limits . 101 6 Mathematical structure of QM mean-field theories 109 6.1 Generalconsiderations .............................. 109 6.2 Spin systems with polynomial local Hamiltonians QN ...............115 6.3 Time evolution in generalized mean-field theories . 132 6.4 Equilibrium states . 144 6.5 An example: The B.C.S. model of superconductivity . 152 5 6 CONTENTS 7 Some models of “quantum measurement” 158 7.1 Introductorynotes ................................ 158 7.2 On ‘philosophy’ of ”models” . 160 7.3 QuantumDomino...................................161 7.4 Particle detection - a “nonideal” measurement . 169 7.5 TheX-Ychainasameasuringdevice . 184 7.6 Radiating finite spin chain . 189 7.7 Onthe“measurementproblem”inQM . 196 Bibliography 201 Index 222 Chapter 1 Introduction 1.1 Motivation and Summary 1.1.1. Successful communication and manipulation with ‘objects’ requires construction of

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