Classical Observables, Measurement, and Quantum Mechanics

Classical Observables, Measurement, and Quantum Mechanics Thomas Breuer Peterhouse Submitted to the University of Cambridge in fulfilment of the requirements for the Doctor of Philosophy in the Department of History and Philosophy of Science September 1994 Agimagimnak´ Contents Introduction 7 1 Self-Reference and the Universal Validity of Quantum Mechanics 13 1.1 Previous arguments against deterministic universally valid theories . 15 1.2 The impossibility of accurate self- measurements . 21 1.3 Universal Validity Revisited . 35 1.4 Application to quantum measurements . 39 1.4.1 EPR-correlations and the universal validity of quantum mechanics . 39 1.4.2 Subjective Decoherence . 43 1.4.3 Wigner’s friend . 51 2 Phase Space Quantisation 55 2.1 Classical Mechanics . 58 2.1.1 Phase space and configuration space as homoge- neous spaces . 58 2.1.2 Lattice theoretic and algebraic formulation of classical mechanics . 63 2.2 Quantisation on Configuration Space . 72 3 4 CONTENTS 2.2.1 Systems of Imprimitivity . 78 2.2.2 The commutation relations . 80 2.2.3 Classification of configuration space quantisations. 82 2.2.4 Algebraic description of systems of imprimitivity 85 2.3 Phase space quantisation of the Weyl system . 87 2.3.1 Phase space quantisation by systems of covariance 88 2.3.2 Phase space representation of the Weyl system . 91 2.3.3 Systems of covariance in geometric quantisation. 105 2.4 Generalised systems of covariance . 117 2.4.1 Phase space quantisation by generalised systems of covariance . 127 2.4.2 Existence of transitive generalised systems of covariance . 129 2.5 An extension and an imprimitivity theorem . 134 2.5.1 An extension theorem . 136 2.5.2 An imprimitivity theorem . 140 3 The Classical Limit 145 3.1 Classical observables in quantum mechanics . 147 3.1.1 Hilbert space formulation . 147 3.1.2 Algebraic formulation . 151 3.1.3 The infinite tensor product . 158 3.2 Simultaneous Sharp Measurability of Position and Mo- mentum in Infinite Quantum Systems . 163 3.2.1 Joint position-momentum observables . 165 3.2.2 Elementary Systems . 167 3.2.3 Finite Systems . 170 3.2.4 Infinite systems . 176 CONTENTS 5 4 The Classical Pointer in Quantum Measurements 183 4.1 Previous arguments for and against classical pointer observables . 184 4.1.1 The projection postulate . 185 4.1.2 The ignorance interpretation . 190 4.2 Inaccurate measurements of non-normal states? . 195 4.2.1 Non-normal states in traditional quantum mechanics . 197 4.2.2 Measurability of non-normal states in algebraic quantum mechanics . 203 4.3 Inaccurate experiments can only be described by a classical pointer . 208 4.3.1 The requirement of finite measurement accuracy . 210 4.3.2 W ¤-version of the proof . 214 4.3.3 C¤-version of the Proof . 217 4.3.4 The continuous classical pointer . 221 4.4 The time-evolution of the classical pointer . 225 4.4.1 Infinite time limit for automorphic measurements with classical pointer . 225 4.4.2 Non-automorphic time evolution . 229 References 237 Acknowlegdements 254 Introduction Are universally valid theories possible? A universally valid theory of material reality seems to be the ultimate goal of scientific enquiry. This was for example expressed in the dream of Laplace’s demon: if a universally valid theory were deterministic, the demon could use it to calculate any future state of the universe from its present state. Chapter 1 addresses an aspect of measurement which is common to all universally valid theories—the problem of self-reference. It will turn out that it is impossible for an observer to distinguish all states of a system in which she or he is contained. With this kind of non- self-measurability it is impossible for us to know the present state of the universe exactly. Therefore, even if the ultimate universally valid theory were deterministic, we could not use it to calculate the exact future state of the universe. The fact that it is impossible for us to distinguish all states of the universe seriously impairs the empirical relevance of universally valid theories. But a physical realist would rather not dismiss a theory just for this reason. In his opinion, there are entities which in some sense are independent of human knowledge. Statements about these entities should not be conflated with statements about knowledge of the entities. From this point of view, the fact that no experiment can, not 7 8 INTRODUCTION even in principle, distinguish all states is not in itself objectionable. Accordingly, a physical realist would not take this argument as suffi- cient reason to exclude the possibility of universally valid theories. We have to accept, however, that a universally valid theory is of ontological relevance rather than empirical. Is quantum theory universally valid? Experimental evidence does not suggest that there is a clear border beyond which quantum theory does not apply. Also, there is nothing in the formalism which makes it inapplicable to 1023 particles. On the other hand, we perceive the macroscopic world around us as being cor- rectly described by classical physics. If quantum theory is universally valid, it should also be able to account for this fact. Is quantum theory consistent with our experience of the macroscopic world? Can we regard macroscopic bodies as consisting of a large num- ber of quantum systems? There are famous historical arguments why this should be impossible. Such an argument is for example Schrödinger’scat paradox. These arguments intend to show that (1) at least some properties of macroscopic bodies (such as a cat being alive or dead) are definite in all states; (I will call such properties classical; and properties for which there are states in which they are not definite, quantum) (2) quantum theory does not allow for classical properties. Assuming that quantum theory is universally valid, (1) and (2) con- tradict each other. If one wants to avoid this contradiction, and at 9 the same time stick to the universal validity of quantum theory, then clearly one has to drop (1) or (2). (1) should not be taken as a stronger claim than it actually is. It is not claimed that all properties of macroscopic bodies are definite in all states. The claim is just that macroscopic systems have some classical properties. It is consistent with (1) that macroscopic bodies display some quantum properties. Superfluidity and superconductivity probably are such macroscopic quantum effects. From a strictly empirical point of view, even this weak claim is vulnerable. For it is probably consistent with our experience of the macroscopic world that states in which macroscopic properties are not definite are rare. If one can, by this or a similar argument, deny (1), classical mechanics does not have any place in a correct physical theory. It is simply a wrong description of the real quantum world, but happens to give on the macroscpic level a good numerical approximation. (1) can be denied in another way, albeit at the cost of modifying the quantum mechanical time evolution. States in which macroscopic properties are not definite need not be rare; but perhaps we do not perceive such states because our observation changes the state of the system into one where the observed property is definite. This observation-induced state change is usually called “reduction of the wave packet”. There is another way in which quantum mechanics can be consistent with macroscopic perception: one can deny (2). The argument for this is that macroscopic quantum systems can have classical properties. This approach will be taken in Chapter 3. There I discuss the emergence of classical properties in quantum systems with infinitely many degrees of freedom. One could, of course, take the point of view that such an emergence 10 INTRODUCTION of classical properties violates the universal validity of quantum theory. I would like to take a different point of view. The emergence of classical properties in infinite systems is natural in the sense that at no point one assumes quantum theory not to be valid anymore. The derivation of classical properties, however, relies on the assumption that the quasilo- cal observables form the algebra of observables of the infinite quantum system. This will be discussed in more detail in section 3.1.3. (Chapter 2 addresses the reverse problem: the transition from classical theories to quantum theories by various quantisation procedures. If quantum theory is universally valid, quantisation is not of immediate ontological relevance. Rather it highlights similarities and differences in the mathematical structure of classical and quantum theories.) The contradiction arising from (1), (2), and the universal validity of quantum theory is part of the measurement problem. In which sense the denial of (2) solves the measurement problem will be discussed in Chapter 4. The approach to the measurement problem presented there violates (2) by using a classical pointer observable belonging to an infinite apparatus or an infinite environment. There is a third alternative: the contradiction arising from (1), (2), and the universal validity of quantum theory can be avoided if we re- nounce the universal validity of quantum theory. This is done for example by insisting—in the spirit, perhaps, of Bohr—that the realm of our immediate experience is by described classical physics and not by quantum physics. One final remark about style. In order to present the problems in a form to which definite answers can be given, I have been at pains to give rigorous mathematical formulations. At the same time I have 11 attempted to keep as clear a distinction as possible between physical motivation and purely mathematical considerations.

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